Subgroups — C₆₇×#34965 #30714216

Order 17152.

3114 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2603	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2566	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2544	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2504	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2502	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2435	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2433	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#2411	C₂	1	4	yes	maximal, complemented
11	C₆₇×#2392	C₂	1	4	yes	maximal, complemented
12	C₄×#127×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₄×#124×C₆₇	C₂	1	2	yes	maximal, complemented
14	C₄×#120×C₆₇	C₂	1	4	yes	maximal, complemented
15	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
18	C₂×#127×C₆₇	C₂²	1	2	yes	—
19	#529×C₆₇	—	2	8	no	—
20	C₂×#126×C₆₇	C₂²	1	2	yes	complemented
21	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
22	C₂×#124×C₆₇	C₂²	1	2	yes	—
23	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic, complemented
24	C₂×#123×C₆₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#120×C₆₇	C₂²	1	4	yes	—
28	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₄×D₄×C₆₇	—	2	8	no	—
30	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
34	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
35	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	complemented
37	C₂×#117×C₆₇	C₂²	1	2	yes	—
38	C₂×#117×C₆₇	C₂²	1	2	yes	complemented
39	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
40	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
41	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
42	#398×C₆₇	C₂²	1	4	yes	complemented
43	#397×C₆₇	C₂²	1	4	yes	complemented
44	#396×C₆₇	C₂²	1	8	yes	complemented
45	#394×C₆₇	C₂²	1	4	yes	—
46	#394×C₆₇	C₂²	1	8	yes	complemented
47	#392×C₆₇	C₂²	1	4	yes	—
48	#391×C₆₇	C₂²	1	2	yes	complemented
49	#390×C₆₇	C₂²	1	2	yes	—
50	#390×C₆₇	C₂²	1	2	yes	—
51	#390×C₆₇	C₂²	1	2	yes	complemented
52	#390×C₆₇	C₂²	1	8	yes	complemented
53	#388×C₆₇	C₂²	1	8	yes	complemented
54	#386×C₆₇	C₂²	1	4	yes	—
55	#386×C₆₇	C₂²	1	8	yes	complemented
56	#385×C₆₇	C₂²	1	4	yes	—
57	#385×C₆₇	C₂²	1	8	yes	complemented
58	#384×C₆₇	C₂²	1	2	yes	complemented
59	#382×C₆₇	C₂²	1	2	yes	complemented
60	#379×C₆₇	C₂²	1	4	yes	—
61	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
62	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
63	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
64	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
65	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
66	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
67	#376×C₆₇	C₂²	1	2	yes	complemented
68	C₄³×C₆₇	C₂²	1	2	yes	complemented
69	C₂⁵×C₆₇	C₂³	1	2	yes	—
70	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×D₄×C₆₇	—	2	8	no	—
72	C₂²×D₄×C₆₇	—	2	4	no	—
73	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
74	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
75	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
76	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
77	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
78	C₂³×C₄×C₆₇	—	2	8	no	—
79	C₂³×C₄×C₆₇	—	4	8	no	—
80	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
81	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
82	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
86	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
87	#127×C₆₇	—	2	8	no	—
88	#126×C₆₇	—	2	8	no	—
89	#125×C₆₇	—	2	4	no	—
90	#124×C₆₇	—	2	8	no	—
91	#124×C₆₇	—	2	4	no	—
92	#123×C₆₇	—	2	8	no	—
