Subgroups — C₂×C₆₇×#3485 #30742025

Order 17152.

11686 conjugacy classes of subgroups.

The subgroup lattice has 532 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	1	yes	maximal, characteristic, complemented
2	C₂³×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3485	C₂	1	32	yes	maximal, direct product factor
4	C₂×#547×C₆₇	C₂	1	8	yes	maximal, complemented
5	C₂×#546×C₆₇	C₂	1	8	yes	maximal, complemented
6	C₂×#536×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#534×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂×#525×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂²×#123×C₆₇	C₂	1	4	yes	maximal, complemented
10	C₂×#518×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×H(5,𝔽₂)×C₆₇	—	2	16	no	—
12	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	16	yes	complemented
13	C₂²×#41×C₆₇	—	2	16	no	—
14	C₂²×#41×C₆₇	C₂²	1	2	yes	—
15	C₂²×#41×C₆₇	C₂²	1	8	yes	complemented
16	C₂²×#41×C₆₇	C₂²	1	4	yes	complemented
17	C₂²×#41×C₆₇	C₂²	1	16	yes	complemented
18	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
19	C₂³×D₄×C₆₇	C₂²	1	4	yes	—
20	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂³×D₄×C₆₇	C₂²	1	2	yes	complemented
22	C₂³×D₄×C₆₇	C₂²	1	4	yes	complemented
23	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	complemented
24	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
25	#547×C₆₇	C₂²	1	128	yes	complemented
26	#546×C₆₇	C₂²	1	128	yes	complemented
27	#536×C₆₇	C₂²	1	32	yes	complemented
28	#534×C₆₇	C₂²	1	64	yes	complemented
29	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
30	C₂×#124×C₆₇	C₂²	1	4	yes	complemented
31	#525×C₆₇	C₂²	1	32	yes	complemented
32	C₂×#123×C₆₇	C₂²	1	4	yes	complemented
33	C₂×#123×C₆₇	C₂²	1	8	yes	complemented
34	C₂×#123×C₆₇	C₂²	1	16	yes	complemented
35	C₂×#123×C₆₇	C₂²	1	8	yes	complemented
36	C₂×#123×C₆₇	C₂²	1	64	yes	complemented
37	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
38	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
39	C₂×#121×C₆₇	C₂²	1	4	yes	complemented
40	C₂×#121×C₆₇	C₂²	1	16	yes	complemented
41	C₂×#120×C₆₇	C₂²	1	4	yes	complemented
42	C₂×#120×C₆₇	C₂²	1	8	yes	complemented
43	#518×C₆₇	C₂²	1	16	yes	complemented
44	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	complemented
45	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	complemented
46	C₂×#117×C₆₇	C₂²	1	2	yes	complemented
47	C₂²×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
48	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
49	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
50	C₂⁵×C₆₇	C₂³	1	4	yes	—
51	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
52	C₂⁵×C₆₇	C₂³	1	2	yes	—
53	H(5,𝔽₂)×C₆₇	—	2	128	no	—
54	C₂×#41×C₆₇	—	2	64	no	—
55	C₂×#41×C₆₇	—	2	32	no	—
56	C₂×#41×C₆₇	—	2	128	no	—
57	C₂×#41×C₆₇	—	2	16	no	—
58	C₂×#41×C₆₇	C₂³	1	16	yes	—
59	C₂×#41×C₆₇	C₂³	1	64	yes	complemented
60	C₂×#41×C₆₇	C₂³	1	32	yes	complemented
61	C₂²×Q₈×C₆₇	—	2	8	no	—
62	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
63	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
