Subgroups — #32911222

Order 18176.

10796 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#585×D₇₁	C₂	1	1	yes	maximal, characteristic
2	#22603053	C₂	1	15	yes	maximal, complemented
3	#22603052	C₂	1	1	yes	maximal, characteristic, complemented
4	#22603051	C₂	1	15	yes	maximal, complemented
5	C₂×#17986764	C₂	1	15	yes	maximal, complemented
6	#568×D₇₁	C₂	1	15	yes	maximal, complemented
7	C₇₁×#3573	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986836	C₄	1	6	yes	—
9	#17986835	C₂²	1	15	yes	—
10	#17986834	C₄	1	10	yes	—
11	2₋¹⁺⁴×D₇₁	C₄	1	6	yes	—
12	H(5,𝔽₂)×D₇₁	C₄	1	10	yes	—
13	C₂×#41×D₇₁	C₂²	1	15	yes	—
14	#17986766	C₂²	1	15	yes	complemented
15	#17986765	C₂²	1	90	yes	complemented
16	#17986764	C₂²	1	60	yes	complemented
17	#17986764	C₂²	1	20	yes	complemented
18	#17986764	C₂²	1	60	yes	complemented
19	#17986764	C₂²	1	60	yes	complemented
20	C₂×#14556136	C₂²	1	45	yes	complemented
21	C₂×#14556135	C₂²	1	15	yes	complemented
22	C₂×#14556134	C₂²	1	45	yes	complemented
23	#131×D₇₁	C₂²	1	120	yes	complemented
24	C₂²×#11765848	C₂²	1	15	yes	complemented
25	C₂×M₁₆×D₇₁	C₂²	1	45	yes	complemented
26	#585×C₇₁	C₂²	1	1	yes	characteristic
27	#568×C₇₁	C₂²	1	15	yes	complemented
28	C₂×#131×C₇₁	C₂²	1	15	yes	complemented
29	#14556176	C₂×C₄	1	30	yes	—
30	#14556175	C₂×C₄	1	60	yes	—
31	#14556174	C₂×C₄	1	60	yes	—
32	#14556173	C₂×C₄	1	90	yes	—
33	C₂×#11765866	C₂×C₄	1	15	yes	—
34	C₂×#11765865	C₂³	1	45	yes	—
35	C₂×#11765864	C₂×C₄	1	45	yes	—
36	#41×D₇₁	C₂×C₄	1	60	yes	—
37	#41×D₇₁	C₂×C₄	1	60	yes	—
38	#41×D₇₁	C₂³	1	20	yes	—
39	#41×D₇₁	C₂³	1	60	yes	—
40	C₂×Q₈×D₇₁	C₂×C₄	1	15	yes	—
41	C₂×D₄×D₇₁	C₂×C₄	1	45	yes	—
42	C₂²×C₄×D₇₁	C₂³	1	15	yes	—
43	#14556136	C₂³	1	180	yes	complemented
44	#14556136	C₂³	1	60	yes	complemented
45	#14556135	C₂³	1	60	yes	complemented
46	#14556135	C₂³	1	20	yes	complemented
47	#14556134	C₂³	1	180	yes	complemented
48	#14556134	C₂³	1	60	yes	complemented
49	C₂×#11765849	C₂³	1	45	yes	complemented
50	C₂×#11765848	C₂³	1	45	yes	—
51	C₂×#11765848	C₂³	1	15	yes	—
