Subgroups — C₂×C₆₇×#3285 #30740296

Order 17152.

2794 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#548×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3285	C₂	1	16	yes	maximal, direct product factor
5	C₂×#495×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#466×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#464×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#452×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#448×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₇	C₂²	1	2	yes	—
14	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂×#135×C₆₇	—	2	8	no	—
16	C₂×#135×C₆₇	C₂²	1	8	yes	complemented
17	C₂²×Q₁₆×C₆₇	—	2	8	no	—
18	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
19	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×Q₁₆×C₆₇	C₂²	1	8	yes	complemented
22	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
23	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
24	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#548×C₆₇	—	2	8	no	—
28	#548×C₆₇	C₂²	1	16	yes	complemented
29	C₂×#124×C₆₇	—	2	2	no	—
30	C₂×#124×C₆₇	C₂²	1	2	yes	—
31	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
32	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
33	C₂×#121×C₆₇	—	2	2	no	—
34	C₂×#121×C₆₇	C₂²	1	2	yes	—
35	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
36	C₂×C₄×D₄×C₆₇	—	2	2	no	—
37	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
38	#495×C₆₇	C₂²	1	8	yes	complemented
39	#467×C₆₇	C₂²	1	8	yes	complemented
40	#466×C₆₇	C₂²	1	8	yes	complemented
41	#464×C₆₇	C₂²	1	16	yes	complemented
42	#452×C₆₇	C₂²	1	16	yes	complemented
43	#448×C₆₇	C₂²	1	16	yes	complemented
44	C₄×Q₁₆×C₆₇	C₂²	1	8	yes	complemented
45	C₈×D₄×C₆₇	C₂²	1	8	yes	complemented
46	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×#103×C₆₇	C₂²	1	2	yes	—
49	C₂×#103×C₆₇	C₂²	1	2	yes	—
50	C₂×#102×C₆₇	C₂²	1	2	yes	—
51	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
52	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
54	C₂×#41×C₆₇	—	2	4	no	—
55	C₂×#41×C₆₇	—	2	8	no	—
56	C₂×#41×C₆₇	—	4	4	no	—
57	C₂×#41×C₆₇	C₂³	1	8	yes	—
58	C₂²×Q₈×C₆₇	—	2	2	no	—
59	C₂²×Q₈×C₆₇	—	2	4	no	—
60	C₂²×Q₈×C₆₇	—	2	8	no	—
61	C₂²×Q₈×C₆₇	—	4	4	no	—
62	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
63	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
64	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
65	C₂²×D₄×C₆₇	—	2	2	no	—
66	C₂²×D₄×C₆₇	—	2	2	no	—
67	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
68	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
69	C₂³×C₄×C₆₇	—	2	2	no	—
70	C₂³×C₄×C₆₇	—	2	2	no	—
71	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
73	#135×C₆₇	—	2	32	no	—
74	C₂×Q₁₆×C₆₇	—	2	8	no	—
75	C₂×Q₁₆×C₆₇	—	2	32	no	—
76	C₂×Q₁₆×C₆₇	—	2	4	no	—
77	C₂×Q₁₆×C₆₇	D₄	1	8	yes	complemented
78	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
79	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
80	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
81	C₂×SD₁₆×C₆₇	—	2	8	no	—
82	C₂×SD₁₆×C₆₇	—	4	4	no	—
83	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
84	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
85	C₂×D₈×C₆₇	—	2	4	no	—
86	C₂×D₈×C₆₇	C₂³	1	4	yes	—
87	C₂²×C₈×C₆₇	—	2	4	no	—
88	C₂²×C₈×C₆₇	—	2	4	no	—
89	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
90	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
91	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	#124×C₆₇	—	2	16	no	—
96	#124×C₆₇	—	2	4	no	—
97	#124×C₆₇	C₂³	1	8	yes	—
98	C₂²⋊Q₈×C₆₇	—	2	4	no	—
99	C₂²⋊Q₈×C₆₇	—	2	16	no	—