93	C₂²⋊Q₈×C₆₇	—	2	4	no	—
94	#121×C₆₇	—	2	4	no	—
95	#120×C₆₇	—	2	16	no	—
96	C₄×Q₈×C₆₇	—	2	4	no	—
97	C₄×D₄×C₆₇	—	2	8	no	—
98	C₄×D₄×C₆₇	—	2	8	no	—
99	C₄×D₄×C₆₇	—	2	8	no	—
100	C₄×D₄×C₆₇	—	2	8	no	—
101	C₄×D₄×C₆₇	—	2	8	no	—
102	C₄×D₄×C₆₇	—	2	4	no	—
103	C₄×D₄×C₆₇	—	4	16	no	—
104	#117×C₆₇	—	2	8	no	—
105	#117×C₆₇	—	2	8	no	—
106	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
117	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
118	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
121	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
123	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
125	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
126	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
127	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
128	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
129	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
130	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄²×C₆₇	D₄	1	8	yes	complemented
132	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
133	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
134	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
135	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
136	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
137	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
138	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
140	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
141	#95×C₆₇	C₂³	1	4	yes	—
142	#95×C₆₇	C₂³	1	2	yes	—
143	#95×C₆₇	C₂³	1	2	yes	—
144	#95×C₆₇	C₂³	1	8	yes	—
145	#95×C₆₇	C₂³	1	8	yes	—
146	C₂⁴×C₆₇	—	2	4	no	—
147	C₂⁴×C₆₇	—	2	8	no	—
148	C₂⁴×C₆₇	—	2	8	no	—
149	C₂⁴×C₆₇	—	4	4	no	—
150	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
151	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
152	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
153	C₂×Q₈×C₆₇	—	2	2	no	—
154	C₂×Q₈×C₆₇	—	2	4	no	—
155	C₂×D₄×C₆₇	—	2	32	no	—
156	C₂×D₄×C₆₇	—	2	8	no	—
157	C₂×D₄×C₆₇	—	2	4	no	—
158	C₂×D₄×C₆₇	—	2	8	no	—
159	C₂×D₄×C₆₇	—	2	16	no	—
160	C₂×D₄×C₆₇	—	2	4	no	—
161	C₂×D₄×C₆₇	—	2	4	no	—
162	C₂×D₄×C₆₇	—	2	2	no	—
163	C₂×D₄×C₆₇	—	2	4	no	—
164	C₂×D₄×C₆₇	—	2	16	no	—
165	C₂×D₄×C₆₇	—	2	4	no	—
166	C₂×D₄×C₆₇	—	4	8	no	—
167	C₂×D₄×C₆₇	#41	1	8	yes	—
168	C₂²×C₄×C₆₇	—	2	4	no	—
169	C₂²×C₄×C₆₇	—	2	4	no	—
170	C₂²×C₄×C₆₇	—	2	4	no	—
171	C₂²×C₄×C₆₇	—	2	4	no	—
172	C₂²×C₄×C₆₇	—	2	4	no	—
173	C₂²×C₄×C₆₇	—	2	8	no	—
174	C₂²×C₄×C₆₇	—	2	8	no	—
175	C₂²×C₄×C₆₇	—	2	8	no	—
176	C₂²×C₄×C₆₇	—	2	16	no	—
177	C₂²×C₄×C₆₇	—	4	8	no	—
178	C₂²×C₄×C₆₇	—	4	4	no	—
179	C₂²×C₄×C₆₇	—	4	32	no	—
180	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
181	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
182	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
183	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
184	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
186	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
187	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
188	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
191	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
192	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
193	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
194	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
195	C₄⋊C₄×C₆₇	—	2	4	no	—
196	C₄⋊C₄×C₆₇	—	2	4	no	—
197	C₄⋊C₄×C₆₇	—	2	4	no	—
198	C₄⋊C₄×C₆₇	—	2	4	no	—
199	C₄⋊C₄×C₆₇	—	2	2	no	—
200	C₄⋊C₄×C₆₇	—	2	4	no	—
201	C₄⋊C₄×C₆₇	—	2	2	no	—
202	C₄⋊C₄×C₆₇	—	2	4	no	—
203	C₄⋊C₄×C₆₇	—	2	8	no	—
204	C₄⋊C₄×C₆₇	—	4	8	no	—
205	C₂²⋊C₄×C₆₇	—	2	4	no	—
206	C₂²⋊C₄×C₆₇	—	2	4	no	—
