Subgroups — C₆₇×#35085 #30714336

Order 17152.

2710 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#525×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2857	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2850	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2663	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2564	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2503	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2501	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2499	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2434	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#2432	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2412	C₂	1	4	yes	maximal, complemented
14	C₆₇×#2322	C₂	1	4	yes	maximal, complemented
15	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
20	C₂×#124×C₆₇	C₂²	1	2	yes	complemented
21	#525×C₆₇	—	2	8	no	—
22	C₂×#123×C₆₇	C₂²	1	4	yes	—
23	C₂×#123×C₆₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊Q₈×C₆₇	—	2	8	no	—
25	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
31	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#120×C₆₇	C₂²	1	2	yes	—
34	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
35	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
36	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
39	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
40	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
41	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
42	#400×C₆₇	C₂²	1	4	yes	complemented
43	#399×C₆₇	C₂²	1	4	yes	complemented
44	#397×C₆₇	C₂²	1	8	yes	—
45	#395×C₆₇	C₂²	1	4	yes	complemented
46	#394×C₆₇	C₂²	1	8	yes	—
47	#393×C₆₇	C₂²	1	4	yes	complemented
48	#391×C₆₇	C₂²	1	2	yes	complemented
49	#391×C₆₇	C₂²	1	2	yes	complemented
50	#391×C₆₇	C₂²	1	2	yes	complemented
51	#390×C₆₇	C₂²	1	2	yes	complemented
52	#389×C₆₇	C₂²	1	8	yes	complemented
53	#388×C₆₇	C₂²	1	8	yes	complemented
54	#387×C₆₇	C₂²	1	8	yes	complemented
55	#386×C₆₇	C₂²	1	8	yes	—
56	#386×C₆₇	C₂²	1	4	yes	—
57	#385×C₆₇	C₂²	1	4	yes	—
58	#385×C₆₇	C₂²	1	8	yes	complemented
59	#384×C₆₇	C₂²	1	2	yes	complemented
60	#382×C₆₇	C₂²	1	2	yes	complemented
61	#381×C₆₇	C₂²	1	4	yes	—
62	#380×C₆₇	C₂²	1	8	yes	—
63	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
64	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
65	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
66	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
67	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
68	C₂×#95×C₆₇	C₂²	1	2	yes	—
69	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×#41×C₆₇	—	2	4	no	—
71	C₂×#41×C₆₇	—	4	4	no	—
72	C₂²×Q₈×C₆₇	—	2	4	no	—
73	C₂²×Q₈×C₆₇	—	4	4	no	—
74	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
75	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
77	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
78	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
79	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
80	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
81	C₂³×C₄×C₆₇	—	4	4	no	—
82	C₂³×C₄×C₆₇	D₄	1	8	yes	—
83	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
84	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
85	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
89	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
90	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
91	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
92	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
93	C₄⋊Q₈×C₆₇	—	4	4	no	—
94	#124×C₆₇	—	4	4	no	—
95	#123×C₆₇	—	2	8	no	—
96	#123×C₆₇	—	2	16	no	—
97	C₂²⋊Q₈×C₆₇	—	2	8	no	—
98	C₂²⋊Q₈×C₆₇	—	2	8	no	—
99	C₂²⋊Q₈×C₆₇	—	2	8	no	—