64	C₂²×Q₈×C₆₇	C₂³	1	8	yes	complemented
65	C₂²×D₄×C₆₇	—	2	32	no	—
66	C₂²×D₄×C₆₇	—	2	16	no	—
67	C₂²×D₄×C₆₇	—	2	32	no	—
68	C₂²×D₄×C₆₇	D₄	1	16	yes	complemented
69	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
70	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
71	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
72	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
73	C₂²×D₄×C₆₇	C₂³	1	32	yes	—
74	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
75	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
76	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
77	C₂²×D₄×C₆₇	C₂³	1	16	yes	complemented
78	C₂²×D₄×C₆₇	C₂³	1	32	yes	complemented
79	C₂³×C₄×C₆₇	—	2	16	no	—
80	C₂³×C₄×C₆₇	—	2	8	no	—
81	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
82	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
83	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
84	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
85	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
87	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
88	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₆₇	C₂³	1	16	yes	complemented
90	C₂³×C₄×C₆₇	C₂³	1	8	yes	complemented
91	C₄⋊Q₈×C₆₇	C₂³	1	32	yes	complemented
92	#124×C₆₇	C₂³	1	32	yes	complemented
93	#123×C₆₇	C₂³	1	32	yes	complemented
94	#123×C₆₇	C₂³	1	64	yes	complemented
95	#123×C₆₇	C₂³	1	128	yes	complemented
96	#123×C₆₇	C₂³	1	64	yes	complemented
97	C₂²⋊Q₈×C₆₇	C₂³	1	32	yes	complemented
98	C₂²⋊Q₈×C₆₇	C₂³	1	64	yes	complemented
99	#121×C₆₇	C₂³	1	32	yes	complemented
100	#121×C₆₇	C₂³	1	128	yes	complemented
101	#120×C₆₇	C₂³	1	32	yes	complemented
102	#120×C₆₇	C₂³	1	64	yes	complemented
103	C₄×D₄×C₆₇	C₂³	1	64	yes	complemented
104	C₄×D₄×C₆₇	C₂³	1	64	yes	complemented
105	#117×C₆₇	C₂³	1	16	yes	complemented
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	64	yes	complemented
110	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
114	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	32	yes	complemented
116	C₂×C₂²⋊C₄×C₆₇	C₂³	1	64	yes	complemented
117	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
118	C₂⁴×C₆₇	—	2	4	no	—
119	C₂⁴×C₆₇	—	2	16	no	—
120	C₂⁴×C₆₇	—	4	8	no	—
121	C₂⁴×C₆₇	—	4	4	no	—
122	C₂⁴×C₆₇	—	4	16	no	—
123	C₂⁴×C₆₇	C₂×D₄	1	16	yes	—
124	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
125	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
126	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
127	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
128	C₂⁴×C₆₇	C₂⁴	1	16	yes	—
129	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
130	C₂⁴×C₆₇	C₂⁴	1	8	yes	—
131	#41×C₆₇	—	2	256	no	—
132	#41×C₆₇	—	2	128	no	—
133	#41×C₆₇	—	2	64	no	—
134	C₂×Q₈×C₆₇	—	2	16	no	—
135	C₂×Q₈×C₆₇	—	2	32	no	—
136	C₂×Q₈×C₆₇	—	2	8	no	—
137	C₂×Q₈×C₆₇	C₂⁴	1	16	yes	—
138	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
139	C₂×D₄×C₆₇	—	2	16	no	—
140	C₂×D₄×C₆₇	—	2	32	no	—
141	C₂×D₄×C₆₇	—	2	32	no	—
142	C₂×D₄×C₆₇	—	2	128	no	—
143	C₂×D₄×C₆₇	—	2	64	no	—
144	C₂×D₄×C₆₇	—	2	128	no	—
145	C₂×D₄×C₆₇	—	2	8	no	—
146	C₂×D₄×C₆₇	—	4	16	no	—