207	C₂²⋊C₄×C₆₇	—	2	16	no	—
208	C₂²⋊C₄×C₆₇	—	2	8	no	—
209	C₂²⋊C₄×C₆₇	—	2	8	no	—
210	C₂²⋊C₄×C₆₇	—	2	4	no	—
211	C₂²⋊C₄×C₆₇	—	2	4	no	—
212	C₂²⋊C₄×C₆₇	—	2	4	no	—
213	C₂²⋊C₄×C₆₇	—	2	4	no	—
214	C₂²⋊C₄×C₆₇	—	2	16	no	—
215	C₂²⋊C₄×C₆₇	—	4	8	no	—
216	C₂²⋊C₄×C₆₇	—	4	8	no	—
217	C₂²⋊C₄×C₆₇	#41	1	8	yes	—
218	C₄²×C₆₇	—	2	4	no	—
219	C₄²×C₆₇	—	2	8	no	—
220	C₄²×C₆₇	—	2	4	no	—
221	C₄²×C₆₇	—	2	4	no	—
222	C₄²×C₆₇	—	2	8	no	—
223	C₄²×C₆₇	C₂×D₄	1	4	yes	—
224	C₄²×C₆₇	C₂×D₄	1	4	yes	—
225	C₄²×C₆₇	C₂×D₄	1	4	yes	—
226	C₄²×C₆₇	C₂×D₄	1	4	yes	—
227	C₂³×C₆₇	—	2	2	no	—
228	C₂³×C₆₇	—	2	2	no	—
229	C₂³×C₆₇	—	2	16	no	—
230	C₂³×C₆₇	—	2	4	no	—
231	C₂³×C₆₇	—	2	8	no	—
232	C₂³×C₆₇	—	2	4	no	—
233	C₂³×C₆₇	—	2	8	no	—
234	C₂³×C₆₇	—	2	2	no	—
235	C₂³×C₆₇	—	2	4	no	—
236	C₂³×C₆₇	—	2	4	no	—
237	C₂³×C₆₇	—	2	4	no	—
238	C₂³×C₆₇	—	4	32	no	—
239	C₂³×C₆₇	—	4	8	no	—
240	C₂³×C₆₇	—	4	8	no	—
241	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
242	C₂³×C₆₇	C₂×#41	1	8	yes	—
243	C₂³×C₆₇	C₂×#41	1	4	yes	—
244	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
245	Q₈×C₆₇	—	4	4	no	—
246	D₄×C₆₇	—	2	16	no	—
247	D₄×C₆₇	—	2	16	no	—
248	D₄×C₆₇	—	4	32	no	—
249	D₄×C₆₇	—	4	8	no	—
250	D₄×C₆₇	—	4	4	no	—
251	C₂×C₄×C₆₇	—	2	8	no	—
252	C₂×C₄×C₆₇	—	2	2	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	4	no	—
257	C₂×C₄×C₆₇	—	2	2	no	—
258	C₂×C₄×C₆₇	—	2	8	no	—
259	C₂×C₄×C₆₇	—	2	8	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	4	no	—
265	C₂×C₄×C₆₇	—	2	4	no	—
266	C₂×C₄×C₆₇	—	2	4	no	—
267	C₂×C₄×C₆₇	—	2	4	no	—
268	C₂×C₄×C₆₇	—	2	4	no	—
269	C₂×C₄×C₆₇	—	2	4	no	—
270	C₂×C₄×C₆₇	—	2	8	no	—
271	C₂×C₄×C₆₇	—	4	16	no	—
272	C₂×C₄×C₆₇	—	4	8	no	—
273	C₂×C₄×C₆₇	—	4	8	no	—
274	C₂×C₄×C₆₇	—	4	16	no	—
275	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
276	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
277	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
278	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
279	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
280	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
281	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
282	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
283	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
284	C₂²×C₆₇	—	2	2	no	—
285	C₂²×C₆₇	—	2	2	no	—
286	C₂²×C₆₇	—	2	8	no	—
287	C₂²×C₆₇	—	2	8	no	—
288	C₂²×C₆₇	—	2	16	no	—
289	C₂²×C₆₇	—	4	32	no	—
290	C₂²×C₆₇	—	4	4	no	—
291	C₂²×C₆₇	—	4	4	no	—
292	C₂²×C₆₇	—	4	8	no	—
293	C₂²×C₆₇	—	4	2	no	—
294	C₂²×C₆₇	—	4	2	no	—
295	C₂²×C₆₇	—	4	2	no	—
296	C₂²×C₆₇	—	4	8	no	—
297	C₂²×C₆₇	#532	1	4	yes	—
298	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
299	C₂²×C₆₇	C₂²×#41	1	2	yes	—
300	C₂²×C₆₇	C₂²×#41	1	2	yes	—
301	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
302	C₄×C₆₇	—	2	2	no	—
303	C₄×C₆₇	—	2	2	no	—
304	C₄×C₆₇	—	2	2	no	—
305	C₄×C₆₇	—	2	2	no	—
306	C₄×C₆₇	—	2	8	no	—
307	C₄×C₆₇	—	4	2	no	—
308	C₄×C₆₇	—	4	2	no	—
309	C₄×C₆₇	—	4	2	no	—
310	C₄×C₆₇	—	4	4	no	—
311	C₄×C₆₇	—	4	4	no	—
312	C₄×C₆₇	#525	1	4	yes	—
313	C₄×C₆₇	#535	1	4	yes	—
314	#34965	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
315	C₂×C₆₇	—	2	4	no	—
316	C₂×C₆₇	—	4	8	no	—
317	C₂×C₆₇	—	8	2	no	—
318	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
319	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
320	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
321	C₂×C₆₇	#3481	1	4	yes	—
322	C₂×#529	C₂×C₆₇	1	1	yes	characteristic, complemented
323	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
324	#2603	C₂×C₆₇	1	4	yes	complemented
325	#2566	C₂×C₆₇	1	2	yes	complemented
326	#2544	C₂×C₆₇	1	2	yes	complemented
327	#2504	C₂×C₆₇	1	1	yes	characteristic, complemented