100	C₂²⋊Q₈×C₆₇	—	2	8	no	—
101	C₂²⋊Q₈×C₆₇	—	2	4	no	—
102	C₂²⋊Q₈×C₆₇	—	2	4	no	—
103	#121×C₆₇	—	2	4	no	—
104	#121×C₆₇	—	2	4	no	—
105	#120×C₆₇	—	2	8	no	—
106	C₄×D₄×C₆₇	—	2	8	no	—
107	C₄×D₄×C₆₇	—	2	8	no	—
108	#117×C₆₇	—	2	4	no	—
109	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
110	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
118	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
121	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
122	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
123	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
124	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
125	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
126	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
127	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
128	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
129	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
130	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
131	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
132	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
133	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
134	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
135	C₂×C₄²×C₆₇	—	4	4	no	—
136	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
137	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
138	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
139	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
140	#95×C₆₇	C₂³	1	4	yes	—
141	#95×C₆₇	C₂³	1	2	yes	—
142	#95×C₆₇	C₂³	1	2	yes	—
143	#95×C₆₇	C₂³	1	8	yes	—
144	#95×C₆₇	C₂³	1	8	yes	—
145	#95×C₆₇	C₂³	1	8	yes	—
146	#95×C₆₇	C₂³	1	8	yes	—
147	C₂⁴×C₆₇	—	2	4	no	—
148	C₂⁴×C₆₇	—	2	4	no	—
149	C₂⁴×C₆₇	—	4	2	no	—
150	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
151	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
152	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
153	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
154	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
155	#41×C₆₇	—	4	16	no	—
156	C₂×Q₈×C₆₇	—	2	4	no	—
157	C₂×Q₈×C₆₇	—	2	4	no	—
158	C₂×Q₈×C₆₇	—	2	2	no	—
159	C₂×Q₈×C₆₇	—	2	4	no	—
160	C₂×Q₈×C₆₇	—	2	4	no	—
161	C₂×Q₈×C₆₇	—	2	4	no	—
162	C₂×Q₈×C₆₇	—	4	4	no	—
163	C₂×Q₈×C₆₇	—	4	16	no	—
164	C₂×D₄×C₆₇	—	2	4	no	—
165	C₂×D₄×C₆₇	—	2	4	no	—
166	C₂×D₄×C₆₇	—	2	4	no	—
167	C₂×D₄×C₆₇	—	2	8	no	—
168	C₂×D₄×C₆₇	—	2	8	no	—
169	C₂×D₄×C₆₇	—	2	2	no	—
170	C₂×D₄×C₆₇	—	2	4	no	—
171	C₂×D₄×C₆₇	—	4	4	no	—
172	C₂²×C₄×C₆₇	—	2	4	no	—
173	C₂²×C₄×C₆₇	—	2	4	no	—
174	C₂²×C₄×C₆₇	—	2	4	no	—
175	C₂²×C₄×C₆₇	—	2	4	no	—
176	C₂²×C₄×C₆₇	—	2	4	no	—
177	C₂²×C₄×C₆₇	—	2	8	no	—
178	C₂²×C₄×C₆₇	—	2	8	no	—
179	C₂²×C₄×C₆₇	—	2	4	no	—
180	C₂²×C₄×C₆₇	—	2	8	no	—
181	C₂²×C₄×C₆₇	—	4	16	no	—
182	C₂²×C₄×C₆₇	—	4	4	no	—
183	C₂²×C₄×C₆₇	—	4	2	no	—
184	C₂²×C₄×C₆₇	—	4	2	no	—
185	C₂²×C₄×C₆₇	—	4	2	no	—
186	C₂²×C₄×C₆₇	—	4	4	no	—
187	C₂²×C₄×C₆₇	—	4	4	no	—
188	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
189	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
190	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
191	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
192	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
193	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
194	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
195	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
196	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
197	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
198	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
199	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
201	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
