Subgroups — C₆₇×#10260 #30689511

Order 17152.

794 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#459×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1954	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#331×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₆₇	—	2	2	no	—
7	C₂²×D₈×C₆₇	—	2	2	no	—
8	C₂×#127×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
10	#493×C₆₇	—	2	4	no	—
11	#459×C₆₇	—	2	4	no	—
12	#459×C₆₇	—	2	4	no	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×#102×C₆₇	—	2	2	no	—
16	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#331×C₆₇	—	2	4	no	—
19	C₂²×D₄×C₆₇	—	2	2	no	—
20	C₂²×D₄×C₆₇	—	2	2	no	—
21	C₂²×D₄×C₆₇	—	2	2	no	—
22	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₆₇	—	2	1	no	—
24	C₂×D₈×C₆₇	—	2	8	no	—
25	C₂×D₈×C₆₇	—	2	8	no	—
26	C₂×D₈×C₆₇	—	4	4	no	—
27	C₂×D₈×C₆₇	—	4	4	no	—
28	C₂×D₈×C₆₇	—	4	4	no	—
29	C₂×D₈×C₆₇	—	4	4	no	—
30	C₂²×C₈×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	—	2	1	no	—
32	C₂²×C₈×C₆₇	D₄	1	2	yes	—
33	#127×C₆₇	—	2	2	no	—
34	#127×C₆₇	—	2	2	no	—
35	#127×C₆₇	—	2	2	no	—
36	#127×C₆₇	D₄	1	4	yes	—
37	C₄×D₄×C₆₇	—	2	2	no	—
38	C₄×D₄×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₇	—	2	2	no	—
43	C₄⋊C₈×C₆₇	—	2	2	no	—
44	C₄⋊C₈×C₆₇	D₄	1	4	yes	—
45	#102×C₆₇	—	2	8	no	—
46	C₂²⋊C₈×C₆₇	—	4	4	no	—
47	C₄×C₈×C₆₇	—	2	2	no	—
48	C₂⁴×C₆₇	—	2	1	no	—
49	C₂⁴×C₆₇	—	4	2	no	—
50	C₂×D₄×C₆₇	—	2	2	no	—
51	C₂×D₄×C₆₇	—	2	4	no	—
52	C₂×D₄×C₆₇	—	2	4	no	—
53	C₂×D₄×C₆₇	—	2	4	no	—
54	C₂×D₄×C₆₇	—	2	1	no	—
55	C₂×D₄×C₆₇	—	2	2	no	—
56	C₂×D₄×C₆₇	—	4	2	no	—
57	C₂×D₄×C₆₇	—	4	2	no	—
58	C₂×D₄×C₆₇	—	4	2	no	—
59	C₂×D₄×C₆₇	—	4	2	no	—
60	C₂×D₄×C₆₇	—	4	4	no	—
61	C₂×D₄×C₆₇	—	4	2	no	—
62	C₂×D₄×C₆₇	—	4	2	no	—
63	C₂×D₄×C₆₇	—	4	4	no	—
64	C₂×D₄×C₆₇	—	4	2	no	—
65	C₂×D₄×C₆₇	—	8	4	no	—
66	C₂×D₄×C₆₇	D₈	1	2	yes	complemented
67	C₂²×C₄×C₆₇	—	2	1	no	—
68	C₂²×C₄×C₆₇	—	4	1	no	—
69	C₂²×C₄×C₆₇	—	4	1	no	—
70	C₂²×C₄×C₆₇	—	4	2	no	—
71	C₂²×C₄×C₆₇	—	4	2	no	—
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
75	D₈×C₆₇	—	4	16	no	—
76	D₈×C₆₇	—	4	16	no	—
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	—	4	2	no	—
81	C₂×C₈×C₆₇	—	4	2	no	—
82	C₂×C₈×C₆₇	—	4	1	no	—
83	C₂×C₈×C₆₇	—	4	2	no	—
84	C₂×C₈×C₆₇	D₈	1	4	yes	complemented
85	C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₂²⋊C₄×C₆₇	—	4	2	no	—
87	C₄²×C₆₇	C₂×D₄	1	2	yes	—
88	C₄²×C₆₇	C₂×D₄	1	2	yes	—
89	C₂³×C₆₇	—	2	2	no	—
90	C₂³×C₆₇	—	4	1	no	—
91	C₂³×C₆₇	—	4	2	no	—
92	C₂³×C₆₇	—	4	1	no	—
93	C₂³×C₆₇	—	4	2	no	—
94	C₂³×C₆₇	—	4	4	no	—
95	C₂³×C₆₇	—	8	4	no	—
96	C₂³×C₆₇	—	8	2	no	—
97	C₂³×C₆₇	—	8	2	no	—
98	C₂³×C₆₇	—	8	4	no	—
99	C₂³×C₆₇	#121	1	1	yes	characteristic
100	D₄×C₆₇	—	2	4	no	—
101	D₄×C₆₇	—	4	4	no	—
102	D₄×C₆₇	—	4	8	no	—
103	D₄×C₆₇	—	4	2	no	—
104	D₄×C₆₇	—	4	4	no	—
105	D₄×C₆₇	—	8	4	no	—
106	D₄×C₆₇	—	8	4	no	—
107	D₄×C₆₇	—	8	2	no	—
108	D₄×C₆₇	—	8	2	no	—
109	D₄×C₆₇	—	8	2	no	—
110	D₄×C₆₇	—	8	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	4	1	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	—	8	2	no	—
