Subgroups — #31013536

Order 17152.

484 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573219	C₂	1	1	yes	maximal, characteristic
2	#22573102	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562442	C₂	1	2	yes	maximal, complemented
4	#22562434	C₂	1	1	yes	maximal, characteristic
5	#22562365	C₂	1	2	yes	maximal, complemented
6	#22562274	C₂	1	2	yes	maximal, complemented
7	#22562250	C₂	1	1	yes	maximal, characteristic
8	#22562111	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14523989	C₂	1	1	yes	maximal, characteristic, complemented
10	#22562060	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864131	—	2	1	no	—
13	#16864121	—	2	1	no	—
14	C₂×#14523989	C₂²	1	1	yes	characteristic
15	C₂×#14523988	C₂²	1	2	yes	—
16	#16863797	C₂²	1	2	yes	—
17	#16863781	C₂²	1	2	yes	—
18	#16863773	C₂²	1	1	yes	characteristic
19	#16863702	—	2	2	no	—
20	#16863678	—	2	1	no	—
21	#16863614	—	2	2	no	—
22	#16863596	—	2	1	no	—
23	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
24	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
25	#16863490	C₂²	1	2	yes	—
26	#16863478	C₂²	1	2	yes	—
27	#16863221	C₂²	1	1	yes	characteristic
28	Dic₆₇⋊C₁₆	C₂²	1	2	yes	—
29	#16863200	C₂²	1	1	yes	characteristic
30	#16863199	C₂²	1	1	yes	characteristic
31	#16863198	C₂²	1	2	yes	—
32	#16863196	C₂²	1	2	yes	—
33	#16863196	C₂²	1	2	yes	—
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×Dic₂₆₈	C₂³	1	1	yes	characteristic
42	C₂×#11761615	—	2	2	no	—
43	C₂×D₅₃₆	—	2	1	no	—
44	C₂×D₅₃₆	C₂³	1	2	yes	—
45	#14524041	—	4	1	no	—
46	#14524040	—	2	1	no	—
47	#14524034	—	2	2	no	—
48	#14524033	—	4	2	no	—
49	#14524032	—	2	2	no	—
50	#14524031	—	4	1	no	—
51	#14524028	—	4	2	no	—
52	Q₈×Dic₆₇	—	4	1	no	—
53	C₄×Dic₁₃₄	—	2	1	no	—
54	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
55	#14524008	—	2	2	no	—
56	C₄×D₂₆₈	—	4	1	no	—
57	#14524005	—	2	2	no	—
58	#14523989	—	2	2	no	—
59	#14523988	—	2	4	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	#14523974	—	2	2	no	—
64	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
65	Dic₆₇⋊C₈	D₄	1	2	yes	—
66	#14523968	—	2	1	no	—
67	#14523966	—	2	2	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	2	yes	—
76	#107×C₆₇	C₂³	1	1	yes	characteristic
77	#103×C₆₇	—	2	1	no	—
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₃₄	—	2	1	no	—
80	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
81	C₂×D₂₆₈	—	2	2	no	—
82	C₂×D₂₆₈	—	4	1	no	—
83	C₂×D₂₆₈	—	4	1	no	—
84	C₂×C₄×D₆₇	—	4	2	no	—
85	Dic₂₆₈	—	2	1	no	—
86	Dic₂₆₈	#41	1	2	yes	—
87	#11761615	—	8	2	no	—
88	D₅₃₆	—	2	2	no	—
89	D₅₃₆	—	2	2	no	—
90	D₅₃₆	—	8	1	no	—
91	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
92	C₄⋊Dic₆₇	—	2	1	no	—
93	C₄⋊Dic₆₇	—	4	2	no	—
94	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
95	Dic₆₇⋊C₄	—	4	1	no	—
96	Dic₆₇⋊C₄	—	4	2	no	—
97	Dic₆₇⋊C₄	—	4	2	no	—
98	Dic₆₇⋊C₄	D₈	1	2	yes	—
99	(C₂×D₆₇)⋊C₄	—	4	2	no	—
100	(C₂×D₆₇)⋊C₄	—	4	2	no	—
101	(C₂×D₆₇)⋊C₄	—	4	2	no	—
102	C₄×Dic₆₇	—	4	1	no	—
103	C₄×Dic₆₇	—	4	2	no	—
104	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
105	C₆₇⋊C₁₆	—	2	1	no	—
106	C₆₇⋊C₁₆	—	2	1	no	—
107	C₂×Q₈×C₆₇	—	2	1	no	—
