Subgroups — C₆₇×#35089 #30714340

Order 17152.

3238 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#534×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#123×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂²×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2854	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2618	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2413	C₂	1	8	yes	maximal, complemented
7	C₆₇×#2411	C₂	1	8	yes	maximal, complemented
8	C₆₇×#2315	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2308	C₂	1	2	yes	maximal, complemented
10	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
12	#534×C₆₇	—	2	8	no	—
13	C₂×#123×C₆₇	—	2	16	no	—
14	C₂×#123×C₆₇	C₂²	1	8	yes	—
15	C₂×#123×C₆₇	C₂²	1	4	yes	—
16	C₂×#123×C₆₇	C₂²	1	4	yes	complemented
17	C₂×#123×C₆₇	C₂²	1	4	yes	complemented
18	C₂×#121×C₆₇	C₂²	1	4	yes	—
19	C₂×#120×C₆₇	C₂²	1	4	yes	—
20	C₂×#117×C₆₇	—	2	8	no	—
21	C₂×#117×C₆₇	C₂²	1	8	yes	complemented
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
24	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
28	C₂²×C₄²×C₆₇	C₂²	1	2	yes	complemented
29	#398×C₆₇	C₂²	1	8	yes	complemented
30	#396×C₆₇	C₂²	1	16	yes	complemented
31	#394×C₆₇	C₂²	1	8	yes	complemented
32	#390×C₆₇	C₂²	1	8	yes	complemented
33	#389×C₆₇	C₂²	1	8	yes	complemented
34	#388×C₆₇	C₂²	1	16	yes	complemented
35	#386×C₆₇	C₂²	1	8	yes	—
36	#386×C₆₇	C₂²	1	8	yes	complemented
37	#385×C₆₇	C₂²	1	8	yes	complemented
38	#381×C₆₇	C₂²	1	4	yes	complemented
39	#380×C₆₇	C₂²	1	4	yes	complemented
40	#379×C₆₇	C₂²	1	4	yes	complemented
41	C₂⁵×C₆₇	C₂³	1	2	yes	—
42	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₆₇	—	2	8	no	—
44	C₂²×D₄×C₆₇	—	2	8	no	—
45	C₂²×D₄×C₆₇	—	2	8	no	—
46	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
47	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
48	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
49	C₂³×C₄×C₆₇	—	2	4	no	—
50	C₂³×C₄×C₆₇	—	2	4	no	—
51	C₂³×C₄×C₆₇	D₄	1	8	yes	complemented
52	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	C₂³	1	4	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	#123×C₆₇	—	2	32	no	—
60	#123×C₆₇	—	2	16	no	—
61	#123×C₆₇	—	2	16	no	—
62	#123×C₆₇	—	2	16	no	—
63	#121×C₆₇	—	2	16	no	—
64	#120×C₆₇	—	2	16	no	—
65	#117×C₆₇	—	2	32	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
76	C₂×C₂²⋊C₄×C₆₇	D₄	1	8	yes	—
77	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
78	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
79	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
80	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
82	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
84	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	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	16	yes	—
89	C₂⁴×C₆₇	—	2	4	no	—
90	C₂⁴×C₆₇	—	2	4	no	—
91	C₂⁴×C₆₇	—	2	4	no	—
92	C₂⁴×C₆₇	—	4	4	no	—
93	C₂⁴×C₆₇	—	4	8	no	—
94	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
95	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
96	C₂⁴×C₆₇	#41	1	4	yes	—
97	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
98	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
99	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₂×D₄×C₆₇	—	2	8	no	—
101	C₂×D₄×C₆₇	—	2	8	no	—
102	C₂×D₄×C₆₇	—	2	8	no	—
103	C₂×D₄×C₆₇	—	2	8	no	—
104	C₂×D₄×C₆₇	—	2	8	no	—
