Subgroups — #30760904

Order 17152.

478 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561015	C₂	1	1	yes	maximal, characteristic
2	#22561014	C₂	1	1	yes	maximal, characteristic
3	#22561014	C₂	1	1	yes	maximal, characteristic
4	#22560336	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560336	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524040	C₂²	1	1	yes	characteristic
9	C₂×#14523965	—	2	2	no	—
10	C₂×#14523963	—	2	2	no	—
11	C₂×#14523963	—	2	2	no	—
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	#16863266	C₂²	1	1	yes	characteristic
15	#16863158	—	2	2	no	—
16	#16863158	—	2	2	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#384×C₆₇	C₂²	1	1	yes	characteristic
19	#384×C₆₇	C₂²	1	1	yes	characteristic
20	#340×C₆₇	—	2	2	no	—
21	C₂²×D₂₆₈	—	2	1	no	—
22	C₂²×D₂₆₈	—	2	1	no	—
23	C₂²×D₂₆₈	—	2	1	no	—
24	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
25	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
26	#14524040	—	2	1	no	—
27	#14524040	—	2	1	no	—
28	#14524040	—	2	1	no	—
29	#14524040	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₆₇	—	2	1	no	—
31	C₂×C₄⋊Dic₆₇	—	2	1	no	—
32	C₂×C₄⋊Dic₆₇	—	2	1	no	—
33	#14523965	—	4	4	no	—
34	#14523963	—	4	4	no	—
35	#14523963	—	4	4	no	—
36	C₄×C₆₇⋊C₈	—	2	1	no	—
37	C₄×C₆₇⋊C₈	—	2	1	no	—
38	C₄×C₆₇⋊C₈	—	2	1	no	—
39	C₄×C₆₇⋊C₈	—	2	1	no	—
40	C₂²×C₈×C₆₇	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₆₇	—	2	1	no	—
49	C₄×C₈×C₆₇	—	2	1	no	—
50	C₂³×D₆₇	—	4	1	no	—
51	C₂×D₂₆₈	—	2	2	no	—
52	C₂×D₂₆₈	—	2	2	no	—
53	C₂×D₂₆₈	—	2	2	no	—
54	C₂×D₂₆₈	—	4	2	no	—
55	C₂×D₂₆₈	—	4	2	no	—
56	C₂×D₂₆₈	—	4	2	no	—
57	C₂×D₂₆₈	—	4	2	no	—
58	C₂×D₂₆₈	—	4	2	no	—
59	C₂×D₂₆₈	—	4	2	no	—
60	C₂×D₂₆₈	—	4	1	no	—
61	C₂×D₂₆₈	—	4	1	no	—
62	C₂×D₂₆₈	—	4	1	no	—
63	C₂²×Dic₆₇	—	4	1	no	—
64	C₂×C₆₇⋊C₈	—	2	2	no	—
65	C₂×C₆₇⋊C₈	—	2	2	no	—
66	C₂×C₆₇⋊C₈	—	2	2	no	—
67	C₂×C₆₇⋊C₈	—	2	2	no	—
68	C₂×C₆₇⋊C₈	—	2	2	no	—
69	C₂×C₆₇⋊C₈	—	2	2	no	—
70	C₄⋊Dic₆₇	—	4	2	no	—
71	C₄⋊Dic₆₇	—	4	2	no	—
72	C₄⋊Dic₆₇	—	4	2	no	—
73	C₂²×C₄×C₆₇	—	4	1	no	—
74	C₂²×C₄×C₆₇	—	4	1	no	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	—	2	2	no	—
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₄⋊C₄×C₆₇	—	4	2	no	—
82	C₄⋊C₄×C₆₇	—	4	2	no	—
83	C₄⋊C₄×C₆₇	—	4	2	no	—
84	C₄⋊C₄×C₆₇	—	4	2	no	—
85	C₄⋊C₄×C₆₇	—	4	2	no	—
86	C₄⋊C₄×C₆₇	—	4	2	no	—
87	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
88	C₄²×C₆₇	#41	1	1	yes	characteristic
89	C₄²×C₆₇	#41	1	1	yes	characteristic
90	C₄²×C₆₇	#41	1	1	yes	characteristic
91	C₂³×C₆₇	#127	1	1	yes	characteristic
92	C₂²×D₆₇	—	4	2	no	—
93	C₂²×D₆₇	—	8	1	no	—
94	C₂²×D₆₇	—	8	1	no	—
95	C₂²×D₆₇	—	8	1	no	—
96	C₂²×D₆₇	—	8	1	no	—
97	C₂²×D₆₇	—	8	1	no	—
98	C₂²×D₆₇	—	8	1	no	—
99	C₂×C₄×C₆₇	—	8	1	no	—
100	C₂×C₄×C₆₇	—	8	1	no	—
101	C₂×C₄×C₆₇	—	8	1	no	—
102	C₂×C₄×C₆₇	—	8	1	no	—
103	C₂×C₄×C₆₇	—	8	1	no	—
104	C₂×C₄×C₆₇	—	8	1	no	—
105	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#135	1	2	yes	—
112	C₂×C₄×C₆₇	#135	1	2	yes	—
113	C₂×C₄×C₆₇	#135	1	2	yes	—
114	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
