Subgroups — C₆₇×#34387 #30713638

Order 17152.

1990 conjugacy classes of subgroups.

The subgroup lattice has 586 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
2	C₆₇×#2378	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2377	C₂	1	2	yes	maximal
4	C₆₇×#2374	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2356	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2353	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2346	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2343	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2337	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2336	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2334	C₂	1	4	yes	maximal, complemented
12	C₆₇×#2319	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2318	C₂	1	2	yes	maximal, complemented
14	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
16	#529×C₆₇	C₄	1	16	yes	complemented
17	C₂×#126×C₆₇	C₂²	1	2	yes	—
18	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#123×C₆₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
26	C₂×#117×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
28	#391×C₆₇	C₂²	1	4	yes	—
29	#390×C₆₇	C₂²	1	4	yes	—
30	#389×C₆₇	C₂²	1	4	yes	—
31	#389×C₆₇	C₂²	1	4	yes	—
32	#388×C₆₇	C₂²	1	4	yes	—
33	#388×C₆₇	C₂²	1	4	yes	—
34	#388×C₆₇	C₂²	1	4	yes	—
35	#388×C₆₇	C₂²	1	4	yes	—
36	#388×C₆₇	C₂²	1	4	yes	—
37	#388×C₆₇	C₂²	1	4	yes	—
38	#387×C₆₇	C₂²	1	4	yes	—
39	#387×C₆₇	C₂²	1	4	yes	—
40	#387×C₆₇	C₂²	1	4	yes	—
41	#387×C₆₇	C₂²	1	4	yes	—
42	#387×C₆₇	C₂²	1	4	yes	—
43	#387×C₆₇	C₂²	1	4	yes	—
44	#386×C₆₇	C₂²	1	4	yes	—
45	#386×C₆₇	C₂²	1	4	yes	—
46	#385×C₆₇	C₂²	1	4	yes	—
47	#385×C₆₇	C₂²	1	4	yes	—
48	#383×C₆₇	C₂²	1	2	yes	—
49	#383×C₆₇	C₂²	1	2	yes	—
50	#383×C₆₇	C₂²	1	2	yes	—
51	#382×C₆₇	C₂²	1	2	yes	—
52	#382×C₆₇	C₂²	1	2	yes	—
53	#382×C₆₇	C₂²	1	2	yes	—
54	#381×C₆₇	C₂²	1	1	yes	characteristic
55	#381×C₆₇	C₂²	1	1	yes	characteristic
56	#381×C₆₇	C₂²	1	2	yes	—
57	#381×C₆₇	C₂²	1	2	yes	—
58	#380×C₆₇	C₂²	1	2	yes	—
59	#380×C₆₇	C₂²	1	2	yes	—
60	#380×C₆₇	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
64	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
65	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
66	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
67	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
68	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
69	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
70	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
71	#376×C₆₇	C₂²	1	1	yes	characteristic
72	#376×C₆₇	C₂²	1	1	yes	characteristic
73	#376×C₆₇	C₂²	1	2	yes	—
74	#376×C₆₇	C₂²	1	2	yes	—
75	C₂×#95×C₆₇	C₂²	1	2	yes	—
76	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
78	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
80	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
84	#126×C₆₇	C₂×C₄	1	16	yes	complemented
85	#125×C₆₇	C₂×C₄	1	8	yes	complemented
86	#124×C₆₇	C₂×C₄	1	8	yes	—
87	#123×C₆₇	C₂×C₄	1	16	yes	complemented
88	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
89	#121×C₆₇	C₂×C₄	1	8	yes	—
90	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
91	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
92	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
93	#117×C₆₇	C₂×C₄	1	16	yes	complemented
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
106	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
116	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
117	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
118	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
119	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
120	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
121	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
122	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
123	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
128	#95×C₆₇	C₂³	1	2	yes	—
129	#95×C₆₇	C₂³	1	2	yes	—
130	#95×C₆₇	C₂³	1	4	yes	—
131	#95×C₆₇	C₂³	1	2	yes	—
132	#95×C₆₇	C₂³	1	2	yes	—
133	#95×C₆₇	C₂³	1	4	yes	—
134	#95×C₆₇	C₂³	1	2	yes	—
135	#95×C₆₇	C₂³	1	2	yes	—
136	#95×C₆₇	C₂³	1	4	yes	—
137	#95×C₆₇	C₂³	1	4	yes	—
138	#95×C₆₇	C₂³	1	4	yes	—
139	#95×C₆₇	C₂³	1	4	yes	—
140	#95×C₆₇	C₂³	1	4	yes	—
