Subgroups — C₆₇×#35097 #30714348

Order 17152.

2586 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#123×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2853	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2623	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2498	C₂	1	8	yes	maximal, complemented
6	C₆₇×#2484	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2482	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2315	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2308	C₂	1	2	yes	maximal
10	C₂×#381×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#123×C₆₇	—	2	8	no	—
14	C₂×#123×C₆₇	C₂²	1	4	yes	—
15	C₂×#123×C₆₇	C₂²	1	2	yes	—
16	C₂×#123×C₆₇	C₂²	1	2	yes	—
17	C₂×#117×C₆₇	—	2	8	no	—
18	C₂×#117×C₆₇	C₂²	1	4	yes	—
19	C₂×#117×C₆₇	C₂²	1	4	yes	complemented
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₄²×C₆₇	C₂²	1	2	yes	complemented
27	#400×C₆₇	C₂²	1	8	yes	complemented
28	#399×C₆₇	C₂²	1	8	yes	complemented
29	#398×C₆₇	C₂²	1	8	yes	complemented
30	#397×C₆₇	C₂²	1	8	yes	complemented
31	#388×C₆₇	C₂²	1	16	yes	complemented
32	#387×C₆₇	C₂²	1	16	yes	complemented
33	#386×C₆₇	C₂²	1	8	yes	—
34	#386×C₆₇	C₂²	1	8	yes	—
35	#385×C₆₇	C₂²	1	8	yes	—
36	#385×C₆₇	C₂²	1	8	yes	complemented
37	#381×C₆₇	C₂²	1	4	yes	—
38	#381×C₆₇	C₂²	1	4	yes	—
39	#381×C₆₇	C₂²	1	2	yes	complemented
40	#381×C₆₇	C₂²	1	2	yes	complemented
41	#380×C₆₇	C₂²	1	8	yes	complemented
42	#379×C₆₇	C₂²	1	4	yes	—
43	C₂×#95×C₆₇	C₂²	1	4	yes	complemented
44	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
45	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₆₇	—	2	4	no	—
47	C₂²×D₄×C₆₇	—	2	8	no	—
48	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
49	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₆₇	—	2	8	no	—
51	C₂³×C₄×C₆₇	D₄	1	8	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
57	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
58	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
59	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
60	#123×C₆₇	—	2	8	no	—
61	#123×C₆₇	—	2	8	no	—
62	#123×C₆₇	—	2	16	no	—
63	#117×C₆₇	—	2	16	no	—
64	#117×C₆₇	—	2	16	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
76	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
77	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
78	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
79	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
80	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
81	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
82	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
83	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
85	C₂×C₄²×C₆₇	—	2	8	no	—
86	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
87	#95×C₆₇	C₂³	1	8	yes	—
88	#95×C₆₇	C₂³	1	8	yes	—
89	#95×C₆₇	C₂³	1	16	yes	—
90	#95×C₆₇	C₂³	1	16	yes	—
91	C₂⁴×C₆₇	—	2	2	no	—
92	C₂⁴×C₆₇	—	2	4	no	—
93	C₂⁴×C₆₇	—	4	4	no	—
94	C₂⁴×C₆₇	—	4	4	no	—
95	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
96	C₂⁴×C₆₇	#41	1	8	yes	—
97	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
98	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
