Subgroups — C₂×C₆₇×#3496 #30742015

Order 17152.

12378 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3496	C₂	1	32	yes	maximal, direct product factor
3	C₂×#547×C₆₇	C₂	1	4	yes	maximal, complemented
4	C₂×#546×C₆₇	C₂	1	8	yes	maximal, complemented
5	C₂×D₄²×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#537×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#534×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂×#532×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂²×#121×C₆₇	C₂	1	4	yes	maximal, complemented
10	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×H(5,𝔽₂)×C₆₇	—	2	32	no	—
12	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	32	yes	complemented
13	C₂²×#41×C₆₇	C₂²	1	4	yes	complemented
14	C₂²×#41×C₆₇	C₂²	1	8	yes	complemented
15	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×D₄×C₆₇	C₂²	1	8	yes	complemented
17	C₂³×D₄×C₆₇	C₂²	1	8	yes	complemented
18	C₂³×D₄×C₆₇	C₂²	1	4	yes	complemented
19	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	complemented
20	#547×C₆₇	C₂²	1	64	yes	complemented
21	#546×C₆₇	C₂²	1	128	yes	complemented
22	D₄²×C₆₇	C₂²	1	64	yes	complemented
23	#537×C₆₇	C₂²	1	32	yes	complemented
24	#534×C₆₇	C₂²	1	64	yes	complemented
25	#532×C₆₇	C₂²	1	32	yes	complemented
26	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
27	C₂×#127×C₆₇	C₂²	1	2	yes	complemented
28	C₂×#124×C₆₇	C₂²	1	4	yes	complemented
29	C₂×#123×C₆₇	C₂²	1	16	yes	complemented
30	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
31	C₂×#121×C₆₇	C₂²	1	4	yes	complemented
32	C₂×#121×C₆₇	C₂²	1	8	yes	complemented
33	C₂×#121×C₆₇	C₂²	1	8	yes	complemented
34	C₂×#121×C₆₇	C₂²	1	8	yes	complemented
35	C₂×#121×C₆₇	C₂²	1	4	yes	complemented
36	C₂×#121×C₆₇	C₂²	1	8	yes	complemented
37	C₂×#121×C₆₇	C₂²	1	64	yes	complemented
38	C₂×#120×C₆₇	C₂²	1	16	yes	complemented
39	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
40	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
41	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
42	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
43	C₂×C₄×D₄×C₆₇	C₂²	1	16	yes	complemented
44	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
46	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	complemented
47	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
48	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic, complemented
49	C₂⁵×C₆₇	C₂³	1	8	yes	—
50	C₂⁵×C₆₇	C₂³	1	2	yes	—
51	H(5,𝔽₂)×C₆₇	—	2	256	no	—
52	C₂×#41×C₆₇	—	2	64	no	—
53	C₂×#41×C₆₇	—	2	32	no	—
54	C₂×#41×C₆₇	C₂³	1	32	yes	complemented
55	C₂×#41×C₆₇	C₂³	1	64	yes	complemented
56	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
57	C₂²×D₄×C₆₇	—	2	64	no	—
58	C₂²×D₄×C₆₇	—	2	64	no	—
59	C₂²×D₄×C₆₇	—	4	16	no	—
60	C₂²×D₄×C₆₇	D₄	1	16	yes	complemented
61	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
62	C₂²×D₄×C₆₇	C₂³	1	16	yes	—
63	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
64	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
65	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
66	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
67	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
68	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
69	C₂²×D₄×C₆₇	C₂³	1	64	yes	complemented