100	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
101	#121×C₆₇	—	2	16	no	—
102	#121×C₆₇	—	2	4	no	—
103	#121×C₆₇	C₂³	1	8	yes	—
104	C₄×Q₈×C₆₇	—	2	4	no	—
105	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
106	C₄×D₄×C₆₇	—	2	16	no	—
107	C₄×D₄×C₆₇	D₄	1	4	yes	—
108	C₄×D₄×C₆₇	C₂³	1	4	yes	—
109	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
111	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
114	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
115	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
118	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₆₇	—	2	2	no	—
120	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
121	#107×C₆₇	C₂³	1	4	yes	—
122	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
123	#103×C₆₇	C₂³	1	8	yes	—
124	#103×C₆₇	C₂³	1	8	yes	—
125	#102×C₆₇	C₂³	1	8	yes	—
126	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
127	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
128	C₄×C₈×C₆₇	C₂³	1	4	yes	—
129	C₂⁴×C₆₇	—	2	2	no	—
130	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
132	#41×C₆₇	—	2	16	no	—
133	#41×C₆₇	—	4	8	no	—
134	#41×C₆₇	—	4	16	no	—
135	C₂×Q₈×C₆₇	—	2	2	no	—
136	C₂×Q₈×C₆₇	—	2	8	no	—
137	C₂×Q₈×C₆₇	—	2	2	no	—
138	C₂×Q₈×C₆₇	—	2	4	no	—
139	C₂×Q₈×C₆₇	—	2	16	no	—
140	C₂×Q₈×C₆₇	—	2	4	no	—
141	C₂×Q₈×C₆₇	—	2	4	no	—
142	C₂×Q₈×C₆₇	—	4	8	no	—
143	C₂×Q₈×C₆₇	—	4	8	no	—
144	C₂×Q₈×C₆₇	—	4	16	no	—
145	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
146	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
147	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
148	C₂×D₄×C₆₇	—	2	4	no	—
149	C₂×D₄×C₆₇	—	2	8	no	—
150	C₂×D₄×C₆₇	—	2	8	no	—
151	C₂×D₄×C₆₇	—	2	4	no	—
152	C₂×D₄×C₆₇	—	2	2	no	—
153	C₂×D₄×C₆₇	—	2	4	no	—
154	C₂×D₄×C₆₇	—	4	8	no	—
155	C₂×D₄×C₆₇	—	4	8	no	—
156	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
157	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₆₇	—	2	2	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	2	2	no	—
161	C₂²×C₄×C₆₇	—	2	2	no	—
162	C₂²×C₄×C₆₇	—	2	2	no	—
163	C₂²×C₄×C₆₇	—	2	8	no	—
164	C₂²×C₄×C₆₇	—	2	2	no	—
165	C₂²×C₄×C₆₇	—	2	2	no	—
166	C₂²×C₄×C₆₇	—	2	8	no	—
167	C₂²×C₄×C₆₇	—	4	8	no	—
168	C₂²×C₄×C₆₇	—	4	8	no	—
169	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
178	Q₁₆×C₆₇	—	2	16	no	—
179	Q₁₆×C₆₇	—	2	8	no	—
180	Q₁₆×C₆₇	C₂×D₄	1	16	yes	complemented
181	SD₁₆×C₆₇	—	2	16	no	—
182	SD₁₆×C₆₇	—	4	8	no	—
183	D₈×C₆₇	—	2	8	no	—
184	C₂×C₈×C₆₇	—	2	2	no	—
185	C₂×C₈×C₆₇	—	2	2	no	—
186	C₂×C₈×C₆₇	—	2	2	no	—
187	C₂×C₈×C₆₇	—	2	8	no	—
188	C₂×C₈×C₆₇	—	2	8	no	—
189	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
190	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
191	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
192	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
193	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
194	C₄⋊C₄×C₆₇	—	2	8	no	—
195	C₄⋊C₄×C₆₇	—	2	8	no	—
196	C₄⋊C₄×C₆₇	—	2	2	no	—
197	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
198	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
199	C₂²⋊C₄×C₆₇	—	2	8	no	—
200	C₂²⋊C₄×C₆₇	—	2	8	no	—
201	C₂²⋊C₄×C₆₇	—	2	8	no	—
202	C₂²⋊C₄×C₆₇	—	2	8	no	—
203	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
204	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
205	C₄²×C₆₇	—	2	8	no	—
206	C₄²×C₆₇	C₂×D₄	1	2	yes	—
207	C₄²×C₆₇	C₂⁴	1	2	yes	—
208	C₂³×C₆₇	—	2	2	no	—
209	C₂³×C₆₇	—	2	2	no	—
210	C₂³×C₆₇	—	2	2	no	—
211	C₂³×C₆₇	—	2	2	no	—
212	C₂³×C₆₇	—	2	2	no	—
213	C₂³×C₆₇	—	2	4	no	—
214	C₂³×C₆₇	—	4	4	no	—
215	C₂³×C₆₇	—	4	4	no	—
216	C₂³×C₆₇	—	4	4	no	—
217	C₂³×C₆₇	#135	1	4	yes	complemented
