Subgroups — C₇₁×#9692 #31830564

Order 18176.

570 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#2868	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#332×C₇₁	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₇₁	—	2	2	no	—
6	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈×C₇₁	—	2	2	no	—
8	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₇₁	—	2	1	no	—
11	#384×C₇₁	—	2	1	no	—
12	#384×C₇₁	—	2	1	no	—
13	#384×C₇₁	C₄	1	4	yes	—
14	C₄³×C₇₁	C₂²	1	1	yes	characteristic
15	#336×C₇₁	—	2	1	no	—
16	#332×C₇₁	—	2	8	no	—
17	C₂²×Q₈×C₇₁	—	2	2	no	—
18	C₂²×C₈×C₇₁	—	2	2	no	—
19	C₂²×C₈×C₇₁	—	2	2	no	—
20	C₄⋊Q₈×C₇₁	—	2	4	no	—
21	C₄⋊Q₈×C₇₁	—	2	4	no	—
22	C₄⋊Q₈×C₇₁	—	4	4	no	—
23	C₄⋊Q₈×C₇₁	—	4	4	no	—
24	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
25	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
26	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₁	D₄	1	2	yes	—
31	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₇₁	—	2	4	no	—
34	C₄⋊C₈×C₇₁	—	2	2	no	—
35	C₄⋊C₈×C₇₁	—	4	4	no	—
36	C₄⋊C₈×C₇₁	D₄	1	4	yes	—
37	C₄×C₈×C₇₁	—	4	2	no	—
38	C₂×Q₈×C₇₁	—	4	2	no	—
39	C₂×Q₈×C₇₁	—	4	2	no	—
40	C₂×Q₈×C₇₁	—	4	4	no	—
41	C₂×Q₈×C₇₁	—	4	4	no	—
42	C₂²×C₄×C₇₁	—	2	4	no	—
43	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	—	2	4	no	—
49	C₂×C₈×C₇₁	—	4	2	no	—
50	C₂×C₈×C₇₁	—	4	2	no	—
51	C₂×C₈×C₇₁	—	4	2	no	—
52	C₂×C₈×C₇₁	—	4	4	no	—
53	C₄⋊C₄×C₇₁	—	2	8	no	—
54	C₄⋊C₄×C₇₁	—	4	8	no	—
55	C₄⋊C₄×C₇₁	—	4	8	no	—
56	C₄⋊C₄×C₇₁	—	4	4	no	—
57	C₄²×C₇₁	—	2	4	no	—
58	C₄²×C₇₁	—	2	2	no	—
59	C₄²×C₇₁	D₈	1	4	yes	—
60	C₄²×C₇₁	SD₁₆	1	4	yes	—
61	C₄²×C₇₁	SD₁₆	1	2	yes	—
62	C₄²×C₇₁	Q₁₆	1	2	yes	—
63	C₄²×C₇₁	C₂×D₄	1	2	yes	—
64	C₄²×C₇₁	#41	1	2	yes	—
65	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	Q₈×C₇₁	—	8	4	no	—
67	Q₈×C₇₁	—	8	2	no	—
68	Q₈×C₇₁	—	8	2	no	—
69	C₂×C₄×C₇₁	—	2	2	no	—
70	C₂×C₄×C₇₁	—	2	2	no	—
71	C₂×C₄×C₇₁	—	2	2	no	—
72	C₂×C₄×C₇₁	—	4	4	no	—
73	C₂×C₄×C₇₁	—	4	8	no	—
74	C₂×C₄×C₇₁	#101	1	4	yes	—
75	C₂×C₄×C₇₁	#102	1	4	yes	—
76	C₂×C₄×C₇₁	#102	1	4	yes	—
77	C₂×C₄×C₇₁	#103	1	4	yes	—
78	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₇₁	#121	1	2	yes	—
80	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
81	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
82	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
85	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄×C₇₁	#137	1	2	yes	—
87	C₈×C₇₁	—	4	4	no	—
88	C₈×C₇₁	—	8	2	no	—
89	C₂²×C₇₁	#390	1	1	yes	characteristic
90	C₂²×C₇₁	C₂×#101	1	2	yes	—
91	C₂²×C₇₁	C₂×#102	1	2	yes	—
92	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
93	C₂²×C₇₁	#419	1	1	yes	characteristic
94	C₄×C₇₁	—	2	4	no	—
95	C₄×C₇₁	—	2	2	no	—
96	C₄×C₇₁	—	8	4	no	—
97	C₄×C₇₁	#332	1	8	yes	complemented
98	C₄×C₇₁	#461	1	2	yes	—
99	C₄×C₇₁	#462	1	2	yes	—
100	C₄×C₇₁	#486	1	2	yes	—
101	C₄×C₇₁	#492	1	1	yes	characteristic
102	C₄×C₇₁	#493	1	1	yes	characteristic
103	#9692	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₇₁	C₂×#332	1	2	yes	—
105	C₂×C₇₁	#1546	1	2	yes	—
106	C₂×C₇₁	#1957	1	1	yes	characteristic
107	C₂×C₇₁	#1966	1	1	yes	characteristic
108	C₂×C₇₁	#1981	1	1	yes	characteristic
109	#2868	C₂×C₇₁	1	1	yes	characteristic
110	#1842	C₂×C₇₁	1	1	yes	characteristic
111	C₄×C₄⋊C₈	C₂×C₇₁	1	1	yes	characteristic
112	C₂×#332	C₂×C₇₁	1	4	yes	—
113	C₇₁	#9692	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×C₄⋊Q₈	—	2	2	no	—