70	C₂²×D₄×C₆₇	C₂³	1	64	yes	complemented
71	C₂²×D₄×C₆₇	C₂³	1	32	yes	complemented
72	C₂³×C₄×C₆₇	—	4	8	no	—
73	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
75	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
76	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
77	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
79	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
80	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₆₇	C₂³	1	16	yes	complemented
82	C₄⋊Q₈×C₆₇	C₂³	1	16	yes	complemented
83	#127×C₆₇	C₂³	1	16	yes	complemented
84	#124×C₆₇	C₂³	1	32	yes	complemented
85	#123×C₆₇	C₂³	1	128	yes	complemented
86	C₂²⋊Q₈×C₆₇	C₂³	1	64	yes	complemented
87	#121×C₆₇	C₂³	1	32	yes	complemented
88	#121×C₆₇	C₂³	1	64	yes	complemented
89	#121×C₆₇	C₂³	1	64	yes	complemented
90	#121×C₆₇	C₂³	1	64	yes	complemented
91	#121×C₆₇	C₂³	1	32	yes	complemented
92	#121×C₆₇	C₂³	1	64	yes	complemented
93	#120×C₆₇	C₂³	1	128	yes	complemented
94	C₄×D₄×C₆₇	C₂³	1	32	yes	complemented
95	C₄×D₄×C₆₇	C₂³	1	32	yes	complemented
96	C₄×D₄×C₆₇	C₂³	1	32	yes	complemented
97	C₄×D₄×C₆₇	C₂³	1	32	yes	complemented
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	complemented
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	32	yes	complemented
104	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
106	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
107	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
108	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	complemented
109	C₂×C₂²⋊C₄×C₆₇	C₂³	1	64	yes	complemented
110	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
112	C₂×C₄²×C₆₇	C₂³	1	8	yes	complemented
113	C₂⁴×C₆₇	—	2	32	no	—
114	C₂⁴×C₆₇	—	4	4	no	—
115	C₂⁴×C₆₇	—	4	4	no	—
116	C₂⁴×C₆₇	—	4	16	no	—
117	C₂⁴×C₆₇	—	4	16	no	—
118	C₂⁴×C₆₇	C₂×D₄	1	16	yes	complemented
119	C₂⁴×C₆₇	C₂⁴	1	8	yes	—
120	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
121	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
122	C₂⁴×C₆₇	C₂⁴	1	32	yes	—
123	C₂⁴×C₆₇	C₂⁴	1	8	yes	—
124	#41×C₆₇	—	2	256	no	—
125	#41×C₆₇	—	2	128	no	—
126	C₂×Q₈×C₆₇	—	2	16	no	—
127	C₂×Q₈×C₆₇	C₂⁴	1	16	yes	—
128	C₂×D₄×C₆₇	—	2	64	no	—
129	C₂×D₄×C₆₇	—	2	32	no	—
130	C₂×D₄×C₆₇	—	2	256	no	—
131	C₂×D₄×C₆₇	—	2	256	no	—
132	C₂×D₄×C₆₇	—	2	16	no	—
133	C₂×D₄×C₆₇	—	2	16	no	—
134	C₂×D₄×C₆₇	—	4	64	no	—
135	C₂×D₄×C₆₇	—	4	16	no	—
136	C₂×D₄×C₆₇	—	4	8	no	—
137	C₂×D₄×C₆₇	C₂×D₄	1	32	yes	complemented
138	C₂×D₄×C₆₇	C₂×D₄	1	64	yes	complemented
139	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
140	C₂×D₄×C₆₇	C₂⁴	1	64	yes	—
141	C₂×D₄×C₆₇	C₂⁴	1	32	yes	—
142	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
143	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
144	C₂×D₄×C₆₇	C₂⁴	1	32	yes	—
145	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
146	C₂²×C₄×C₆₇	—	2	32	no	—
147	C₂²×C₄×C₆₇	—	2	32	no	—
148	C₂²×C₄×C₆₇	—	2	4	no	—
149	C₂²×C₄×C₆₇	—	2	16	no	—
150	C₂²×C₄×C₆₇	—	4	32	no	—
151	C₂²×C₄×C₆₇	—	4	8	no	—
152	C₂²×C₄×C₆₇	—	4	4	no	—
153	C₂²×C₄×C₆₇	—	4	2	no	—
154	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	complemented
155	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₆₇	C₂⁴	1	32	yes	—
157	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
158	C₂²×C₄×C₆₇	C₂⁴	1	32	yes	—
