Subgroups — #31013482

Order 17152.

506 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573214	C₂	1	1	yes	maximal, characteristic, complemented
2	#22573061	C₂	1	1	yes	maximal, characteristic
3	#22562438	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562435	C₂	1	2	yes	maximal, complemented
5	#22562350	C₂	1	2	yes	maximal, complemented
6	#22562255	C₂	1	1	yes	maximal, characteristic
7	#22562253	C₂	1	2	yes	maximal
8	#22562111	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14523986	C₂	1	1	yes	maximal, characteristic
10	#22562060	C₂	1	2	yes	maximal
11	C₆₇×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864124	—	2	1	no	—
13	#16864061	—	2	1	no	—
14	C₂×#14523988	C₂²	1	2	yes	—
15	C₂×#14523986	C₂²	1	1	yes	characteristic
16	#16863800	C₂²	1	2	yes	—
17	#16863776	C₂²	1	1	yes	characteristic
18	#16863774	C₂²	1	2	yes	—
19	#16863675	—	2	2	no	—
20	#16863673	—	2	1	no	—
21	#16863606	—	2	1	no	—
22	#16863605	—	2	2	no	—
23	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
24	#16863490	C₂²	1	2	yes	—
25	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
26	#16863478	C₂²	1	2	yes	—
27	#16863223	C₂²	1	1	yes	characteristic
28	Dic₆₇⋊C₁₆	C₂²	1	2	yes	—
29	#16863201	C₂²	1	2	yes	—
30	#16863200	C₂²	1	1	yes	characteristic
31	#16863199	C₂²	1	2	yes	—
32	#16863199	C₂²	1	2	yes	—
33	#16863196	C₂²	1	1	yes	characteristic
34	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₆₇	C₂²	1	2	yes	—
36	#500×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
38	#367×C₆₇	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	#358×C₆₇	C₂²	1	2	yes	—
41	C₂×Q₈×D₆₇	—	4	1	no	—
42	C₂×Dic₂₆₈	—	2	1	no	—
43	C₂×Dic₂₆₈	C₂³	1	2	yes	—
44	C₂×#11761618	—	2	2	no	—
45	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
46	#14524043	—	2	1	no	—
47	#14524042	—	2	2	no	—
48	#14524041	—	2	2	no	—
49	#14524035	—	4	1	no	—
50	#14524033	—	4	2	no	—
51	#14524021	—	4	2	no	—
52	#14524019	—	4	1	no	—
53	C₄×Dic₁₃₄	—	4	1	no	—
54	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
55	#14524008	—	2	2	no	—
56	C₄×D₂₆₈	—	2	1	no	—
57	#14524005	—	2	2	no	—
58	#14523988	—	2	4	no	—
59	#14523986	—	2	2	no	—
60	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
62	#14523977	C₂³	1	1	yes	characteristic
63	#14523976	—	2	2	no	—
64	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
65	Dic₆₇⋊C₈	D₄	1	2	yes	—
66	#14523966	—	2	2	no	—
67	#14523965	—	2	1	no	—
68	#14523963	—	2	2	no	—
69	C₈×Dic₆₇	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₆₇	—	2	1	no	—
72	C₄×Q₈×C₆₇	—	2	1	no	—
73	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
74	#107×C₆₇	—	2	1	no	—
75	#107×C₆₇	C₂³	1	1	yes	characteristic
76	#107×C₆₇	C₂³	1	2	yes	—
77	#103×C₆₇	—	2	1	no	—
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₃₄	—	2	2	no	—
80	C₂×Dic₁₃₄	—	4	2	no	—
81	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
82	Q₈×D₆₇	—	8	4	no	—
83	C₂×D₂₆₈	—	2	1	no	—
84	C₂×C₄×D₆₇	—	4	1	no	—
85	C₂×C₄×D₆₇	—	4	2	no	—
86	Dic₂₆₈	—	2	2	no	—
87	Dic₂₆₈	—	2	2	no	—
88	Dic₂₆₈	—	8	1	no	—
89	#11761618	—	8	2	no	—
90	D₅₃₆	—	2	1	no	—
91	D₅₃₆	#41	1	2	yes	—
92	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
93	C₄⋊Dic₆₇	—	2	1	no	—
94	C₄⋊Dic₆₇	—	4	1	no	—
95	C₄⋊Dic₆₇	—	4	1	no	—
96	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
97	Dic₆₇⋊C₄	—	4	2	no	—
98	Dic₆₇⋊C₄	—	4	2	no	—
99	Dic₆₇⋊C₄	—	4	2	no	—