52	C₂×#11765848	C₂³	1	45	yes	complemented
53	C₂×#11765848	C₂³	1	45	yes	complemented
54	M₁₆×D₇₁	C₂³	1	60	yes	—
55	M₁₆×D₇₁	C₂³	1	180	yes	complemented
56	C₂²×C₇₁⋊C₈	C₂³	1	15	yes	complemented
57	C₂×C₈×D₇₁	C₂³	1	90	yes	complemented
58	2₋¹⁺⁴×C₇₁	C₂×C₄	1	6	yes	—
59	H(5,𝔽₂)×C₇₁	C₂×C₄	1	10	yes	—
60	C₂×#41×C₇₁	C₂³	1	15	yes	—
61	#131×C₇₁	C₂³	1	20	yes	complemented
62	#131×C₇₁	C₂³	1	60	yes	complemented
63	C₂×M₁₆×C₇₁	C₂³	1	45	yes	complemented
64	C₂²×C₈×C₇₁	C₂³	1	15	yes	complemented
65	C₂³×D₇₁	C₂²×C₄	1	15	yes	—
66	#11765866	C₂²×C₄	1	20	yes	—
67	#11765866	C₂²×C₄	1	60	yes	—
68	#11765865	C₂²×C₄	1	180	yes	—
69	#11765865	C₂⁴	1	60	yes	—
70	#11765864	C₂²×C₄	1	60	yes	—
71	#11765864	C₂²×C₄	1	180	yes	—
72	C₂×Dic₁₄₂	C₂²×C₄	1	45	yes	—
73	Q₈×D₇₁	C₂²×C₄	1	20	yes	—
74	Q₈×D₇₁	C₂²×C₄	1	60	yes	—
75	C₂×D₂₈₄	C₂²×C₄	1	45	yes	—
76	C₂×C₂²⋊D₇₁	C₂²×C₄	1	90	yes	—
77	D₄×D₇₁	C₂²×C₄	1	60	yes	—
78	D₄×D₇₁	C₂²×C₄	1	180	yes	—
79	C₂²×Dic₇₁	C₂²×C₄	1	15	yes	—
80	C₂×C₄×D₇₁	C₂²×C₄	1	45	yes	—
81	C₂×C₄×D₇₁	C₂²×C₄	1	45	yes	—
82	C₂×C₄×D₇₁	C₂⁴	1	15	yes	—
83	C₂×C₄×D₇₁	C₂⁴	1	45	yes	—
84	#11765849	C₂⁴	1	60	yes	—
85	#11765848	C₂⁴	1	15	yes	—
86	#11765848	C₂⁴	1	90	yes	—
87	#11765848	C₂⁴	1	1	yes	characteristic
88	#11765848	C₂⁴	1	15	yes	—
89	#11765848	C₂⁴	1	15	yes	—
90	C₂×C₇₁⋊C₈	C₂⁴	1	45	yes	—
91	C₂×C₇₁⋊C₈	C₂⁴	1	15	yes	—
92	C₈×D₇₁	C₂⁴	1	120	yes	—
93	#41×C₇₁	C₂²×C₄	1	60	yes	—
94	#41×C₇₁	C₂⁴	1	20	yes	—
95	C₂×Q₈×C₇₁	C₂²×C₄	1	15	yes	—
96	C₂×D₄×C₇₁	C₂²×C₄	1	45	yes	—
97	C₂²×C₄×C₇₁	C₂⁴	1	15	yes	—
98	M₁₆×C₇₁	C₂⁴	1	60	yes	—
99	C₂×C₈×C₇₁	C₂⁴	1	15	yes	—
100	C₂×C₈×C₇₁	C₂⁴	1	45	yes	—
101	C₂³×C₇₁	C₂³×C₄	1	15	yes	—
102	C₂²×D₇₁	—	8	15	no	—
103	C₂²×D₇₁	C₂³×C₄	1	15	yes	—
104	C₂²×D₇₁	C₂³×C₄	1	45	yes	—
105	Q₈×C₇₁	C₂³×C₄	1	20	yes	—
106	D₄×C₇₁	C₂³×C₄	1	60	yes	—
107	C₂×C₄×C₇₁	C₂³×C₄	1	45	yes	—
108	C₂×C₄×C₇₁	C₂⁵	1	15	yes	—
109	C₂²⋊D₇₁	C₂³×C₄	1	120	yes	—
110	C₂×Dic₇₁	C₂³×C₄	1	15	yes	—
111	C₂×Dic₇₁	C₂³×C₄	1	45	yes	—
112	D₂₈₄	C₂³×C₄	1	60	yes	—
113	C₄×D₇₁	C₂³×C₄	1	15	yes	—
114	C₄×D₇₁	C₂³×C₄	1	15	yes	—
115	C₄×D₇₁	C₂³×C₄	1	90	yes	—