99	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
101	C₂×D₄×C₆₇	—	2	4	no	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	—	2	8	no	—
104	C₂×D₄×C₆₇	—	2	8	no	—
105	C₂×D₄×C₆₇	—	4	16	no	—
106	C₂²×C₄×C₆₇	—	2	32	no	—
107	C₂²×C₄×C₆₇	—	2	8	no	—
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	2	4	no	—
110	C₂²×C₄×C₆₇	—	2	8	no	—
111	C₂²×C₄×C₆₇	—	2	8	no	—
112	C₂²×C₄×C₆₇	—	2	4	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	4	no	—
115	C₂²×C₄×C₆₇	—	2	8	no	—
116	C₂²×C₄×C₆₇	—	4	8	no	—
117	C₂²×C₄×C₆₇	—	4	4	no	—
118	C₂²×C₄×C₆₇	—	4	8	no	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
122	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
123	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
125	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
126	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
127	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
128	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
129	C₄⋊C₄×C₆₇	—	2	8	no	—
130	C₄⋊C₄×C₆₇	—	2	8	no	—
131	C₄⋊C₄×C₆₇	—	2	8	no	—
132	C₄⋊C₄×C₆₇	—	2	8	no	—
133	C₄⋊C₄×C₆₇	—	4	16	no	—
134	C₂²⋊C₄×C₆₇	—	2	8	no	—
135	C₂²⋊C₄×C₆₇	—	2	8	no	—
136	C₂²⋊C₄×C₆₇	—	2	4	no	—
137	C₂²⋊C₄×C₆₇	—	2	4	no	—
138	C₂²⋊C₄×C₆₇	—	2	8	no	—
139	C₂²⋊C₄×C₆₇	—	2	8	no	—
140	C₂²⋊C₄×C₆₇	—	2	8	no	—
141	C₂²⋊C₄×C₆₇	—	2	8	no	—
142	C₂²⋊C₄×C₆₇	—	4	16	no	—
143	C₄²×C₆₇	—	2	16	no	—
144	C₂³×C₆₇	—	2	4	no	—
145	C₂³×C₆₇	—	2	4	no	—
146	C₂³×C₆₇	—	2	4	no	—
147	C₂³×C₆₇	—	2	2	no	—
148	C₂³×C₆₇	—	2	4	no	—
149	C₂³×C₆₇	—	2	8	no	—
150	C₂³×C₆₇	—	4	8	no	—
151	C₂³×C₆₇	—	4	16	no	—
152	C₂³×C₆₇	—	4	8	no	—
153	C₂³×C₆₇	—	4	4	no	—
154	C₂³×C₆₇	—	8	8	no	—
155	C₂³×C₆₇	#123	1	16	yes	—
156	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
157	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
158	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
159	C₂³×C₆₇	C₂×#41	1	4	yes	—
160	C₂³×C₆₇	C₂×#41	1	4	yes	—
161	C₂³×C₆₇	C₂×#41	1	4	yes	—
162	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
163	D₄×C₆₇	—	4	8	no	—
164	D₄×C₆₇	—	4	8	no	—
165	C₂×C₄×C₆₇	—	2	16	no	—
166	C₂×C₄×C₆₇	—	2	8	no	—
167	C₂×C₄×C₆₇	—	2	8	no	—
168	C₂×C₄×C₆₇	—	2	16	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	2	8	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	8	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	2	16	no	—
177	C₂×C₄×C₆₇	—	2	16	no	—
178	C₂×C₄×C₆₇	—	2	8	no	—
179	C₂×C₄×C₆₇	—	4	16	no	—
180	C₂×C₄×C₆₇	—	4	8	no	—
181	C₂×C₄×C₆₇	—	4	4	no	—
182	C₂×C₄×C₆₇	—	8	8	no	—
183	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
184	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
185	C₂²×C₆₇	—	2	4	no	—
186	C₂²×C₆₇	—	2	4	no	—
187	C₂²×C₆₇	—	2	4	no	—
188	C₂²×C₆₇	—	2	8	no	—
189	C₂²×C₆₇	—	2	8	no	—
190	C₂²×C₆₇	—	2	8	no	—
191	C₂²×C₆₇	—	4	16	no	—
192	C₂²×C₆₇	—	4	2	no	—
193	C₂²×C₆₇	—	4	4	no	—
194	C₂²×C₆₇	—	4	8	no	—
195	C₂²×C₆₇	—	8	8	no	—
196	C₂²×C₆₇	C₂×#123	1	4	yes	—
197	C₂²×C₆₇	C₂×#123	1	4	yes	—
198	C₂²×C₆₇	C₂×#123	1	4	yes	—
199	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
200	C₂²×C₆₇	C₂²×#41	1	2	yes	—
201	C₂²×C₆₇	#585	1	4	yes	—
202	C₄×C₆₇	—	2	4	no	—
203	C₄×C₆₇	—	2	4	no	—
204	C₄×C₆₇	—	4	2	no	—
205	C₄×C₆₇	—	4	8	no	—
206	C₄×C₆₇	—	4	4	no	—