159	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
169	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
170	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
171	C₄⋊C₄×C₆₇	C₂⁴	1	32	yes	—
172	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
173	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
174	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
175	C₂²⋊C₄×C₆₇	C₂⁴	1	64	yes	—
176	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
177	C₄²×C₆₇	C₂⁴	1	8	yes	—
178	C₄²×C₆₇	C₂⁴	1	8	yes	—
179	C₂³×C₆₇	—	2	2	no	—
180	C₂³×C₆₇	—	2	8	no	—
181	C₂³×C₆₇	—	2	2	no	—
182	C₂³×C₆₇	—	2	8	no	—
183	C₂³×C₆₇	—	2	16	no	—
184	C₂³×C₆₇	—	2	16	no	—
185	C₂³×C₆₇	—	2	16	no	—
186	C₂³×C₆₇	—	2	64	no	—
187	C₂³×C₆₇	—	4	8	no	—
188	C₂³×C₆₇	—	4	32	no	—
189	C₂³×C₆₇	—	4	8	no	—
190	C₂³×C₆₇	—	4	32	no	—
191	C₂³×C₆₇	—	4	32	no	—
192	C₂³×C₆₇	—	8	64	no	—
193	C₂³×C₆₇	C₂²×D₄	1	16	yes	—
194	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
195	C₂³×C₆₇	C₂²×D₄	1	32	yes	complemented
196	C₂³×C₆₇	C₂⁵	1	16	yes	—
197	C₂³×C₆₇	C₂⁵	1	8	yes	—
198	C₂³×C₆₇	C₂⁵	1	2	yes	—
199	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
200	Q₈×C₆₇	—	2	32	no	—
201	D₄×C₆₇	—	2	128	no	—
202	D₄×C₆₇	—	2	64	no	—
203	D₄×C₆₇	—	2	32	no	—
204	D₄×C₆₇	—	2	32	no	—
205	D₄×C₆₇	—	4	32	no	—
206	D₄×C₆₇	—	4	16	no	—
207	D₄×C₆₇	C₂²×D₄	1	64	yes	complemented
208	C₂×C₄×C₆₇	—	2	16	no	—
209	C₂×C₄×C₆₇	—	2	64	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	8	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	64	no	—
214	C₂×C₄×C₆₇	—	2	8	no	—
215	C₂×C₄×C₆₇	—	2	32	no	—
216	C₂×C₄×C₆₇	—	4	16	no	—
217	C₂×C₄×C₆₇	—	4	8	no	—
218	C₂×C₄×C₆₇	—	4	4	no	—
219	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
220	C₂×C₄×C₆₇	C₂²×D₄	1	16	yes	complemented
221	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	complemented
222	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
223	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
224	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
225	C₂×C₄×C₆₇	C₂⁵	1	16	yes	—
226	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
227	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
228	C₂²×C₆₇	—	2	2	no	—
229	C₂²×C₆₇	—	2	8	no	—
230	C₂²×C₆₇	—	2	2	no	—
231	C₂²×C₆₇	—	2	8	no	—
232	C₂²×C₆₇	—	2	8	no	—
233	C₂²×C₆₇	—	2	32	no	—
234	C₂²×C₆₇	—	2	16	no	—
235	C₂²×C₆₇	—	2	16	no	—
236	C₂²×C₆₇	—	2	16	no	—
237	C₂²×C₆₇	—	4	32	no	—
238	C₂²×C₆₇	—	4	32	no	—
239	C₂²×C₆₇	—	8	64	no	—
240	C₂²×C₆₇	C₂³×D₄	1	16	yes	—
241	C₂²×C₆₇	C₂³×D₄	1	4	yes	—
242	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
243	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
244	C₂²×C₆₇	#585	1	2	yes	—
245	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
246	C₄×C₆₇	—	2	16	no	—
247	C₄×C₆₇	—	2	2	no	—
248	C₄×C₆₇	—	2	8	no	—
249	C₄×C₆₇	—	2	4	no	—
250	C₄×C₆₇	C₂³×D₄	1	8	yes	—
251	C₄×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	complemented
252	C₂×#3496	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
253	C₂×C₆₇	—	2	4	no	—
254	C₂×C₆₇	—	2	16	no	—
255	C₂×C₆₇	—	4	16	no	—
256	C₂×C₆₇	#3496	1	4	yes	direct product factor
257	C₂×C₆₇	C₂⁴×D₄	1	1	yes	characteristic
258	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
259	C₂×C₆₇	C₂×#585	1	1	yes	characteristic
260	C₂²×H(5,𝔽₂)	C₂×C₆₇	1	2	yes	complemented
