Subgroups — C₇₁×#19596 #31840468

Order 18176.

814 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#3167	C₂	1	4	yes	maximal, complemented
2	C₇₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#3147	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1708	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1707	C₂	1	2	yes	maximal, complemented
6	C₇₁×#1706	C₂	1	2	yes	maximal, complemented
7	C₇₁×#1705	C₂	1	2	yes	maximal, complemented
8	C₂×C₈²×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₇₁	—	2	2	no	—
10	#533×C₇₁	C₂²	1	2	yes	—
11	#532×C₇₁	C₂²	1	2	yes	—
12	#500×C₇₁	C₂²	1	2	yes	—
13	#499×C₇₁	—	2	2	no	—
14	#498×C₇₁	C₂²	1	2	yes	—
15	#498×C₇₁	C₂²	1	4	yes	—
16	#495×C₇₁	—	2	2	no	—
17	#494×C₇₁	C₂²	1	1	yes	characteristic
18	#493×C₇₁	C₂²	1	1	yes	characteristic
19	#492×C₇₁	C₂²	1	2	yes	—
20	#492×C₇₁	C₂²	1	4	yes	—
21	#487×C₇₁	C₂²	1	4	yes	—
22	#486×C₇₁	C₂²	1	4	yes	—
23	#465×C₇₁	—	2	8	no	—
24	C₄×SD₁₆×C₇₁	—	2	8	no	—
25	#427×C₇₁	—	2	2	no	—
26	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
28	C₈²×C₇₁	C₂²	1	1	yes	characteristic
29	C₈²×C₇₁	C₂²	1	2	yes	—
30	C₈²×C₇₁	C₂²	1	1	yes	characteristic
31	C₂×#41×C₇₁	—	2	2	no	—
32	#135×C₇₁	—	4	8	no	—
33	C₂×Q₁₆×C₇₁	—	2	2	no	—
34	C₂×SD₁₆×C₇₁	—	2	8	no	—
35	C₂×SD₁₆×C₇₁	—	2	4	no	—
36	C₂×D₈×C₇₁	—	2	2	no	—
37	C₂²×C₈×C₇₁	D₄	1	4	yes	—
38	C₂²×C₈×C₇₁	D₄	1	2	yes	—
39	C₄⋊Q₈×C₇₁	C₂³	1	4	yes	—
40	C₄⋊Q₈×C₇₁	C₂³	1	2	yes	—
41	#127×C₇₁	C₂³	1	2	yes	—
42	#125×C₇₁	—	2	2	no	—
43	#124×C₇₁	—	2	2	no	—
44	C₂²⋊Q₈×C₇₁	—	2	4	no	—
45	#121×C₇₁	—	2	4	no	—
46	C₄×Q₈×C₇₁	—	2	4	no	—
47	C₄×D₄×C₇₁	—	2	4	no	—
48	#117×C₇₁	—	2	2	no	—
49	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
50	#107×C₇₁	—	2	4	no	—
51	#106×C₇₁	—	2	4	no	—
52	#106×C₇₁	—	2	8	no	—
53	#103×C₇₁	—	2	8	no	—
54	#102×C₇₁	—	2	8	no	—
55	C₄×C₈×C₇₁	D₄	1	4	yes	—
56	C₄×C₈×C₇₁	D₄	1	2	yes	—
57	C₄×C₈×C₇₁	C₂³	1	2	yes	—
58	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₇₁	C₂³	1	2	yes	—
60	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
61	#41×C₇₁	—	4	8	no	—
62	C₂×Q₈×C₇₁	—	2	4	no	—
63	C₂×Q₈×C₇₁	—	2	2	no	—
64	C₂×D₄×C₇₁	—	2	4	no	—
65	C₂×D₄×C₇₁	—	2	2	no	—
66	C₂×D₄×C₇₁	—	4	2	no	—
67	C₂²×C₄×C₇₁	—	4	2	no	—
68	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
70	Q₁₆×C₇₁	—	4	4	no	—
71	SD₁₆×C₇₁	—	4	8	no	—
72	SD₁₆×C₇₁	—	4	16	no	—
73	D₈×C₇₁	—	4	4	no	—
74	C₂×C₈×C₇₁	—	2	8	no	—
75	C₂×C₈×C₇₁	SD₁₆	1	8	yes	complemented
76	C₂×C₈×C₇₁	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₇₁	C₂×D₄	1	4	yes	—
80	C₄⋊C₄×C₇₁	—	2	4	no	—
81	C₄⋊C₄×C₇₁	—	2	4	no	—
82	C₄⋊C₄×C₇₁	—	2	4	no	—
83	C₄⋊C₄×C₇₁	—	4	4	no	—
84	C₂²⋊C₄×C₇₁	—	4	2	no	—
85	C₂²⋊C₄×C₇₁	—	4	4	no	—
86	C₄²×C₇₁	—	4	2	no	—
87	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₇₁	C₂×D₄	1	2	yes	—
89	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
