Subgroups — #31013548

Order 17152.

532 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573184	C₂	1	1	yes	maximal, characteristic, complemented
2	#22562452	C₂	1	2	yes	maximal, complemented
3	#22562434	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562363	C₂	1	2	yes	maximal, complemented
5	#22562353	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562244	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562244	C₂	1	2	yes	maximal, complemented
8	#22562118	C₂	1	1	yes	maximal, characteristic, complemented
9	#22562083	C₂	1	2	yes	maximal, complemented
10	C₄×#14523984	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3380	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864068	—	2	1	no	—
13	C₂×#14523984	C₂²	1	1	yes	characteristic
14	C₂×#14523984	C₂²	1	2	yes	—
15	#16863773	C₂²	1	1	yes	characteristic
16	#16863773	C₂²	1	2	yes	—
17	#16863592	—	2	1	no	—
18	#16863592	—	2	2	no	—
19	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
20	C₄×D₅₃₆	C₂²	1	2	yes	—
21	#16863223	C₂²	1	2	yes	—
22	#16863221	C₂²	1	1	yes	characteristic
23	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
25	#16863197	C₂²	1	1	yes	characteristic
26	#16863196	C₂²	1	1	yes	characteristic
27	#16863196	C₂²	1	2	yes	—
28	#16863196	C₂²	1	2	yes	—
29	#16863196	C₂²	1	2	yes	—
30	#16863196	C₂²	1	2	yes	—
31	#16863172	C₂²	1	2	yes	—
32	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	#554×C₆₇	—	2	1	no	—
34	#500×C₆₇	C₂²	1	1	yes	characteristic
35	#499×C₆₇	C₂²	1	2	yes	—
36	#478×C₆₇	—	2	2	no	—
37	#476×C₆₇	—	2	1	no	—
38	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₆₇	C₂²	1	2	yes	—
41	C₂×#11761629	—	4	1	no	—
42	C₂×D₅₃₆	—	2	2	no	—
43	C₂×D₅₃₆	—	2	1	no	—
44	C₂×D₅₃₆	C₂³	1	2	yes	—
45	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
46	#14524040	—	2	2	no	—
47	#14524040	—	2	1	no	—
48	#14524018	—	4	1	no	—
49	#14524018	—	4	2	no	—
50	C₄×D₂₆₈	—	2	2	no	—
51	C₄×D₂₆₈	—	2	1	no	—
52	#14523984	—	2	2	no	—
53	#14523984	—	2	4	no	—
54	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
56	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
57	#14523977	C₂³	1	1	yes	characteristic
58	#14523965	—	2	2	no	—
59	#14523965	—	2	1	no	—
60	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₆₇	—	2	1	no	—
62	#125×C₆₇	—	2	2	no	—
63	C₂²⋊Q₈×C₆₇	—	4	1	no	—
64	C₂²⋊Q₈×C₆₇	—	4	2	no	—
65	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
66	C₄×D₄×C₆₇	—	2	1	no	—
67	C₄×D₄×C₆₇	—	2	2	no	—
68	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
69	#107×C₆₇	—	2	1	no	—
70	#107×C₆₇	C₂³	1	1	yes	characteristic
71	#107×C₆₇	C₂³	1	2	yes	—
72	#106×C₆₇	—	2	2	no	—
73	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
75	#102×C₆₇	—	2	2	no	—
76	#102×C₆₇	—	2	1	no	—