100	Dic₆₇⋊C₄	—	4	2	no	—
101	Dic₆₇⋊C₄	D₈	1	2	yes	—
102	(C₂×D₆₇)⋊C₄	—	4	1	no	—
103	(C₂×D₆₇)⋊C₄	—	4	2	no	—
104	C₄×Dic₆₇	—	4	2	no	—
105	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
106	C₆₇⋊C₁₆	—	2	1	no	—
107	C₆₇⋊C₁₆	—	2	1	no	—
108	C₂×Q₈×C₆₇	—	2	1	no	—
109	C₂×Q₈×C₆₇	—	4	1	no	—
110	Q₁₆×C₆₇	—	2	1	no	—
111	Q₁₆×C₆₇	—	2	1	no	—
112	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₆₇	—	2	1	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	4	1	no	—
117	C₄⋊C₄×C₆₇	—	4	1	no	—
118	C₄²×C₆₇	—	4	1	no	—
119	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₁₆×C₆₇	—	2	2	no	—
121	C₂²×D₆₇	—	4	1	no	—
122	Q₈×C₆₇	—	4	1	no	—
123	Q₈×C₆₇	—	4	1	no	—
124	Q₈×C₆₇	—	8	2	no	—
125	C₂×C₄×C₆₇	—	4	1	no	—
126	C₂×C₄×C₆₇	—	4	1	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₆₇	—	4	2	no	—
132	C₂×Dic₆₇	—	4	1	no	—
133	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
134	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
135	D₂₆₈	—	2	2	no	—
136	D₂₆₈	—	4	1	no	—
137	C₄×D₆₇	—	8	2	no	—
138	C₄×D₆₇	—	8	4	no	—
139	Dic₁₃₄	—	4	2	no	—
140	Dic₁₃₄	—	4	2	no	—
141	Dic₁₃₄	—	8	4	no	—
142	Dic₁₃₄	SD₃₂	1	4	yes	complemented
143	C₈×C₆₇	—	2	1	no	—
144	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
145	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
146	C₆₇⋊C₈	—	2	2	no	—
147	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₆₇	—	4	2	no	—
149	C₂×D₆₇	—	8	1	no	—
150	C₂×D₆₇	—	8	1	no	—
151	C₄×C₆₇	—	2	1	no	—
152	C₄×C₆₇	—	8	1	no	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	—	8	1	no	—
155	C₄×C₆₇	C₂×SD₃₂	1	2	yes	—
156	C₄×C₆₇	#548	1	1	yes	characteristic
157	C₄×C₆₇	#577	1	1	yes	characteristic
158	Dic₆₇	—	2	2	no	—
159	Dic₆₇	—	8	2	no	—
160	Dic₆₇	—	8	1	no	—
161	Dic₆₇	C₂×SD₃₂	1	4	yes	—
162	#33926	—	67	1	no	maximal, Sylow
163	C₂×C₆₇	#3358	1	1	yes	characteristic
164	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
165	C₂×C₆₇	#3417	1	1	yes	characteristic
166	D₆₇	—	8	2	no	—
167	#3395	—	67	1	no	—
168	Q₈×D₈	—	67	1	no	—
169	#2241	—	67	1	no	—
170	#2241	—	67	2	no	—
171	#2225	—	67	2	no	—
172	#2225	D₆₇	1	1	yes	characteristic, complemented
173	#2208	—	67	1	no	—
174	#2208	—	67	2	no	—
175	Q₈×C₁₆	—	67	1	no	—
176	C₄×SD₃₂	—	67	1	no	—
177	C₄×SD₃₂	—	67	2	no	—
178	C₆₇	#33926	1	1	yes	characteristic, complemented, Sylow
179	#557	—	134	1	no	—
180	D₄×Q₈	—	134	1	no	—
181	C₂×SD₃₂	—	67	1	no	—
182	C₂×SD₃₂	—	67	2	no	—
183	#500	—	67	2	no	—
184	#500	C₂×D₆₇	1	1	yes	characteristic
185	#494	—	67	2	no	—
186	#494	—	67	1	no	—
187	#474	—	134	2	no	—
188	#474	—	134	1	no	—
189	#462	—	134	2	no	—
190	#462	—	134	1	no	—
191	C₈×Q₈	—	67	1	no	—
192	C₄×Q₁₆	—	67	2	no	—
193	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
194	C₄×D₈	—	67	2	no	—
195	C₄×D₈	—	67	1	no	—
196	#367	—	67	1	no	—
197	#367	C₂×D₆₇	1	2	yes	—
198	C₄⋊C₁₆	—	67	2	no	—
199	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
200	#358	—	67	1	no	—
201	#358	—	67	2	no	—
202	#358	—	67	2	no	—
203	#358	—	67	2	no	—
204	#358	C₂×D₆₇	1	2	yes	—
205	#357	—	67	1	no	—
206	C₄×C₁₆	—	67	2	no	—
207	C₄×C₁₆	—	67	1	no	—
208	C₂²×Q₈	—	268	1	no	—
209	C₂×Q₁₆	—	67	2	no	—
210	C₂×Q₁₆	—	134	2	no	—
211	C₂×Q₁₆	—	134	1	no	—
212	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
213	C₂×D₈	—	67	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	134	2	no	—
216	C₄⋊Q₈	—	134	2	no	—
217	C₄⋊Q₈	—	134	1	no	—