202	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
203	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
204	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
205	C₄⋊C₄×C₆₇	—	2	4	no	—
206	C₄⋊C₄×C₆₇	—	2	4	no	—
207	C₄⋊C₄×C₆₇	—	2	8	no	—
208	C₄⋊C₄×C₆₇	—	2	8	no	—
209	C₄⋊C₄×C₆₇	—	2	2	no	—
210	C₄⋊C₄×C₆₇	—	2	2	no	—
211	C₄⋊C₄×C₆₇	—	2	4	no	—
212	C₄⋊C₄×C₆₇	—	2	4	no	—
213	C₄⋊C₄×C₆₇	—	2	16	no	—
214	C₄⋊C₄×C₆₇	—	4	8	no	—
215	C₂²⋊C₄×C₆₇	—	2	8	no	—
216	C₂²⋊C₄×C₆₇	—	2	8	no	—
217	C₂²⋊C₄×C₆₇	—	2	4	no	—
218	C₂²⋊C₄×C₆₇	—	2	4	no	—
219	C₂²⋊C₄×C₆₇	—	2	4	no	—
220	C₂²⋊C₄×C₆₇	—	2	4	no	—
221	C₂²⋊C₄×C₆₇	—	2	4	no	—
222	C₂²⋊C₄×C₆₇	—	2	4	no	—
223	C₂²⋊C₄×C₆₇	—	2	8	no	—
224	C₂²⋊C₄×C₆₇	—	2	16	no	—
225	C₂²⋊C₄×C₆₇	—	4	8	no	—
226	C₂²⋊C₄×C₆₇	#41	1	8	yes	complemented
227	C₄²×C₆₇	—	2	4	no	—
228	C₄²×C₆₇	—	2	4	no	—
229	C₄²×C₆₇	—	2	4	no	—
230	C₄²×C₆₇	—	4	4	no	—
231	C₂³×C₆₇	—	2	2	no	—
232	C₂³×C₆₇	—	2	2	no	—
233	C₂³×C₆₇	—	2	2	no	—
234	C₂³×C₆₇	—	2	2	no	—
235	C₂³×C₆₇	—	2	8	no	—
236	C₂³×C₆₇	—	2	8	no	—
237	C₂³×C₆₇	—	2	2	no	—
238	C₂³×C₆₇	—	2	2	no	—
239	C₂³×C₆₇	—	2	2	no	—
240	C₂³×C₆₇	—	2	2	no	—
241	C₂³×C₆₇	—	2	4	no	—
242	C₂³×C₆₇	—	2	4	no	—
243	C₂³×C₆₇	—	2	4	no	—
244	C₂³×C₆₇	—	4	4	no	—
245	C₂³×C₆₇	—	4	16	no	—
246	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
247	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
248	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
249	C₂³×C₆₇	C₂×#41	1	4	yes	—
250	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
251	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
252	Q₈×C₆₇	—	4	8	no	—
253	Q₈×C₆₇	—	4	8	no	—
254	Q₈×C₆₇	—	4	4	no	—
255	D₄×C₆₇	—	4	8	no	—
256	D₄×C₆₇	—	4	8	no	—
257	D₄×C₆₇	—	4	8	no	—
258	D₄×C₆₇	—	4	4	no	—
259	C₂×C₄×C₆₇	—	2	4	no	—
260	C₂×C₄×C₆₇	—	2	2	no	—
261	C₂×C₄×C₆₇	—	2	2	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	4	no	—
264	C₂×C₄×C₆₇	—	2	8	no	—
265	C₂×C₄×C₆₇	—	2	8	no	—
266	C₂×C₄×C₆₇	—	2	8	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	—	2	2	no	—
269	C₂×C₄×C₆₇	—	2	4	no	—
270	C₂×C₄×C₆₇	—	2	2	no	—
271	C₂×C₄×C₆₇	—	2	4	no	—
272	C₂×C₄×C₆₇	—	2	4	no	—
273	C₂×C₄×C₆₇	—	2	8	no	—
274	C₂×C₄×C₆₇	—	2	4	no	—
275	C₂×C₄×C₆₇	—	2	4	no	—
276	C₂×C₄×C₆₇	—	2	4	no	—
277	C₂×C₄×C₆₇	—	2	4	no	—
278	C₂×C₄×C₆₇	—	2	4	no	—
279	C₂×C₄×C₆₇	—	2	4	no	—
280	C₂×C₄×C₆₇	—	2	4	no	—
281	C₂×C₄×C₆₇	—	4	4	no	—
282	C₂×C₄×C₆₇	—	4	4	no	—
283	C₂×C₄×C₆₇	—	4	4	no	—
284	C₂×C₄×C₆₇	—	4	16	no	—
285	C₂×C₄×C₆₇	—	4	8	no	—
286	C₂×C₄×C₆₇	—	4	8	no	—
287	C₂×C₄×C₆₇	—	4	8	no	—
288	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
289	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
290	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
291	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
292	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
293	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
294	C₂²×C₆₇	—	2	2	no	—
295	C₂²×C₆₇	—	2	2	no	—
296	C₂²×C₆₇	—	2	2	no	—
297	C₂²×C₆₇	—	2	2	no	—
298	C₂²×C₆₇	—	2	8	no	—
299	C₂²×C₆₇	—	2	8	no	—
300	C₂²×C₆₇	—	2	2	no	—
301	C₂²×C₆₇	—	2	2	no	—
302	C₂²×C₆₇	—	2	2	no	—
303	C₂²×C₆₇	—	4	16	no	—
304	C₂²×C₆₇	—	4	4	no	—
305	C₂²×C₆₇	—	4	2	no	—
306	C₂²×C₆₇	—	4	2	no	—
307	C₂²×C₆₇	—	4	2	no	—
308	C₂²×C₆₇	—	4	8	no	—
309	C₂²×C₆₇	#532	1	4	yes	—
310	C₂²×C₆₇	#536	1	4	yes	—
311	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
312	C₂²×C₆₇	C₂²×#41	1	2	yes	—
313	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
314	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
315	C₄×C₆₇	—	2	2	no	—
316	C₄×C₆₇	—	2	2	no	—
317	C₄×C₆₇	—	2	2	no	—
318	C₄×C₆₇	—	2	8	no	—
319	C₄×C₆₇	—	4	4	no	—
320	C₄×C₆₇	—	4	2	no	—
321	C₄×C₆₇	—	4	2	no	—
322	C₄×C₆₇	—	4	4	no	—
323	C₄×C₆₇	—	4	2	no	—
324	C₄×C₆₇	—	4	4	no	—
325	C₄×C₆₇	#534	1	4	yes	—