141	#95×C₆₇	C₂³	1	2	yes	—
142	#95×C₆₇	C₂³	1	4	yes	—
143	#95×C₆₇	C₂³	1	2	yes	—
144	#95×C₆₇	C₂³	1	4	yes	—
145	#95×C₆₇	C₂³	1	4	yes	—
146	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
147	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
148	C₂×Q₈×C₆₇	—	4	2	no	—
149	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
150	C₂×D₄×C₆₇	—	4	4	no	—
151	C₂×D₄×C₆₇	—	4	2	no	—
152	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
153	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
154	C₂²×C₄×C₆₇	—	4	2	no	—
155	C₂²×C₄×C₆₇	—	4	4	no	—
156	C₂²×C₄×C₆₇	—	4	2	no	—
157	C₂²×C₄×C₆₇	—	4	2	no	—
158	C₂²×C₄×C₆₇	—	4	4	no	—
159	C₂²×C₄×C₆₇	—	4	4	no	—
160	C₂²×C₄×C₆₇	—	4	4	no	—
161	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
162	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
163	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
164	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
178	C₄⋊C₄×C₆₇	—	4	4	no	—
179	C₄⋊C₄×C₆₇	—	4	4	no	—
180	C₄⋊C₄×C₆₇	—	4	4	no	—
181	C₄⋊C₄×C₆₇	—	4	4	no	—
182	C₄⋊C₄×C₆₇	—	4	4	no	—
183	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
184	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
185	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
186	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
187	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
188	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
189	C₂²⋊C₄×C₆₇	—	4	4	no	—
190	C₂²⋊C₄×C₆₇	—	4	4	no	—
191	C₂²⋊C₄×C₆₇	—	4	2	no	—
192	C₂²⋊C₄×C₆₇	—	4	4	no	—
193	C₂²⋊C₄×C₆₇	—	4	2	no	—
194	C₂²⋊C₄×C₆₇	—	4	4	no	—
195	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
196	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
197	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
198	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
199	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
200	C₄²×C₆₇	—	4	2	no	—
201	C₄²×C₆₇	—	4	4	no	—
202	C₄²×C₆₇	—	4	2	no	—
203	C₄²×C₆₇	—	4	4	no	—
204	C₄²×C₆₇	—	4	4	no	—
205	C₄²×C₆₇	—	4	4	no	—
206	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
207	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
208	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
209	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
210	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
211	C₂³×C₆₇	—	2	2	no	—
212	C₂³×C₆₇	—	2	2	no	—
213	C₂³×C₆₇	—	2	2	no	—
214	C₂³×C₆₇	—	2	4	no	—
215	C₂³×C₆₇	—	2	4	no	—
216	C₂³×C₆₇	—	2	2	no	—
217	C₂³×C₆₇	—	2	2	no	—
218	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
219	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
220	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
221	Q₈×C₆₇	—	4	4	no	—
222	D₄×C₆₇	—	4	4	no	—
223	D₄×C₆₇	—	4	8	no	—
224	C₂×C₄×C₆₇	—	2	2	no	—
225	C₂×C₄×C₆₇	—	2	2	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	4	no	—
228	C₂×C₄×C₆₇	—	2	4	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	2	2	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	2	2	no	—
233	C₂×C₄×C₆₇	—	2	2	no	—
234	C₂×C₄×C₆₇	—	2	4	no	—
235	C₂×C₄×C₆₇	—	2	4	no	—
236	C₂×C₄×C₆₇	—	2	4	no	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	2	4	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	4	no	—
243	C₂×C₄×C₆₇	—	2	4	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	4	4	no	—
247	C₂×C₄×C₆₇	—	4	8	no	—
248	C₂×C₄×C₆₇	—	4	4	no	—
249	C₂×C₄×C₆₇	—	4	4	no	—
250	C₂×C₄×C₆₇	—	8	8	no	—
251	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
252	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
253	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
254	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
255	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
256	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
257	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
258	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
259	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
260	C₂²×C₆₇	—	2	2	no	—
261	C₂²×C₆₇	—	2	2	no	—
262	C₂²×C₆₇	—	2	2	no	—
263	C₂²×C₆₇	—	2	4	no	—
264	C₂²×C₆₇	—	2	4	no	—
265	C₂²×C₆₇	—	2	4	no	—
266	C₂²×C₆₇	—	4	4	no	—
267	C₂²×C₆₇	—	4	4	no	—
268	C₂²×C₆₇	—	4	4	no	—
269	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
270	C₂²×C₆₇	#585	1	2	yes	—
271	C₂²×C₆₇	#585	1	2	yes	—