261	#3496	C₂×C₆₇	1	32	yes	direct product factor
262	C₂×#547	C₂×C₆₇	1	4	yes	complemented
263	C₂×#546	C₂×C₆₇	1	8	yes	complemented
264	C₂×D₄²	C₂×C₆₇	1	4	yes	complemented
265	C₂×#537	C₂×C₆₇	1	2	yes	complemented
266	C₂×#534	C₂×C₆₇	1	4	yes	complemented
267	C₂×#532	C₂×C₆₇	1	2	yes	complemented
268	C₂²×#121	C₂×C₆₇	1	4	yes	complemented
269	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
270	C₆₇	C₂×#3496	1	1	yes	characteristic, direct product factor, Sylow
271	C₂×H(5,𝔽₂)	—	2	32	no	—
272	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	32	yes	complemented
273	C₂²×#41	C₂²×C₆₇	1	4	yes	complemented
274	C₂²×#41	C₂²×C₆₇	1	8	yes	complemented
275	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
276	C₂³×D₄	C₂²×C₆₇	1	8	yes	complemented
277	C₂³×D₄	C₂²×C₆₇	1	8	yes	complemented
278	C₂³×D₄	C₂²×C₆₇	1	4	yes	complemented
279	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	complemented
280	#547	C₂²×C₆₇	1	64	yes	complemented
281	#546	C₂²×C₆₇	1	128	yes	complemented
282	D₄²	C₂²×C₆₇	1	64	yes	complemented
283	#537	C₂²×C₆₇	1	32	yes	complemented
284	#534	C₂²×C₆₇	1	64	yes	complemented
285	#532	C₂²×C₆₇	1	32	yes	complemented
286	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
287	C₂×#127	C₂²×C₆₇	1	2	yes	complemented
288	C₂×#124	C₂²×C₆₇	1	4	yes	complemented
289	C₂×#123	C₂²×C₆₇	1	16	yes	complemented
290	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
291	C₂×#121	C₂²×C₆₇	1	4	yes	complemented
292	C₂×#121	C₂²×C₆₇	1	8	yes	complemented
293	C₂×#121	C₂²×C₆₇	1	8	yes	complemented
294	C₂×#121	C₂²×C₆₇	1	8	yes	complemented
295	C₂×#121	C₂²×C₆₇	1	4	yes	complemented
296	C₂×#121	C₂²×C₆₇	1	8	yes	complemented
297	C₂×#121	C₂²×C₆₇	1	64	yes	complemented
298	C₂×#120	C₂²×C₆₇	1	16	yes	complemented
299	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
300	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
301	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
302	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
303	C₂×C₄×D₄	C₂²×C₆₇	1	16	yes	complemented
304	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
305	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
306	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	complemented
307	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
308	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic, complemented
309	C₂⁵	C₂³×C₆₇	1	8	yes	—
310	C₂⁵	C₂³×C₆₇	1	2	yes	—
311	H(5,𝔽₂)	—	2	256	no	—
312	C₂×#41	—	2	64	no	—
313	C₂×#41	—	2	32	no	—
314	C₂×#41	C₂³×C₆₇	1	32	yes	complemented
315	C₂×#41	C₂³×C₆₇	1	64	yes	complemented
316	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
317	C₂²×D₄	—	2	64	no	—
318	C₂²×D₄	—	2	64	no	—
319	C₂²×D₄	—	4	16	no	—
320	C₂²×D₄	D₄×C₆₇	1	16	yes	complemented
321	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
322	C₂²×D₄	C₂³×C₆₇	1	16	yes	—
323	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
324	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
325	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
326	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
327	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
328	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
329	C₂²×D₄	C₂³×C₆₇	1	64	yes	complemented
