Subgroups — #2262×D₇₁ #32155576

Order 18176.

470 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#508×D₇₁	C₂	1	2	yes	maximal, complemented
2	#22590650	C₂	1	1	yes	maximal, characteristic, complemented
3	#22590649	C₂	1	1	yes	maximal, characteristic, complemented
4	#22590648	C₂	1	2	yes	maximal, complemented
5	#22590647	C₂	1	2	yes	maximal, complemented
6	#22590646	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₆₄×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	SD₆₄×D₇₁	C₂	1	2	yes	maximal, complemented
9	D₃₂×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₂×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#2262	C₂	1	1	yes	maximal, characteristic, complemented
12	#135×D₇₁	—	2	2	no	—
13	#17986432	C₂²	1	2	yes	complemented
14	#17986431	C₂²	1	2	yes	complemented
15	#17986430	—	2	2	no	—
16	#17986429	—	2	2	no	—
17	#17986428	C₂²	1	2	yes	complemented
18	Q₃₂×D₇₁	C₂²	1	2	yes	—
19	SD₃₂×D₇₁	—	2	2	no	—
20	D₁₆×D₇₁	C₂²	1	2	yes	—
21	C₂×C₁₆×D₇₁	C₂²	1	1	yes	characteristic
22	Dic₁₁₃₆	C₂²	1	1	yes	characteristic, complemented
23	#17985839	C₂²	1	2	yes	complemented
24	#17985838	C₂²	1	2	yes	complemented
25	#17985837	C₂²	1	2	yes	complemented
26	#17985836	C₂²	1	2	yes	complemented
27	D₂₂₇₂	C₂²	1	1	yes	characteristic, complemented
28	#17985834	C₂²	1	2	yes	complemented
29	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic, complemented
30	C₃₂×D₇₁	C₂²	1	1	yes	characteristic
31	C₃₂×D₇₁	C₂²	1	1	yes	characteristic
32	C₃₂×D₇₁	C₂²	1	1	yes	characteristic, complemented
33	C₃₂×D₇₁	C₂²	1	1	yes	characteristic, complemented
34	#508×C₇₁	C₂²	1	2	yes	complemented
35	Q₆₄×C₇₁	C₂²	1	1	yes	characteristic, complemented
36	SD₆₄×C₇₁	C₂²	1	2	yes	complemented
37	D₃₂×C₇₁	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic, complemented
39	#41×D₇₁	—	4	2	no	—
40	#14556151	—	2	2	no	—
41	#14556150	—	2	2	no	—
42	#14556149	—	4	2	no	—
43	#14556148	—	4	2	no	—
44	#14556147	—	2	2	no	—
45	Q₁₆×D₇₁	—	2	2	no	—
46	SD₁₆×D₇₁	—	4	2	no	—
47	D₈×D₇₁	—	2	2	no	—
48	C₂×C₈×D₇₁	D₄	1	1	yes	characteristic
49	Dic₅₆₈	C₂³	1	2	yes	—
50	#14556074	—	2	2	no	—
51	#14556073	—	2	2	no	—
52	#14556072	—	2	2	no	—
53	#14556071	—	2	2	no	—
54	D₁₁₃₆	C₂³	1	2	yes	—
55	#14556069	—	2	2	no	—
56	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
57	C₁₆×D₇₁	D₄	1	1	yes	characteristic
58	C₁₆×D₇₁	D₄	1	1	yes	characteristic
59	C₁₆×D₇₁	C₂³	1	1	yes	characteristic
60	C₁₆×D₇₁	C₂³	1	1	yes	characteristic
61	C₇₁⋊C₃₂	C₂³	1	1	yes	characteristic
62	C₇₁⋊C₃₂	C₂³	1	1	yes	characteristic
63	#135×C₇₁	—	2	2	no	—
64	Q₃₂×C₇₁	C₂³	1	2	yes	—
65	SD₃₂×C₇₁	—	2	2	no	—
66	D₁₆×C₇₁	C₂³	1	2	yes	—
67	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