147	C₂×D₄×C₆₇	C₂×D₄	1	32	yes	—
148	C₂×D₄×C₆₇	C₂×D₄	1	64	yes	complemented
149	C₂×D₄×C₆₇	C₂⁴	1	32	yes	—
150	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
151	C₂×D₄×C₆₇	C₂⁴	1	32	yes	—
152	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
153	C₂×D₄×C₆₇	C₂⁴	1	32	yes	—
154	C₂×D₄×C₆₇	C₂⁴	1	16	yes	—
155	C₂²×C₄×C₆₇	—	2	4	no	—
156	C₂²×C₄×C₆₇	—	2	8	no	—
157	C₂²×C₄×C₆₇	—	2	32	no	—
158	C₂²×C₄×C₆₇	—	2	64	no	—
159	C₂²×C₄×C₆₇	—	2	16	no	—
160	C₂²×C₄×C₆₇	—	2	16	no	—
161	C₂²×C₄×C₆₇	—	2	32	no	—
162	C₂²×C₄×C₆₇	—	2	16	no	—
163	C₂²×C₄×C₆₇	—	4	16	no	—
164	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	32	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	16	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
172	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	32	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
178	C₄⋊C₄×C₆₇	C₂⁴	1	32	yes	—
179	C₄⋊C₄×C₆₇	C₂⁴	1	64	yes	—
180	C₄⋊C₄×C₆₇	C₂⁴	1	16	yes	—
181	C₂²⋊C₄×C₆₇	C₂⁴	1	16	yes	—
182	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
183	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
184	C₂²⋊C₄×C₆₇	C₂⁴	1	64	yes	—
185	C₂²⋊C₄×C₆₇	C₂⁴	1	32	yes	—
186	C₄²×C₆₇	C₂⁴	1	16	yes	—
187	C₂³×C₆₇	—	2	16	no	—
188	C₂³×C₆₇	—	2	4	no	—
189	C₂³×C₆₇	—	2	8	no	—
190	C₂³×C₆₇	—	2	4	no	—
191	C₂³×C₆₇	—	2	8	no	—
192	C₂³×C₆₇	—	2	8	no	—
193	C₂³×C₆₇	—	2	16	no	—
194	C₂³×C₆₇	—	2	32	no	—
195	C₂³×C₆₇	—	4	32	no	—
196	C₂³×C₆₇	—	4	16	no	—
197	C₂³×C₆₇	—	4	8	no	—
198	C₂³×C₆₇	—	4	32	no	—
199	C₂³×C₆₇	—	4	16	no	—
200	C₂³×C₆₇	—	8	32	no	—
201	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
202	C₂³×C₆₇	C₂²×D₄	1	16	yes	—
203	C₂³×C₆₇	C₂²×D₄	1	32	yes	—
204	C₂³×C₆₇	C₂⁵	1	8	yes	—
205	C₂³×C₆₇	C₂⁵	1	4	yes	—
206	C₂³×C₆₇	C₂⁵	1	2	yes	—
207	C₂³×C₆₇	C₂⁵	1	8	yes	—
208	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
209	Q₈×C₆₇	—	2	32	no	—
210	Q₈×C₆₇	—	2	16	no	—
211	D₄×C₆₇	—	2	32	no	—
212	D₄×C₆₇	—	2	64	no	—
213	D₄×C₆₇	—	2	64	no	—
214	D₄×C₆₇	—	2	16	no	—
215	D₄×C₆₇	—	4	32	no	—
216	D₄×C₆₇	C₂²×D₄	1	64	yes	—
217	C₂×C₄×C₆₇	—	2	8	no	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	2	8	no	—
220	C₂×C₄×C₆₇	—	2	16	no	—
221	C₂×C₄×C₆₇	—	2	8	no	—
222	C₂×C₄×C₆₇	—	2	64	no	—
223	C₂×C₄×C₆₇	—	2	32	no	—
224	C₂×C₄×C₆₇	—	2	16	no	—
225	C₂×C₄×C₆₇	—	2	32	no	—
226	C₂×C₄×C₆₇	—	2	32	no	—
227	C₂×C₄×C₆₇	—	4	32	no	—
228	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
229	C₂×C₄×C₆₇	C₂²×D₄	1	16	yes	—
230	C₂×C₄×C₆₇	C₂⁵	1	16	yes	—
231	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
232	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
233	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
234	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
235	C₂²×C₆₇	—	2	16	no	—
236	C₂²×C₆₇	—	2	4	no	—
237	C₂²×C₆₇	—	2	8	no	—
238	C₂²×C₆₇	—	2	32	no	—
239	C₂²×C₆₇	—	2	4	no	—
240	C₂²×C₆₇	—	2	8	no	—
241	C₂²×C₆₇	—	2	8	no	—
242	C₂²×C₆₇	—	2	16	no	—
243	C₂²×C₆₇	—	4	32	no	—
244	C₂²×C₆₇	—	4	16	no	—
245	C₂²×C₆₇	—	4	8	no	—
246	C₂²×C₆₇	—	4	16	no	—
247	C₂²×C₆₇	—	8	32	no	—
248	C₂²×C₆₇	C₂³×D₄	1	4	yes	—
249	C₂²×C₆₇	C₂³×D₄	1	16	yes	—