115	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×Dic₆₇	—	8	1	no	—
118	C₂×Dic₆₇	—	8	1	no	—
119	C₂×Dic₆₇	—	8	1	no	—
120	D₂₆₈	—	4	4	no	—
121	D₂₆₈	—	4	4	no	—
122	D₂₆₈	—	4	4	no	—
123	D₂₆₈	—	8	2	no	—
124	D₂₆₈	—	8	2	no	—
125	D₂₆₈	—	8	2	no	—
126	C₈×C₆₇	—	4	2	no	—
127	C₆₇⋊C₈	—	4	2	no	—
128	C₆₇⋊C₈	—	4	2	no	—
129	C₂²×C₆₇	#399	1	1	yes	characteristic
130	C₂²×C₆₇	#495	1	1	yes	characteristic
131	C₂²×C₆₇	#495	1	1	yes	characteristic
132	C₂²×C₆₇	#495	1	1	yes	characteristic
133	C₂²×C₆₇	#496	1	1	yes	characteristic
134	C₂²×C₆₇	#496	1	1	yes	characteristic
135	C₂²×C₆₇	#496	1	1	yes	characteristic
136	C₂×D₆₇	—	8	2	no	—
137	C₂×D₆₇	—	8	2	no	—
138	C₂×D₆₇	—	8	2	no	—
139	C₂×D₆₇	—	16	1	no	—
140	C₂×D₆₇	—	16	1	no	—
141	C₂×D₆₇	—	16	1	no	—
142	C₂×D₆₇	—	16	1	no	—
143	C₄×C₆₇	—	16	1	no	—
144	C₄×C₆₇	—	16	1	no	—
145	C₄×C₆₇	#478	1	2	yes	—
146	C₄×C₆₇	#478	1	2	yes	—
147	C₄×C₆₇	#478	1	2	yes	—
148	C₄×C₆₇	#482	1	2	yes	—
149	C₄×C₆₇	#482	1	2	yes	—
150	C₄×C₆₇	#482	1	2	yes	—
151	Dic₆₇	—	16	1	no	—
152	#10678	—	67	1	no	maximal, Sylow
153	C₂×C₆₇	#1711	1	1	yes	characteristic
154	C₂×C₆₇	#1717	1	1	yes	characteristic
155	C₂×C₆₇	#1717	1	1	yes	characteristic
156	C₂×C₆₇	#1717	1	1	yes	characteristic
157	C₂×C₆₇	#2092	1	1	yes	characteristic
158	C₂×C₆₇	#2092	1	1	yes	characteristic
159	C₂×C₆₇	#2092	1	1	yes	characteristic
160	D₆₇	—	16	2	no	—
161	#1957	—	67	1	no	—
162	#1957	—	67	1	no	—
163	#1957	—	67	1	no	—
164	#1834	—	67	1	no	—
165	#1834	—	67	1	no	—
166	#1834	D₆₇	1	1	yes	characteristic, complemented
167	#1273	—	67	1	no	—
168	C₆₇	#10678	1	1	yes	characteristic, complemented, Sylow
169	C₂×#127	—	67	1	no	—
170	C₂×#102	—	134	2	no	—
171	C₂×#102	—	134	2	no	—
172	C₂×#102	—	134	2	no	—
173	C₂×C₄×C₈	—	67	1	no	—
174	C₂×C₄×C₈	—	67	1	no	—
175	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
176	#384	—	67	1	no	—
177	#384	C₂×D₆₇	1	1	yes	characteristic
178	#384	C₂×D₆₇	1	1	yes	characteristic
179	#340	—	2	2	no	—
180	#340	—	134	2	no	—
181	#340	—	134	2	no	—
182	C₂²×D₄	—	134	1	no	—
183	C₂²×D₄	—	134	1	no	—
184	C₂²×D₄	—	134	1	no	—
185	C₂²×C₈	—	67	2	no	—
186	C₂²×C₈	—	67	2	no	—
187	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
188	#127	—	67	2	no	—
189	#127	—	134	1	no	—
190	#127	—	134	1	no	—
191	#127	—	134	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	134	1	no	—
199	C₂×C₄⋊C₄	—	134	1	no	—
200	C₂×C₄⋊C₄	—	134	1	no	—
201	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
202	#102	—	268	4	no	—
203	#102	—	268	4	no	—
204	#102	—	268	4	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	134	1	no	—
208	C₄×C₈	—	134	1	no	—
209	C₄×C₈	—	134	1	no	—
210	C₄×C₈	—	134	1	no	—
211	C₂⁴	—	268	1	no	—
212	C₂×D₄	—	268	1	no	—
213	C₂×D₄	—	268	1	no	—
214	C₂×D₄	—	268	1	no	—
215	C₂×D₄	—	268	2	no	—
216	C₂×D₄	—	268	2	no	—
217	C₂×D₄	—	268	2	no	—
218	C₂×D₄	—	268	2	no	—
219	C₂×D₄	—	268	2	no	—
220	C₂×D₄	—	268	2	no	—
221	C₂×D₄	—	134	2	no	—
222	C₂×D₄	—	134	2	no	—
223	C₂×D₄	—	134	2	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	268	1	no	—
227	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