68	C₃₂×C₇₁	C₂³	1	1	yes	characteristic
69	C₃₂×C₇₁	C₂³	1	1	yes	characteristic
70	#11765866	—	4	2	no	—
71	#11765865	—	4	2	no	—
72	#11765865	—	8	2	no	—
73	#11765864	—	4	2	no	—
74	#11765864	—	8	2	no	—
75	Q₈×D₇₁	—	4	2	no	—
76	D₄×D₇₁	—	4	2	no	—
77	D₄×D₇₁	—	8	2	no	—
78	C₂×C₄×D₇₁	—	8	2	no	—
79	C₂×C₄×D₇₁	D₈	1	1	yes	characteristic
80	Dic₂₈₄	—	2	2	no	—
81	#11765855	—	4	2	no	—
82	#11765854	—	4	2	no	—
83	#11765853	—	4	2	no	—
84	#11765852	—	4	2	no	—
85	D₅₆₈	—	2	2	no	—
86	#11765850	—	4	2	no	—
87	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
88	C₈×D₇₁	D₈	1	1	yes	characteristic
89	C₈×D₇₁	D₈	1	1	yes	characteristic
90	C₈×D₇₁	C₂×D₄	1	1	yes	characteristic
91	C₈×D₇₁	C₂×D₄	1	1	yes	characteristic
92	C₇₁⋊C₁₆	C₂×D₄	1	1	yes	characteristic
93	C₇₁⋊C₁₆	C₂×D₄	1	1	yes	characteristic
94	#41×C₇₁	—	4	2	no	—
95	Q₁₆×C₇₁	—	2	2	no	—
96	SD₁₆×C₇₁	—	4	2	no	—
97	D₈×C₇₁	—	2	2	no	—
98	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
99	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
100	C₁₆×C₇₁	C₂⁴	1	1	yes	characteristic
101	C₂²×D₇₁	—	8	2	no	—
102	C₂²×D₇₁	D₁₆	1	1	yes	characteristic
103	Q₈×C₇₁	—	4	2	no	—
104	D₄×C₇₁	—	4	2	no	—
105	D₄×C₇₁	—	8	2	no	—
106	C₂×C₄×C₇₁	—	8	2	no	—
107	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
108	C₂²⋊D₇₁	—	8	2	no	—
109	C₂²⋊D₇₁	—	8	2	no	—
110	C₂²⋊D₇₁	—	8	2	no	—
111	C₂×Dic₇₁	—	8	2	no	—
112	C₂×Dic₇₁	D₁₆	1	1	yes	characteristic
113	D₂₈₄	—	4	2	no	—
114	D₂₈₄	—	8	2	no	—
115	C₄×D₇₁	—	8	2	no	—
116	C₄×D₇₁	—	8	2	no	—
117	C₄×D₇₁	—	8	2	no	—
118	C₄×D₇₁	D₁₆	1	1	yes	characteristic
119	C₄×D₇₁	D₁₆	1	1	yes	characteristic
120	C₄×D₇₁	C₂×D₈	1	1	yes	characteristic
121	C₄×D₇₁	C₂×D₈	1	1	yes	characteristic
122	Dic₁₄₂	—	4	2	no	—
123	Dic₁₄₂	—	8	2	no	—
124	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
125	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
126	C₇₁⋊C₈	C₂×D₈	1	1	yes	characteristic
127	C₇₁⋊C₈	C₂×D₈	1	1	yes	characteristic
128	C₂²×C₇₁	—	8	2	no	—
129	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
130	C₂×D₇₁	—	2	2	no	—
131	C₂×D₇₁	—	8	2	no	—
132	C₂×D₇₁	—	16	4	no	—
133	C₂×D₇₁	C₂×D₁₆	1	1	yes	characteristic
134	C₂×D₇₁	C₂×D₁₆	1	1	yes	characteristic
135	C₄×C₇₁	—	8	2	no	—
136	C₄×C₇₁	C₂×D₁₆	1	1	yes	characteristic
137	C₄×C₇₁	C₂²×D₈	1	1	yes	characteristic
138	Dic₇₁	—	8	2	no	—
139	Dic₇₁	C₂×D₁₆	1	1	yes	characteristic
140	Dic₇₁	C₂×D₁₆	1	1	yes	characteristic
141	C₂×#2262	—	71	1	no	maximal, Sylow
142	C₂×C₇₁	—	2	1	no	—
143	C₂×C₇₁	—	16	2	no	—
144	C₂×C₇₁	C₂²×D₁₆	1	1	yes	characteristic
145	D₇₁	—	2	1	no	—
146	D₇₁	—	16	2	no	—
147	D₇₁	#2262	1	2	yes	direct product factor
148	C₂×#508	—	71	2	no	—
149	#2262	—	71	2	no	—
150	#2262	—	71	1	no	—
151	#2262	—	71	2	no	—
152	#2262	—	71	1	no	—