272	C₂²×C₆₇	#585	1	2	yes	—
273	C₄×C₆₇	—	2	2	no	—
274	C₄×C₆₇	—	2	2	no	—
275	C₄×C₆₇	—	2	2	no	—
276	C₄×C₆₇	—	2	4	no	—
277	C₄×C₆₇	—	2	2	no	—
278	C₄×C₆₇	—	2	2	no	—
279	C₄×C₆₇	—	2	2	no	—
280	C₄×C₆₇	—	2	4	no	—
281	C₄×C₆₇	—	2	4	no	—
282	C₄×C₆₇	—	4	4	no	—
283	C₄×C₆₇	—	4	4	no	—
284	C₄×C₆₇	—	4	4	no	—
285	C₄×C₆₇	—	4	4	no	—
286	#34387	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
287	C₂×C₆₇	—	4	2	no	—
288	C₂×C₆₇	—	4	4	no	—
289	C₂×C₆₇	#3428	1	1	yes	characteristic
290	C₂×C₆₇	#3428	1	1	yes	characteristic
291	C₂×C₆₇	#3428	1	1	yes	characteristic
292	C₂×C₆₇	#3566	1	4	yes	—
293	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
294	#2378	C₂×C₆₇	1	2	yes	complemented
295	#2377	C₂×C₆₇	1	2	yes	—
296	#2374	C₂×C₆₇	1	4	yes	complemented
297	#2356	C₂×C₆₇	1	2	yes	complemented
298	#2353	C₂×C₆₇	1	2	yes	complemented
299	#2346	C₂×C₆₇	1	2	yes	complemented
300	#2343	C₂×C₆₇	1	2	yes	complemented
301	#2337	C₂×C₆₇	1	2	yes	complemented
302	#2336	C₂×C₆₇	1	2	yes	complemented
303	#2334	C₂×C₆₇	1	4	yes	complemented
304	#2319	C₂×C₆₇	1	2	yes	complemented
305	#2318	C₂×C₆₇	1	2	yes	complemented
306	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
307	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
308	C₆₇	#34387	1	1	yes	characteristic, direct product factor, Sylow
309	#529	C₄×C₆₇	1	16	yes	complemented
310	C₂×#126	C₂²×C₆₇	1	2	yes	—
311	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
312	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×#123	C₂²×C₆₇	1	2	yes	—
314	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
315	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
316	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
317	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
318	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
319	C₂×#117	C₂²×C₆₇	1	2	yes	—
320	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
321	#391	C₂²×C₆₇	1	4	yes	—
322	#390	C₂²×C₆₇	1	4	yes	—
323	#389	C₂²×C₆₇	1	4	yes	—
324	#389	C₂²×C₆₇	1	4	yes	—
325	#388	C₂²×C₆₇	1	4	yes	—
326	#388	C₂²×C₆₇	1	4	yes	—
327	#388	C₂²×C₆₇	1	4	yes	—
328	#388	C₂²×C₆₇	1	4	yes	—
329	#388	C₂²×C₆₇	1	4	yes	—
330	#388	C₂²×C₆₇	1	4	yes	—
331	#387	C₂²×C₆₇	1	4	yes	—
332	#387	C₂²×C₆₇	1	4	yes	—
333	#387	C₂²×C₆₇	1	4	yes	—
334	#387	C₂²×C₆₇	1	4	yes	—
335	#387	C₂²×C₆₇	1	4	yes	—
336	#387	C₂²×C₆₇	1	4	yes	—
337	#386	C₂²×C₆₇	1	4	yes	—
338	#386	C₂²×C₆₇	1	4	yes	—
339	#385	C₂²×C₆₇	1	4	yes	—
340	#385	C₂²×C₆₇	1	4	yes	—
341	#383	C₂²×C₆₇	1	2	yes	—
342	#383	C₂²×C₆₇	1	2	yes	—
343	#383	C₂²×C₆₇	1	2	yes	—
344	#382	C₂²×C₆₇	1	2	yes	—
345	#382	C₂²×C₆₇	1	2	yes	—
346	#382	C₂²×C₆₇	1	2	yes	—
347	#381	C₂²×C₆₇	1	1	yes	characteristic
348	#381	C₂²×C₆₇	1	1	yes	characteristic
349	#381	C₂²×C₆₇	1	2	yes	—
350	#381	C₂²×C₆₇	1	2	yes	—
351	#380	C₂²×C₆₇	1	2	yes	—
352	#380	C₂²×C₆₇	1	2	yes	—
353	#380	C₂²×C₆₇	1	2	yes	—
354	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
355	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
356	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
357	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
358	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
359	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
360	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
361	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
362	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
363	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
364	#376	C₂²×C₆₇	1	2	yes	—
365	#376	C₂²×C₆₇	1	1	yes	characteristic
366	#376	C₂²×C₆₇	1	1	yes	characteristic
367	#376	C₂²×C₆₇	1	2	yes	—
368	C₂×#95	C₂²×C₆₇	1	2	yes	—
369	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
370	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
371	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
372	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
373	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
374	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
375	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
376	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
377	#126	C₂×C₄×C₆₇	1	16	yes	complemented
378	#125	C₂×C₄×C₆₇	1	8	yes	complemented
379	#124	C₂×C₄×C₆₇	1	8	yes	—
380	#123	C₂×C₄×C₆₇	1	16	yes	complemented