250	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
251	C₂²×C₆₇	#585	1	4	yes	—
252	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
253	C₄×C₆₇	—	2	4	no	—
254	C₄×C₆₇	—	2	8	no	—
255	C₄×C₆₇	—	2	8	no	—
256	C₄×C₆₇	—	2	16	no	—
257	C₄×C₆₇	C₂³×D₄	1	8	yes	—
258	C₂×#3485	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
259	C₂×C₆₇	—	2	4	no	—
260	C₂×C₆₇	—	2	16	no	—
261	C₂×C₆₇	—	4	4	no	—
262	C₂×C₆₇	—	4	8	no	—
263	C₂×C₆₇	#3485	1	4	yes	direct product factor
264	C₂×C₆₇	C₂⁴×D₄	1	1	yes	characteristic
265	C₂×C₆₇	C₂×#585	1	2	yes	—
266	C₂²×H(5,𝔽₂)	C₂×C₆₇	1	1	yes	characteristic, complemented
267	C₂³×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
268	#3485	C₂×C₆₇	1	32	yes	direct product factor
269	C₂×#547	C₂×C₆₇	1	8	yes	complemented
270	C₂×#546	C₂×C₆₇	1	8	yes	complemented
271	C₂×#536	C₂×C₆₇	1	2	yes	complemented
272	C₂×#534	C₂×C₆₇	1	4	yes	complemented
273	C₂×#525	C₂×C₆₇	1	2	yes	complemented
274	C₂²×#123	C₂×C₆₇	1	4	yes	complemented
275	C₂×#518	C₂×C₆₇	1	1	yes	characteristic, complemented
276	C₆₇	C₂×#3485	1	1	yes	characteristic, direct product factor, Sylow
277	C₂×H(5,𝔽₂)	—	2	16	no	—
278	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	16	yes	complemented
279	C₂²×#41	—	2	16	no	—
280	C₂²×#41	C₂²×C₆₇	1	2	yes	—
281	C₂²×#41	C₂²×C₆₇	1	8	yes	complemented
282	C₂²×#41	C₂²×C₆₇	1	4	yes	complemented
283	C₂²×#41	C₂²×C₆₇	1	16	yes	complemented
284	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
285	C₂³×D₄	C₂²×C₆₇	1	4	yes	—
286	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
287	C₂³×D₄	C₂²×C₆₇	1	2	yes	complemented
288	C₂³×D₄	C₂²×C₆₇	1	4	yes	complemented
289	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	complemented
290	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
291	#547	C₂²×C₆₇	1	128	yes	complemented
292	#546	C₂²×C₆₇	1	128	yes	complemented
293	#536	C₂²×C₆₇	1	32	yes	complemented
294	#534	C₂²×C₆₇	1	64	yes	complemented
295	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
296	C₂×#124	C₂²×C₆₇	1	4	yes	complemented
297	#525	C₂²×C₆₇	1	32	yes	complemented
298	C₂×#123	C₂²×C₆₇	1	4	yes	complemented
299	C₂×#123	C₂²×C₆₇	1	8	yes	complemented
300	C₂×#123	C₂²×C₆₇	1	16	yes	complemented
301	C₂×#123	C₂²×C₆₇	1	8	yes	complemented
302	C₂×#123	C₂²×C₆₇	1	64	yes	complemented
303	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
304	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
305	C₂×#121	C₂²×C₆₇	1	4	yes	complemented
306	C₂×#121	C₂²×C₆₇	1	16	yes	complemented
307	C₂×#120	C₂²×C₆₇	1	4	yes	complemented
308	C₂×#120	C₂²×C₆₇	1	8	yes	complemented
309	#518	C₂²×C₆₇	1	16	yes	complemented
310	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	complemented
311	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	complemented
312	C₂×#117	C₂²×C₆₇	1	2	yes	complemented
313	C₂²×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
314	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
315	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
316	C₂⁵	C₂³×C₆₇	1	4	yes	—
317	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
318	C₂⁵	C₂³×C₆₇	1	2	yes	—