328	#2502	C₂×C₆₇	1	1	yes	characteristic, complemented
329	#2435	C₂×C₆₇	1	1	yes	characteristic, complemented
330	#2433	C₂×C₆₇	1	1	yes	characteristic
331	#2411	C₂×C₆₇	1	4	yes	complemented
332	#2392	C₂×C₆₇	1	4	yes	complemented
333	C₄×#127	C₂×C₆₇	1	2	yes	complemented
334	C₄×#124	C₂×C₆₇	1	2	yes	complemented
335	C₄×#120	C₂×C₆₇	1	4	yes	complemented
336	C₄×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
337	C₆₇	#34965	1	1	yes	characteristic, direct product factor, Sylow
338	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
339	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
340	C₂×#127	C₂²×C₆₇	1	2	yes	—
341	#529	—	2	8	no	—
342	C₂×#126	C₂²×C₆₇	1	2	yes	complemented
343	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
344	C₂×#124	C₂²×C₆₇	1	2	yes	—
345	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic, complemented
346	C₂×#123	C₂²×C₆₇	1	2	yes	—
347	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
348	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
349	C₂×#120	C₂²×C₆₇	1	4	yes	—
350	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
351	C₂×C₄×D₄	—	2	8	no	—
352	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
353	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
354	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
355	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
356	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
357	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
358	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	complemented
359	C₂×#117	C₂²×C₆₇	1	2	yes	—
360	C₂×#117	C₂²×C₆₇	1	2	yes	complemented
361	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
362	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
363	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
364	#398	C₂²×C₆₇	1	4	yes	complemented
365	#397	C₂²×C₆₇	1	4	yes	complemented
366	#396	C₂²×C₆₇	1	8	yes	complemented
367	#394	C₂²×C₆₇	1	4	yes	—
368	#394	C₂²×C₆₇	1	8	yes	complemented
369	#392	C₂²×C₆₇	1	4	yes	—
370	#391	C₂²×C₆₇	1	2	yes	complemented
371	#390	C₂²×C₆₇	1	2	yes	—
372	#390	C₂²×C₆₇	1	2	yes	—
373	#390	C₂²×C₆₇	1	8	yes	complemented
374	#390	C₂²×C₆₇	1	2	yes	complemented
375	#388	C₂²×C₆₇	1	8	yes	complemented
376	#386	C₂²×C₆₇	1	4	yes	—
377	#386	C₂²×C₆₇	1	8	yes	complemented
378	#385	C₂²×C₆₇	1	4	yes	—
379	#385	C₂²×C₆₇	1	8	yes	complemented
380	#384	C₂²×C₆₇	1	2	yes	complemented
381	#382	C₂²×C₆₇	1	2	yes	complemented
382	#379	C₂²×C₆₇	1	4	yes	—
383	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
384	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
385	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
386	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
387	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
388	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
389	#376	C₂²×C₆₇	1	2	yes	complemented
390	C₄³	C₂²×C₆₇	1	2	yes	complemented
391	C₂⁵	C₂³×C₆₇	1	2	yes	—
392	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
393	C₂²×D₄	—	2	8	no	—
394	C₂²×D₄	—	2	4	no	—
395	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
396	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
397	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
398	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
399	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
400	C₂³×C₄	—	2	8	no	—
401	C₂³×C₄	—	4	8	no	—
402	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
403	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
404	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
405	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