228	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	134	2	no	—
234	C₂×C₈	—	134	2	no	—
235	C₂×C₈	—	134	2	no	—
236	C₂×C₈	—	134	2	no	—
237	C₂×C₈	—	134	2	no	—
238	C₂×C₈	—	134	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	268	2	no	—
246	C₄⋊C₄	—	268	2	no	—
247	C₄⋊C₄	—	268	2	no	—
248	C₄²	Q₈×D₆₇	1	1	yes	characteristic
249	C₄²	#11761628	1	1	yes	characteristic
250	C₄²	#11761628	1	1	yes	characteristic
251	C₄²	#11761629	1	1	yes	characteristic
252	C₂³	—	268	2	no	—
253	C₂³	—	536	1	no	—
254	C₂³	—	536	1	no	—
255	C₂³	—	536	1	no	—
256	C₂³	—	536	1	no	—
257	C₂³	—	536	1	no	—
258	C₂³	—	536	1	no	—
259	C₂³	#14524039	1	1	yes	characteristic
260	D₄	—	268	4	no	—
261	D₄	—	268	4	no	—
262	D₄	—	268	4	no	—
263	D₄	—	536	2	no	—
264	D₄	—	536	2	no	—
265	D₄	—	536	2	no	—
266	C₂×C₄	—	8	1	no	—
267	C₂×C₄	—	8	1	no	—
268	C₂×C₄	—	8	1	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	536	1	no	—
273	C₂×C₄	—	536	1	no	—
274	C₂×C₄	—	536	1	no	—
275	C₂×C₄	#14524019	1	1	yes	characteristic
276	C₂×C₄	#14524021	1	1	yes	characteristic
277	C₂×C₄	#14524021	1	1	yes	characteristic
278	C₂×C₄	#14524026	1	1	yes	characteristic
279	C₂×C₄	#14524028	1	1	yes	characteristic
280	C₂×C₄	#14524028	1	1	yes	characteristic
281	C₂×C₄	#14524063	1	2	yes	—
282	C₂×C₄	#14524064	1	2	yes	—
283	C₂×C₄	#14524066	1	2	yes	—
284	C₂×C₄	#14524067	1	2	yes	—
285	C₂×C₄	#14524068	1	2	yes	—
286	C₂×C₄	#14524070	1	2	yes	—
287	C₈	—	4	2	no	—
288	C₈	—	268	2	no	—
289	C₈	—	268	2	no	—
290	C₂²	—	536	2	no	—
291	C₂²	—	536	2	no	—
292	C₂²	—	536	2	no	—
293	C₂²	—	1072	1	no	—
294	C₂²	—	1072	1	no	—
295	C₂²	—	1072	1	no	—
296	C₂²	—	1072	1	no	—
297	C₂²	#16863331	1	1	yes	characteristic
298	C₂²	#16863777	1	1	yes	characteristic
299	C₂²	#16863778	1	1	yes	characteristic
300	C₂²	#16863781	1	1	yes	characteristic
301	C₂²	#16863782	1	1	yes	characteristic
302	C₂²	#16863784	1	1	yes	characteristic
303	C₂²	#16863786	1	1	yes	characteristic
304	C₄	—	16	1	no	—
305	C₄	—	16	1	no	—
306	C₄	—	1072	1	no	—
307	C₄	#16863693	1	2	yes	—
308	C₄	#16863695	1	2	yes	—
309	C₄	#16863699	1	2	yes	—
310	C₄	#16863720	1	2	yes	—
311	C₄	#16863722	1	2	yes	—
312	C₄	#16863723	1	2	yes	—
313	C₂	—	1072	2	no	—
314	C₂	#22559822	1	1	yes	characteristic
315	C₂	#22559849	1	1	yes	characteristic
316	C₂	#22559850	1	1	yes	characteristic
317	C₂	#22559851	1	1	yes	characteristic
318	C₂	#22561727	1	1	yes	characteristic
319	C₂	#22561731	1	1	yes	characteristic
320	C₂	#22561733	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524039
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1834	C₂
socle	C₂³×C₆₇	#127
prime radical	C₂×#14524040	C₂²
hypercenter	#1834	D₆₇
hypocenter	C₆₇	#10678
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
4. #22560336	C₂	semidirect
5. #22560336	C₂	semidirect
6. C₆₇⋊#1273	C₂	semidirect
7. C₆₇×#1834	C₂	semidirect
166. #1834	D₆₇	semidirect
168. C₆₇	#10678	semidirect

Maximal subgroups

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

subgroup	index
#22561015	2
#22561014	2
#22560336	2
C₆₇⋊#1273	2
C₆₇×#1834	2
#10678	67