319	H(5,𝔽₂)	—	2	128	no	—
320	C₂×#41	—	2	64	no	—
321	C₂×#41	—	2	32	no	—
322	C₂×#41	—	2	128	no	—
323	C₂×#41	—	2	16	no	—
324	C₂×#41	C₂³×C₆₇	1	16	yes	—
325	C₂×#41	C₂³×C₆₇	1	64	yes	complemented
326	C₂×#41	C₂³×C₆₇	1	32	yes	complemented
327	C₂²×Q₈	—	2	8	no	—
328	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
329	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
330	C₂²×Q₈	C₂³×C₆₇	1	8	yes	complemented
331	C₂²×D₄	—	2	32	no	—
332	C₂²×D₄	—	2	16	no	—
333	C₂²×D₄	—	2	32	no	—
334	C₂²×D₄	D₄×C₆₇	1	16	yes	complemented
335	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
336	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
337	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
338	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
339	C₂²×D₄	C₂³×C₆₇	1	32	yes	—
340	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
341	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
342	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
343	C₂²×D₄	C₂³×C₆₇	1	16	yes	complemented
344	C₂²×D₄	C₂³×C₆₇	1	32	yes	complemented
345	C₂³×C₄	—	2	16	no	—
346	C₂³×C₄	—	2	8	no	—
347	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
348	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
349	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
350	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
351	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
352	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
353	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
354	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₆₇	1	16	yes	complemented
356	C₂³×C₄	C₂³×C₆₇	1	8	yes	complemented
357	C₄⋊Q₈	C₂³×C₆₇	1	32	yes	complemented
358	#124	C₂³×C₆₇	1	32	yes	complemented
359	#123	C₂³×C₆₇	1	32	yes	complemented
360	#123	C₂³×C₆₇	1	64	yes	complemented
361	#123	C₂³×C₆₇	1	128	yes	complemented
362	#123	C₂³×C₆₇	1	64	yes	complemented
363	C₂²⋊Q₈	C₂³×C₆₇	1	32	yes	complemented
364	C₂²⋊Q₈	C₂³×C₆₇	1	64	yes	complemented
365	#121	C₂³×C₆₇	1	32	yes	complemented
366	#121	C₂³×C₆₇	1	128	yes	complemented
367	#120	C₂³×C₆₇	1	32	yes	complemented
368	#120	C₂³×C₆₇	1	64	yes	complemented
369	C₄×D₄	C₂³×C₆₇	1	64	yes	complemented
370	C₄×D₄	C₂³×C₆₇	1	64	yes	complemented
371	#117	C₂³×C₆₇	1	16	yes	complemented
372	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
373	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
374	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
375	C₂×C₄⋊C₄	C₂³×C₆₇	1	64	yes	complemented
376	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₆₇	1	32	yes	complemented
382	C₂×C₂²⋊C₄	C₂³×C₆₇	1	64	yes	complemented
383	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
384	C₂⁴	—	2	4	no	—
385	C₂⁴	—	2	16	no	—
386	C₂⁴	—	4	8	no	—
387	C₂⁴	—	4	4	no	—
388	C₂⁴	—	4	16	no	—
389	C₂⁴	C₂×D₄×C₆₇	1	16	yes	—
390	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
391	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
392	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
393	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
394	C₂⁴	C₂⁴×C₆₇	1	16	yes	—