381	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
382	#121	C₂×C₄×C₆₇	1	8	yes	—
383	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
384	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
385	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
386	#117	C₂×C₄×C₆₇	1	16	yes	complemented
387	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
388	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
389	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
390	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
391	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
392	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
393	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
394	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
395	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
396	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
397	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
409	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	complemented
410	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
411	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
412	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
413	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
414	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
415	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
416	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
418	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
420	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
421	#95	C₂³×C₆₇	1	2	yes	—
422	#95	C₂³×C₆₇	1	2	yes	—
423	#95	C₂³×C₆₇	1	4	yes	—
424	#95	C₂³×C₆₇	1	2	yes	—
425	#95	C₂³×C₆₇	1	2	yes	—
426	#95	C₂³×C₆₇	1	4	yes	—
427	#95	C₂³×C₆₇	1	2	yes	—
428	#95	C₂³×C₆₇	1	2	yes	—
429	#95	C₂³×C₆₇	1	4	yes	—
430	#95	C₂³×C₆₇	1	4	yes	—
431	#95	C₂³×C₆₇	1	4	yes	—
432	#95	C₂³×C₆₇	1	4	yes	—
433	#95	C₂³×C₆₇	1	4	yes	—
434	#95	C₂³×C₆₇	1	2	yes	—
435	#95	C₂³×C₆₇	1	4	yes	—
436	#95	C₂³×C₆₇	1	2	yes	—
437	#95	C₂³×C₆₇	1	4	yes	—
438	#95	C₂³×C₆₇	1	4	yes	—
439	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
440	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
441	C₂×Q₈	—	4	2	no	—
442	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
443	C₂×D₄	—	4	4	no	—
444	C₂×D₄	—	4	2	no	—
445	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
446	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
447	C₂²×C₄	—	4	2	no	—
448	C₂²×C₄	—	4	4	no	—
449	C₂²×C₄	—	4	2	no	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	—	4	4	no	—
452	C₂²×C₄	—	4	4	no	—
453	C₂²×C₄	—	4	4	no	—
454	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
455	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
456	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
457	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
458	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
459	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
460	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
461	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
462	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
463	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
464	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
465	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
467	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
468	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
469	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
470	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
471	C₄⋊C₄	—	4	4	no	—
472	C₄⋊C₄	—	4	4	no	—
473	C₄⋊C₄	—	4	4	no	—
474	C₄⋊C₄	—	4	4	no	—
475	C₄⋊C₄	—	4	4	no	—
476	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
477	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
478	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
479	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
480	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
481	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
482	C₂²⋊C₄	—	4	4	no	—
483	C₂²⋊C₄	—	4	4	no	—
484	C₂²⋊C₄	—	4	2	no	—
485	C₂²⋊C₄	—	4	4	no	—
486	C₂²⋊C₄	—	4	2	no	—
487	C₂²⋊C₄	—	4	4	no	—
488	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
489	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