90	C₂³×C₇₁	—	4	2	no	—
91	C₂³×C₇₁	#127	1	1	yes	characteristic
92	Q₈×C₇₁	—	4	4	no	—
93	Q₈×C₇₁	—	4	8	no	—
94	D₄×C₇₁	—	4	4	no	—
95	D₄×C₇₁	—	4	8	no	—
96	D₄×C₇₁	—	8	4	no	—
97	C₂×C₄×C₇₁	—	2	4	no	—
98	C₂×C₄×C₇₁	—	4	4	no	—
99	C₂×C₄×C₇₁	—	4	2	no	—
100	C₂×C₄×C₇₁	—	8	4	no	—
101	C₂×C₄×C₇₁	#127	1	2	yes	—
102	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
103	C₂×C₄×C₇₁	C₂×SD₁₆	1	4	yes	—
104	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
106	C₈×C₇₁	—	2	4	no	—
107	C₈×C₇₁	C₂×SD₁₆	1	4	yes	—
108	C₈×C₇₁	C₂×SD₁₆	1	4	yes	—
109	C₈×C₇₁	#135	1	8	yes	complemented
110	C₂²×C₇₁	—	2	2	no	—
111	C₂²×C₇₁	—	8	2	no	—
112	C₂²×C₇₁	—	8	4	no	—
113	C₂²×C₇₁	#492	1	2	yes	—
114	C₂²×C₇₁	C₂×#127	1	1	yes	characteristic
115	C₄×C₇₁	—	2	2	no	—
116	C₄×C₇₁	—	8	2	no	—
117	C₄×C₇₁	—	8	4	no	—
118	C₄×C₇₁	#492	1	2	yes	—
119	C₄×C₇₁	C₂²×SD₁₆	1	2	yes	—
120	C₄×C₇₁	C₂×#135	1	4	yes	—
121	#19596	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂×C₇₁	—	2	2	no	—
123	C₂×C₇₁	—	16	2	no	—
124	C₂×C₇₁	C₂×#492	1	1	yes	characteristic
125	C₂×C₇₁	#3147	1	2	yes	—
126	#3167	C₂×C₇₁	1	4	yes	complemented
127	#3160	C₂×C₇₁	1	1	yes	characteristic, complemented
128	#3147	C₂×C₇₁	1	1	yes	characteristic
129	#1708	C₂×C₇₁	1	2	yes	complemented
130	#1707	C₂×C₇₁	1	2	yes	complemented
131	#1706	C₂×C₇₁	1	2	yes	complemented
132	#1705	C₂×C₇₁	1	2	yes	complemented
133	C₂×C₈²	C₂×C₇₁	1	1	yes	characteristic, complemented
134	C₇₁	#19596	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×#135	—	2	2	no	—
136	#533	C₂²×C₇₁	1	2	yes	—
137	#532	C₂²×C₇₁	1	2	yes	—
138	#500	C₂²×C₇₁	1	2	yes	—
139	#499	—	2	2	no	—
140	#498	C₂²×C₇₁	1	2	yes	—
141	#498	C₂²×C₇₁	1	4	yes	—
142	#495	—	2	2	no	—
143	#494	C₂²×C₇₁	1	1	yes	characteristic
144	#493	C₂²×C₇₁	1	1	yes	characteristic
145	#492	C₂²×C₇₁	1	2	yes	—
146	#492	C₂²×C₇₁	1	4	yes	—
147	#487	C₂²×C₇₁	1	4	yes	—
148	#486	C₂²×C₇₁	1	4	yes	—
149	#465	—	2	8	no	—
150	C₄×SD₁₆	—	2	8	no	—
151	#427	—	2	2	no	—
152	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₇₁	1	2	yes	—
154	C₈²	C₂²×C₇₁	1	1	yes	characteristic
155	C₈²	C₂²×C₇₁	1	2	yes	—
156	C₈²	C₂²×C₇₁	1	1	yes	characteristic
157	C₂×#41	—	2	2	no	—
158	#135	—	4	8	no	—
159	C₂×Q₁₆	—	2	2	no	—
160	C₂×SD₁₆	—	2	4	no	—
161	C₂×SD₁₆	—	2	8	no	—
162	C₂×D₈	—	2	2	no	—
163	C₂²×C₈	D₄×C₇₁	1	4	yes	—
164	C₂²×C₈	D₄×C₇₁	1	2	yes	—
165	C₄⋊Q₈	C₂³×C₇₁	1	2	yes	—
166	C₄⋊Q₈	C₂³×C₇₁	1	4	yes	—
167	#127	C₂³×C₇₁	1	2	yes	—
168	#125	—	2	2	no	—
169	#124	—	2	2	no	—
170	C₂²⋊Q₈	—	2	4	no	—
171	#121	—	2	4	no	—
172	C₄×Q₈	—	2	4	no	—
173	C₄×D₄	—	2	4	no	—
174	#117	—	2	2	no	—
175	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
176	#107	—	2	4	no	—
177	#106	—	2	4	no	—
178	#106	—	2	8	no	—
179	#103	—	2	8	no	—
180	#102	—	2	8	no	—
181	C₄×C₈	D₄×C₇₁	1	4	yes	—
182	C₄×C₈	D₄×C₇₁	1	2	yes	—
