Subgroups — C₇₁×#11941 #31832813

Order 18176.

334 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2150	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1564	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1365	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊Q₈×C₇₁	—	2	1	no	—
6	C₂×M₃₂×C₇₁	—	2	1	no	—
7	#500×C₇₁	—	2	1	no	—
8	#500×C₇₁	—	2	1	no	—
9	#500×C₇₁	C₄	1	2	yes	—
10	#500×C₇₁	C₂²	1	2	yes	—
11	#431×C₇₁	—	2	1	no	—
12	C₂×#107×C₇₁	—	2	1	no	—
13	C₂×#107×C₇₁	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₇₁	—	2	1	no	—
16	C₄⋊C₁₆×C₇₁	C₂²	1	2	yes	—
17	#335×C₇₁	C₂²	1	2	yes	—
18	#334×C₇₁	—	2	1	no	—
19	C₂²×Q₈×C₇₁	—	4	1	no	—
20	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
22	C₄⋊Q₈×C₇₁	—	2	2	no	—
23	C₄⋊Q₈×C₇₁	—	2	2	no	—
24	C₄⋊Q₈×C₇₁	—	4	1	no	—
25	C₄⋊Q₈×C₇₁	—	4	1	no	—
26	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
28	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
29	M₃₂×C₇₁	—	4	2	no	—
30	C₂×C₁₆×C₇₁	—	2	2	no	—
31	#107×C₇₁	—	2	2	no	—
32	#107×C₇₁	—	2	2	no	—
33	#107×C₇₁	—	2	2	no	—
34	#107×C₇₁	C₂×C₄	1	4	yes	—
35	C₄⋊C₈×C₇₁	—	2	2	no	—
36	C₂²⋊C₈×C₇₁	—	4	1	no	—
37	C₄×C₈×C₇₁	—	4	1	no	—
38	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
42	C₂×Q₈×C₇₁	—	4	2	no	—
43	C₂×Q₈×C₇₁	—	4	2	no	—
44	C₂×Q₈×C₇₁	—	8	4	no	—
45	C₂²×C₄×C₇₁	—	4	2	no	—
46	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₇₁	—	2	2	no	—
50	C₂×C₈×C₇₁	—	4	2	no	—
51	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
54	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
55	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄×C₇₁	—	2	4	no	—
58	C₄⋊C₄×C₇₁	—	4	2	no	—
59	C₄⋊C₄×C₇₁	—	4	2	no	—
60	C₄⋊C₄×C₇₁	—	4	2	no	—
61	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₇₁	D₈	1	1	yes	characteristic
63	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
64	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₁₆×C₇₁	—	4	2	no	—
66	C₂³×C₇₁	#102	1	1	yes	characteristic
67	Q₈×C₇₁	—	8	4	no	—
68	Q₈×C₇₁	—	8	4	no	—
69	C₂×C₄×C₇₁	—	2	2	no	—
70	C₂×C₄×C₇₁	—	2	2	no	—
71	C₂×C₄×C₇₁	—	4	4	no	—
72	C₂×C₄×C₇₁	—	8	4	no	—
73	C₂×C₄×C₇₁	#101	1	2	yes	—
74	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
75	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
76	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
77	C₂×C₄×C₇₁	#103	1	2	yes	—
78	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
81	C₈×C₇₁	—	2	1	no	—
82	C₈×C₇₁	—	2	1	no	—
83	C₈×C₇₁	—	8	2	no	—
84	C₈×C₇₁	#103	1	2	yes	—
85	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
86	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
87	C₂²×C₇₁	—	2	1	no	—
88	C₂²×C₇₁	—	2	1	no	—
89	C₂²×C₇₁	#332	1	2	yes	—
90	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
91	C₄×C₇₁	—	2	1	no	—
92	C₄×C₇₁	—	2	1	no	—
93	C₄×C₇₁	—	8	4	no	—
94	C₄×C₇₁	#332	1	2	yes	—
95	C₄×C₇₁	#361	1	2	yes	—
96	C₄×C₇₁	C₂×#101	1	1	yes	characteristic
97	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
98	C₄×C₇₁	#510	1	2	yes	—