153	#2262	—	71	1	no	—
154	#2262	D₇₁	1	1	yes	characteristic, direct product factor
155	C₂×Q₆₄	—	71	1	no	—
156	C₂×SD₆₄	—	71	2	no	—
157	C₂×D₃₂	—	71	1	no	—
158	C₂²×C₃₂	—	71	1	no	—
159	C₇₁	C₂×#2262	1	1	yes	characteristic, complemented, Sylow
160	C₂×#135	—	142	2	no	—
161	#508	—	71	2	no	—
162	#508	—	71	2	no	—
163	#508	—	71	2	no	—
164	#508	—	142	2	no	—
165	#508	—	142	2	no	—
166	#508	C₂×D₇₁	1	2	yes	complemented
167	C₂×Q₃₂	—	71	2	no	—
168	C₂×SD₃₂	—	142	2	no	—
169	C₂×D₁₆	—	71	2	no	—
170	C₂²×C₁₆	—	71	1	no	—
171	Q₆₄	—	71	2	no	—
172	Q₆₄	—	71	1	no	—
173	Q₆₄	C₂×D₇₁	1	1	yes	characteristic, complemented
174	SD₆₄	—	71	2	no	—
175	SD₆₄	—	71	2	no	—
176	SD₆₄	—	71	2	no	—
177	SD₆₄	C₂×D₇₁	1	2	yes	complemented
178	D₃₂	—	71	2	no	—
179	D₃₂	—	71	1	no	—
180	D₃₂	C₂×D₇₁	1	1	yes	characteristic, complemented
181	C₂×C₃₂	—	71	1	no	—
182	C₂×C₃₂	—	71	1	no	—
183	C₂×C₃₂	—	71	1	no	—
184	C₂×C₃₂	—	71	1	no	—
185	C₂×C₃₂	—	71	1	no	—
186	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic, complemented
187	C₂×#41	—	284	2	no	—
188	#135	—	2	2	no	—
189	#135	—	284	2	no	—
190	#135	—	284	2	no	—
191	#135	—	142	2	no	—
192	#135	—	142	2	no	—
193	#135	—	142	2	no	—
194	C₂×Q₁₆	—	142	2	no	—
195	C₂×SD₁₆	—	284	2	no	—
196	C₂×D₈	—	142	2	no	—
197	C₂²×C₈	—	71	1	no	—
198	Q₃₂	—	71	2	no	—
199	Q₃₂	—	142	2	no	—
200	Q₃₂	C₂²×D₇₁	1	2	yes	—
201	SD₃₂	—	2	2	no	—
202	SD₃₂	—	142	2	no	—
203	SD₃₂	—	142	2	no	—
204	SD₃₂	—	142	2	no	—
205	D₁₆	—	71	2	no	—
206	D₁₆	—	142	2	no	—
207	D₁₆	C₂²×D₇₁	1	2	yes	—
208	C₂×C₁₆	—	71	1	no	—
209	C₂×C₁₆	—	71	1	no	—
210	C₂×C₁₆	—	71	1	no	—
211	C₂×C₁₆	—	71	1	no	—
212	C₂×C₁₆	—	71	1	no	—
213	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
214	C₃₂	—	71	1	no	—
215	C₃₂	—	71	1	no	—
216	C₃₂	C₂²×D₇₁	1	1	yes	characteristic
217	C₃₂	C₂²×D₇₁	1	1	yes	characteristic
218	#41	—	4	2	no	—
219	#41	—	284	2	no	—
220	#41	—	284	2	no	—
221	#41	—	284	2	no	—
222	#41	—	568	2	no	—
223	#41	—	568	2	no	—
224	C₂×Q₈	—	284	2	no	—
225	C₂×D₄	—	284	2	no	—
226	C₂×D₄	—	568	2	no	—
227	C₂²×C₄	—	568	2	no	—
228	C₂²×C₄	—	71	1	no	—
229	Q₁₆	—	2	2	no	—
230	Q₁₆	—	284	2	no	—
231	Q₁₆	—	142	2	no	—
232	SD₁₆	—	4	2	no	—
233	SD₁₆	—	284	2	no	—
234	SD₁₆	—	284	2	no	—
235	SD₁₆	—	284	2	no	—
236	D₈	—	2	2	no	—
237	D₈	—	284	2	no	—
238	D₈	—	142	2	no	—
239	C₂×C₈	—	71	1	no	—
240	C₂×C₈	—	71	1	no	—
241	C₂×C₈	—	71	1	no	—
242	C₂×C₈	—	71	1	no	—
243	C₂×C₈	—	71	1	no	—
244	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
245	C₁₆	—	71	1	no	—
246	C₁₆	—	71	1	no	—
247	C₁₆	D₄×D₇₁	1	1	yes	characteristic
248	C₁₆	C₂³×D₇₁	1	1	yes	characteristic
249	C₂³	—	568	2	no	—
250	C₂³	—	71	1	no	—
251	Q₈	—	4	2	no	—
252	Q₈	—	284	2	no	—
253	Q₈	—	568	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	8	2	no	—