116	C₄×D₇₁	C₂⁵	1	15	yes	—
117	C₄×D₇₁	C₂⁵	1	1	yes	characteristic
118	Dic₁₄₂	C₂³×C₄	1	60	yes	—
119	C₈×C₇₁	C₂⁵	1	1	yes	characteristic
120	C₈×C₇₁	C₂⁵	1	15	yes	—
121	C₇₁⋊C₈	C₂⁵	1	15	yes	—
122	C₇₁⋊C₈	C₂⁵	1	1	yes	characteristic
123	C₂²×C₇₁	—	4	15	no	—
124	C₂²×C₇₁	C₂⁴×C₄	1	15	yes	—
125	C₂×D₇₁	—	4	15	no	—
126	C₂×D₇₁	—	4	45	no	—
127	C₂×D₇₁	C₂⁴×C₄	1	1	yes	characteristic
128	C₂×D₇₁	C₂⁴×C₄	1	15	yes	—
129	C₄×C₇₁	C₂⁴×C₄	1	15	yes	—
130	C₄×C₇₁	C₂⁶	1	1	yes	characteristic
131	Dic₇₁	C₂⁴×C₄	1	15	yes	—
132	Dic₇₁	C₂⁴×C₄	1	1	yes	characteristic
133	#63076	—	71	1	no	maximal, Sylow
134	C₂×C₇₁	—	2	15	no	—
135	C₂×C₇₁	C₂⁵×C₄	1	1	yes	characteristic
136	D₇₁	—	2	1	no	—
137	D₇₁	—	2	15	no	—
138	C₂×#585	—	71	1	no	—
139	#3573	—	71	1	no	—
140	#3573	—	71	15	no	—
141	#3573	—	71	15	no	—
142	#3573	D₇₁	1	1	yes	characteristic, complemented
143	C₂×#568	—	71	15	no	—
144	C₂×#568	—	71	15	no	—
145	C₇₁	#63076	1	1	yes	characteristic, complemented, Sylow
146	#585	—	71	6	no	—
147	#585	—	71	10	no	—
148	#585	—	71	15	no	—
149	#585	C₂×D₇₁	1	1	yes	characteristic
150	C₂×2₋¹⁺⁴	—	71	6	no	—
151	C₂×H(5,𝔽₂)	—	71	10	no	—
152	C₂²×#41	—	71	15	no	—
153	#568	—	71	15	no	—
154	#568	—	71	60	no	—
155	#568	—	71	20	no	—
156	#568	—	71	60	no	—
157	#568	—	71	60	no	—
158	#568	—	71	90	no	—
159	#568	C₂×D₇₁	1	15	yes	complemented
160	C₂×#131	—	71	15	no	—
161	C₂×#131	—	71	120	no	—
162	C₂×#131	—	71	45	no	—
163	C₂×#131	—	71	45	no	—
164	C₂×#131	C₂×D₇₁	1	15	yes	complemented
165	C₂²×M₁₆	—	71	45	no	—
166	C₂²×M₁₆	—	71	15	no	—
167	2₋¹⁺⁴	—	71	60	no	—
168	2₋¹⁺⁴	—	71	30	no	—
169	2₋¹⁺⁴	C₄×D₇₁	1	6	yes	—
170	H(5,𝔽₂)	—	71	60	no	—
171	H(5,𝔽₂)	—	71	90	no	—
172	H(5,𝔽₂)	C₄×D₇₁	1	10	yes	—
173	C₂×#41	—	71	60	no	—
174	C₂×#41	—	71	60	no	—
175	C₂×#41	—	71	60	no	—
176	C₂×#41	—	71	20	no	—
177	C₂×#41	—	71	15	no	—
178	C₂×#41	—	71	45	no	—
179	C₂×#41	—	71	45	no	—
180	C₂×#41	C₂²×D₇₁	1	15	yes	—
181	C₂²×Q₈	—	71	15	no	—
182	C₂²×D₄	—	71	45	no	—
183	C₂³×C₄	—	71	15	no	—
184	#131	—	71	20	no	—
185	#131	—	71	60	no	—
186	#131	—	71	180	no	—