77	C₄×C₈×C₆₇	C₂³	1	2	yes	—
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	#11761629	—	8	4	no	—
80	C₂×D₂₆₈	—	2	2	no	—
81	C₂×D₂₆₈	—	2	1	no	—
82	C₂×D₂₆₈	—	4	2	no	—
83	C₂×D₂₆₈	—	4	1	no	—
84	C₂×D₂₆₈	—	4	1	no	—
85	C₂×D₂₆₈	—	4	2	no	—
86	C₂×C₄×D₆₇	—	4	2	no	—
87	C₂×C₄×D₆₇	—	4	1	no	—
88	D₅₃₆	—	2	1	no	—
89	D₅₃₆	—	2	2	no	—
90	D₅₃₆	—	2	2	no	—
91	D₅₃₆	—	2	1	no	—
92	D₅₃₆	—	8	2	no	—
93	D₅₃₆	—	8	1	no	—
94	C₄⋊Dic₆₇	—	2	1	no	—
95	(C₂×D₆₇)⋊C₄	—	4	1	no	—
96	(C₂×D₆₇)⋊C₄	—	4	2	no	—
97	C₆₇⋊C₁₆	—	2	1	no	—
98	C₆₇⋊C₁₆	—	2	2	no	—
99	C₆₇⋊C₁₆	—	2	1	no	—
100	C₂×Q₈×C₆₇	—	4	1	no	—
101	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
102	C₂×D₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	4	1	no	—
104	C₂²×C₄×C₆₇	—	4	2	no	—
105	D₈×C₆₇	—	2	1	no	—
106	D₈×C₆₇	#41	1	2	yes	complemented
107	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄×C₆₇	—	2	1	no	—
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	—	4	2	no	—
113	C₄⋊C₄×C₆₇	—	4	1	no	—
114	C₄⋊C₄×C₆₇	—	4	2	no	—
115	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
116	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
117	C₂²⋊C₄×C₆₇	—	4	1	no	—
118	C₂²⋊C₄×C₆₇	—	4	2	no	—
119	C₄²×C₆₇	C₂×D₄	1	2	yes	—
120	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂³×C₆₇	—	4	1	no	—
122	C₂²×D₆₇	—	4	2	no	—
123	C₂²×D₆₇	—	4	1	no	—
124	Q₈×C₆₇	—	2	2	no	—
125	Q₈×C₆₇	—	16	1	no	—
126	D₄×C₆₇	—	2	2	no	—
127	D₄×C₆₇	—	4	1	no	—
128	C₂×C₄×C₆₇	—	4	1	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	8	2	no	—
131	C₂×C₄×C₆₇	—	16	2	no	—
132	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₆₇	—	4	1	no	—
138	D₂₆₈	—	4	1	no	—
139	D₂₆₈	—	4	2	no	—
140	D₂₆₈	—	4	1	no	—
141	D₂₆₈	—	4	2	no	—
142	D₂₆₈	—	8	4	no	—
143	D₂₆₈	—	8	2	no	—
144	D₂₆₈	—	16	2	no	—
145	D₂₆₈	—	16	1	no	—
146	C₄×D₆₇	—	8	4	no	—
147	C₄×D₆₇	—	8	2	no	—
148	C₈×C₆₇	—	2	2	no	—
149	C₈×C₆₇	—	2	1	no	—
150	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
151	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
152	C₂²×C₆₇	—	4	2	no	—
153	C₂²×C₆₇	—	8	1	no	—
154	C₂²×C₆₇	—	8	1	no	—
155	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₆₇	—	8	2	no	—
157	C₂×D₆₇	—	8	1	no	—
158	C₂×D₆₇	—	8	2	no	—
159	C₂×D₆₇	—	8	1	no	—
160	C₂×D₆₇	—	8	1	no	—
161	C₂×D₆₇	—	8	2	no	—
162	C₄×C₆₇	—	2	2	no	—
163	C₄×C₆₇	—	2	2	no	—
164	C₄×C₆₇	—	2	1	no	—
165	C₄×C₆₇	—	8	1	no	—
166	C₄×C₆₇	—	8	2	no	—
167	C₄×C₆₇	#509	1	2	yes	—
168	C₄×C₆₇	#548	1	1	yes	characteristic
169	C₄×C₆₇	#577	1	1	yes	characteristic
170	Dic₆₇	—	8	1	no	—
171	#33933	—	67	1	no	maximal, Sylow
172	C₂×C₆₇	—	8	2	no	—
173	C₂×C₆₇	#3358	1	1	yes	characteristic
174	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
175	C₂×C₆₇	#3415	1	1	yes	characteristic
176	D₆₇	—	16	2	no	—
177	D₆₇	—	16	1	no	—
178	#3391	—	67	1	no	—