115	C₂×C₄×C₆₇	—	8	1	no	—
116	C₂×C₄×C₆₇	—	8	1	no	—
117	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#120	1	2	yes	—
119	C₂×C₄×C₆₇	#121	1	2	yes	—
120	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
124	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
126	C₈×C₆₇	—	2	4	no	—
127	C₈×C₆₇	—	4	4	no	—
128	C₈×C₆₇	—	8	1	no	—
129	C₂²×C₆₇	—	4	4	no	—
130	C₂²×C₆₇	—	4	2	no	—
131	C₂²×C₆₇	—	8	2	no	—
132	C₂²×C₆₇	—	8	1	no	—
133	C₂²×C₆₇	—	8	1	no	—
134	C₂²×C₆₇	—	8	4	no	—
135	C₂²×C₆₇	—	8	4	no	—
136	C₂²×C₆₇	—	8	4	no	—
137	C₂²×C₆₇	—	16	4	no	—
138	C₂²×C₆₇	—	16	4	no	—
139	C₂²×C₆₇	#392	1	1	yes	characteristic
140	C₂²×C₆₇	#459	1	2	yes	—
141	C₂²×C₆₇	#466	1	1	yes	characteristic
142	C₂²×C₆₇	#469	1	1	yes	characteristic
143	C₂²×C₆₇	#472	1	2	yes	—
144	C₄×C₆₇	—	2	2	no	—
145	C₄×C₆₇	—	8	1	no	—
146	C₄×C₆₇	#447	1	2	yes	—
147	C₄×C₆₇	#447	1	2	yes	—
148	C₄×C₆₇	#449	1	2	yes	—
149	C₄×C₆₇	#493	1	1	yes	characteristic
150	C₄×C₆₇	#496	1	1	yes	characteristic
151	#10260	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂×C₆₇	—	8	2	no	—
153	C₂×C₆₇	—	16	4	no	—
154	C₂×C₆₇	#1667	1	2	yes	—
155	C₂×C₆₇	#1687	1	2	yes	—
156	C₂×C₆₇	#1999	1	1	yes	characteristic
157	C₂×C₆₇	#2011	1	1	yes	characteristic
158	C₂×C₆₇	#2018	1	1	yes	characteristic
159	C₂×#493	C₂×C₆₇	1	1	yes	characteristic, complemented
160	C₂×#459	C₂×C₆₇	1	2	yes	complemented
161	#1954	C₂×C₆₇	1	1	yes	characteristic, complemented
162	C₂×#331	C₂×C₆₇	1	2	yes	complemented
163	#1277	C₂×C₆₇	1	1	yes	characteristic, complemented
164	C₆₇	#10260	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×D₈	—	2	2	no	—
166	C₂²×D₈	—	2	2	no	—
167	C₂×#127	C₂²×C₆₇	1	2	yes	—
168	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
169	#493	—	2	4	no	—
170	#459	—	2	4	no	—
171	#459	—	2	4	no	—
172	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
173	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
174	C₂×#102	—	2	2	no	—
175	C₂×C₂²⋊C₈	—	2	2	no	—
176	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
177	#331	—	2	4	no	—
178	C₂²×D₄	—	2	2	no	—
179	C₂²×D₄	—	2	2	no	—
180	C₂²×D₄	—	2	2	no	—
181	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
182	C₂³×C₄	—	2	1	no	—
183	C₂×D₈	—	2	8	no	—
184	C₂×D₈	—	2	8	no	—
185	C₂×D₈	—	4	4	no	—
186	C₂×D₈	—	4	4	no	—
187	C₂×D₈	—	4	4	no	—
188	C₂×D₈	—	4	4	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	D₄×C₆₇	1	2	yes	—
192	#127	—	2	2	no	—
193	#127	—	2	2	no	—
194	#127	—	2	2	no	—
195	#127	D₄×C₆₇	1	4	yes	—
196	C₄×D₄	—	2	2	no	—
197	C₄×D₄	—	2	2	no	—
198	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
199	C₂×C₂²⋊C₄	—	2	1	no	—
200	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	D₄×C₆₇	1	4	yes	—
204	#102	—	2	8	no	—
205	C₂²⋊C₈	—	4	4	no	—
206	C₄×C₈	—	2	2	no	—
207	C₂⁴	—	2	1	no	—
208	C₂⁴	—	4	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	2	4	no	—
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	4	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	4	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	8	4	no	—