218	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
219	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
220	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
221	Q₈×C₆₇	—	2	8	no	—
222	Q₈×C₆₇	—	2	4	no	—
223	Q₈×C₆₇	—	2	4	no	—
224	Q₈×C₆₇	—	2	4	no	—
225	Q₈×C₆₇	—	4	16	no	—
226	Q₈×C₆₇	—	4	4	no	—
227	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
228	D₄×C₆₇	—	2	8	no	—
229	D₄×C₆₇	—	2	4	no	—
230	D₄×C₆₇	—	2	4	no	—
231	D₄×C₆₇	—	4	16	no	—
232	D₄×C₆₇	—	4	16	no	—
233	D₄×C₆₇	—	4	4	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	2	no	—
236	C₂×C₄×C₆₇	—	2	8	no	—
237	C₂×C₄×C₆₇	—	2	2	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	4	16	no	—
249	C₂×C₄×C₆₇	—	4	16	no	—
250	C₂×C₄×C₆₇	—	4	4	no	—
251	C₂×C₄×C₆₇	—	4	4	no	—
252	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
253	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
258	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
259	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
260	C₈×C₆₇	—	2	2	no	—
261	C₈×C₆₇	—	2	2	no	—
262	C₈×C₆₇	—	2	2	no	—
263	C₈×C₆₇	C₂²×D₄	1	4	yes	—
264	C₂²×C₆₇	—	2	2	no	—
265	C₂²×C₆₇	—	2	8	no	—
266	C₂²×C₆₇	—	2	2	no	—
267	C₂²×C₆₇	—	2	2	no	—
268	C₂²×C₆₇	—	2	2	no	—
269	C₂²×C₆₇	—	2	2	no	—
270	C₂²×C₆₇	—	4	4	no	—
271	C₂²×C₆₇	—	4	4	no	—
272	C₂²×C₆₇	—	4	4	no	—
273	C₂²×C₆₇	—	4	4	no	—
274	C₂²×C₆₇	—	8	8	no	—
275	C₂²×C₆₇	D₄²	1	2	yes	—
276	C₂²×C₆₇	C₂×#135	1	2	yes	—
277	C₂²×C₆₇	C₂×#135	1	4	yes	complemented
278	C₂²×C₆₇	#578	1	2	yes	—
279	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
280	C₄×C₆₇	—	2	2	no	—
281	C₄×C₆₇	—	2	2	no	—
282	C₄×C₆₇	—	2	8	no	—
283	C₄×C₆₇	—	2	2	no	—
284	C₄×C₆₇	—	2	2	no	—
285	C₄×C₆₇	—	4	4	no	—
286	C₄×C₆₇	—	4	4	no	—
287	C₄×C₆₇	D₄²	1	2	yes	—
288	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
289	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
290	C₂×#3285	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
291	C₂×C₆₇	—	2	4	no	—
292	C₂×C₆₇	—	4	2	no	—
293	C₂×C₆₇	—	8	4	no	—
294	C₂×C₆₇	#3285	1	4	yes	direct product factor
295	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
296	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
297	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
298	C₂²×#135	C₂×C₆₇	1	1	yes	characteristic, complemented
299	C₂³×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
300	C₂×#548	C₂×C₆₇	1	2	yes	complemented
301	#3285	C₂×C₆₇	1	16	yes	direct product factor
302	C₂×#495	C₂×C₆₇	1	1	yes	characteristic, complemented
303	C₂×#467	C₂×C₆₇	1	1	yes	characteristic, complemented
304	C₂×#466	C₂×C₆₇	1	1	yes	characteristic, complemented
305	C₂×#464	C₂×C₆₇	1	2	yes	complemented
306	C₂×#452	C₂×C₆₇	1	2	yes	complemented
307	C₂×#448	C₂×C₆₇	1	2	yes	complemented
308	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
309	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
310	C₆₇	C₂×#3285	1	1	yes	characteristic, direct product factor, Sylow
311	C₂²×#41	C₂²×C₆₇	1	2	yes	—
312	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
313	C₂×#135	—	2	8	no	—
314	C₂×#135	C₂²×C₆₇	1	8	yes	complemented
315	C₂²×Q₁₆	—	2	8	no	—
316	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
317	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
318	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
319	C₂²×Q₁₆	C₂²×C₆₇	1	8	yes	complemented
320	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
321	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
322	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
323	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
324	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