108	C₂×Q₈×C₆₇	—	4	1	no	—
109	Q₁₆×C₆₇	—	2	1	no	—
110	Q₁₆×C₆₇	—	2	1	no	—
111	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₆₇	—	2	1	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	4	1	no	—
116	C₄⋊C₄×C₆₇	—	4	1	no	—
117	C₄²×C₆₇	—	4	1	no	—
118	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₆₇	—	2	2	no	—
120	C₂²×D₆₇	—	4	2	no	—
121	Q₈×C₆₇	—	4	1	no	—
122	Q₈×C₆₇	—	4	1	no	—
123	Q₈×C₆₇	—	8	2	no	—
124	C₂×C₄×C₆₇	—	4	1	no	—
125	C₂×C₄×C₆₇	—	4	1	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₆₇	—	4	1	no	—
131	C₂×Dic₆₇	—	4	1	no	—
132	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
133	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
134	D₂₆₈	—	4	2	no	—
135	D₂₆₈	—	4	2	no	—
136	D₂₆₈	—	8	2	no	—
137	D₂₆₈	—	16	1	no	—
138	C₄×D₆₇	—	16	2	no	—
139	Dic₁₃₄	—	2	2	no	—
140	Dic₁₃₄	—	2	2	no	—
141	Dic₁₃₄	—	4	1	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
144	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
145	C₆₇⋊C₈	—	2	2	no	—
146	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₆₇	—	8	2	no	—
148	C₂×D₆₇	—	8	2	no	—
149	C₂×D₆₇	—	8	2	no	—
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	8	1	no	—
152	C₄×C₆₇	—	8	2	no	—
153	C₄×C₆₇	—	8	1	no	—
154	C₄×C₆₇	#508	1	2	yes	—
155	C₄×C₆₇	#548	1	1	yes	characteristic
156	C₄×C₆₇	#577	1	1	yes	characteristic
157	Dic₆₇	—	2	2	no	—
158	Dic₆₇	—	2	2	no	—
159	Dic₆₇	—	4	2	no	—
160	Dic₆₇	—	8	1	no	—
161	#33932	—	67	1	no	maximal, Sylow
162	C₂×C₆₇	#3358	1	1	yes	characteristic
163	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
164	C₂×C₆₇	#3416	1	1	yes	characteristic
165	D₆₇	—	16	2	no	—
166	#3396	—	67	1	no	—
167	#3383	—	67	1	no	—
168	#2242	—	67	2	no	—
169	#2240	—	67	1	no	—
170	#2228	—	67	2	no	—
171	#2225	D₆₇	1	1	yes	characteristic, complemented
172	#2211	—	67	2	no	—
173	#2207	—	67	1	no	—
174	Q₈×C₁₆	—	67	1	no	—
175	C₄×Q₃₂	—	67	1	no	—
176	C₄×SD₃₂	—	67	2	no	—
177	C₆₇	#33932	1	1	yes	characteristic, complemented, Sylow
178	#559	—	134	1	no	—
179	#557	—	134	1	no	—
180	C₂×Q₃₂	—	67	1	no	—
181	C₂×SD₃₂	—	67	2	no	—
182	#500	C₂×D₆₇	1	1	yes	characteristic
183	#499	—	67	2	no	—
184	#495	—	67	2	no	—
185	#493	—	67	1	no	—
186	#479	—	134	2	no	—
187	#475	—	134	1	no	—
188	#463	—	134	2	no	—
189	#460	—	134	1	no	—
190	C₈×Q₈	—	67	1	no	—
191	C₄×Q₁₆	—	67	2	no	—
192	C₄×Q₁₆	—	67	1	no	—
193	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
194	C₄×D₈	—	67	2	no	—
195	#367	C₂×D₆₇	1	2	yes	—
196	#366	—	67	1	no	—
197	C₄⋊C₁₆	—	67	2	no	—
198	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
199	#358	—	67	1	no	—
200	#358	—	67	1	no	—
201	#358	C₂×D₆₇	1	2	yes	—
202	#357	—	67	2	no	—
203	#357	—	67	2	no	—
204	#357	—	67	2	no	—
205	C₄×C₁₆	—	67	2	no	—
206	C₄×C₁₆	—	67	1	no	—
207	C₂×Q₁₆	—	67	1	no	—
208	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
209	C₂×SD₁₆	—	134	2	no	—
210	C₂×D₈	—	67	2	no	—
211	C₂×D₈	—	134	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	268	1	no	—
214	#127	—	134	1	no	—
215	#125	—	268	2	no	—
216	#125	—	134	2	no	—
217	#124	—	268	1	no	—
218	#124	—	134	2	no	—
219	#123	—	268	2	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	268	1	no	—
222	C₄×Q₈	—	67	1	no	—