330	C₂²×D₄	C₂³×C₆₇	1	64	yes	complemented
331	C₂²×D₄	C₂³×C₆₇	1	32	yes	complemented
332	C₂³×C₄	—	4	8	no	—
333	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
334	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
335	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
336	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
337	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
338	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
339	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
340	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
341	C₂³×C₄	C₂³×C₆₇	1	16	yes	complemented
342	C₄⋊Q₈	C₂³×C₆₇	1	16	yes	complemented
343	#127	C₂³×C₆₇	1	16	yes	complemented
344	#124	C₂³×C₆₇	1	32	yes	complemented
345	#123	C₂³×C₆₇	1	128	yes	complemented
346	C₂²⋊Q₈	C₂³×C₆₇	1	64	yes	complemented
347	#121	C₂³×C₆₇	1	32	yes	complemented
348	#121	C₂³×C₆₇	1	64	yes	complemented
349	#121	C₂³×C₆₇	1	64	yes	complemented
350	#121	C₂³×C₆₇	1	64	yes	complemented
351	#121	C₂³×C₆₇	1	32	yes	complemented
352	#121	C₂³×C₆₇	1	64	yes	complemented
353	#120	C₂³×C₆₇	1	128	yes	complemented
354	C₄×D₄	C₂³×C₆₇	1	32	yes	complemented
355	C₄×D₄	C₂³×C₆₇	1	32	yes	complemented
356	C₄×D₄	C₂³×C₆₇	1	32	yes	complemented
357	C₄×D₄	C₂³×C₆₇	1	32	yes	complemented
358	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
359	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
360	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
361	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
362	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	complemented
363	C₂×C₄⋊C₄	C₂³×C₆₇	1	32	yes	complemented
364	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
365	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
366	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
367	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
368	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	complemented
369	C₂×C₂²⋊C₄	C₂³×C₆₇	1	64	yes	complemented
370	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
371	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
372	C₂×C₄²	C₂³×C₆₇	1	8	yes	complemented
373	C₂⁴	—	2	32	no	—
374	C₂⁴	—	4	4	no	—
375	C₂⁴	—	4	4	no	—
376	C₂⁴	—	4	16	no	—
377	C₂⁴	—	4	16	no	—
378	C₂⁴	C₂×D₄×C₆₇	1	16	yes	complemented
379	C₂⁴	C₂⁴×C₆₇	1	8	yes	—
380	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
381	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
382	C₂⁴	C₂⁴×C₆₇	1	32	yes	—
383	C₂⁴	C₂⁴×C₆₇	1	8	yes	—
384	#41	—	2	256	no	—
385	#41	—	2	128	no	—
386	C₂×Q₈	—	2	16	no	—
387	C₂×Q₈	C₂⁴×C₆₇	1	16	yes	—
388	C₂×D₄	—	2	64	no	—
389	C₂×D₄	—	2	32	no	—
390	C₂×D₄	—	2	256	no	—
391	C₂×D₄	—	2	256	no	—
392	C₂×D₄	—	2	16	no	—
393	C₂×D₄	—	2	16	no	—
394	C₂×D₄	—	4	64	no	—
395	C₂×D₄	—	4	16	no	—
396	C₂×D₄	—	4	8	no	—
397	C₂×D₄	C₂×D₄×C₆₇	1	32	yes	complemented
398	C₂×D₄	C₂×D₄×C₆₇	1	64	yes	complemented
399	C₂×D₄	C₂⁴×C₆₇	1	16	yes	—
400	C₂×D₄	C₂⁴×C₆₇	1	64	yes	—
401	C₂×D₄	C₂⁴×C₆₇	1	32	yes	—
402	C₂×D₄	C₂⁴×C₆₇	1	16	yes	—
403	C₂×D₄	C₂⁴×C₆₇	1	16	yes	—
404	C₂×D₄	C₂⁴×C₆₇	1	32	yes	—
405	C₂×D₄	C₂⁴×C₆₇	1	16	yes	—
406	C₂²×C₄	—	2	32	no	—
407	C₂²×C₄	—	2	32	no	—
408	C₂²×C₄	—	2	4	no	—
409	C₂²×C₄	—	2	16	no	—
410	C₂²×C₄	—	4	32	no	—
411	C₂²×C₄	—	4	8	no	—
412	C₂²×C₄	—	4	4	no	—
413	C₂²×C₄	—	4	2	no	—
