Subgroups — #32153005

Order 18176.

526 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598910	C₂	1	2	yes	maximal, complemented
2	#22590619	C₂	1	1	yes	maximal, characteristic, complemented
3	#22590601	C₂	1	1	yes	maximal, characteristic, complemented
4	#22590572	C₂	1	2	yes	maximal, complemented
5	#22590436	C₂	1	2	yes	maximal, complemented
6	#22590431	C₂	1	1	yes	maximal, characteristic, complemented
7	#22590426	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₅₆₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14556073	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₁₃₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986533	—	2	2	no	—
13	C₂×Dic₅₆₈	C₂²	1	1	yes	characteristic
14	C₂×#14556073	C₂²	1	2	yes	—
15	C₂×D₁₁₃₆	C₂²	1	1	yes	characteristic
16	#17986401	C₂²	1	2	yes	—
17	#17986383	C₂²	1	2	yes	—
18	#17986354	—	2	2	no	—
19	#17986240	C₂²	1	2	yes	—
20	#17986235	C₂²	1	2	yes	—
21	#17986228	—	2	2	no	—
22	C₄×Dic₂₈₄	C₂²	1	2	yes	—
23	C₄×#11765854	—	2	2	no	—
24	C₄×D₅₆₈	C₂²	1	2	yes	—
25	#17985827	C₂²	1	2	yes	—
26	#17985825	C₂²	1	1	yes	characteristic
27	#17985825	C₂²	1	1	yes	characteristic
28	#17985804	C₂²	1	2	yes	—
29	#17985804	C₂²	1	2	yes	—
30	#17985801	C₂²	1	2	yes	—
31	#17985801	C₂²	1	2	yes	—
32	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
33	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
35	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
36	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
37	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
39	C₂×Dic₂₈₄	C₂³	1	2	yes	—
40	C₂×#11765854	—	2	2	no	—
41	C₂×D₅₆₈	C₂³	1	2	yes	—
42	#14556122	—	4	2	no	—
43	#14556120	—	2	2	no	—
44	#14556116	—	2	2	no	—
45	#14556111	—	4	2	no	—
46	#14556107	—	2	2	no	—
47	#14556102	—	2	2	no	—
48	C₄×Dic₁₄₂	—	2	2	no	—
49	C₄×D₂₈₄	—	2	2	no	—
50	C₄×C₂²⋊D₇₁	—	4	2	no	—
51	#14556085	—	4	2	no	—
52	Dic₅₆₈	—	2	2	no	—
53	#14556073	—	2	4	no	—
54	D₁₁₃₆	—	2	2	no	—
55	#14556062	C₂³	1	2	yes	—
56	#14556062	C₂³	1	2	yes	—
57	#14556060	—	2	2	no	—
58	#14556053	—	2	2	no	—
59	#14556053	—	2	2	no	—
60	#14556050	—	2	2	no	—
61	#14556050	—	2	2	no	—
62	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
65	C₂×C₁₆×C₇₁	—	2	2	no	—
66	C₂×C₁₆×C₇₁	—	2	2	no	—
67	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
72	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
73	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₄₂	—	2	2	no	—
76	C₂×D₂₈₄	—	2	2	no	—
77	C₂×C₂²⋊D₇₁	—	4	2	no	—
78	C₂×C₄×D₇₁	—	4	2	no	—
79	Dic₂₈₄	—	2	2	no	—
80	Dic₂₈₄	—	2	2	no	—
81	#11765854	—	4	4	no	—
82	D₅₆₈	—	2	2	no	—
83	D₅₆₈	—	2	2	no	—
84	C₄⋊Dic₇₁	—	2	2	no	—
85	C₄⋊Dic₇₁	—	2	2	no	—
86	Dic₇₁⋊C₄	—	4	2	no	—
87	Dic₇₁⋊C₄	—	4	2	no	—
88	Dic₇₁⋊C₄	—	4	2	no	—
89	Dic₇₁⋊C₄	—	4	2	no	—
90	C₂²⋊Dic₇₁	—	4	2	no	—
91	(C₂×D₇₁)⋊C₄	—	4	2	no	—
92	(C₂×D₇₁)⋊C₄	—	4	2	no	—
93	(C₂×D₇₁)⋊C₄	—	4	2	no	—
94	(C₂×D₇₁)⋊C₄	—	4	2	no	—
95	C₄×Dic₇₁	—	4	2	no	—
96	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
98	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₇₁	—	2	2	no	—
100	C₂×C₈×C₇₁	—	2	1	no	—
101	C₂×C₈×C₇₁	—	2	1	no	—
102	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
106	C₄²×C₇₁	D₈	1	1	yes	characteristic
107	C₄²×C₇₁	D₈	1	1	yes	characteristic
108	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
110	C₁₆×C₇₁	—	2	1	no	—
111	C₁₆×C₇₁	—	2	1	no	—
112	C₁₆×C₇₁	#41	1	2	yes	—