406	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
407	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
408	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
409	#127	—	2	8	no	—
410	#126	—	2	8	no	—
411	#125	—	2	4	no	—
412	#124	—	2	8	no	—
413	#124	—	2	4	no	—
414	#123	—	2	8	no	—
415	C₂²⋊Q₈	—	2	4	no	—
416	#121	—	2	4	no	—
417	#120	—	2	16	no	—
418	C₄×Q₈	—	2	4	no	—
419	C₄×D₄	—	2	8	no	—
420	C₄×D₄	—	2	8	no	—
421	C₄×D₄	—	2	8	no	—
422	C₄×D₄	—	2	8	no	—
423	C₄×D₄	—	2	8	no	—
424	C₄×D₄	—	2	4	no	—
425	C₄×D₄	—	4	16	no	—
426	#117	—	2	8	no	—
427	#117	—	2	8	no	—
428	C₂×C₄⋊C₄	—	2	4	no	—
429	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
430	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
431	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
432	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
433	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
434	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
435	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
436	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
437	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
438	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
439	C₂×C₂²⋊C₄	—	2	8	no	—
440	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
441	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
442	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
443	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
444	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
445	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
446	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
447	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
448	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
449	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
450	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
451	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
452	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
453	C₂×C₄²	D₄×C₆₇	1	8	yes	complemented
454	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
455	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
456	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
457	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
458	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
459	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
460	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
461	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
462	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
463	#95	C₂³×C₆₇	1	4	yes	—
464	#95	C₂³×C₆₇	1	2	yes	—
465	#95	C₂³×C₆₇	1	2	yes	—
466	#95	C₂³×C₆₇	1	8	yes	—
467	#95	C₂³×C₆₇	1	8	yes	—
468	C₂⁴	—	2	4	no	—
469	C₂⁴	—	2	8	no	—
470	C₂⁴	—	2	8	no	—
471	C₂⁴	—	4	4	no	—
472	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
473	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
474	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
475	C₂×Q₈	—	2	2	no	—
476	C₂×Q₈	—	2	4	no	—
477	C₂×D₄	—	2	32	no	—
478	C₂×D₄	—	2	8	no	—
479	C₂×D₄	—	2	4	no	—
480	C₂×D₄	—	2	8	no	—
481	C₂×D₄	—	2	16	no	—
482	C₂×D₄	—	2	4	no	—
483	C₂×D₄	—	2	4	no	—
484	C₂×D₄	—	2	2	no	—
485	C₂×D₄	—	2	4	no	—
486	C₂×D₄	—	2	16	no	—
487	C₂×D₄	—	2	4	no	—
488	C₂×D₄	—	4	8	no	—
489	C₂×D₄	#41×C₆₇	1	8	yes	—
490	C₂²×C₄	—	2	4	no	—
491	C₂²×C₄	—	2	4	no	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	—	2	4	no	—
494	C₂²×C₄	—	2	4	no	—
495	C₂²×C₄	—	2	8	no	—
496	C₂²×C₄	—	2	8	no	—
497	C₂²×C₄	—	2	8	no	—
498	C₂²×C₄	—	2	16	no	—
499	C₂²×C₄	—	4	8	no	—
500	C₂²×C₄	—	4	4	no	—
501	C₂²×C₄	—	4	32	no	—