225	C₂×D₄	D₈×C₆₇	1	2	yes	complemented
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
233	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
234	D₈	—	4	16	no	—
235	D₈	—	4	16	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	D₈×C₆₇	1	4	yes	complemented
244	C₄⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₄²	C₂×D₄×C₆₇	1	2	yes	—
247	C₄²	C₂×D₄×C₆₇	1	2	yes	—
248	C₂³	—	2	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	4	1	no	—
253	C₂³	—	4	4	no	—
254	C₂³	—	8	4	no	—
255	C₂³	—	8	2	no	—
256	C₂³	—	8	2	no	—
257	C₂³	—	8	4	no	—
258	C₂³	#121×C₆₇	1	1	yes	characteristic
259	D₄	—	2	4	no	—
260	D₄	—	4	2	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	4	no	—
263	D₄	—	4	8	no	—
264	D₄	—	8	4	no	—
265	D₄	—	8	4	no	—
266	D₄	—	8	2	no	—
267	D₄	—	8	2	no	—
268	D₄	—	8	2	no	—
269	D₄	—	8	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	2	no	—
276	C₂×C₄	#120×C₆₇	1	2	yes	—
277	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
278	C₂×C₄	#121×C₆₇	1	2	yes	—
279	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
280	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
281	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
282	C₂×C₄	#135×C₆₇	1	1	yes	characteristic
283	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
284	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
285	C₈	—	2	4	no	—
286	C₈	—	4	4	no	—
287	C₈	—	8	1	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	8	1	no	—
291	C₂²	—	8	1	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	4	no	—
294	C₂²	—	8	4	no	—
295	C₂²	—	8	4	no	—
296	C₂²	—	16	4	no	—
297	C₂²	—	16	4	no	—
298	C₂²	#392×C₆₇	1	1	yes	characteristic
299	C₂²	#459×C₆₇	1	2	yes	—
300	C₂²	#466×C₆₇	1	1	yes	characteristic
301	C₂²	#469×C₆₇	1	1	yes	characteristic
302	C₂²	#472×C₆₇	1	2	yes	—
303	C₄	—	2	2	no	—
304	C₄	—	8	1	no	—
305	C₄	#447×C₆₇	1	2	yes	—
306	C₄	#447×C₆₇	1	2	yes	—
307	C₄	#449×C₆₇	1	2	yes	—
308	C₄	#493×C₆₇	1	1	yes	characteristic
309	C₄	#496×C₆₇	1	1	yes	characteristic
310	C₂	—	8	2	no	—
311	C₂	—	16	4	no	—
312	C₂	C₆₇×#1667	1	2	yes	—
313	C₂	C₆₇×#1687	1	2	yes	—
314	C₂	C₆₇×#1999	1	1	yes	characteristic
315	C₂	C₆₇×#2011	1	1	yes	characteristic
316	C₂	C₆₇×#2018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#493×C₆₇	C₂	semidirect
2. C₂×#459×C₆₇	C₂	semidirect
3. C₆₇×#1954	C₂	semidirect
4. C₂×#331×C₆₇	C₂	semidirect
5. C₆₇×#1277	C₂	semidirect
66. C₂×D₄×C₆₇	D₈	semidirect
84. C₂×C₈×C₆₇	D₈	semidirect
151. #10260	C₆₇	direct (N × H)
159. C₂×#493	C₂×C₆₇	semidirect
160. C₂×#459	C₂×C₆₇	semidirect
161. #1954	C₂×C₆₇	semidirect
162. C₂×#331	C₂×C₆₇	semidirect
163. #1277	C₂×C₆₇	semidirect
164. C₆₇	#10260	direct (N × H)
225. C₂×D₄	D₈×C₆₇	semidirect
243. C₂×C₈	D₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₆₇	2
C₂×#459×C₆₇	2
C₆₇×#1954	2
C₂×#331×C₆₇	2
C₆₇×#1277	2
#10260	67

Subgroups — C67×#10260 #30689511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10260 #30689511