113	C₁₆×C₇₁	#41	1	2	yes	—
114	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₇₁	—	4	2	no	—
116	C₂×C₄×C₇₁	—	2	2	no	—
117	C₂×C₄×C₇₁	—	2	1	no	—
118	C₂×C₄×C₇₁	—	2	1	no	—
119	C₂×C₄×C₇₁	—	2	2	no	—
120	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
126	C₂²⋊D₇₁	—	8	4	no	—
127	C₂×Dic₇₁	—	4	2	no	—
128	C₂×Dic₇₁	—	4	2	no	—
129	C₂×Dic₇₁	—	4	2	no	—
130	D₂₈₄	—	4	2	no	—
131	D₂₈₄	—	4	2	no	—
132	C₄×D₇₁	—	8	4	no	—
133	Dic₁₄₂	—	4	2	no	—
134	Dic₁₄₂	—	4	2	no	—
135	C₈×C₇₁	—	2	1	no	—
136	C₈×C₇₁	—	2	1	no	—
137	C₈×C₇₁	#135	1	2	yes	—
138	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
139	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
140	C₂²×C₇₁	—	2	1	no	—
141	C₂²×C₇₁	—	2	1	no	—
142	C₂²×C₇₁	#508	1	2	yes	—
143	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
144	C₂×D₇₁	—	8	2	no	—
145	C₂×D₇₁	—	8	2	no	—
146	C₂×D₇₁	—	8	2	no	—
147	C₄×C₇₁	—	2	1	no	—
148	C₄×C₇₁	—	2	1	no	—
149	C₄×C₇₁	—	2	1	no	—
150	C₄×C₇₁	—	2	1	no	—
151	C₄×C₇₁	#508	1	2	yes	—
152	C₄×C₇₁	#532	1	1	yes	characteristic
153	C₄×C₇₁	C₂×#135	1	1	yes	characteristic
154	Dic₇₁	—	8	2	no	—
155	Dic₇₁	—	8	2	no	—
156	Dic₇₁	—	8	2	no	—
157	#33735	—	71	1	no	maximal, Sylow
158	C₂×C₇₁	—	2	2	no	—
159	C₂×C₇₁	#3169	1	1	yes	characteristic
160	C₂×C₇₁	C₂×#508	1	1	yes	characteristic
161	C₂×C₇₁	C₂×#508	1	1	yes	characteristic
162	D₇₁	—	16	2	no	—
163	#3168	—	71	2	no	—
164	#2253	—	71	1	no	—
165	#2249	—	71	1	no	—
166	#2242	—	71	2	no	—
167	#2217	—	71	2	no	—
168	#2216	—	71	1	no	—
169	#2215	—	71	1	no	—
170	C₄×Q₃₂	—	71	1	no	—
171	C₄×SD₃₂	—	71	2	no	—
172	C₄×D₁₆	—	71	1	no	—
173	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic, complemented
174	C₇₁	#33735	1	1	yes	characteristic, complemented, Sylow
175	#529	—	142	2	no	—
176	C₂×Q₃₂	—	71	1	no	—
177	C₂×SD₃₂	—	71	2	no	—
178	C₂×D₁₆	—	71	1	no	—
179	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
180	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
181	#499	—	71	2	no	—
182	#495	—	71	2	no	—
183	#488	—	142	2	no	—
184	#467	—	71	2	no	—
185	#466	—	71	2	no	—
186	#465	—	142	2	no	—
187	C₄×Q₁₆	—	71	2	no	—
188	C₄×SD₁₆	—	142	2	no	—
189	C₄×D₈	—	71	2	no	—
190	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
191	#367	—	71	2	no	—
192	#366	—	71	1	no	—
193	#366	—	71	1	no	—
194	#358	—	71	2	no	—
195	#358	—	71	2	no	—
196	#357	—	71	2	no	—
197	#357	—	71	2	no	—
198	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
199	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
200	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
201	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
202	C₂×Q₁₆	—	71	2	no	—
203	C₂×SD₁₆	—	142	2	no	—
204	C₂×D₈	—	71	2	no	—
205	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
207	#126	—	284	2	no	—
208	#125	—	142	2	no	—
209	#124	—	142	2	no	—
210	#123	—	284	2	no	—
211	C₂²⋊Q₈	—	142	2	no	—
212	#121	—	142	2	no	—
213	C₄×Q₈	—	142	2	no	—
214	C₄×D₄	—	284	2	no	—
215	C₄×D₄	—	142	2	no	—
216	#117	—	284	2	no	—
217	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
218	Q₃₂	—	142	2	no	—
219	SD₃₂	—	142	4	no	—
220	D₁₆	—	142	2	no	—
221	C₂×C₁₆	—	2	2	no	—
222	C₂×C₁₆	—	2	2	no	—
223	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
224	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
227	#107	—	71	2	no	—
228	#107	—	71	2	no	—
229	#106	—	142	2	no	—
230	#103	—	142	2	no	—
231	#103	—	142	2	no	—