325	#548	—	2	8	no	—
326	#548	C₂²×C₆₇	1	16	yes	complemented
327	C₂×#124	—	2	2	no	—
328	C₂×#124	C₂²×C₆₇	1	2	yes	—
329	C₂×C₂²⋊Q₈	—	2	2	no	—
330	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
331	C₂×#121	—	2	2	no	—
332	C₂×#121	C₂²×C₆₇	1	2	yes	—
333	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
334	C₂×C₄×D₄	—	2	2	no	—
335	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
336	#495	C₂²×C₆₇	1	8	yes	complemented
337	#467	C₂²×C₆₇	1	8	yes	complemented
338	#466	C₂²×C₆₇	1	8	yes	complemented
339	#464	C₂²×C₆₇	1	16	yes	complemented
340	#452	C₂²×C₆₇	1	16	yes	complemented
341	#448	C₂²×C₆₇	1	16	yes	complemented
342	C₄×Q₁₆	C₂²×C₆₇	1	8	yes	complemented
343	C₈×D₄	C₂²×C₆₇	1	8	yes	complemented
344	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
345	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
346	C₂×#103	C₂²×C₆₇	1	2	yes	—
347	C₂×#103	C₂²×C₆₇	1	2	yes	—
348	C₂×#102	C₂²×C₆₇	1	2	yes	—
349	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
350	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
351	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
352	C₂×#41	—	2	4	no	—
353	C₂×#41	—	2	8	no	—
354	C₂×#41	—	4	4	no	—
355	C₂×#41	C₂³×C₆₇	1	8	yes	—
356	C₂²×Q₈	—	2	2	no	—
357	C₂²×Q₈	—	2	4	no	—
358	C₂²×Q₈	—	2	8	no	—
359	C₂²×Q₈	—	4	4	no	—
360	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
361	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
362	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
363	C₂²×D₄	—	2	2	no	—
364	C₂²×D₄	—	2	2	no	—
365	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
366	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
367	C₂³×C₄	—	2	2	no	—
368	C₂³×C₄	—	2	2	no	—
369	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
370	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
371	#135	—	2	32	no	—
372	C₂×Q₁₆	—	2	8	no	—
373	C₂×Q₁₆	—	2	32	no	—
374	C₂×Q₁₆	—	2	4	no	—
375	C₂×Q₁₆	D₄×C₆₇	1	8	yes	complemented
376	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
377	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
378	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
379	C₂×SD₁₆	—	2	8	no	—
380	C₂×SD₁₆	—	4	4	no	—
381	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
382	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
383	C₂×D₈	—	2	4	no	—
384	C₂×D₈	C₂³×C₆₇	1	4	yes	—
385	C₂²×C₈	—	2	4	no	—
386	C₂²×C₈	—	2	4	no	—
387	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
388	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
389	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
390	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
391	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
392	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
393	#124	—	2	16	no	—
394	#124	—	2	4	no	—
395	#124	C₂³×C₆₇	1	8	yes	—
396	C₂²⋊Q₈	—	2	4	no	—
397	C₂²⋊Q₈	—	2	16	no	—
398	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
399	#121	—	2	16	no	—
400	#121	—	2	4	no	—
401	#121	C₂³×C₆₇	1	8	yes	—
402	C₄×Q₈	—	2	4	no	—
403	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
404	C₄×D₄	—	2	16	no	—
405	C₄×D₄	D₄×C₆₇	1	4	yes	—
406	C₄×D₄	C₂³×C₆₇	1	4	yes	—
407	C₂×C₄⋊C₄	—	2	2	no	—
408	C₂×C₄⋊C₄	—	2	2	no	—
409	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
411	C₂×C₂²⋊C₄	—	2	2	no	—
412	C₂×C₂²⋊C₄	—	2	2	no	—
413	C₂×C₂²⋊C₄	—	2	2	no	—
414	C₂×C₂²⋊C₄	—	2	2	no	—
415	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
416	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
417	C₂×C₄²	—	2	2	no	—
418	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
419	#107	C₂³×C₆₇	1	4	yes	—