187	#131	—	71	60	no	—
188	#131	—	71	180	no	—
189	#131	—	71	60	no	—
190	#131	C₂²×D₇₁	1	60	yes	complemented
191	#131	C₂²×D₇₁	1	20	yes	complemented
192	C₂×M₁₆	—	71	45	no	—
193	C₂×M₁₆	—	71	60	no	—
194	C₂×M₁₆	—	71	180	no	—
195	C₂×M₁₆	—	71	45	no	—
196	C₂×M₁₆	—	71	15	no	—
197	C₂×M₁₆	—	71	45	no	—
198	C₂×M₁₆	—	71	45	no	—
199	C₂×M₁₆	C₂²×D₇₁	1	45	yes	complemented
200	C₂²×C₈	—	71	15	no	—
201	C₂²×C₈	—	71	90	no	—
202	C₂²×C₈	C₂²×D₇₁	1	15	yes	complemented
203	C₂⁴	—	71	15	no	—
204	#41	—	71	20	no	—
205	#41	—	71	60	no	—
206	#41	—	71	60	no	—
207	#41	—	71	180	no	—
208	#41	—	71	180	no	—
209	#41	—	71	60	no	—
210	#41	C₂×C₄×D₇₁	1	60	yes	—
211	#41	C₂³×D₇₁	1	20	yes	—
212	C₂×Q₈	—	71	20	no	—
213	C₂×Q₈	—	71	60	no	—
214	C₂×Q₈	—	71	45	no	—
215	C₂×Q₈	C₂×C₄×D₇₁	1	15	yes	—
216	C₂×D₄	—	71	90	no	—
217	C₂×D₄	—	71	60	no	—
218	C₂×D₄	—	71	180	no	—
219	C₂×D₄	—	71	45	no	—
220	C₂×D₄	C₂×C₄×D₇₁	1	45	yes	—
221	C₂²×C₄	—	71	15	no	—
222	C₂²×C₄	—	71	45	no	—
223	C₂²×C₄	—	71	45	no	—
224	C₂²×C₄	—	71	45	no	—
225	C₂²×C₄	—	71	15	no	—
226	C₂²×C₄	C₂³×D₇₁	1	15	yes	—
227	M₁₆	—	71	60	no	—
228	M₁₆	—	71	15	no	—
229	M₁₆	—	71	1	no	—
230	M₁₆	—	71	90	no	—
231	M₁₆	—	71	15	no	—
232	M₁₆	—	71	15	no	—
233	M₁₆	C₂³×D₇₁	1	60	yes	—
234	C₂×C₈	—	71	45	no	—
235	C₂×C₈	—	71	15	no	—
236	C₂×C₈	—	71	120	no	—
237	C₂×C₈	C₂³×D₇₁	1	45	yes	—
238	C₂×C₈	C₂³×D₇₁	1	15	yes	—
239	C₂³	—	568	15	no	—
240	C₂³	—	71	15	no	—
241	C₂³	—	71	45	no	—
242	C₂³	C₂²×C₄×D₇₁	1	15	yes	—
243	Q₈	—	71	60	no	—
244	Q₈	C₂²×C₄×D₇₁	1	20	yes	—
245	D₄	—	71	120	no	—
246	D₄	—	71	60	no	—
247	D₄	C₂²×C₄×D₇₁	1	60	yes	—
248	C₂×C₄	—	71	15	no	—
249	C₂×C₄	—	71	45	no	—
250	C₂×C₄	—	71	15	no	—
251	C₂×C₄	—	71	15	no	—
252	C₂×C₄	—	71	90	no	—
253	C₂×C₄	—	71	15	no	—
254	C₂×C₄	—	71	1	no	—
255	C₂×C₄	C₂²×C₄×D₇₁	1	45	yes	—
256	C₂×C₄	C₂⁴×D₇₁	1	15	yes	—
257	C₈	—	71	1	no	—
258	C₈	—	71	15	no	—
259	C₈	C₂⁴×D₇₁	1	15	yes	—
260	C₈	C₂⁴×D₇₁	1	1	yes	characteristic
261	C₂²	—	4	15	no	—
262	C₂²	—	284	15	no	—
263	C₂²	—	284	45	no	—
264	C₂²	—	71	1	no	—
265	C₂²	—	71	15	no	—
266	C₂²	C₂³×C₄×D₇₁	1	15	yes	—