326	#35085	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
327	C₂×C₆₇	—	2	4	no	—
328	C₂×C₆₇	—	2	4	no	—
329	C₂×C₆₇	—	4	2	no	—
330	C₂×C₆₇	—	8	2	no	—
331	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
332	C₂×C₆₇	C₂×#534	1	1	yes	characteristic
333	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
334	C₂×C₆₇	#3520	1	4	yes	—
335	C₂×#525	C₂×C₆₇	1	1	yes	characteristic
336	C₂²×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
337	#2857	C₂×C₆₇	1	2	yes	complemented
338	#2850	C₂×C₆₇	1	2	yes	complemented
339	#2663	C₂×C₆₇	1	4	yes	complemented
340	#2564	C₂×C₆₇	1	4	yes	complemented
341	#2503	C₂×C₆₇	1	1	yes	characteristic, complemented
342	#2501	C₂×C₆₇	1	1	yes	characteristic, complemented
343	#2499	C₂×C₆₇	1	2	yes	complemented
344	#2436	C₂×C₆₇	1	1	yes	characteristic, complemented
345	#2434	C₂×C₆₇	1	1	yes	characteristic, complemented
346	#2432	C₂×C₆₇	1	2	yes	complemented
347	#2412	C₂×C₆₇	1	4	yes	complemented
348	#2322	C₂×C₆₇	1	4	yes	complemented
349	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
350	C₆₇	#35085	1	1	yes	characteristic, direct product factor, Sylow
351	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
352	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
353	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
354	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
355	C₂×#124	C₂²×C₆₇	1	2	yes	complemented
356	#525	—	2	8	no	—
357	C₂×#123	C₂²×C₆₇	1	4	yes	—
358	C₂×#123	C₂²×C₆₇	1	2	yes	—
359	C₂×C₂²⋊Q₈	—	2	8	no	—
360	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
361	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
362	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
363	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
364	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
365	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
366	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
367	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
368	C₂×#120	C₂²×C₆₇	1	2	yes	—
369	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
370	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
371	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
372	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
373	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
374	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
375	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
376	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
377	#400	C₂²×C₆₇	1	4	yes	complemented
378	#399	C₂²×C₆₇	1	4	yes	complemented
379	#397	C₂²×C₆₇	1	8	yes	—
380	#395	C₂²×C₆₇	1	4	yes	complemented
381	#394	C₂²×C₆₇	1	8	yes	—
382	#393	C₂²×C₆₇	1	4	yes	complemented
383	#391	C₂²×C₆₇	1	2	yes	complemented
384	#391	C₂²×C₆₇	1	2	yes	complemented
385	#391	C₂²×C₆₇	1	2	yes	complemented
386	#390	C₂²×C₆₇	1	2	yes	complemented
387	#389	C₂²×C₆₇	1	8	yes	complemented
388	#388	C₂²×C₆₇	1	8	yes	complemented
389	#387	C₂²×C₆₇	1	8	yes	complemented
390	#386	C₂²×C₆₇	1	8	yes	—
391	#386	C₂²×C₆₇	1	4	yes	—
392	#385	C₂²×C₆₇	1	4	yes	—
393	#385	C₂²×C₆₇	1	8	yes	complemented
394	#384	C₂²×C₆₇	1	2	yes	complemented
395	#382	C₂²×C₆₇	1	2	yes	complemented
396	#381	C₂²×C₆₇	1	4	yes	—
397	#380	C₂²×C₆₇	1	8	yes	—
398	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
399	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
400	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
401	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
402	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
403	C₂×#95	C₂²×C₆₇	1	2	yes	—
404	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
405	C₂×#41	—	2	4	no	—
406	C₂×#41	—	4	4	no	—
407	C₂²×Q₈	—	2	4	no	—
408	C₂²×Q₈	—	4	4	no	—
409	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