105	C₂×D₄×C₆₇	—	2	8	no	—
106	C₂×D₄×C₆₇	—	2	8	no	—
107	C₂×D₄×C₆₇	—	4	8	no	—
108	C₂×D₄×C₆₇	—	4	32	no	—
109	C₂²×C₄×C₆₇	—	2	32	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	—	2	8	no	—
113	C₂²×C₄×C₆₇	—	2	8	no	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	2	8	no	—
118	C₂²×C₄×C₆₇	—	2	4	no	—
119	C₂²×C₄×C₆₇	—	2	4	no	—
120	C₂²×C₄×C₆₇	—	2	4	no	—
121	C₂²×C₄×C₆₇	—	4	4	no	—
122	C₂²×C₄×C₆₇	—	4	4	no	—
123	C₂²×C₄×C₆₇	—	4	8	no	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
127	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
128	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
130	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
131	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
132	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
134	C₄⋊C₄×C₆₇	—	2	16	no	—
135	C₄⋊C₄×C₆₇	—	2	16	no	—
136	C₄⋊C₄×C₆₇	—	2	16	no	—
137	C₄⋊C₄×C₆₇	—	2	8	no	—
138	C₄⋊C₄×C₆₇	—	4	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	—	2	16	no	—
143	C₂²⋊C₄×C₆₇	—	2	16	no	—
144	C₂²⋊C₄×C₆₇	—	2	16	no	—
145	C₂²⋊C₄×C₆₇	—	4	16	no	—
146	C₂²⋊C₄×C₆₇	C₂×D₄	1	16	yes	—
147	C₄²×C₆₇	—	2	16	no	—
148	C₂³×C₆₇	—	2	4	no	—
149	C₂³×C₆₇	—	2	8	no	—
150	C₂³×C₆₇	—	2	4	no	—
151	C₂³×C₆₇	—	2	4	no	—
152	C₂³×C₆₇	—	2	4	no	—
153	C₂³×C₆₇	—	2	16	no	—
154	C₂³×C₆₇	—	4	8	no	—
155	C₂³×C₆₇	—	4	8	no	—
156	C₂³×C₆₇	—	4	8	no	—
157	C₂³×C₆₇	—	4	16	no	—
158	C₂³×C₆₇	—	4	8	no	—
159	C₂³×C₆₇	—	8	8	no	—
160	C₂³×C₆₇	—	8	8	no	—
161	C₂³×C₆₇	#121	1	16	yes	—
162	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
163	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
164	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
165	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
166	C₂³×C₆₇	C₂×#41	1	4	yes	—
167	C₂³×C₆₇	C₂×#41	1	2	yes	—
168	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
169	D₄×C₆₇	—	4	16	no	—
170	D₄×C₆₇	—	4	16	no	—
171	D₄×C₆₇	—	8	8	no	—
172	C₂×C₄×C₆₇	—	2	32	no	—
173	C₂×C₄×C₆₇	—	2	8	no	—
174	C₂×C₄×C₆₇	—	2	8	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	8	no	—
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	2	8	no	—
182	C₂×C₄×C₆₇	—	2	4	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	8	no	—
186	C₂×C₄×C₆₇	—	2	16	no	—
187	C₂×C₄×C₆₇	—	4	8	no	—
188	C₂×C₄×C₆₇	—	4	16	no	—
189	C₂×C₄×C₆₇	—	4	8	no	—
190	C₂×C₄×C₆₇	—	4	8	no	—
191	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
192	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
193	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
194	C₂²×C₆₇	—	2	8	no	—
195	C₂²×C₆₇	—	2	4	no	—
196	C₂²×C₆₇	—	2	4	no	—
197	C₂²×C₆₇	—	2	8	no	—
198	C₂²×C₆₇	—	2	8	no	—
199	C₂²×C₆₇	—	2	4	no	—
200	C₂²×C₆₇	—	4	16	no	—
201	C₂²×C₆₇	—	4	4	no	—
202	C₂²×C₆₇	—	4	8	no	—
203	C₂²×C₆₇	—	4	8	no	—
204	C₂²×C₆₇	—	4	8	no	—
205	C₂²×C₆₇	—	8	16	no	—
206	C₂²×C₆₇	C₂×#121	1	8	yes	—
207	C₂²×C₆₇	C₂×#121	1	4	yes	—
208	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
209	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
210	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
211	C₂²×C₆₇	#585	1	4	yes	—
212	C₄×C₆₇	—	2	4	no	—
213	C₄×C₆₇	—	2	4	no	—
214	C₄×C₆₇	—	2	8	no	—
215	C₄×C₆₇	—	4	4	no	—
216	C₄×C₆₇	—	4	8	no	—
217	C₄×C₆₇	—	4	4	no	—
218	#35089	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂×C₆₇	—	2	8	no	—
220	C₂×C₆₇	—	2	4	no	—
221	C₂×C₆₇	—	8	4	no	—