207	C₄×C₆₇	—	4	8	no	—
208	#35097	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂×C₆₇	—	2	4	no	—
210	C₂×C₆₇	—	2	4	no	—
211	C₂×C₆₇	—	2	4	no	—
212	C₂×C₆₇	—	8	2	no	—
213	C₂×C₆₇	C₂²×#123	1	1	yes	characteristic
214	C₂×C₆₇	#3449	1	2	yes	—
215	C₂×C₆₇	#3491	1	4	yes	—
216	C₂²×#123	C₂×C₆₇	1	1	yes	characteristic
217	C₂²×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
218	#2853	C₂×C₆₇	1	4	yes	complemented
219	#2623	C₂×C₆₇	1	4	yes	complemented
220	#2498	C₂×C₆₇	1	8	yes	complemented
221	#2484	C₂×C₆₇	1	4	yes	complemented
222	#2482	C₂×C₆₇	1	4	yes	complemented
223	#2315	C₂×C₆₇	1	2	yes	complemented
224	#2308	C₂×C₆₇	1	2	yes	—
225	C₂×#381	C₂×C₆₇	1	1	yes	characteristic, complemented
226	C₆₇	#35097	1	1	yes	characteristic, direct product factor, Sylow
227	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
228	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
229	C₂×#123	—	2	8	no	—
230	C₂×#123	C₂²×C₆₇	1	2	yes	—
231	C₂×#123	C₂²×C₆₇	1	2	yes	—
232	C₂×#123	C₂²×C₆₇	1	4	yes	—
233	C₂×#117	—	2	8	no	—
234	C₂×#117	C₂²×C₆₇	1	4	yes	—
235	C₂×#117	C₂²×C₆₇	1	4	yes	complemented
236	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
237	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
238	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
239	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
240	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
241	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
242	C₂²×C₄²	C₂²×C₆₇	1	2	yes	complemented
243	#400	C₂²×C₆₇	1	8	yes	complemented
244	#399	C₂²×C₆₇	1	8	yes	complemented
245	#398	C₂²×C₆₇	1	8	yes	complemented
246	#397	C₂²×C₆₇	1	8	yes	complemented
247	#388	C₂²×C₆₇	1	16	yes	complemented
248	#387	C₂²×C₆₇	1	16	yes	complemented
249	#386	C₂²×C₆₇	1	8	yes	—
250	#386	C₂²×C₆₇	1	8	yes	—
251	#385	C₂²×C₆₇	1	8	yes	—
252	#385	C₂²×C₆₇	1	8	yes	complemented
253	#381	C₂²×C₆₇	1	4	yes	—
254	#381	C₂²×C₆₇	1	4	yes	—
255	#381	C₂²×C₆₇	1	2	yes	complemented
256	#381	C₂²×C₆₇	1	2	yes	complemented
257	#380	C₂²×C₆₇	1	8	yes	complemented
258	#379	C₂²×C₆₇	1	4	yes	—
259	C₂×#95	C₂²×C₆₇	1	4	yes	complemented
260	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
261	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
262	C₂²×D₄	—	2	4	no	—
263	C₂²×D₄	—	2	8	no	—
264	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
265	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
266	C₂³×C₄	—	2	8	no	—
267	C₂³×C₄	D₄×C₆₇	1	8	yes	—
268	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
269	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
270	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
271	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
272	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
273	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
274	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
275	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
276	#123	—	2	8	no	—
277	#123	—	2	8	no	—
278	#123	—	2	16	no	—
279	#117	—	2	16	no	—
280	#117	—	2	16	no	—
281	C₂×C₄⋊C₄	—	2	8	no	—
282	C₂×C₄⋊C₄	—	2	8	no	—
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
285	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