395	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
396	C₂⁴	C₂⁴×C₆₇	1	8	yes	—
397	#41	—	2	256	no	—
398	#41	—	2	128	no	—
399	#41	—	2	64	no	—
400	C₂×Q₈	—	2	16	no	—
401	C₂×Q₈	—	2	32	no	—
402	C₂×Q₈	—	2	8	no	—
403	C₂×Q₈	C₂⁴×C₆₇	1	16	yes	—
404	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
405	C₂×D₄	—	2	16	no	—
406	C₂×D₄	—	2	32	no	—
407	C₂×D₄	—	2	32	no	—
408	C₂×D₄	—	2	128	no	—
409	C₂×D₄	—	2	64	no	—
410	C₂×D₄	—	2	128	no	—
411	C₂×D₄	—	2	8	no	—
412	C₂×D₄	—	4	16	no	—
413	C₂×D₄	C₂×D₄×C₆₇	1	32	yes	—
414	C₂×D₄	C₂×D₄×C₆₇	1	64	yes	complemented
415	C₂×D₄	C₂⁴×C₆₇	1	32	yes	—
416	C₂×D₄	C₂⁴×C₆₇	1	16	yes	—
417	C₂×D₄	C₂⁴×C₆₇	1	32	yes	—
418	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
419	C₂×D₄	C₂⁴×C₆₇	1	32	yes	—
420	C₂×D₄	C₂⁴×C₆₇	1	16	yes	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	8	no	—
423	C₂²×C₄	—	2	32	no	—
424	C₂²×C₄	—	2	64	no	—
425	C₂²×C₄	—	2	16	no	—
426	C₂²×C₄	—	2	16	no	—
427	C₂²×C₄	—	2	32	no	—
428	C₂²×C₄	—	2	16	no	—
429	C₂²×C₄	—	4	16	no	—
430	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
431	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
432	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
433	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
434	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
435	C₂²×C₄	C₂⁴×C₆₇	1	32	yes	—
436	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
437	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
438	C₂²×C₄	C₂⁴×C₆₇	1	16	yes	—
439	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
441	C₂²×C₄	C₂⁴×C₆₇	1	32	yes	—
442	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
444	C₄⋊C₄	C₂⁴×C₆₇	1	32	yes	—
445	C₄⋊C₄	C₂⁴×C₆₇	1	64	yes	—
446	C₄⋊C₄	C₂⁴×C₆₇	1	16	yes	—
447	C₂²⋊C₄	C₂⁴×C₆₇	1	16	yes	—
448	C₂²⋊C₄	C₂⁴×C₆₇	1	32	yes	—
449	C₂²⋊C₄	C₂⁴×C₆₇	1	32	yes	—
450	C₂²⋊C₄	C₂⁴×C₆₇	1	64	yes	—
451	C₂²⋊C₄	C₂⁴×C₆₇	1	32	yes	—
452	C₄²	C₂⁴×C₆₇	1	16	yes	—
453	C₂³	—	2	16	no	—
454	C₂³	—	2	4	no	—
455	C₂³	—	2	8	no	—
456	C₂³	—	2	4	no	—
457	C₂³	—	2	8	no	—
458	C₂³	—	2	8	no	—
459	C₂³	—	2	16	no	—
460	C₂³	—	2	32	no	—
461	C₂³	—	4	32	no	—
462	C₂³	—	4	16	no	—
463	C₂³	—	4	8	no	—
464	C₂³	—	4	32	no	—
465	C₂³	—	4	16	no	—
466	C₂³	—	8	32	no	—
467	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
468	C₂³	C₂²×D₄×C₆₇	1	16	yes	—
469	C₂³	C₂²×D₄×C₆₇	1	32	yes	—
470	C₂³	C₂⁵×C₆₇	1	8	yes	—
471	C₂³	C₂⁵×C₆₇	1	4	yes	—
472	C₂³	C₂⁵×C₆₇	1	2	yes	—
473	C₂³	C₂⁵×C₆₇	1	8	yes	—
474	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
475	Q₈	—	2	32	no	—
476	Q₈	—	2	16	no	—
477	D₄	—	2	32	no	—
478	D₄	—	2	64	no	—
479	D₄	—	2	64	no	—
480	D₄	—	2	16	no	—
481	D₄	—	4	32	no	—
482	D₄	C₂²×D₄×C₆₇	1	64	yes	—
483	C₂×C₄	—	2	8	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	8	no	—
486	C₂×C₄	—	2	16	no	—
487	C₂×C₄	—	2	8	no	—
488	C₂×C₄	—	2	64	no	—
489	C₂×C₄	—	2	32	no	—
490	C₂×C₄	—	2	16	no	—
491	C₂×C₄	—	2	32	no	—
492	C₂×C₄	—	2	32	no	—
493	C₂×C₄	—	4	32	no	—
494	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
495	C₂×C₄	C₂²×D₄×C₆₇	1	16	yes	—
496	C₂×C₄	C₂⁵×C₆₇	1	16	yes	—
497	C₂×C₄	C₂⁵×C₆₇	1	4	yes	—
498	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