267	C₄	—	71	1	no	—
268	C₄	—	71	15	no	—
269	C₄	C₂³×C₄×D₇₁	1	15	yes	—
270	C₄	C₂⁵×D₇₁	1	1	yes	characteristic
271	C₂	—	2	15	no	—
272	C₂	—	142	1	no	—
273	C₂	—	142	15	no	—
274	C₂	C₂⁴×C₄×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂⁵×D₇₁
commutator, socle	C₂×C₇₁	C₂⁵×C₄
Fitting	C₇₁×#3573	C₂
prime radical	#585×D₇₁	C₂
cosocle	C₄×C₇₁	C₂⁶
hypercenter	#3573	D₇₁
hypocenter	C₇₁	#63076
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ #3573

Complements

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

N	H ≅ G/N	type
2. #22603053	C₂	semidirect
3. #22603052	C₂	semidirect
4. #22603051	C₂	semidirect
5. C₂×#17986764	C₂	semidirect
6. #568×D₇₁	C₂	semidirect
7. C₇₁×#3573	C₂	semidirect
14. #17986766	C₂²	semidirect
15. #17986765	C₂²	semidirect
16. #17986764	C₂²	semidirect
17. #17986764	C₂²	semidirect
18. #17986764	C₂²	semidirect
19. #17986764	C₂²	semidirect
20. C₂×#14556136	C₂²	semidirect
21. C₂×#14556135	C₂²	semidirect
22. C₂×#14556134	C₂²	semidirect
23. #131×D₇₁	C₂²	semidirect
24. C₂²×#11765848	C₂²	semidirect
25. C₂×M₁₆×D₇₁	C₂²	semidirect
27. #568×C₇₁	C₂²	semidirect
28. C₂×#131×C₇₁	C₂²	semidirect
43. #14556136	C₂³	semidirect
44. #14556136	C₂³	semidirect
45. #14556135	C₂³	semidirect
46. #14556135	C₂³	semidirect
47. #14556134	C₂³	semidirect
48. #14556134	C₂³	semidirect
49. C₂×#11765849	C₂³	semidirect
52. C₂×#11765848	C₂³	semidirect
53. C₂×#11765848	C₂³	semidirect
55. M₁₆×D₇₁	C₂³	semidirect
56. C₂²×C₇₁⋊C₈	C₂³	semidirect
57. C₂×C₈×D₇₁	C₂³	semidirect
61. #131×C₇₁	C₂³	semidirect
62. #131×C₇₁	C₂³	semidirect
63. C₂×M₁₆×C₇₁	C₂³	semidirect
64. C₂²×C₈×C₇₁	C₂³	semidirect
142. #3573	D₇₁	semidirect
145. C₇₁	#63076	semidirect
159. #568	C₂×D₇₁	semidirect
164. C₂×#131	C₂×D₇₁	semidirect
190. #131	C₂²×D₇₁	semidirect
191. #131	C₂²×D₇₁	semidirect
199. C₂×M₁₆	C₂²×D₇₁	semidirect
202. C₂²×C₈	C₂²×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
#585×D₇₁	2
#22603053	2
#22603052	2
#22603051	2
C₂×#17986764	2
#568×D₇₁	2
C₇₁×#3573	2
#63076	71