256	D₄	—	284	2	no	—
257	D₄	—	568	2	no	—
258	D₄	—	568	2	no	—
259	D₄	—	568	2	no	—
260	D₄	—	568	2	no	—
261	C₂×C₄	—	8	2	no	—
262	C₂×C₄	—	568	2	no	—
263	C₂×C₄	—	568	2	no	—
264	C₂×C₄	—	568	2	no	—
265	C₂×C₄	—	568	2	no	—
266	C₂×C₄	—	71	1	no	—
267	C₂×C₄	—	71	1	no	—
268	C₂×C₄	—	71	1	no	—
269	C₂×C₄	—	71	1	no	—
270	C₂×C₄	—	71	1	no	—
271	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
272	C₈	—	71	1	no	—
273	C₈	—	71	1	no	—
274	C₈	D₈×D₇₁	1	1	yes	characteristic
275	C₈	C₂×D₄×D₇₁	1	1	yes	characteristic
276	C₂²	—	8	2	no	—
277	C₂²	—	568	2	no	—
278	C₂²	—	71	1	no	—
279	C₂²	—	71	1	no	—
280	C₂²	—	1136	4	no	—
281	C₂²	—	142	2	no	—
282	C₂²	D₁₆×D₇₁	1	1	yes	characteristic
283	C₄	—	8	2	no	—
284	C₄	—	568	2	no	—
285	C₄	—	71	1	no	—
286	C₄	—	71	1	no	—
287	C₄	D₁₆×D₇₁	1	1	yes	characteristic
288	C₄	C₂×D₈×D₇₁	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	16	2	no	—
291	C₂	—	71	2	no	—
292	C₂	—	1136	2	no	—
293	C₂	—	142	1	no	—
294	C₂	C₂×D₁₆×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₆×D₇₁
commutator, cosocle	C₁₆×C₇₁	C₂⁴
Frattini	C₁₆	C₂³×D₇₁
Fitting	C₇₁×#2262	C₂
socle	C₂×C₇₁	C₂²×D₁₆
hypercenter	#2262	D₇₁
hypocenter	C₇₁	C₂×#2262
LCS	G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2262

Complements

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

N	H ≅ G/N	type
1. #508×D₇₁	C₂	semidirect
2. #22590650	C₂	semidirect
3. #22590649	C₂	semidirect
4. #22590648	C₂	semidirect
5. #22590647	C₂	semidirect
6. #22590646	C₂	semidirect
7. Q₆₄×D₇₁	C₂	semidirect
8. SD₆₄×D₇₁	C₂	semidirect
9. D₃₂×D₇₁	C₂	semidirect
10. C₂×C₃₂×D₇₁	C₂	semidirect
11. C₇₁×#2262	C₂	semidirect
13. #17986432	C₂²	semidirect
14. #17986431	C₂²	semidirect
17. #17986428	C₂²	semidirect
22. Dic₁₁₃₆	C₂²	semidirect
23. #17985839	C₂²	semidirect
24. #17985838	C₂²	semidirect
25. #17985837	C₂²	semidirect
26. #17985836	C₂²	semidirect
27. D₂₂₇₂	C₂²	semidirect
28. #17985834	C₂²	semidirect
29. C₂×C₇₁⋊C₃₂	C₂²	semidirect
32. C₃₂×D₇₁	C₂²	semidirect
33. C₃₂×D₇₁	C₂²	semidirect
34. #508×C₇₁	C₂²	semidirect
35. Q₆₄×C₇₁	C₂²	semidirect
36. SD₆₄×C₇₁	C₂²	semidirect
37. D₃₂×C₇₁	C₂²	semidirect
38. C₂×C₃₂×C₇₁	C₂²	semidirect
147. D₇₁	#2262	direct (N × H)
154. #2262	D₇₁	direct (N × H)
159. C₇₁	C₂×#2262	semidirect
166. #508	C₂×D₇₁	semidirect
173. Q₆₄	C₂×D₇₁	semidirect
177. SD₆₄	C₂×D₇₁	semidirect
180. D₃₂	C₂×D₇₁	semidirect
186. C₂×C₃₂	C₂×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
#508×D₇₁	2
#22590650	2
#22590649	2
#22590648	2
#22590647	2
#22590646	2
Q₆₄×D₇₁	2
SD₆₄×D₇₁	2
D₃₂×D₇₁	2
C₂×C₃₂×D₇₁	2
C₇₁×#2262	2
C₂×#2262	71

Subgroups — #2262×D71 #32155576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2262×D₇₁ #32155576