499	C₂×C₄	C₂⁵×C₆₇	1	8	yes	—
500	C₂×C₄	C₂⁵×C₆₇	1	4	yes	—
501	C₂²	—	2	16	no	—
502	C₂²	—	2	4	no	—
503	C₂²	—	2	8	no	—
504	C₂²	—	2	32	no	—
505	C₂²	—	2	4	no	—
506	C₂²	—	2	8	no	—
507	C₂²	—	2	8	no	—
508	C₂²	—	2	16	no	—
509	C₂²	—	4	32	no	—
510	C₂²	—	4	16	no	—
511	C₂²	—	4	8	no	—
512	C₂²	—	4	16	no	—
513	C₂²	—	8	32	no	—
514	C₂²	C₂³×D₄×C₆₇	1	4	yes	—
515	C₂²	C₂³×D₄×C₆₇	1	16	yes	—
516	C₂²	C₂³×D₄×C₆₇	1	2	yes	—
517	C₂²	#585×C₆₇	1	4	yes	—
518	C₂²	C₂⁶×C₆₇	1	1	yes	characteristic
519	C₄	—	2	4	no	—
520	C₄	—	2	8	no	—
521	C₄	—	2	8	no	—
522	C₄	—	2	16	no	—
523	C₄	C₂³×D₄×C₆₇	1	8	yes	—
524	C₂	—	2	4	no	—
525	C₂	—	2	16	no	—
526	C₂	—	4	4	no	—
527	C₂	—	4	8	no	—
528	C₂	C₆₇×#3485	1	4	yes	direct product factor
529	C₂	C₂⁴×D₄×C₆₇	1	1	yes	characteristic
530	C₂	C₂×#585×C₆₇	1	2	yes	—

	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. C₂³×#41×C₆₇	C₂	semidirect
3. C₆₇×#3485	C₂	direct (N × H)
4. C₂×#547×C₆₇	C₂	semidirect
5. C₂×#546×C₆₇	C₂	semidirect
6. C₂×#536×C₆₇	C₂	semidirect
7. C₂×#534×C₆₇	C₂	semidirect
8. C₂×#525×C₆₇	C₂	semidirect
9. C₂²×#123×C₆₇	C₂	semidirect
10. C₂×#518×C₆₇	C₂	semidirect
12. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
15. C₂²×#41×C₆₇	C₂²	semidirect
16. C₂²×#41×C₆₇	C₂²	semidirect
17. C₂²×#41×C₆₇	C₂²	semidirect
18. C₂³×Q₈×C₆₇	C₂²	semidirect
21. C₂³×D₄×C₆₇	C₂²	semidirect
22. C₂³×D₄×C₆₇	C₂²	semidirect
23. C₂⁴×C₄×C₆₇	C₂²	semidirect
24. C₂⁴×C₄×C₆₇	C₂²	semidirect
25. #547×C₆₇	C₂²	semidirect
26. #546×C₆₇	C₂²	semidirect
27. #536×C₆₇	C₂²	semidirect
28. #534×C₆₇	C₂²	semidirect
29. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
30. C₂×#124×C₆₇	C₂²	semidirect
31. #525×C₆₇	C₂²	semidirect
32. C₂×#123×C₆₇	C₂²	semidirect
33. C₂×#123×C₆₇	C₂²	semidirect
34. C₂×#123×C₆₇	C₂²	semidirect
35. C₂×#123×C₆₇	C₂²	semidirect
36. C₂×#123×C₆₇	C₂²	semidirect
37. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
38. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
39. C₂×#121×C₆₇	C₂²	semidirect
40. C₂×#121×C₆₇	C₂²	semidirect
41. C₂×#120×C₆₇	C₂²	semidirect
42. C₂×#120×C₆₇	C₂²	semidirect
43. #518×C₆₇	C₂²	semidirect
44. C₂×C₄×D₄×C₆₇	C₂²	semidirect
45. C₂×C₄×D₄×C₆₇	C₂²	semidirect
46. C₂×#117×C₆₇	C₂²	semidirect
47. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
48. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
49. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
59. C₂×#41×C₆₇	C₂³	semidirect
60. C₂×#41×C₆₇	C₂³	semidirect
64. C₂²×Q₈×C₆₇	C₂³	semidirect
68. C₂²×D₄×C₆₇	D₄	semidirect
77. C₂²×D₄×C₆₇	C₂³	semidirect
78. C₂²×D₄×C₆₇	C₂³	semidirect
89. C₂³×C₄×C₆₇	C₂³	semidirect
90. C₂³×C₄×C₆₇	C₂³	semidirect
91. C₄⋊Q₈×C₆₇	C₂³	semidirect
92. #124×C₆₇	C₂³	semidirect
93. #123×C₆₇	C₂³	semidirect
94. #123×C₆₇	C₂³	semidirect
95. #123×C₆₇	C₂³	semidirect
96. #123×C₆₇	C₂³	semidirect
97. C₂²⋊Q₈×C₆₇	C₂³	semidirect
98. C₂²⋊Q₈×C₆₇	C₂³	semidirect
99. #121×C₆₇	C₂³	semidirect
100. #121×C₆₇	C₂³	semidirect
101. #120×C₆₇	C₂³	semidirect
102. #120×C₆₇	C₂³	semidirect
103. C₄×D₄×C₆₇	C₂³	semidirect
104. C₄×D₄×C₆₇	C₂³	semidirect
105. #117×C₆₇	C₂³	semidirect
109. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
115. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
116. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
148. C₂×D₄×C₆₇	C₂×D₄	semidirect
258. C₂×#3485	C₆₇	direct (N × H)
263. C₂×C₆₇	#3485	direct (N × H)
266. C₂²×H(5,𝔽₂)	C₂×C₆₇	semidirect
267. C₂³×#41	C₂×C₆₇	semidirect
268. #3485	C₂×C₆₇	direct (N × H)
269. C₂×#547	C₂×C₆₇	semidirect
270. C₂×#546	C₂×C₆₇	semidirect
271. C₂×#536	C₂×C₆₇	semidirect
272. C₂×#534	C₂×C₆₇	semidirect
273. C₂×#525	C₂×C₆₇	semidirect
274. C₂²×#123	C₂×C₆₇	semidirect
275. C₂×#518	C₂×C₆₇	semidirect
276. C₆₇	C₂×#3485	direct (N × H)
278. C₂×H(5,𝔽₂)	C₂²×C₆₇	semidirect
281. C₂²×#41	C₂²×C₆₇	semidirect
282. C₂²×#41	C₂²×C₆₇	semidirect
283. C₂²×#41	C₂²×C₆₇	semidirect
284. C₂³×Q₈	C₂²×C₆₇	semidirect
287. C₂³×D₄	C₂²×C₆₇	semidirect
288. C₂³×D₄	C₂²×C₆₇	semidirect
289. C₂⁴×C₄	C₂²×C₆₇	semidirect
290. C₂⁴×C₄	C₂²×C₆₇	semidirect
291. #547	C₂²×C₆₇	semidirect
292. #546	C₂²×C₆₇	semidirect
293. #536	C₂²×C₆₇	semidirect
294. #534	C₂²×C₆₇	semidirect
295. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
296. C₂×#124	C₂²×C₆₇	semidirect
297. #525	C₂²×C₆₇	semidirect
298. C₂×#123	C₂²×C₆₇	semidirect
299. C₂×#123	C₂²×C₆₇	semidirect
300. C₂×#123	C₂²×C₆₇	semidirect
301. C₂×#123	C₂²×C₆₇	semidirect
302. C₂×#123	C₂²×C₆₇	semidirect
303. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
304. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
305. C₂×#121	C₂²×C₆₇	semidirect
306. C₂×#121	C₂²×C₆₇	semidirect
307. C₂×#120	C₂²×C₆₇	semidirect
308. C₂×#120	C₂²×C₆₇	semidirect
309. #518	C₂²×C₆₇	semidirect
310. C₂×C₄×D₄	C₂²×C₆₇	semidirect
311. C₂×C₄×D₄	C₂²×C₆₇	semidirect
312. C₂×#117	C₂²×C₆₇	semidirect
313. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
314. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
315. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
325. C₂×#41	C₂³×C₆₇	semidirect
326. C₂×#41	C₂³×C₆₇	semidirect
330. C₂²×Q₈	C₂³×C₆₇	semidirect
334. C₂²×D₄	D₄×C₆₇	semidirect
343. C₂²×D₄	C₂³×C₆₇	semidirect
344. C₂²×D₄	C₂³×C₆₇	semidirect
355. C₂³×C₄	C₂³×C₆₇	semidirect
356. C₂³×C₄	C₂³×C₆₇	semidirect
357. C₄⋊Q₈	C₂³×C₆₇	semidirect
358. #124	C₂³×C₆₇	semidirect
359. #123	C₂³×C₆₇	semidirect
360. #123	C₂³×C₆₇	semidirect
361. #123	C₂³×C₆₇	semidirect
362. #123	C₂³×C₆₇	semidirect
363. C₂²⋊Q₈	C₂³×C₆₇	semidirect
364. C₂²⋊Q₈	C₂³×C₆₇	semidirect
365. #121	C₂³×C₆₇	semidirect
366. #121	C₂³×C₆₇	semidirect
367. #120	C₂³×C₆₇	semidirect
368. #120	C₂³×C₆₇	semidirect
369. C₄×D₄	C₂³×C₆₇	semidirect
370. C₄×D₄	C₂³×C₆₇	semidirect
371. #117	C₂³×C₆₇	semidirect
375. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
381. C₂×C₂²⋊C₄	C₂³×C₆₇	semidirect
382. C₂×C₂²⋊C₄	C₂³×C₆₇	semidirect
414. C₂×D₄	C₂×D₄×C₆₇	semidirect
528. C₂	C₆₇×#3485	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
C₂³×#41×C₆₇	2
C₆₇×#3485	2
C₂×#547×C₆₇	2
C₂×#546×C₆₇	2
C₂×#536×C₆₇	2
C₂×#534×C₆₇	2
C₂×#525×C₆₇	2
C₂²×#123×C₆₇	2
C₂×#518×C₆₇	2
C₂×#3485	67

Subgroups — C2×C67×#3485 #30742025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3485 #30742025