286	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
287	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
288	C₂×C₂²⋊C₄	—	2	8	no	—
289	C₂×C₂²⋊C₄	—	2	4	no	—
290	C₂×C₂²⋊C₄	—	2	8	no	—
291	C₂×C₂²⋊C₄	—	2	8	no	—
292	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
293	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
294	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
295	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
296	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
297	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
298	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
299	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
300	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
301	C₂×C₄²	—	2	8	no	—
302	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
303	#95	C₂³×C₆₇	1	8	yes	—
304	#95	C₂³×C₆₇	1	8	yes	—
305	#95	C₂³×C₆₇	1	16	yes	—
306	#95	C₂³×C₆₇	1	16	yes	—
307	C₂⁴	—	2	2	no	—
308	C₂⁴	—	2	4	no	—
309	C₂⁴	—	4	4	no	—
310	C₂⁴	—	4	4	no	—
311	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
312	C₂⁴	#41×C₆₇	1	8	yes	—
313	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
314	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
315	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
316	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	2	4	no	—
319	C₂×D₄	—	2	8	no	—
320	C₂×D₄	—	2	8	no	—
321	C₂×D₄	—	4	16	no	—
322	C₂²×C₄	—	2	32	no	—
323	C₂²×C₄	—	2	8	no	—
324	C₂²×C₄	—	2	4	no	—
325	C₂²×C₄	—	2	4	no	—
326	C₂²×C₄	—	2	8	no	—
327	C₂²×C₄	—	2	8	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	4	no	—
331	C₂²×C₄	—	2	8	no	—
332	C₂²×C₄	—	4	8	no	—
333	C₂²×C₄	—	4	4	no	—
334	C₂²×C₄	—	4	8	no	—
335	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
336	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
337	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
338	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
339	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
340	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
341	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
342	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
343	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
344	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
345	C₄⋊C₄	—	2	8	no	—
346	C₄⋊C₄	—	2	8	no	—
347	C₄⋊C₄	—	2	8	no	—
348	C₄⋊C₄	—	2	8	no	—
349	C₄⋊C₄	—	4	16	no	—
350	C₂²⋊C₄	—	2	8	no	—
351	C₂²⋊C₄	—	2	8	no	—
352	C₂²⋊C₄	—	2	4	no	—
353	C₂²⋊C₄	—	2	4	no	—
354	C₂²⋊C₄	—	2	8	no	—
355	C₂²⋊C₄	—	2	8	no	—
356	C₂²⋊C₄	—	2	8	no	—
357	C₂²⋊C₄	—	2	8	no	—
358	C₂²⋊C₄	—	4	16	no	—
359	C₄²	—	2	16	no	—
360	C₂³	—	2	4	no	—
361	C₂³	—	2	4	no	—
362	C₂³	—	2	4	no	—
363	C₂³	—	2	2	no	—
364	C₂³	—	2	4	no	—
365	C₂³	—	2	8	no	—
366	C₂³	—	4	8	no	—
367	C₂³	—	4	16	no	—
368	C₂³	—	4	8	no	—
369	C₂³	—	4	4	no	—
370	C₂³	—	8	8	no	—
371	C₂³	#123×C₆₇	1	16	yes	—
372	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
373	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
374	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
375	C₂³	C₂×#41×C₆₇	1	4	yes	—
376	C₂³	C₂×#41×C₆₇	1	4	yes	—
377	C₂³	C₂×#41×C₆₇	1	4	yes	—