414	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	complemented
415	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
416	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
417	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
418	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
419	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
420	C₂²×C₄	C₂⁴×C₆₇	1	32	yes	—
421	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
422	C₂²×C₄	C₂⁴×C₆₇	1	32	yes	—
423	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
424	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
425	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
426	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
427	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
428	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
429	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
430	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
431	C₄⋊C₄	C₂⁴×C₆₇	1	32	yes	—
432	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
433	C₂²⋊C₄	C₂⁴×C₆₇	1	32	yes	—
434	C₂²⋊C₄	C₂⁴×C₆₇	1	32	yes	—
435	C₂²⋊C₄	C₂⁴×C₆₇	1	64	yes	—
436	C₂²⋊C₄	C₂⁴×C₆₇	1	32	yes	—
437	C₄²	C₂⁴×C₆₇	1	8	yes	—
438	C₄²	C₂⁴×C₆₇	1	8	yes	—
439	C₂³	—	2	2	no	—
440	C₂³	—	2	8	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	8	no	—
443	C₂³	—	2	16	no	—
444	C₂³	—	2	16	no	—
445	C₂³	—	2	16	no	—
446	C₂³	—	2	64	no	—
447	C₂³	—	4	8	no	—
448	C₂³	—	4	32	no	—
449	C₂³	—	4	8	no	—
450	C₂³	—	4	32	no	—
451	C₂³	—	4	32	no	—
452	C₂³	—	8	64	no	—
453	C₂³	C₂²×D₄×C₆₇	1	16	yes	—
454	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
455	C₂³	C₂²×D₄×C₆₇	1	32	yes	complemented
456	C₂³	C₂⁵×C₆₇	1	16	yes	—
457	C₂³	C₂⁵×C₆₇	1	8	yes	—
458	C₂³	C₂⁵×C₆₇	1	2	yes	—
459	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
460	Q₈	—	2	32	no	—
461	D₄	—	2	128	no	—
462	D₄	—	2	64	no	—
463	D₄	—	2	32	no	—
464	D₄	—	2	32	no	—
465	D₄	—	4	32	no	—
466	D₄	—	4	16	no	—
467	D₄	C₂²×D₄×C₆₇	1	64	yes	complemented
468	C₂×C₄	—	2	16	no	—
469	C₂×C₄	—	2	64	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	64	no	—
474	C₂×C₄	—	2	8	no	—
475	C₂×C₄	—	2	32	no	—
476	C₂×C₄	—	4	16	no	—
477	C₂×C₄	—	4	8	no	—
478	C₂×C₄	—	4	4	no	—
479	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
480	C₂×C₄	C₂²×D₄×C₆₇	1	16	yes	complemented
481	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	complemented
482	C₂×C₄	C₂⁵×C₆₇	1	4	yes	—
483	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
484	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
485	C₂×C₄	C₂⁵×C₆₇	1	16	yes	—
486	C₂×C₄	C₂⁵×C₆₇	1	4	yes	—
487	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	8	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	8	no	—
492	C₂²	—	2	8	no	—
493	C₂²	—	2	32	no	—
494	C₂²	—	2	16	no	—
495	C₂²	—	2	16	no	—
496	C₂²	—	2	16	no	—
497	C₂²	—	4	32	no	—
498	C₂²	—	4	32	no	—
499	C₂²	—	8	64	no	—
500	C₂²	C₂³×D₄×C₆₇	1	16	yes	—
501	C₂²	C₂³×D₄×C₆₇	1	4	yes	—
502	C₂²	C₂³×D₄×C₆₇	1	2	yes	—
503	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
504	C₂²	#585×C₆₇	1	2	yes	—
505	C₂²	C₂⁶×C₆₇	1	1	yes	characteristic
506	C₄	—	2	16	no	—
507	C₄	—	2	2	no	—
508	C₄	—	2	8	no	—
509	C₄	—	2	4	no	—
510	C₄	C₂³×D₄×C₆₇	1	8	yes	—
511	C₄	C₂×H(5,𝔽₂)×C₆₇	1	4	yes	complemented
512	C₂	—	2	4	no	—
513	C₂	—	2	16	no	—