115	C₂×C₄⋊Q₈	C₂²×C₇₁	1	2	yes	—
116	C₂×C₄⋊C₈	—	2	2	no	—
117	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
118	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
119	C₂×C₄×C₈	—	2	1	no	—
120	#384	—	2	1	no	—
121	#384	—	2	1	no	—
122	#384	C₄×C₇₁	1	4	yes	—
123	C₄³	C₂²×C₇₁	1	1	yes	characteristic
124	#336	—	2	1	no	—
125	#332	—	2	8	no	—
126	C₂²×Q₈	—	2	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	4	no	—
130	C₄⋊Q₈	—	2	4	no	—
131	C₄⋊Q₈	—	4	4	no	—
132	C₄⋊Q₈	—	4	4	no	—
133	C₂×C₄⋊C₄	—	2	4	no	—
134	C₂×C₄⋊C₄	—	2	4	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	4	yes	—
137	C₂×C₄²	C₂×C₄×C₇₁	1	2	yes	—
138	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
139	C₂×C₄²	D₄×C₇₁	1	2	yes	—
140	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
141	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
142	C₄⋊C₈	—	2	4	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	4	4	no	—
145	C₄⋊C₈	D₄×C₇₁	1	4	yes	—
146	C₄×C₈	—	4	2	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	4	4	no	—
150	C₂×Q₈	—	4	4	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
153	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
154	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	4	4	no	—
162	C₄⋊C₄	—	2	8	no	—
163	C₄⋊C₄	—	4	8	no	—
164	C₄⋊C₄	—	4	8	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄²	—	2	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	D₈×C₇₁	1	4	yes	—
169	C₄²	SD₁₆×C₇₁	1	4	yes	—
170	C₄²	SD₁₆×C₇₁	1	2	yes	—
171	C₄²	Q₁₆×C₇₁	1	2	yes	—
172	C₄²	C₂×D₄×C₇₁	1	2	yes	—
173	C₄²	#41×C₇₁	1	2	yes	—
174	C₂³	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
175	Q₈	—	8	4	no	—
176	Q₈	—	8	2	no	—
177	Q₈	—	8	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	8	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	#101×C₇₁	1	4	yes	—
184	C₂×C₄	#102×C₇₁	1	4	yes	—
185	C₂×C₄	#102×C₇₁	1	4	yes	—
186	C₂×C₄	#103×C₇₁	1	4	yes	—
187	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
188	C₂×C₄	#121×C₇₁	1	2	yes	—
189	C₂×C₄	#124×C₇₁	1	1	yes	characteristic
190	C₂×C₄	#127×C₇₁	1	1	yes	characteristic
191	C₂×C₄	C₂×D₈×C₇₁	1	2	yes	—
192	C₂×C₄	C₂×SD₁₆×C₇₁	1	1	yes	characteristic
193	C₂×C₄	C₂×SD₁₆×C₇₁	1	2	yes	—
194	C₂×C₄	C₂×Q₁₆×C₇₁	1	1	yes	characteristic
195	C₂×C₄	#137×C₇₁	1	2	yes	—
196	C₈	—	4	4	no	—
197	C₈	—	8	2	no	—
198	C₂²	#390×C₇₁	1	1	yes	characteristic
199	C₂²	C₂×#101×C₇₁	1	2	yes	—
200	C₂²	C₂×#102×C₇₁	1	2	yes	—
201	C₂²	C₂×#103×C₇₁	1	1	yes	characteristic
202	C₂²	#419×C₇₁	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	2	no	—
205	C₄	—	8	4	no	—
206	C₄	#332×C₇₁	1	8	yes	complemented
207	C₄	#461×C₇₁	1	2	yes	—
208	C₄	#462×C₇₁	1	2	yes	—
209	C₄	#486×C₇₁	1	2	yes	—
210	C₄	#492×C₇₁	1	1	yes	characteristic
211	C₄	#493×C₇₁	1	1	yes	characteristic
212	C₂	C₂×#332×C₇₁	1	2	yes	—
213	C₂	C₇₁×#1546	1	2	yes	—
214	C₂	C₇₁×#1957	1	1	yes	characteristic
215	C₂	C₇₁×#1966	1	1	yes	characteristic
216	C₂	C₇₁×#1981	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₄³×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
97. C₄×C₇₁	#332	semidirect
103. #9692	C₇₁	direct (N × H)
113. C₇₁	#9692	direct (N × H)
206. C₄	#332×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#2868	2
C₇₁×#1842	2
C₄×C₄⋊C₈×C₇₁	2
C₂×#332×C₇₁	2
#9692	71

Subgroups — C71×#9692 #31830564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#9692 #31830564