218	#125	—	268	1	no	—
219	#125	—	268	2	no	—
220	C₂²⋊Q₈	—	268	2	no	—
221	C₂²⋊Q₈	—	268	1	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	268	1	no	—
224	C₄×Q₈	—	67	1	no	—
225	C₄×Q₈	—	134	2	no	—
226	C₄×D₄	—	134	2	no	—
227	C₄×D₄	—	134	1	no	—
228	SD₃₂	—	134	4	no	—
229	SD₃₂	—	134	2	no	—
230	C₂×C₁₆	—	67	1	no	—
231	C₂×C₁₆	—	67	1	no	—
232	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
233	#107	—	2	1	no	—
234	#107	—	67	1	no	—
235	#107	—	134	2	no	—
236	#107	C₂²×D₆₇	1	1	yes	characteristic
237	#107	C₂²×D₆₇	1	2	yes	—
238	C₄⋊C₈	—	67	2	no	—
239	C₄⋊C₈	—	67	1	no	—
240	#103	—	2	1	no	—
241	#103	—	134	2	no	—
242	#102	—	134	2	no	—
243	#102	—	134	1	no	—
244	C₄×C₈	—	67	2	no	—
245	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	268	2	no	—
249	C₂×Q₈	—	536	4	no	—
250	C₂×Q₈	—	67	1	no	—
251	C₂×Q₈	—	134	2	no	—
252	C₂×D₄	—	134	1	no	—
253	C₂²×C₄	—	268	2	no	—
254	C₂²×C₄	—	268	1	no	—
255	Q₁₆	—	2	1	no	—
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	536	1	no	—
258	Q₁₆	—	536	2	no	—
259	Q₁₆	—	134	2	no	—
260	Q₁₆	—	134	2	no	—
261	D₈	—	67	2	no	—
262	D₈	—	134	1	no	—
263	C₂×C₈	—	67	2	no	—
264	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	268	2	no	—
271	C₄⋊C₄	—	268	2	no	—
272	C₄⋊C₄	—	268	2	no	—
273	C₄⋊C₄	—	268	2	no	—
274	C₄⋊C₄	—	268	1	no	—
275	C₄⋊C₄	—	268	1	no	—
276	C₄⋊C₄	—	67	2	no	—
277	C₄⋊C₄	—	67	1	no	—
278	C₄⋊C₄	—	134	1	no	—
279	C₂²⋊C₄	—	268	2	no	—
280	C₂²⋊C₄	—	268	1	no	—
281	C₄²	—	4	1	no	—
282	C₄²	—	268	2	no	—
283	C₄²	—	67	2	no	—
284	C₄²	D₄×D₆₇	1	1	yes	characteristic
285	C₁₆	—	2	2	no	—
286	C₁₆	—	134	1	no	—
287	C₁₆	—	134	1	no	—
288	C₂³	—	268	1	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	8	2	no	—
292	Q₈	—	268	2	no	—
293	Q₈	—	268	2	no	—
294	Q₈	—	536	4	no	—
295	Q₈	—	67	4	no	—
296	D₄	—	268	1	no	—
297	D₄	—	134	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	268	2	no	—
302	C₂×C₄	—	268	1	no	—
303	C₂×C₄	—	536	4	no	—
304	C₂×C₄	—	536	2	no	—
305	C₂×C₄	—	67	2	no	—
306	C₂×C₄	—	67	2	no	—
307	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
308	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	134	2	no	—
312	C₈	#41×D₆₇	1	1	yes	characteristic
313	C₈	#14524091	1	1	yes	characteristic
314	C₂²	—	268	2	no	—
315	C₂²	—	536	1	no	—
316	C₂²	—	536	1	no	—
317	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	1	no	—
322	C₄	—	536	1	no	—
323	C₄	—	536	2	no	—
324	C₄	—	67	4	no	—
325	C₄	—	134	2	no	—
326	C₄	SD₃₂×D₆₇	1	2	yes	—
327	C₄	#16864056	1	1	yes	characteristic
328	C₄	#16864205	1	1	yes	characteristic
329	C₂	—	536	2	no	—
330	C₂	#22572843	1	1	yes	characteristic
331	C₂	C₂×SD₃₂×D₆₇	1	1	yes	characteristic
332	C₂	#22573361	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2225	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×D₅₃₆	C₂³
hypercenter	#2225	D₆₇
hypocenter	C₆₇	#33926
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
1. #22573214	C₂	semidirect
3. #22562438	C₂	semidirect
4. #22562435	C₂	semidirect
5. #22562350	C₂	semidirect
8. #22562111	C₂	semidirect
11. C₆₇×#2225	C₂	semidirect
142. Dic₁₃₄	SD₃₂	semidirect
172. #2225	D₆₇	semidirect
178. C₆₇	#33926	semidirect

Maximal subgroups

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

subgroup	index
#22573214	2
#22573061	2
#22562438	2
#22562435	2
#22562350	2
#22562255	2
#22562253	2
#22562111	2
C₄×#14523986	2
#22562060	2
C₆₇×#2225	2
#33926	67