378	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
379	D₄	—	4	8	no	—
380	D₄	—	4	8	no	—
381	C₂×C₄	—	2	16	no	—
382	C₂×C₄	—	2	8	no	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	2	16	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	2	4	no	—
387	C₂×C₄	—	2	8	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	8	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	16	no	—
393	C₂×C₄	—	2	16	no	—
394	C₂×C₄	—	2	8	no	—
395	C₂×C₄	—	4	16	no	—
396	C₂×C₄	—	4	8	no	—
397	C₂×C₄	—	4	4	no	—
398	C₂×C₄	—	8	8	no	—
399	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
400	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
401	C₂²	—	2	4	no	—
402	C₂²	—	2	4	no	—
403	C₂²	—	2	4	no	—
404	C₂²	—	2	8	no	—
405	C₂²	—	2	8	no	—
406	C₂²	—	2	8	no	—
407	C₂²	—	4	16	no	—
408	C₂²	—	4	2	no	—
409	C₂²	—	4	4	no	—
410	C₂²	—	4	8	no	—
411	C₂²	—	8	8	no	—
412	C₂²	C₂×#123×C₆₇	1	4	yes	—
413	C₂²	C₂×#123×C₆₇	1	4	yes	—
414	C₂²	C₂×#123×C₆₇	1	4	yes	—
415	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
416	C₂²	C₂²×#41×C₆₇	1	2	yes	—
417	C₂²	#585×C₆₇	1	4	yes	—
418	C₄	—	2	4	no	—
419	C₄	—	2	4	no	—
420	C₄	—	4	2	no	—
421	C₄	—	4	8	no	—
422	C₄	—	4	4	no	—
423	C₄	—	4	8	no	—
424	C₂	—	2	4	no	—
425	C₂	—	2	4	no	—
426	C₂	—	2	4	no	—
427	C₂	—	8	2	no	—
428	C₂	C₂²×#123×C₆₇	1	1	yes	characteristic
429	C₂	C₆₇×#3449	1	2	yes	—
430	C₂	C₆₇×#3491	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
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×#117×C₆₇	C₂	semidirect
3. C₆₇×#2853	C₂	semidirect
4. C₆₇×#2623	C₂	semidirect
5. C₆₇×#2498	C₂	semidirect
6. C₆₇×#2484	C₂	semidirect
7. C₆₇×#2482	C₂	semidirect
8. C₆₇×#2315	C₂	semidirect
10. C₂×#381×C₆₇	C₂	semidirect
19. C₂×#117×C₆₇	C₂²	semidirect
21. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
26. C₂²×C₄²×C₆₇	C₂²	semidirect
27. #400×C₆₇	C₂²	semidirect
28. #399×C₆₇	C₂²	semidirect
29. #398×C₆₇	C₂²	semidirect
30. #397×C₆₇	C₂²	semidirect
31. #388×C₆₇	C₂²	semidirect
32. #387×C₆₇	C₂²	semidirect
36. #385×C₆₇	C₂²	semidirect
39. #381×C₆₇	C₂²	semidirect
40. #381×C₆₇	C₂²	semidirect
41. #380×C₆₇	C₂²	semidirect
43. C₂×#95×C₆₇	C₂²	semidirect
208. #35097	C₆₇	direct (N × H)
217. C₂²×#117	C₂×C₆₇	semidirect
218. #2853	C₂×C₆₇	semidirect
219. #2623	C₂×C₆₇	semidirect
220. #2498	C₂×C₆₇	semidirect
221. #2484	C₂×C₆₇	semidirect
222. #2482	C₂×C₆₇	semidirect
223. #2315	C₂×C₆₇	semidirect
225. C₂×#381	C₂×C₆₇	semidirect
226. C₆₇	#35097	direct (N × H)
235. C₂×#117	C₂²×C₆₇	semidirect
237. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
242. C₂²×C₄²	C₂²×C₆₇	semidirect
243. #400	C₂²×C₆₇	semidirect
244. #399	C₂²×C₆₇	semidirect
245. #398	C₂²×C₆₇	semidirect
246. #397	C₂²×C₆₇	semidirect
247. #388	C₂²×C₆₇	semidirect
248. #387	C₂²×C₆₇	semidirect
252. #385	C₂²×C₆₇	semidirect
255. #381	C₂²×C₆₇	semidirect
256. #381	C₂²×C₆₇	semidirect
257. #380	C₂²×C₆₇	semidirect
259. C₂×#95	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#123×C₆₇	2
C₂²×#117×C₆₇	2
C₆₇×#2853	2
C₆₇×#2623	2
C₆₇×#2498	2
C₆₇×#2484	2
C₆₇×#2482	2
C₆₇×#2315	2
C₆₇×#2308	2
C₂×#381×C₆₇	2
#35097	67

Subgroups — C67×#35097 #30714348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#35097 #30714348