514	C₂	—	4	16	no	—
515	C₂	C₆₇×#3496	1	4	yes	direct product factor
516	C₂	C₂⁴×D₄×C₆₇	1	1	yes	characteristic
517	C₂	C₂²×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
518	C₂	C₂×#585×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₆₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×H(5,𝔽₂)×C₆₇	C₂	semidirect
2. C₆₇×#3496	C₂	direct (N × H)
3. C₂×#547×C₆₇	C₂	semidirect
4. C₂×#546×C₆₇	C₂	semidirect
5. C₂×D₄²×C₆₇	C₂	semidirect
6. C₂×#537×C₆₇	C₂	semidirect
7. C₂×#534×C₆₇	C₂	semidirect
8. C₂×#532×C₆₇	C₂	semidirect
9. C₂²×#121×C₆₇	C₂	semidirect
10. C₂²×C₄×D₄×C₆₇	C₂	semidirect
12. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
13. C₂²×#41×C₆₇	C₂²	semidirect
14. C₂²×#41×C₆₇	C₂²	semidirect
16. C₂³×D₄×C₆₇	C₂²	semidirect
17. C₂³×D₄×C₆₇	C₂²	semidirect
18. C₂³×D₄×C₆₇	C₂²	semidirect
19. C₂⁴×C₄×C₆₇	C₂²	semidirect
20. #547×C₆₇	C₂²	semidirect
21. #546×C₆₇	C₂²	semidirect
22. D₄²×C₆₇	C₂²	semidirect
23. #537×C₆₇	C₂²	semidirect
24. #534×C₆₇	C₂²	semidirect
25. #532×C₆₇	C₂²	semidirect
26. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
27. C₂×#127×C₆₇	C₂²	semidirect
28. C₂×#124×C₆₇	C₂²	semidirect
29. C₂×#123×C₆₇	C₂²	semidirect
30. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
31. C₂×#121×C₆₇	C₂²	semidirect
32. C₂×#121×C₆₇	C₂²	semidirect
33. C₂×#121×C₆₇	C₂²	semidirect
34. C₂×#121×C₆₇	C₂²	semidirect
35. C₂×#121×C₆₇	C₂²	semidirect
36. C₂×#121×C₆₇	C₂²	semidirect
37. C₂×#121×C₆₇	C₂²	semidirect
38. C₂×#120×C₆₇	C₂²	semidirect
39. C₂×C₄×D₄×C₆₇	C₂²	semidirect
40. C₂×C₄×D₄×C₆₇	C₂²	semidirect
41. C₂×C₄×D₄×C₆₇	C₂²	semidirect
42. C₂×C₄×D₄×C₆₇	C₂²	semidirect
43. C₂×C₄×D₄×C₆₇	C₂²	semidirect
44. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
45. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
46. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
47. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
48. C₂²×C₄²×C₆₇	C₂²	semidirect
54. C₂×#41×C₆₇	C₂³	semidirect
55. C₂×#41×C₆₇	C₂³	semidirect
60. C₂²×D₄×C₆₇	D₄	semidirect
69. C₂²×D₄×C₆₇	C₂³	semidirect
70. C₂²×D₄×C₆₇	C₂³	semidirect
71. C₂²×D₄×C₆₇	C₂³	semidirect
81. C₂³×C₄×C₆₇	C₂³	semidirect
82. C₄⋊Q₈×C₆₇	C₂³	semidirect
83. #127×C₆₇	C₂³	semidirect
84. #124×C₆₇	C₂³	semidirect
85. #123×C₆₇	C₂³	semidirect
86. C₂²⋊Q₈×C₆₇	C₂³	semidirect
87. #121×C₆₇	C₂³	semidirect
88. #121×C₆₇	C₂³	semidirect
89. #121×C₆₇	C₂³	semidirect
90. #121×C₆₇	C₂³	semidirect
91. #121×C₆₇	C₂³	semidirect
92. #121×C₆₇	C₂³	semidirect
93. #120×C₆₇	C₂³	semidirect
94. C₄×D₄×C₆₇	C₂³	semidirect
95. C₄×D₄×C₆₇	C₂³	semidirect
96. C₄×D₄×C₆₇	C₂³	semidirect
97. C₄×D₄×C₆₇	C₂³	semidirect
102. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
103. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
108. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
109. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
112. C₂×C₄²×C₆₇	C₂³	semidirect
118. C₂⁴×C₆₇	C₂×D₄	semidirect
137. C₂×D₄×C₆₇	C₂×D₄	semidirect
138. C₂×D₄×C₆₇	C₂×D₄	semidirect
154. C₂²×C₄×C₆₇	C₂×D₄	semidirect
195. C₂³×C₆₇	C₂²×D₄	semidirect
207. D₄×C₆₇	C₂²×D₄	semidirect
220. C₂×C₄×C₆₇	C₂²×D₄	semidirect
221. C₂×C₄×C₆₇	H(5,𝔽₂)	semidirect
251. C₄×C₆₇	C₂×H(5,𝔽₂)	semidirect
252. C₂×#3496	C₆₇	direct (N × H)
256. C₂×C₆₇	#3496	direct (N × H)
260. C₂²×H(5,𝔽₂)	C₂×C₆₇	semidirect