420	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
421	#103	C₂³×C₆₇	1	8	yes	—
422	#103	C₂³×C₆₇	1	8	yes	—
423	#102	C₂³×C₆₇	1	8	yes	—
424	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
425	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
426	C₄×C₈	C₂³×C₆₇	1	4	yes	—
427	C₂⁴	—	2	2	no	—
428	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
429	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
430	#41	—	2	16	no	—
431	#41	—	4	8	no	—
432	#41	—	4	16	no	—
433	C₂×Q₈	—	2	2	no	—
434	C₂×Q₈	—	2	8	no	—
435	C₂×Q₈	—	2	2	no	—
436	C₂×Q₈	—	2	4	no	—
437	C₂×Q₈	—	2	16	no	—
438	C₂×Q₈	—	2	4	no	—
439	C₂×Q₈	—	2	4	no	—
440	C₂×Q₈	—	4	8	no	—
441	C₂×Q₈	—	4	8	no	—
442	C₂×Q₈	—	4	16	no	—
443	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
444	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
445	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
446	C₂×D₄	—	2	4	no	—
447	C₂×D₄	—	2	8	no	—
448	C₂×D₄	—	2	8	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	4	8	no	—
453	C₂×D₄	—	4	8	no	—
454	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
455	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	8	no	—
464	C₂²×C₄	—	2	8	no	—
465	C₂²×C₄	—	4	8	no	—
466	C₂²×C₄	—	4	8	no	—
467	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
469	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
470	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
472	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
473	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
474	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
475	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
476	Q₁₆	—	2	16	no	—
477	Q₁₆	—	2	8	no	—
478	Q₁₆	C₂×D₄×C₆₇	1	16	yes	complemented
479	SD₁₆	—	2	16	no	—
480	SD₁₆	—	4	8	no	—
481	D₈	—	2	8	no	—
482	C₂×C₈	—	2	2	no	—
483	C₂×C₈	—	2	2	no	—
484	C₂×C₈	—	2	2	no	—
485	C₂×C₈	—	2	8	no	—
486	C₂×C₈	—	2	8	no	—
487	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
488	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
489	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
490	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
491	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
492	C₄⋊C₄	—	2	8	no	—
493	C₄⋊C₄	—	2	8	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
496	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
497	C₂²⋊C₄	—	2	8	no	—
498	C₂²⋊C₄	—	2	8	no	—
499	C₂²⋊C₄	—	2	8	no	—
500	C₂²⋊C₄	—	2	8	no	—
501	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
502	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
503	C₄²	—	2	8	no	—
504	C₄²	C₂×D₄×C₆₇	1	2	yes	—
505	C₄²	C₂⁴×C₆₇	1	2	yes	—
506	C₂³	—	2	2	no	—
507	C₂³	—	2	2	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	2	4	no	—
512	C₂³	—	4	4	no	—
513	C₂³	—	4	4	no	—
514	C₂³	—	4	4	no	—
515	C₂³	#135×C₆₇	1	4	yes	complemented
516	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
517	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
518	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
519	Q₈	—	2	8	no	—
520	Q₈	—	2	4	no	—
521	Q₈	—	2	4	no	—
522	Q₈	—	2	4	no	—
523	Q₈	—	4	16	no	—
524	Q₈	—	4	4	no	—
525	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
526	D₄	—	2	8	no	—
527	D₄	—	2	4	no	—
528	D₄	—	2	4	no	—
529	D₄	—	4	16	no	—
530	D₄	—	4	16	no	—
531	D₄	—	4	4	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	8	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	4	16	no	—
547	C₂×C₄	—	4	16	no	—
548	C₂×C₄	—	4	4	no	—
549	C₂×C₄	—	4	4	no	—
550	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
551	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