502	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
503	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
504	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
505	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
506	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
507	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
508	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
509	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
510	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
511	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
512	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
513	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
514	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
515	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
516	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
517	C₄⋊C₄	—	2	4	no	—
518	C₄⋊C₄	—	2	4	no	—
519	C₄⋊C₄	—	2	4	no	—
520	C₄⋊C₄	—	2	4	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	—	2	4	no	—
523	C₄⋊C₄	—	2	2	no	—
524	C₄⋊C₄	—	2	4	no	—
525	C₄⋊C₄	—	2	8	no	—
526	C₄⋊C₄	—	4	8	no	—
527	C₂²⋊C₄	—	2	4	no	—
528	C₂²⋊C₄	—	2	4	no	—
529	C₂²⋊C₄	—	2	16	no	—
530	C₂²⋊C₄	—	2	8	no	—
531	C₂²⋊C₄	—	2	8	no	—
532	C₂²⋊C₄	—	2	4	no	—
533	C₂²⋊C₄	—	2	4	no	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₂²⋊C₄	—	2	16	no	—
537	C₂²⋊C₄	—	4	8	no	—
538	C₂²⋊C₄	—	4	8	no	—
539	C₂²⋊C₄	#41×C₆₇	1	8	yes	—
540	C₄²	—	2	4	no	—
541	C₄²	—	2	8	no	—
542	C₄²	—	2	4	no	—
543	C₄²	—	2	4	no	—
544	C₄²	—	2	8	no	—
545	C₄²	C₂×D₄×C₆₇	1	4	yes	—
546	C₄²	C₂×D₄×C₆₇	1	4	yes	—
547	C₄²	C₂×D₄×C₆₇	1	4	yes	—
548	C₄²	C₂×D₄×C₆₇	1	4	yes	—
549	C₂³	—	2	2	no	—
550	C₂³	—	2	2	no	—
551	C₂³	—	2	16	no	—
552	C₂³	—	2	4	no	—
553	C₂³	—	2	8	no	—
554	C₂³	—	2	4	no	—
555	C₂³	—	2	8	no	—
556	C₂³	—	2	2	no	—
557	C₂³	—	2	4	no	—
558	C₂³	—	2	4	no	—
559	C₂³	—	2	4	no	—
560	C₂³	—	4	32	no	—
561	C₂³	—	4	8	no	—
562	C₂³	—	4	8	no	—
563	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
564	C₂³	C₂×#41×C₆₇	1	8	yes	—
565	C₂³	C₂×#41×C₆₇	1	4	yes	—
566	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
567	Q₈	—	4	4	no	—
568	D₄	—	2	16	no	—
569	D₄	—	2	16	no	—
570	D₄	—	4	32	no	—
571	D₄	—	4	8	no	—
572	D₄	—	4	4	no	—
573	C₂×C₄	—	2	8	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	8	no	—
581	C₂×C₄	—	2	8	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	4	no	—
586	C₂×C₄	—	2	4	no	—
587	C₂×C₄	—	2	4	no	—
588	C₂×C₄	—	2	4	no	—
589	C₂×C₄	—	2	4	no	—
590	C₂×C₄	—	2	4	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	8	no	—
593	C₂×C₄	—	4	16	no	—
594	C₂×C₄	—	4	8	no	—
595	C₂×C₄	—	4	16	no	—
596	C₂×C₄	—	4	8	no	—
597	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
598	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
599	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
600	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
601	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
602	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
603	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
604	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
605	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
606	C₂²	—	2	2	no	—
607	C₂²	—	2	2	no	—
608	C₂²	—	2	8	no	—
609	C₂²	—	2	8	no	—
610	C₂²	—	2	16	no	—
611	C₂²	—	4	32	no	—
612	C₂²	—	4	4	no	—
613	C₂²	—	4	4	no	—
614	C₂²	—	4	8	no	—
615	C₂²	—	4	2	no	—
616	C₂²	—	4	2	no	—
617	C₂²	—	4	2	no	—
618	C₂²	—	4	8	no	—
619	C₂²	#532×C₆₇	1	4	yes	—
620	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
621	C₂²	C₂²×#41×C₆₇	1	2	yes	—
622	C₂²	C₂²×#41×C₆₇	1	2	yes	—
623	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