179	#3380	D₆₇	1	1	yes	characteristic, complemented
180	#2244	—	67	2	no	—
181	#2240	—	67	1	no	—
182	#2228	—	67	2	no	—
183	#2226	—	67	1	no	—
184	#2206	—	67	2	no	—
185	#2206	—	67	1	no	—
186	#2183	—	67	1	no	—
187	#2177	—	67	2	no	—
188	C₄×D₁₆	—	67	1	no	—
189	C₆₇	#33933	1	1	yes	characteristic, complemented, Sylow
190	#554	—	2	1	no	—
191	#550	—	134	1	no	—
192	C₂×D₁₆	—	67	1	no	—
193	C₂×D₁₆	—	67	2	no	—
194	#500	C₂×D₆₇	1	1	yes	characteristic
195	#499	C₂×D₆₇	1	2	yes	—
196	#493	—	67	1	no	—
197	#493	—	67	2	no	—
198	#478	—	2	2	no	—
199	#476	—	2	1	no	—
200	#459	—	134	1	no	—
201	#459	—	134	2	no	—
202	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
203	C₄×D₈	—	67	1	no	—
204	C₄×D₈	—	67	2	no	—
205	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
206	C₄×D₈	C₂×D₆₇	1	2	yes	—
207	#367	—	67	2	no	—
208	#366	—	67	1	no	—
209	C₄⋊C₁₆	—	67	1	no	—
210	C₄⋊C₁₆	—	67	2	no	—
211	#357	—	67	1	no	—
212	#357	—	67	1	no	—
213	#357	—	67	2	no	—
214	#357	—	67	2	no	—
215	#357	—	67	2	no	—
216	#357	—	67	2	no	—
217	#346	—	67	2	no	—
218	C₄×C₁₆	—	67	1	no	—
219	C₂×#41	—	268	1	no	—
220	C₂×D₈	—	67	1	no	—
221	C₂×D₈	—	67	2	no	—
222	C₂×D₈	—	134	2	no	—
223	C₂×D₈	—	134	1	no	—
224	C₂×D₈	C₂²×D₆₇	1	1	yes	characteristic
225	C₄⋊Q₈	—	2	1	no	—
226	#127	—	134	2	no	—
227	#127	—	134	1	no	—
228	#125	—	2	2	no	—
229	C₂²⋊Q₈	—	4	1	no	—
230	C₂²⋊Q₈	—	4	2	no	—
231	#121	—	268	1	no	—
232	#121	—	268	2	no	—
233	C₄×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
234	C₄×D₄	—	2	1	no	—
235	C₄×D₄	—	2	2	no	—
236	C₄×D₄	—	134	2	no	—
237	C₄×D₄	—	134	1	no	—
238	C₂×C₄⋊C₄	—	4	1	no	—
239	D₁₆	—	134	2	no	—
240	D₁₆	—	134	4	no	—
241	C₂×C₁₆	—	67	1	no	—
242	C₂×C₁₆	—	67	1	no	—
243	C₂×C₁₆	—	67	2	no	—
244	#107	—	2	1	no	—
245	#107	—	67	1	no	—
246	#107	C₂²×D₆₇	1	1	yes	characteristic
247	#107	C₂²×D₆₇	1	2	yes	—
248	#106	—	2	2	no	—
249	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
250	C₄⋊C₈	C₂²⋊D₆₇	1	2	yes	—
251	#102	—	2	2	no	—
252	#102	—	2	1	no	—
253	#102	—	134	2	no	—
254	#102	—	134	1	no	—
255	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
256	C₄×C₈	C₂²×D₆₇	1	2	yes	—
257	#41	—	536	4	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	#11761613	1	1	yes	characteristic
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	268	2	no	—
262	C₂×D₄	—	268	1	no	—
263	C₂×D₄	—	268	1	no	—
264	C₂×D₄	—	268	2	no	—
265	C₂×D₄	—	134	2	no	—
266	C₂×D₄	—	134	1	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	—	268	2	no	—
270	C₂²×C₄	—	268	1	no	—
271	D₈	—	2	1	no	—
272	D₈	—	536	2	no	—
273	D₈	—	536	1	no	—
274	D₈	—	134	1	no	—
275	D₈	—	134	2	no	—
276	D₈	—	134	1	no	—
277	D₈	—	134	2	no	—
278	D₈	#11761629	1	2	yes	complemented
279	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