222	C₂×C₆₇	C₂²×#121	1	1	yes	characteristic
223	C₂×C₆₇	#3449	1	2	yes	—
224	C₂×C₆₇	#3485	1	2	yes	—
225	C₂×C₆₇	#3491	1	2	yes	—
226	C₂×#534	C₂×C₆₇	1	1	yes	characteristic
227	C₂²×#123	C₂×C₆₇	1	2	yes	complemented
228	C₂²×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
229	#2854	C₂×C₆₇	1	4	yes	complemented
230	#2618	C₂×C₆₇	1	4	yes	complemented
231	#2413	C₂×C₆₇	1	8	yes	complemented
232	#2411	C₂×C₆₇	1	8	yes	complemented
233	#2315	C₂×C₆₇	1	1	yes	characteristic, complemented
234	#2308	C₂×C₆₇	1	2	yes	complemented
235	C₆₇	#35089	1	1	yes	characteristic, direct product factor, Sylow
236	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
237	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
238	#534	—	2	8	no	—
239	C₂×#123	—	2	16	no	—
240	C₂×#123	C₂²×C₆₇	1	8	yes	—
241	C₂×#123	C₂²×C₆₇	1	4	yes	—
242	C₂×#123	C₂²×C₆₇	1	4	yes	complemented
243	C₂×#123	C₂²×C₆₇	1	4	yes	complemented
244	C₂×#121	C₂²×C₆₇	1	4	yes	—
245	C₂×#120	C₂²×C₆₇	1	4	yes	—
246	C₂×#117	—	2	8	no	—
247	C₂×#117	C₂²×C₆₇	1	8	yes	complemented
248	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
249	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
250	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
251	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
252	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
253	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
254	C₂²×C₄²	C₂²×C₆₇	1	2	yes	complemented
255	#398	C₂²×C₆₇	1	8	yes	complemented
256	#396	C₂²×C₆₇	1	16	yes	complemented
257	#394	C₂²×C₆₇	1	8	yes	complemented
258	#390	C₂²×C₆₇	1	8	yes	complemented
259	#389	C₂²×C₆₇	1	8	yes	complemented
260	#388	C₂²×C₆₇	1	16	yes	complemented
261	#386	C₂²×C₆₇	1	8	yes	—
262	#386	C₂²×C₆₇	1	8	yes	complemented
263	#385	C₂²×C₆₇	1	8	yes	complemented
264	#381	C₂²×C₆₇	1	4	yes	complemented
265	#380	C₂²×C₆₇	1	4	yes	complemented
266	#379	C₂²×C₆₇	1	4	yes	complemented
267	C₂⁵	C₂³×C₆₇	1	2	yes	—
268	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
269	C₂²×D₄	—	2	8	no	—
270	C₂²×D₄	—	2	8	no	—
271	C₂²×D₄	—	2	8	no	—
272	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
273	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
274	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
275	C₂³×C₄	—	2	4	no	—
276	C₂³×C₄	—	2	4	no	—
277	C₂³×C₄	D₄×C₆₇	1	8	yes	complemented
278	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
279	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
280	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
281	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
282	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
283	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
284	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
285	#123	—	2	32	no	—
286	#123	—	2	16	no	—
287	#123	—	2	16	no	—
288	#123	—	2	16	no	—
289	#121	—	2	16	no	—
290	#120	—	2	16	no	—
291	#117	—	2	32	no	—
292	C₂×C₄⋊C₄	—	2	8	no	—
293	C₂×C₄⋊C₄	—	2	16	no	—
294	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
295	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
296	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
297	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
298	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
299	C₂×C₂²⋊C₄	—	2	4	no	—
300	C₂×C₂²⋊C₄	—	2	8	no	—
301	C₂×C₂²⋊C₄	—	2	16	no	—
302	C₂×C₂²⋊C₄	D₄×C₆₇	1	8	yes	—