183	C₄×C₈	C₂³×C₇₁	1	2	yes	—
184	C₄×C₈	C₂³×C₇₁	1	1	yes	characteristic
185	C₄×C₈	C₂³×C₇₁	1	2	yes	—
186	C₄×C₈	C₂³×C₇₁	1	1	yes	characteristic
187	#41	—	4	8	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
196	Q₁₆	—	4	4	no	—
197	SD₁₆	—	4	8	no	—
198	SD₁₆	—	4	16	no	—
199	D₈	—	4	4	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	SD₁₆×C₇₁	1	8	yes	complemented
202	C₂×C₈	C₂×D₄×C₇₁	1	4	yes	—
203	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
204	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
205	C₂×C₈	C₂×D₄×C₇₁	1	4	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	4	4	no	—
212	C₄²	—	4	2	no	—
213	C₄²	C₂×D₄×C₇₁	1	2	yes	—
214	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
215	C₄²	C₂⁴×C₇₁	1	1	yes	characteristic
216	C₂³	—	4	2	no	—
217	C₂³	#127×C₇₁	1	1	yes	characteristic
218	Q₈	—	4	4	no	—
219	Q₈	—	4	8	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	8	no	—
222	D₄	—	8	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	8	4	no	—
227	C₂×C₄	#127×C₇₁	1	2	yes	—
228	C₂×C₄	#127×C₇₁	1	1	yes	characteristic
229	C₂×C₄	C₂×SD₁₆×C₇₁	1	4	yes	—
230	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
231	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
232	C₈	—	2	4	no	—
233	C₈	C₂×SD₁₆×C₇₁	1	4	yes	—
234	C₈	C₂×SD₁₆×C₇₁	1	4	yes	—
235	C₈	#135×C₇₁	1	8	yes	complemented
236	C₂²	—	2	2	no	—
237	C₂²	—	8	2	no	—
238	C₂²	—	8	4	no	—
239	C₂²	#492×C₇₁	1	2	yes	—
240	C₂²	C₂×#127×C₇₁	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	8	4	no	—
244	C₄	#492×C₇₁	1	2	yes	—
245	C₄	C₂²×SD₁₆×C₇₁	1	2	yes	—
246	C₄	C₂×#135×C₇₁	1	4	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	16	2	no	—
249	C₂	C₂×#492×C₇₁	1	1	yes	characteristic
250	C₂	C₇₁×#3147	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇₁	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇₁
socle	C₂²×C₇₁	C₂×#127
prime radical	C₇₁×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₁×#3167	C₂	semidirect
2. C₇₁×#3160	C₂	semidirect
4. C₇₁×#1708	C₂	semidirect
5. C₇₁×#1707	C₂	semidirect
6. C₇₁×#1706	C₂	semidirect
7. C₇₁×#1705	C₂	semidirect
8. C₂×C₈²×C₇₁	C₂	semidirect
75. C₂×C₈×C₇₁	SD₁₆	semidirect
109. C₈×C₇₁	#135	semidirect
121. #19596	C₇₁	direct (N × H)
126. #3167	C₂×C₇₁	semidirect
127. #3160	C₂×C₇₁	semidirect
129. #1708	C₂×C₇₁	semidirect
130. #1707	C₂×C₇₁	semidirect
131. #1706	C₂×C₇₁	semidirect
132. #1705	C₂×C₇₁	semidirect
133. C₂×C₈²	C₂×C₇₁	semidirect
134. C₇₁	#19596	direct (N × H)
201. C₂×C₈	SD₁₆×C₇₁	semidirect
235. C₈	#135×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#3167	2
C₇₁×#3160	2
C₇₁×#3147	2
C₇₁×#1708	2
C₇₁×#1707	2
C₇₁×#1706	2
C₇₁×#1705	2
C₂×C₈²×C₇₁	2
#19596	71

Subgroups — C71×#19596 #31840468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#19596 #31840468