410	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
411	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
412	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
413	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
414	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
415	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
416	C₂³×C₄	—	4	4	no	—
417	C₂³×C₄	D₄×C₆₇	1	8	yes	—
418	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
419	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
420	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
421	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
422	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
423	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
424	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
425	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
426	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
427	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
428	C₄⋊Q₈	—	4	4	no	—
429	#124	—	4	4	no	—
430	#123	—	2	8	no	—
431	#123	—	2	16	no	—
432	C₂²⋊Q₈	—	2	8	no	—
433	C₂²⋊Q₈	—	2	8	no	—
434	C₂²⋊Q₈	—	2	8	no	—
435	C₂²⋊Q₈	—	2	8	no	—
436	C₂²⋊Q₈	—	2	4	no	—
437	C₂²⋊Q₈	—	2	4	no	—
438	#121	—	2	4	no	—
439	#121	—	2	4	no	—
440	#120	—	2	8	no	—
441	C₄×D₄	—	2	8	no	—
442	C₄×D₄	—	2	8	no	—
443	#117	—	2	4	no	—
444	C₂×C₄⋊C₄	—	2	4	no	—
445	C₂×C₄⋊C₄	—	2	8	no	—
446	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
447	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
448	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
449	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
450	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
451	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
452	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
453	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
454	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
455	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
456	C₂×C₂²⋊C₄	—	2	8	no	—
457	C₂×C₂²⋊C₄	—	2	8	no	—
458	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
459	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
460	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
461	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
462	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
463	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
464	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
465	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
466	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
467	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
468	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
469	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
470	C₂×C₄²	—	4	4	no	—
471	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
472	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
473	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
474	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
475	#95	C₂³×C₆₇	1	4	yes	—
476	#95	C₂³×C₆₇	1	2	yes	—
477	#95	C₂³×C₆₇	1	2	yes	—
478	#95	C₂³×C₆₇	1	8	yes	—
479	#95	C₂³×C₆₇	1	8	yes	—
480	#95	C₂³×C₆₇	1	8	yes	—
481	#95	C₂³×C₆₇	1	8	yes	—
482	C₂⁴	—	2	4	no	—
483	C₂⁴	—	2	4	no	—
484	C₂⁴	—	4	2	no	—
485	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
486	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
487	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
488	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
489	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
490	#41	—	4	16	no	—
491	C₂×Q₈	—	2	4	no	—
492	C₂×Q₈	—	2	4	no	—
493	C₂×Q₈	—	2	2	no	—
494	C₂×Q₈	—	2	4	no	—
495	C₂×Q₈	—	2	4	no	—