303	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
304	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
305	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
306	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
307	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
308	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
309	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
310	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
311	C₂×C₄²	—	2	8	no	—
312	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
313	#95	C₂³×C₆₇	1	8	yes	—
314	#95	C₂³×C₆₇	1	16	yes	—
315	C₂⁴	—	2	4	no	—
316	C₂⁴	—	2	4	no	—
317	C₂⁴	—	2	4	no	—
318	C₂⁴	—	4	4	no	—
319	C₂⁴	—	4	8	no	—
320	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
321	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
322	C₂⁴	#41×C₆₇	1	4	yes	—
323	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
324	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
325	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
326	C₂×D₄	—	2	8	no	—
327	C₂×D₄	—	2	8	no	—
328	C₂×D₄	—	2	8	no	—
329	C₂×D₄	—	2	8	no	—
330	C₂×D₄	—	2	8	no	—
331	C₂×D₄	—	2	8	no	—
332	C₂×D₄	—	2	8	no	—
333	C₂×D₄	—	4	8	no	—
334	C₂×D₄	—	4	32	no	—
335	C₂²×C₄	—	2	32	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	8	no	—
339	C₂²×C₄	—	2	8	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	2	8	no	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	4	4	no	—
348	C₂²×C₄	—	4	4	no	—
349	C₂²×C₄	—	4	8	no	—
350	C₂²×C₄	C₂×D₄×C₆₇	1	16	yes	—
351	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
352	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
353	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
354	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
355	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
356	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
357	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
358	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
359	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
360	C₄⋊C₄	—	2	16	no	—
361	C₄⋊C₄	—	2	16	no	—
362	C₄⋊C₄	—	2	16	no	—
363	C₄⋊C₄	—	2	8	no	—
364	C₄⋊C₄	—	4	8	no	—
365	C₂²⋊C₄	—	2	8	no	—
366	C₂²⋊C₄	—	2	8	no	—
367	C₂²⋊C₄	—	2	8	no	—
368	C₂²⋊C₄	—	2	16	no	—
369	C₂²⋊C₄	—	2	16	no	—
370	C₂²⋊C₄	—	2	16	no	—
371	C₂²⋊C₄	—	4	16	no	—
372	C₂²⋊C₄	C₂×D₄×C₆₇	1	16	yes	—
373	C₄²	—	2	16	no	—
374	C₂³	—	2	4	no	—
375	C₂³	—	2	8	no	—
376	C₂³	—	2	4	no	—
377	C₂³	—	2	4	no	—
378	C₂³	—	2	4	no	—
379	C₂³	—	2	16	no	—
380	C₂³	—	4	8	no	—
381	C₂³	—	4	8	no	—
382	C₂³	—	4	8	no	—
383	C₂³	—	4	16	no	—
384	C₂³	—	4	8	no	—
385	C₂³	—	8	8	no	—
386	C₂³	—	8	8	no	—
387	C₂³	#121×C₆₇	1	16	yes	—
388	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
389	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
390	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
391	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
392	C₂³	C₂×#41×C₆₇	1	4	yes	—
393	C₂³	C₂×#41×C₆₇	1	2	yes	—
394	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
395	D₄	—	4	16	no	—
396	D₄	—	4	16	no	—
397	D₄	—	8	8	no	—
398	C₂×C₄	—	2	32	no	—
399	C₂×C₄	—	2	8	no	—
400	C₂×C₄	—	2	8	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	8	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	8	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	8	no	—