552	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
553	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
554	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
555	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
556	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
557	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
558	C₈	—	2	2	no	—
559	C₈	—	2	2	no	—
560	C₈	—	2	2	no	—
561	C₈	C₂²×D₄×C₆₇	1	4	yes	—
562	C₂²	—	2	8	no	—
563	C₂²	—	2	2	no	—
564	C₂²	—	2	2	no	—
565	C₂²	—	2	2	no	—
566	C₂²	—	2	2	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	4	4	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	4	4	no	—
572	C₂²	—	8	8	no	—
573	C₂²	D₄²×C₆₇	1	2	yes	—
574	C₂²	C₂×#135×C₆₇	1	2	yes	—
575	C₂²	C₂×#135×C₆₇	1	4	yes	complemented
576	C₂²	#578×C₆₇	1	2	yes	—
577	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
578	C₄	—	2	2	no	—
579	C₄	—	2	2	no	—
580	C₄	—	2	2	no	—
581	C₄	—	2	2	no	—
582	C₄	—	2	8	no	—
583	C₄	—	4	4	no	—
584	C₄	—	4	4	no	—
585	C₄	D₄²×C₆₇	1	2	yes	—
586	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
587	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
588	C₂	—	2	4	no	—
589	C₂	—	4	2	no	—
590	C₂	—	8	4	no	—
591	C₂	C₆₇×#3285	1	4	yes	direct product factor
592	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
593	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
594	C₂	C₂×#578×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#135×C₆₇	C₂	semidirect
2. C₂³×Q₁₆×C₆₇	C₂	semidirect
3. C₂×#548×C₆₇	C₂	semidirect
4. C₆₇×#3285	C₂	direct (N × H)
5. C₂×#495×C₆₇	C₂	semidirect
6. C₂×#467×C₆₇	C₂	semidirect
7. C₂×#466×C₆₇	C₂	semidirect
8. C₂×#464×C₆₇	C₂	semidirect
9. C₂×#452×C₆₇	C₂	semidirect
10. C₂×#448×C₆₇	C₂	semidirect
11. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
12. C₂×C₈×D₄×C₆₇	C₂	semidirect
16. C₂×#135×C₆₇	C₂²	semidirect
21. C₂²×Q₁₆×C₆₇	C₂²	semidirect
28. #548×C₆₇	C₂²	semidirect
38. #495×C₆₇	C₂²	semidirect
39. #467×C₆₇	C₂²	semidirect
40. #466×C₆₇	C₂²	semidirect
41. #464×C₆₇	C₂²	semidirect
42. #452×C₆₇	C₂²	semidirect
43. #448×C₆₇	C₂²	semidirect
44. C₄×Q₁₆×C₆₇	C₂²	semidirect
45. C₈×D₄×C₆₇	C₂²	semidirect
77. C₂×Q₁₆×C₆₇	D₄	semidirect
180. Q₁₆×C₆₇	C₂×D₄	semidirect
217. C₂³×C₆₇	#135	semidirect
277. C₂²×C₆₇	C₂×#135	semidirect
290. C₂×#3285	C₆₇	direct (N × H)
294. C₂×C₆₇	#3285	direct (N × H)
298. C₂²×#135	C₂×C₆₇	semidirect
299. C₂³×Q₁₆	C₂×C₆₇	semidirect
300. C₂×#548	C₂×C₆₇	semidirect
301. #3285	C₂×C₆₇	direct (N × H)
302. C₂×#495	C₂×C₆₇	semidirect
303. C₂×#467	C₂×C₆₇	semidirect
304. C₂×#466	C₂×C₆₇	semidirect
305. C₂×#464	C₂×C₆₇	semidirect
306. C₂×#452	C₂×C₆₇	semidirect
307. C₂×#448	C₂×C₆₇	semidirect
308. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
309. C₂×C₈×D₄	C₂×C₆₇	semidirect
310. C₆₇	C₂×#3285	direct (N × H)
314. C₂×#135	C₂²×C₆₇	semidirect
319. C₂²×Q₁₆	C₂²×C₆₇	semidirect
326. #548	C₂²×C₆₇	semidirect
336. #495	C₂²×C₆₇	semidirect
337. #467	C₂²×C₆₇	semidirect
338. #466	C₂²×C₆₇	semidirect
339. #464	C₂²×C₆₇	semidirect
340. #452	C₂²×C₆₇	semidirect
341. #448	C₂²×C₆₇	semidirect
342. C₄×Q₁₆	C₂²×C₆₇	semidirect
343. C₈×D₄	C₂²×C₆₇	semidirect
375. C₂×Q₁₆	D₄×C₆₇	semidirect
478. Q₁₆	C₂×D₄×C₆₇	semidirect
515. C₂³	#135×C₆₇	semidirect
575. C₂²	C₂×#135×C₆₇	semidirect
591. C₂	C₆₇×#3285	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×#135×C₆₇	2
C₂³×Q₁₆×C₆₇	2
C₂×#548×C₆₇	2
C₆₇×#3285	2
C₂×#495×C₆₇	2
C₂×#467×C₆₇	2
C₂×#466×C₆₇	2
C₂×#464×C₆₇	2
C₂×#452×C₆₇	2
C₂×#448×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×#3285	67

Subgroups — C2×C67×#3285 #30740296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3285 #30740296