280	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
281	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	134	1	no	—
288	C₄⋊C₄	#11761613	1	1	yes	characteristic
289	C₄⋊C₄	#11761613	1	2	yes	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	268	1	no	—
293	C₂²⋊C₄	—	268	2	no	—
294	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
295	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
296	C₁₆	—	134	1	no	—
297	C₁₆	—	134	2	no	—
298	C₁₆	—	134	1	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	268	2	no	—
301	C₂³	—	268	1	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	16	1	no	—
304	D₄	—	2	2	no	—
305	D₄	—	4	1	no	—
306	D₄	—	268	1	no	—
307	D₄	—	268	2	no	—
308	D₄	—	268	1	no	—
309	D₄	—	268	2	no	—
310	D₄	—	536	4	no	—
311	D₄	—	536	2	no	—
312	D₄	—	1072	2	no	—
313	D₄	—	1072	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	—	268	1	no	—
318	C₂×C₄	—	16	2	no	—
319	C₂×C₄	—	536	4	no	—
320	C₂×C₄	—	536	2	no	—
321	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
322	C₂×C₄	C₂×#11761613	1	2	yes	—
323	C₂×C₄	C₂×#11761613	1	2	yes	—
324	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	2	1	no	—
328	C₈	C₂×#11761629	1	1	yes	characteristic
329	C₈	#14524091	1	1	yes	characteristic
330	C₂²	—	4	2	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	536	2	no	—
334	C₂²	—	536	1	no	—
335	C₂²	—	536	1	no	—
336	C₂²	—	536	2	no	—
337	C₂²	—	536	2	no	—
338	C₂²	—	536	1	no	—
339	C₂²	C₂²×#11761613	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	2	no	—
345	C₄	—	536	1	no	—
346	C₄	#16863831	1	2	yes	—
347	C₄	#16864057	1	1	yes	characteristic
348	C₄	#16864204	1	1	yes	characteristic
349	C₂	—	8	2	no	—
350	C₂	—	1072	2	no	—
351	C₂	—	1072	1	no	—
352	C₂	#22572844	1	1	yes	characteristic
353	C₂	C₂×#16863831	1	1	yes	characteristic
354	C₂	#22573348	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#3380	C₂
socle	C₂²×C₆₇	C₂²×D₈
hypercenter	#3380	D₆₇
hypocenter	C₆₇	#33933
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3380

Complements

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

N	H ≅ G/N	type
1. #22573184	C₂	semidirect
2. #22562452	C₂	semidirect
3. #22562434	C₂	semidirect
4. #22562363	C₂	semidirect
5. #22562353	C₂	semidirect
6. #22562244	C₂	semidirect
7. #22562244	C₂	semidirect
8. #22562118	C₂	semidirect
9. #22562083	C₂	semidirect
10. C₄×#14523984	C₂	semidirect
11. C₆₇×#3380	C₂	semidirect
106. D₈×C₆₇	#41	semidirect
179. #3380	D₆₇	semidirect
189. C₆₇	#33933	semidirect
278. D₈	#11761629	semidirect

Maximal subgroups

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

subgroup	index
#22573184	2
#22562452	2
#22562434	2
#22562363	2
#22562353	2
#22562244	2
#22562118	2
#22562083	2
C₄×#14523984	2
C₆₇×#3380	2
#33933	67