496	C₂×Q₈	—	2	4	no	—
497	C₂×Q₈	—	4	4	no	—
498	C₂×Q₈	—	4	16	no	—
499	C₂×D₄	—	2	4	no	—
500	C₂×D₄	—	2	4	no	—
501	C₂×D₄	—	2	4	no	—
502	C₂×D₄	—	2	8	no	—
503	C₂×D₄	—	2	8	no	—
504	C₂×D₄	—	2	2	no	—
505	C₂×D₄	—	2	4	no	—
506	C₂×D₄	—	4	4	no	—
507	C₂²×C₄	—	2	4	no	—
508	C₂²×C₄	—	2	4	no	—
509	C₂²×C₄	—	2	4	no	—
510	C₂²×C₄	—	2	4	no	—
511	C₂²×C₄	—	2	4	no	—
512	C₂²×C₄	—	2	8	no	—
513	C₂²×C₄	—	2	8	no	—
514	C₂²×C₄	—	2	4	no	—
515	C₂²×C₄	—	2	8	no	—
516	C₂²×C₄	—	4	16	no	—
517	C₂²×C₄	—	4	4	no	—
518	C₂²×C₄	—	4	2	no	—
519	C₂²×C₄	—	4	2	no	—
520	C₂²×C₄	—	4	2	no	—
521	C₂²×C₄	—	4	4	no	—
522	C₂²×C₄	—	4	4	no	—
523	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
524	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
525	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
526	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
527	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
528	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
529	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
530	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
531	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
532	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
533	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
534	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
535	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
536	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
537	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
538	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
539	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
540	C₄⋊C₄	—	2	4	no	—
541	C₄⋊C₄	—	2	4	no	—
542	C₄⋊C₄	—	2	8	no	—
543	C₄⋊C₄	—	2	8	no	—
544	C₄⋊C₄	—	2	2	no	—
545	C₄⋊C₄	—	2	2	no	—
546	C₄⋊C₄	—	2	4	no	—
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	16	no	—
549	C₄⋊C₄	—	4	8	no	—
550	C₂²⋊C₄	—	2	8	no	—
551	C₂²⋊C₄	—	2	8	no	—
552	C₂²⋊C₄	—	2	4	no	—
553	C₂²⋊C₄	—	2	4	no	—
554	C₂²⋊C₄	—	2	4	no	—
555	C₂²⋊C₄	—	2	4	no	—
556	C₂²⋊C₄	—	2	4	no	—
557	C₂²⋊C₄	—	2	4	no	—
558	C₂²⋊C₄	—	2	8	no	—
559	C₂²⋊C₄	—	2	16	no	—
560	C₂²⋊C₄	—	4	8	no	—
561	C₂²⋊C₄	#41×C₆₇	1	8	yes	complemented
562	C₄²	—	2	4	no	—
563	C₄²	—	2	4	no	—
564	C₄²	—	2	4	no	—
565	C₄²	—	4	4	no	—
566	C₂³	—	2	2	no	—
567	C₂³	—	2	2	no	—
568	C₂³	—	2	2	no	—
569	C₂³	—	2	2	no	—
570	C₂³	—	2	8	no	—
571	C₂³	—	2	8	no	—
572	C₂³	—	2	2	no	—
573	C₂³	—	2	2	no	—
574	C₂³	—	2	2	no	—
575	C₂³	—	2	2	no	—
576	C₂³	—	2	4	no	—
577	C₂³	—	2	4	no	—
578	C₂³	—	2	4	no	—
579	C₂³	—	4	4	no	—
580	C₂³	—	4	16	no	—
581	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
582	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
583	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
584	C₂³	C₂×#41×C₆₇	1	4	yes	—
585	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
586	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
587	Q₈	—	4	8	no	—
588	Q₈	—	4	8	no	—
589	Q₈	—	4	4	no	—
590	D₄	—	4	8	no	—
591	D₄	—	4	8	no	—
592	D₄	—	4	8	no	—
593	D₄	—	4	4	no	—
594	C₂×C₄	—	2	4	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	4	no	—
599	C₂×C₄	—	2	8	no	—
600	C₂×C₄	—	2	8	no	—
601	C₂×C₄	—	2	8	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	4	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	8	no	—
609	C₂×C₄	—	2	4	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	4	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	2	4	no	—
614	C₂×C₄	—	2	4	no	—
615	C₂×C₄	—	2	4	no	—
616	C₂×C₄	—	4	4	no	—
617	C₂×C₄	—	4	4	no	—
618	C₂×C₄	—	4	4	no	—
619	C₂×C₄	—	4	16	no	—
620	C₂×C₄	—	4	8	no	—
621	C₂×C₄	—	4	8	no	—
622	C₂×C₄	—	4	8	no	—
623	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
624	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
625	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