412	C₂×C₄	—	2	16	no	—
413	C₂×C₄	—	4	8	no	—
414	C₂×C₄	—	4	16	no	—
415	C₂×C₄	—	4	8	no	—
416	C₂×C₄	—	4	8	no	—
417	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
418	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
419	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
420	C₂²	—	2	8	no	—
421	C₂²	—	2	4	no	—
422	C₂²	—	2	4	no	—
423	C₂²	—	2	8	no	—
424	C₂²	—	2	8	no	—
425	C₂²	—	2	4	no	—
426	C₂²	—	4	16	no	—
427	C₂²	—	4	4	no	—
428	C₂²	—	4	8	no	—
429	C₂²	—	4	8	no	—
430	C₂²	—	4	8	no	—
431	C₂²	—	8	16	no	—
432	C₂²	C₂×#121×C₆₇	1	8	yes	—
433	C₂²	C₂×#121×C₆₇	1	4	yes	—
434	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
435	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
436	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
437	C₂²	#585×C₆₇	1	4	yes	—
438	C₄	—	2	4	no	—
439	C₄	—	2	4	no	—
440	C₄	—	2	8	no	—
441	C₄	—	4	4	no	—
442	C₄	—	4	8	no	—
443	C₄	—	4	4	no	—
444	C₂	—	2	8	no	—
445	C₂	—	2	4	no	—
446	C₂	—	8	4	no	—
447	C₂	C₂²×#121×C₆₇	1	1	yes	characteristic
448	C₂	C₆₇×#3449	1	2	yes	—
449	C₂	C₆₇×#3485	1	2	yes	—
450	C₂	C₆₇×#3491	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#534×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×#123×C₆₇	C₂	semidirect
3. C₂²×#117×C₆₇	C₂	semidirect
4. C₆₇×#2854	C₂	semidirect
5. C₆₇×#2618	C₂	semidirect
6. C₆₇×#2413	C₂	semidirect
7. C₆₇×#2411	C₂	semidirect
8. C₆₇×#2315	C₂	semidirect
9. C₆₇×#2308	C₂	semidirect
16. C₂×#123×C₆₇	C₂²	semidirect
17. C₂×#123×C₆₇	C₂²	semidirect
21. C₂×#117×C₆₇	C₂²	semidirect
22. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
23. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
27. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
28. C₂²×C₄²×C₆₇	C₂²	semidirect
29. #398×C₆₇	C₂²	semidirect
30. #396×C₆₇	C₂²	semidirect
31. #394×C₆₇	C₂²	semidirect
32. #390×C₆₇	C₂²	semidirect
33. #389×C₆₇	C₂²	semidirect
34. #388×C₆₇	C₂²	semidirect
36. #386×C₆₇	C₂²	semidirect
37. #385×C₆₇	C₂²	semidirect
38. #381×C₆₇	C₂²	semidirect
39. #380×C₆₇	C₂²	semidirect
40. #379×C₆₇	C₂²	semidirect
51. C₂³×C₄×C₆₇	D₄	semidirect
218. #35089	C₆₇	direct (N × H)
227. C₂²×#123	C₂×C₆₇	semidirect
228. C₂²×#117	C₂×C₆₇	semidirect
229. #2854	C₂×C₆₇	semidirect
230. #2618	C₂×C₆₇	semidirect
231. #2413	C₂×C₆₇	semidirect
232. #2411	C₂×C₆₇	semidirect
233. #2315	C₂×C₆₇	semidirect
234. #2308	C₂×C₆₇	semidirect
235. C₆₇	#35089	direct (N × H)
242. C₂×#123	C₂²×C₆₇	semidirect
243. C₂×#123	C₂²×C₆₇	semidirect
247. C₂×#117	C₂²×C₆₇	semidirect
248. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
249. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
253. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
254. C₂²×C₄²	C₂²×C₆₇	semidirect
255. #398	C₂²×C₆₇	semidirect
256. #396	C₂²×C₆₇	semidirect
257. #394	C₂²×C₆₇	semidirect
258. #390	C₂²×C₆₇	semidirect
259. #389	C₂²×C₆₇	semidirect
260. #388	C₂²×C₆₇	semidirect
262. #386	C₂²×C₆₇	semidirect
263. #385	C₂²×C₆₇	semidirect
264. #381	C₂²×C₆₇	semidirect
265. #380	C₂²×C₆₇	semidirect
266. #379	C₂²×C₆₇	semidirect
277. C₂³×C₄	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×#534×C₆₇	2
C₂²×#123×C₆₇	2
C₂²×#117×C₆₇	2
C₆₇×#2854	2
C₆₇×#2618	2
C₆₇×#2413	2
C₆₇×#2411	2
C₆₇×#2315	2
C₆₇×#2308	2
#35089	67

Subgroups — C67×#35089 #30714340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#35089 #30714340