232	#102	—	142	2	no	—
233	#102	—	142	2	no	—
234	C₄×C₈	D₂₈₄	1	1	yes	characteristic
235	C₄×C₈	D₂₈₄	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
238	C₂×Q₈	—	142	2	no	—
239	C₂×D₄	—	284	2	no	—
240	C₂×D₄	—	142	2	no	—
241	C₂²×C₄	—	284	2	no	—
242	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
243	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
245	Q₁₆	—	142	2	no	—
246	Q₁₆	—	142	2	no	—
247	SD₁₆	—	284	4	no	—
248	D₈	—	142	2	no	—
249	D₈	—	142	2	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
254	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
255	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
256	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
257	C₄⋊C₄	—	284	2	no	—
258	C₄⋊C₄	—	284	2	no	—
259	C₄⋊C₄	—	284	2	no	—
260	C₄⋊C₄	—	284	2	no	—
261	C₄⋊C₄	—	142	2	no	—
262	C₄⋊C₄	—	142	2	no	—
263	C₂²⋊C₄	—	284	2	no	—
264	C₂²⋊C₄	—	284	2	no	—
265	C₂²⋊C₄	—	284	2	no	—
266	C₂²⋊C₄	—	284	2	no	—
267	C₂²⋊C₄	—	284	2	no	—
268	C₄²	—	284	2	no	—
269	C₄²	D₅₆₈	1	1	yes	characteristic
270	C₄²	D₅₆₈	1	1	yes	characteristic
271	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
272	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
273	C₁₆	—	2	1	no	—
274	C₁₆	—	2	1	no	—
275	C₁₆	#11765865	1	2	yes	—
276	C₁₆	#11765865	1	2	yes	—
277	C₂³	—	284	2	no	—
278	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
279	Q₈	—	284	2	no	—
280	Q₈	—	284	2	no	—
281	D₄	—	284	2	no	—
282	D₄	—	284	2	no	—
283	D₄	—	568	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	284	2	no	—
289	C₂×C₄	—	284	2	no	—
290	C₂×C₄	—	284	2	no	—
291	C₂×C₄	—	568	4	no	—
292	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
293	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
294	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
295	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
296	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	#14556150	1	2	yes	—
301	C₈	C₂×#11765865	1	1	yes	characteristic
302	C₈	C₂×#11765865	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	568	2	no	—
306	C₂²	—	568	2	no	—
307	C₂²	—	568	2	no	—
308	C₂²	#17986431	1	2	yes	—
309	C₂²	C₂²×D₅₆₈	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	568	2	no	—
315	C₄	—	568	2	no	—
316	C₄	—	568	2	no	—
317	C₄	#17986431	1	2	yes	—
318	C₄	#17986549	1	1	yes	characteristic
319	C₄	C₂×#14556150	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	1136	2	no	—
322	C₂	#22598919	1	1	yes	characteristic
323	C₂	C₂×#17986431	1	1	yes	characteristic
324	C₂	C₂×#17986431	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₆₈
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	#22590426	C₂
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	#33735
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #22598910	C₂	semidirect
2. #22590619	C₂	semidirect
3. #22590601	C₂	semidirect
4. #22590572	C₂	semidirect
5. #22590436	C₂	semidirect
6. #22590431	C₂	semidirect
8. C₄×Dic₅₆₈	C₂	semidirect
9. C₄×#14556073	C₂	semidirect
10. C₄×D₁₁₃₆	C₂	semidirect
11. C₂×C₄×C₁₆×C₇₁	C₂	semidirect
173. C₂×C₄×C₁₆	D₇₁	semidirect
174. C₇₁	#33735	semidirect

Maximal subgroups

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

subgroup	index
#22598910	2
#22590619	2
#22590601	2
#22590572	2
#22590436	2
#22590431	2
#22590426	2
C₄×Dic₅₆₈	2
C₄×#14556073	2
C₄×D₁₁₃₆	2
C₂×C₄×C₁₆×C₇₁	2
#33735	71