626	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
627	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
628	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	2	no	—
631	C₂²	—	2	2	no	—
632	C₂²	—	2	2	no	—
633	C₂²	—	2	8	no	—
634	C₂²	—	2	8	no	—
635	C₂²	—	2	2	no	—
636	C₂²	—	2	2	no	—
637	C₂²	—	2	2	no	—
638	C₂²	—	4	16	no	—
639	C₂²	—	4	4	no	—
640	C₂²	—	4	2	no	—
641	C₂²	—	4	2	no	—
642	C₂²	—	4	2	no	—
643	C₂²	—	4	8	no	—
644	C₂²	#532×C₆₇	1	4	yes	—
645	C₂²	#536×C₆₇	1	4	yes	—
646	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
647	C₂²	C₂²×#41×C₆₇	1	2	yes	—
648	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
649	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
650	C₄	—	2	2	no	—
651	C₄	—	2	2	no	—
652	C₄	—	2	2	no	—
653	C₄	—	2	8	no	—
654	C₄	—	4	4	no	—
655	C₄	—	4	2	no	—
656	C₄	—	4	2	no	—
657	C₄	—	4	4	no	—
658	C₄	—	4	2	no	—
659	C₄	—	4	4	no	—
660	C₄	#534×C₆₇	1	4	yes	—
661	C₂	—	2	4	no	—
662	C₂	—	2	4	no	—
663	C₂	—	4	2	no	—
664	C₂	—	8	2	no	—
665	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
666	C₂	C₂×#534×C₆₇	1	1	yes	characteristic
667	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
668	C₂	C₆₇×#3520	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#525×C₆₇	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂²⋊Q₈×C₆₇	C₂	semidirect
3. C₆₇×#2857	C₂	semidirect
4. C₆₇×#2850	C₂	semidirect
5. C₆₇×#2663	C₂	semidirect
6. C₆₇×#2564	C₂	semidirect
7. C₆₇×#2503	C₂	semidirect
8. C₆₇×#2501	C₂	semidirect
9. C₆₇×#2499	C₂	semidirect
10. C₆₇×#2436	C₂	semidirect
11. C₆₇×#2434	C₂	semidirect
12. C₆₇×#2432	C₂	semidirect
13. C₆₇×#2412	C₂	semidirect
14. C₆₇×#2322	C₂	semidirect
15. C₂×C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
19. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
20. C₂×#124×C₆₇	C₂²	semidirect
42. #400×C₆₇	C₂²	semidirect
43. #399×C₆₇	C₂²	semidirect
45. #395×C₆₇	C₂²	semidirect
47. #393×C₆₇	C₂²	semidirect
48. #391×C₆₇	C₂²	semidirect
49. #391×C₆₇	C₂²	semidirect
50. #391×C₆₇	C₂²	semidirect
51. #390×C₆₇	C₂²	semidirect
52. #389×C₆₇	C₂²	semidirect
53. #388×C₆₇	C₂²	semidirect
54. #387×C₆₇	C₂²	semidirect
58. #385×C₆₇	C₂²	semidirect
59. #384×C₆₇	C₂²	semidirect
60. #382×C₆₇	C₂²	semidirect
226. C₂²⋊C₄×C₆₇	#41	semidirect
326. #35085	C₆₇	direct (N × H)
336. C₂²×C₂²⋊Q₈	C₂×C₆₇	semidirect
337. #2857	C₂×C₆₇	semidirect
338. #2850	C₂×C₆₇	semidirect
339. #2663	C₂×C₆₇	semidirect
340. #2564	C₂×C₆₇	semidirect
341. #2503	C₂×C₆₇	semidirect
342. #2501	C₂×C₆₇	semidirect
343. #2499	C₂×C₆₇	semidirect
344. #2436	C₂×C₆₇	semidirect
345. #2434	C₂×C₆₇	semidirect
346. #2432	C₂×C₆₇	semidirect
347. #2412	C₂×C₆₇	semidirect
348. #2322	C₂×C₆₇	semidirect
349. C₂×C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
350. C₆₇	#35085	direct (N × H)
354. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
355. C₂×#124	C₂²×C₆₇	semidirect
377. #400	C₂²×C₆₇	semidirect
378. #399	C₂²×C₆₇	semidirect
380. #395	C₂²×C₆₇	semidirect
382. #393	C₂²×C₆₇	semidirect
383. #391	C₂²×C₆₇	semidirect
384. #391	C₂²×C₆₇	semidirect
385. #391	C₂²×C₆₇	semidirect
386. #390	C₂²×C₆₇	semidirect
387. #389	C₂²×C₆₇	semidirect
388. #388	C₂²×C₆₇	semidirect
389. #387	C₂²×C₆₇	semidirect
393. #385	C₂²×C₆₇	semidirect
394. #384	C₂²×C₆₇	semidirect
395. #382	C₂²×C₆₇	semidirect
561. C₂²⋊C₄	#41×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#525×C₆₇	2
C₂²×C₂²⋊Q₈×C₆₇	2
C₆₇×#2857	2
C₆₇×#2850	2
C₆₇×#2663	2
C₆₇×#2564	2
C₆₇×#2503	2
C₆₇×#2501	2
C₆₇×#2499	2
C₆₇×#2436	2
C₆₇×#2434	2
C₆₇×#2432	2
C₆₇×#2412	2
C₆₇×#2322	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
#35085	67

Subgroups — C67×#35085 #30714336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#35085 #30714336