261. #3496	C₂×C₆₇	direct (N × H)
262. C₂×#547	C₂×C₆₇	semidirect
263. C₂×#546	C₂×C₆₇	semidirect
264. C₂×D₄²	C₂×C₆₇	semidirect
265. C₂×#537	C₂×C₆₇	semidirect
266. C₂×#534	C₂×C₆₇	semidirect
267. C₂×#532	C₂×C₆₇	semidirect
268. C₂²×#121	C₂×C₆₇	semidirect
269. C₂²×C₄×D₄	C₂×C₆₇	semidirect
270. C₆₇	C₂×#3496	direct (N × H)
272. C₂×H(5,𝔽₂)	C₂²×C₆₇	semidirect
273. C₂²×#41	C₂²×C₆₇	semidirect
274. C₂²×#41	C₂²×C₆₇	semidirect
276. C₂³×D₄	C₂²×C₆₇	semidirect
277. C₂³×D₄	C₂²×C₆₇	semidirect
278. C₂³×D₄	C₂²×C₆₇	semidirect
279. C₂⁴×C₄	C₂²×C₆₇	semidirect
280. #547	C₂²×C₆₇	semidirect
281. #546	C₂²×C₆₇	semidirect
282. D₄²	C₂²×C₆₇	semidirect
283. #537	C₂²×C₆₇	semidirect
284. #534	C₂²×C₆₇	semidirect
285. #532	C₂²×C₆₇	semidirect
286. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
287. C₂×#127	C₂²×C₆₇	semidirect
288. C₂×#124	C₂²×C₆₇	semidirect
289. C₂×#123	C₂²×C₆₇	semidirect
290. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
291. C₂×#121	C₂²×C₆₇	semidirect
292. C₂×#121	C₂²×C₆₇	semidirect
293. C₂×#121	C₂²×C₆₇	semidirect
294. C₂×#121	C₂²×C₆₇	semidirect
295. C₂×#121	C₂²×C₆₇	semidirect
296. C₂×#121	C₂²×C₆₇	semidirect
297. C₂×#121	C₂²×C₆₇	semidirect
298. C₂×#120	C₂²×C₆₇	semidirect
299. C₂×C₄×D₄	C₂²×C₆₇	semidirect
300. C₂×C₄×D₄	C₂²×C₆₇	semidirect
301. C₂×C₄×D₄	C₂²×C₆₇	semidirect
302. C₂×C₄×D₄	C₂²×C₆₇	semidirect
303. C₂×C₄×D₄	C₂²×C₆₇	semidirect
304. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
305. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
306. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
307. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
308. C₂²×C₄²	C₂²×C₆₇	semidirect
314. C₂×#41	C₂³×C₆₇	semidirect
315. C₂×#41	C₂³×C₆₇	semidirect
320. C₂²×D₄	D₄×C₆₇	semidirect
329. C₂²×D₄	C₂³×C₆₇	semidirect
330. C₂²×D₄	C₂³×C₆₇	semidirect
331. C₂²×D₄	C₂³×C₆₇	semidirect
341. C₂³×C₄	C₂³×C₆₇	semidirect
342. C₄⋊Q₈	C₂³×C₆₇	semidirect
343. #127	C₂³×C₆₇	semidirect
344. #124	C₂³×C₆₇	semidirect
345. #123	C₂³×C₆₇	semidirect
346. C₂²⋊Q₈	C₂³×C₆₇	semidirect
347. #121	C₂³×C₆₇	semidirect
348. #121	C₂³×C₆₇	semidirect
349. #121	C₂³×C₆₇	semidirect
350. #121	C₂³×C₆₇	semidirect
351. #121	C₂³×C₆₇	semidirect
352. #121	C₂³×C₆₇	semidirect
353. #120	C₂³×C₆₇	semidirect
354. C₄×D₄	C₂³×C₆₇	semidirect
355. C₄×D₄	C₂³×C₆₇	semidirect
356. C₄×D₄	C₂³×C₆₇	semidirect
357. C₄×D₄	C₂³×C₆₇	semidirect
362. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
363. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
368. C₂×C₂²⋊C₄	C₂³×C₆₇	semidirect
369. C₂×C₂²⋊C₄	C₂³×C₆₇	semidirect
372. C₂×C₄²	C₂³×C₆₇	semidirect
378. C₂⁴	C₂×D₄×C₆₇	semidirect
397. C₂×D₄	C₂×D₄×C₆₇	semidirect
398. C₂×D₄	C₂×D₄×C₆₇	semidirect
414. C₂²×C₄	C₂×D₄×C₆₇	semidirect
455. C₂³	C₂²×D₄×C₆₇	semidirect
467. D₄	C₂²×D₄×C₆₇	semidirect
480. C₂×C₄	C₂²×D₄×C₆₇	semidirect
481. C₂×C₄	H(5,𝔽₂)×C₆₇	semidirect
511. C₄	C₂×H(5,𝔽₂)×C₆₇	semidirect
515. C₂	C₆₇×#3496	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×H(5,𝔽₂)×C₆₇	2
C₆₇×#3496	2
C₂×#547×C₆₇	2
C₂×#546×C₆₇	2
C₂×D₄²×C₆₇	2
C₂×#537×C₆₇	2
C₂×#534×C₆₇	2
C₂×#532×C₆₇	2
C₂²×#121×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₂×#3496	67

Subgroups — C2×C67×#3496 #30742015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3496 #30742015