223	C₄×Q₈	—	134	2	no	—
224	C₄×Q₈	—	134	1	no	—
225	C₄×D₄	—	268	1	no	—
226	C₄×D₄	—	134	2	no	—
227	Q₃₂	—	134	2	no	—
228	SD₃₂	—	134	4	no	—
229	C₂×C₁₆	—	67	1	no	—
230	C₂×C₁₆	—	67	1	no	—
231	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
232	#107	—	2	1	no	—
233	#107	—	67	1	no	—
234	#107	C₂²×D₆₇	1	2	yes	—
235	#107	C₂²×D₆₇	1	1	yes	characteristic
236	#106	—	134	2	no	—
237	C₄⋊C₈	—	67	2	no	—
238	C₄⋊C₈	—	67	1	no	—
239	#103	—	2	1	no	—
240	#103	—	134	2	no	—
241	#103	—	134	1	no	—
242	#102	—	134	2	no	—
243	C₄×C₈	—	67	2	no	—
244	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	67	1	no	—
248	C₂×Q₈	—	134	1	no	—
249	C₂×D₄	—	268	1	no	—
250	C₂×D₄	—	268	1	no	—
251	C₂×D₄	—	134	2	no	—
252	C₂²×C₄	—	268	2	no	—
253	Q₁₆	—	2	1	no	—
254	Q₁₆	—	2	1	no	—
255	Q₁₆	—	67	2	no	—
256	Q₁₆	—	134	1	no	—
257	SD₁₆	—	536	2	no	—
258	D₈	—	536	1	no	—
259	D₈	—	134	2	no	—
260	D₈	—	134	2	no	—
261	C₂×C₈	—	67	2	no	—
262	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	268	2	no	—
269	C₄⋊C₄	—	268	2	no	—
270	C₄⋊C₄	—	268	2	no	—
271	C₄⋊C₄	—	268	1	no	—
272	C₄⋊C₄	—	67	1	no	—
273	C₄⋊C₄	—	67	2	no	—
274	C₄⋊C₄	—	134	1	no	—
275	C₂²⋊C₄	—	268	2	no	—
276	C₂²⋊C₄	—	268	2	no	—
277	C₂²⋊C₄	—	268	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	268	2	no	—
280	C₄²	—	268	1	no	—
281	C₄²	—	67	2	no	—
282	C₄²	D₄×D₆₇	1	1	yes	characteristic
283	C₁₆	—	2	2	no	—
284	C₁₆	—	134	1	no	—
285	C₁₆	—	134	1	no	—
286	C₂³	—	268	2	no	—
287	Q₈	—	4	1	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	8	2	no	—
290	Q₈	—	268	1	no	—
291	Q₈	—	134	2	no	—
292	Q₈	—	134	2	no	—
293	D₄	—	268	2	no	—
294	D₄	—	268	2	no	—
295	D₄	—	536	2	no	—
296	D₄	—	1072	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	268	1	no	—
301	C₂×C₄	—	268	1	no	—
302	C₂×C₄	—	1072	2	no	—
303	C₂×C₄	—	67	2	no	—
304	C₂×C₄	—	67	2	no	—
305	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
306	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	134	2	no	—
310	C₈	#41×D₆₇	1	1	yes	characteristic
311	C₈	#14524091	1	1	yes	characteristic
312	C₂²	—	536	2	no	—
313	C₂²	—	536	2	no	—
314	C₂²	—	536	2	no	—
315	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	268	2	no	—
321	C₄	—	536	1	no	—
322	C₄	—	134	2	no	—
323	C₄	—	134	2	no	—
324	C₄	#16863827	1	2	yes	—
325	C₄	#16864056	1	1	yes	characteristic
326	C₄	#16864205	1	1	yes	characteristic
327	C₂	—	1072	2	no	—
328	C₂	#22572843	1	1	yes	characteristic
329	C₂	C₂×#16863827	1	1	yes	characteristic
330	C₂	#22573354	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	#16863773	C₂²
hypercenter	#2225	D₆₇
hypocenter	C₆₇	#33932
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
2. #22573102	C₂	semidirect
3. #22562442	C₂	semidirect
5. #22562365	C₂	semidirect
6. #22562274	C₂	semidirect
8. #22562111	C₂	semidirect
9. C₄×#14523989	C₂	semidirect
10. #22562060	C₂	semidirect
11. C₆₇×#2225	C₂	semidirect
171. #2225	D₆₇	semidirect
177. C₆₇	#33932	semidirect

Maximal subgroups

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

subgroup	index
#22573219	2
#22573102	2
#22562442	2
#22562434	2
#22562365	2
#22562274	2
#22562250	2
#22562111	2
C₄×#14523989	2
#22562060	2
C₆₇×#2225	2
#33932	67