490	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
491	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
492	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
493	C₄²	—	4	2	no	—
494	C₄²	—	4	4	no	—
495	C₄²	—	4	2	no	—
496	C₄²	—	4	4	no	—
497	C₄²	—	4	4	no	—
498	C₄²	—	4	4	no	—
499	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
500	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
501	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
502	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
503	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
504	C₂³	—	2	2	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	2	2	no	—
507	C₂³	—	2	4	no	—
508	C₂³	—	2	4	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
512	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
513	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
514	Q₈	—	4	4	no	—
515	D₄	—	4	4	no	—
516	D₄	—	4	8	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	4	4	no	—
540	C₂×C₄	—	4	8	no	—
541	C₂×C₄	—	4	4	no	—
542	C₂×C₄	—	4	4	no	—
543	C₂×C₄	—	8	8	no	—
544	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
545	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
546	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
547	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
548	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
549	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
550	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
551	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
552	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	2	no	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	4	no	—
557	C₂²	—	2	4	no	—
558	C₂²	—	2	4	no	—
559	C₂²	—	4	4	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	4	4	no	—
562	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
563	C₂²	#585×C₆₇	1	2	yes	—
564	C₂²	#585×C₆₇	1	2	yes	—
565	C₂²	#585×C₆₇	1	2	yes	—
566	C₄	—	2	2	no	—
567	C₄	—	2	2	no	—
568	C₄	—	2	2	no	—
569	C₄	—	2	2	no	—
570	C₄	—	2	4	no	—
571	C₄	—	2	2	no	—
572	C₄	—	2	2	no	—
573	C₄	—	2	4	no	—
574	C₄	—	2	4	no	—
575	C₄	—	4	4	no	—
576	C₄	—	4	4	no	—
577	C₄	—	4	4	no	—
578	C₄	—	4	4	no	—
579	C₂	—	4	2	no	—
580	C₂	—	4	4	no	—
581	C₂	C₆₇×#3428	1	1	yes	characteristic
582	C₂	C₆₇×#3428	1	1	yes	characteristic
583	C₂	C₆₇×#3428	1	1	yes	characteristic
584	C₂	C₆₇×#3566	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
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
2. C₆₇×#2378	C₂	semidirect
4. C₆₇×#2374	C₂	semidirect
5. C₆₇×#2356	C₂	semidirect
6. C₆₇×#2353	C₂	semidirect
7. C₆₇×#2346	C₂	semidirect
8. C₆₇×#2343	C₂	semidirect
9. C₆₇×#2337	C₂	semidirect
10. C₆₇×#2336	C₂	semidirect
11. C₆₇×#2334	C₂	semidirect
12. C₆₇×#2319	C₂	semidirect
13. C₆₇×#2318	C₂	semidirect
14. C₆₇×#2283	C₂	semidirect
15. C₆₇×#2283	C₂	semidirect
16. #529×C₆₇	C₄	semidirect
84. #126×C₆₇	C₂×C₄	semidirect
85. #125×C₆₇	C₂×C₄	semidirect
87. #123×C₆₇	C₂×C₄	semidirect
88. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
90. C₄×Q₈×C₆₇	C₂×C₄	semidirect
92. C₄×D₄×C₆₇	C₂×C₄	semidirect
93. #117×C₆₇	C₂×C₄	semidirect
116. C₂×C₄²×C₆₇	C₂×C₄	semidirect
286. #34387	C₆₇	direct (N × H)
294. #2378	C₂×C₆₇	semidirect
296. #2374	C₂×C₆₇	semidirect
297. #2356	C₂×C₆₇	semidirect
298. #2353	C₂×C₆₇	semidirect
299. #2346	C₂×C₆₇	semidirect
300. #2343	C₂×C₆₇	semidirect
301. #2337	C₂×C₆₇	semidirect
302. #2336	C₂×C₆₇	semidirect
303. #2334	C₂×C₆₇	semidirect
304. #2319	C₂×C₆₇	semidirect
305. #2318	C₂×C₆₇	semidirect
306. #2283	C₂×C₆₇	semidirect
307. #2283	C₂×C₆₇	semidirect
308. C₆₇	#34387	direct (N × H)
309. #529	C₄×C₆₇	semidirect
377. #126	C₂×C₄×C₆₇	semidirect
378. #125	C₂×C₄×C₆₇	semidirect
380. #123	C₂×C₄×C₆₇	semidirect
381. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
383. C₄×Q₈	C₂×C₄×C₆₇	semidirect
385. C₄×D₄	C₂×C₄×C₆₇	semidirect
386. #117	C₂×C₄×C₆₇	semidirect
409. C₂×C₄²	C₂×C₄×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
C₆₇×#2378	2
C₆₇×#2377	2
C₆₇×#2374	2
C₆₇×#2356	2
C₆₇×#2353	2
C₆₇×#2346	2
C₆₇×#2343	2
C₆₇×#2337	2
C₆₇×#2336	2
C₆₇×#2334	2
C₆₇×#2319	2
C₆₇×#2318	2
C₆₇×#2283	2
#34387	67

Subgroups — C67×#34387 #30713638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34387 #30713638