624	C₄	—	2	2	no	—
625	C₄	—	2	2	no	—
626	C₄	—	2	2	no	—
627	C₄	—	2	2	no	—
628	C₄	—	2	8	no	—
629	C₄	—	4	2	no	—
630	C₄	—	4	2	no	—
631	C₄	—	4	4	no	—
632	C₄	—	4	2	no	—
633	C₄	—	4	4	no	—
634	C₄	#525×C₆₇	1	4	yes	—
635	C₄	#535×C₆₇	1	4	yes	—
636	C₂	—	2	4	no	—
637	C₂	—	4	8	no	—
638	C₂	—	8	2	no	—
639	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
640	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
641	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
642	C₂	C₆₇×#3481	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₆₇×#2433	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#529×C₆₇	C₂	semidirect
2. C₂²×C₄×D₄×C₆₇	C₂	semidirect
3. C₆₇×#2603	C₂	semidirect
4. C₆₇×#2566	C₂	semidirect
5. C₆₇×#2544	C₂	semidirect
6. C₆₇×#2504	C₂	semidirect
7. C₆₇×#2502	C₂	semidirect
8. C₆₇×#2435	C₂	semidirect
10. C₆₇×#2411	C₂	semidirect
11. C₆₇×#2392	C₂	semidirect
12. C₄×#127×C₆₇	C₂	semidirect
13. C₄×#124×C₆₇	C₂	semidirect
14. C₄×#120×C₆₇	C₂	semidirect
15. C₄×#117×C₆₇	C₂	semidirect
20. C₂×#126×C₆₇	C₂²	semidirect
21. C₂×#125×C₆₇	C₂²	semidirect
23. C₂×#124×C₆₇	C₂²	semidirect
28. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
35. C₂×C₄×D₄×C₆₇	C₂²	semidirect
36. C₂×C₄×D₄×C₆₇	C₂²	semidirect
38. C₂×#117×C₆₇	C₂²	semidirect
42. #398×C₆₇	C₂²	semidirect
43. #397×C₆₇	C₂²	semidirect
44. #396×C₆₇	C₂²	semidirect
46. #394×C₆₇	C₂²	semidirect
48. #391×C₆₇	C₂²	semidirect
51. #390×C₆₇	C₂²	semidirect
52. #390×C₆₇	C₂²	semidirect
53. #388×C₆₇	C₂²	semidirect
55. #386×C₆₇	C₂²	semidirect
57. #385×C₆₇	C₂²	semidirect
58. #384×C₆₇	C₂²	semidirect
59. #382×C₆₇	C₂²	semidirect
61. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
62. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
66. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
67. #376×C₆₇	C₂²	semidirect
68. C₄³×C₆₇	C₂²	semidirect
131. C₂×C₄²×C₆₇	D₄	semidirect
314. #34965	C₆₇	direct (N × H)
322. C₂×#529	C₂×C₆₇	semidirect
323. C₂²×C₄×D₄	C₂×C₆₇	semidirect
324. #2603	C₂×C₆₇	semidirect
325. #2566	C₂×C₆₇	semidirect
326. #2544	C₂×C₆₇	semidirect
327. #2504	C₂×C₆₇	semidirect
328. #2502	C₂×C₆₇	semidirect
329. #2435	C₂×C₆₇	semidirect
331. #2411	C₂×C₆₇	semidirect
332. #2392	C₂×C₆₇	semidirect
333. C₄×#127	C₂×C₆₇	semidirect
334. C₄×#124	C₂×C₆₇	semidirect
335. C₄×#120	C₂×C₆₇	semidirect
336. C₄×#117	C₂×C₆₇	semidirect
337. C₆₇	#34965	direct (N × H)
342. C₂×#126	C₂²×C₆₇	semidirect
343. C₂×#125	C₂²×C₆₇	semidirect
345. C₂×#124	C₂²×C₆₇	semidirect
350. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
357. C₂×C₄×D₄	C₂²×C₆₇	semidirect
358. C₂×C₄×D₄	C₂²×C₆₇	semidirect
360. C₂×#117	C₂²×C₆₇	semidirect
364. #398	C₂²×C₆₇	semidirect
365. #397	C₂²×C₆₇	semidirect
366. #396	C₂²×C₆₇	semidirect
368. #394	C₂²×C₆₇	semidirect
370. #391	C₂²×C₆₇	semidirect
373. #390	C₂²×C₆₇	semidirect
374. #390	C₂²×C₆₇	semidirect
375. #388	C₂²×C₆₇	semidirect
377. #386	C₂²×C₆₇	semidirect
379. #385	C₂²×C₆₇	semidirect
380. #384	C₂²×C₆₇	semidirect
381. #382	C₂²×C₆₇	semidirect
383. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
384. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
388. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
389. #376	C₂²×C₆₇	semidirect
390. C₄³	C₂²×C₆₇	semidirect
453. C₂×C₄²	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#529×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2603	2
C₆₇×#2566	2
C₆₇×#2544	2
C₆₇×#2504	2
C₆₇×#2502	2
C₆₇×#2435	2
C₆₇×#2433	2
C₆₇×#2411	2
C₆₇×#2392	2
C₄×#127×C₆₇	2
C₄×#124×C₆₇	2
C₄×#120×C₆₇	2
C₄×#117×C₆₇	2
#34965	67

Subgroups — C67×#34965 #30714216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34965 #30714216