99	#11941	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂×C₇₁	—	2	2	no	—
101	C₂×C₇₁	C₂×#332	1	1	yes	characteristic
102	C₂×C₇₁	#2141	1	1	yes	characteristic
103	C₂×C₇₁	C₂×#361	1	1	yes	characteristic
104	C₂×#500	C₂×C₇₁	1	1	yes	characteristic
105	#2150	C₂×C₇₁	1	1	yes	characteristic
106	#1564	C₂×C₇₁	1	1	yes	characteristic
107	#1365	C₂×C₇₁	1	4	yes	complemented
108	C₇₁	#11941	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×C₄⋊Q₈	—	2	1	no	—
110	C₂×M₃₂	—	2	1	no	—
111	#500	—	2	1	no	—
112	#500	—	2	1	no	—
113	#500	C₄×C₇₁	1	2	yes	—
114	#500	C₂²×C₇₁	1	2	yes	—
115	#431	—	2	1	no	—
116	C₂×#107	—	2	1	no	—
117	C₂×#107	C₄×C₇₁	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
119	C₄⋊C₁₆	—	2	1	no	—
120	C₄⋊C₁₆	C₂²×C₇₁	1	2	yes	—
121	#335	C₂²×C₇₁	1	2	yes	—
122	#334	—	2	1	no	—
123	C₂²×Q₈	—	4	1	no	—
124	C₂²×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
125	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	4	1	no	—
129	C₄⋊Q₈	—	4	1	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
133	M₃₂	—	4	2	no	—
134	C₂×C₁₆	—	2	2	no	—
135	#107	—	2	2	no	—
136	#107	—	2	2	no	—
137	#107	—	2	2	no	—
138	#107	C₂×C₄×C₇₁	1	4	yes	—
139	C₄⋊C₈	—	2	2	no	—
140	C₂²⋊C₈	—	4	1	no	—
141	C₄×C₈	—	4	1	no	—
142	C₄×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
143	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
144	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
145	C₄×C₈	C₂³×C₇₁	1	1	yes	characteristic
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	8	4	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
151	C₂²×C₄	D₈×C₇₁	1	1	yes	characteristic
152	C₂²×C₄	SD₁₆×C₇₁	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
156	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
157	C₂×C₈	SD₁₆×C₇₁	1	2	yes	—
158	C₂×C₈	Q₁₆×C₇₁	1	2	yes	—
159	C₂×C₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
160	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄²	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
166	C₄²	D₈×C₇₁	1	1	yes	characteristic
167	C₄²	SD₁₆×C₇₁	1	1	yes	characteristic
168	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
169	C₁₆	—	4	2	no	—
170	C₂³	#102×C₇₁	1	1	yes	characteristic
171	Q₈	—	8	4	no	—
172	Q₈	—	8	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	4	no	—
177	C₂×C₄	#101×C₇₁	1	2	yes	—
178	C₂×C₄	#102×C₇₁	1	1	yes	characteristic
179	C₂×C₄	#102×C₇₁	1	1	yes	characteristic
180	C₂×C₄	#102×C₇₁	1	1	yes	characteristic
181	C₂×C₄	#103×C₇₁	1	2	yes	—
182	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
183	C₂×C₄	C₂×D₈×C₇₁	1	1	yes	characteristic
184	C₂×C₄	C₂×SD₁₆×C₇₁	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	8	2	no	—
188	C₈	#103×C₇₁	1	2	yes	—
189	C₈	C₂×SD₁₆×C₇₁	1	1	yes	characteristic
190	C₈	C₂×Q₁₆×C₇₁	1	1	yes	characteristic
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	#332×C₇₁	1	2	yes	—
194	C₂²	C₂×#102×C₇₁	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	8	4	no	—
198	C₄	#332×C₇₁	1	2	yes	—
199	C₄	#361×C₇₁	1	2	yes	—
200	C₄	C₂×#101×C₇₁	1	1	yes	characteristic
201	C₄	C₂×#103×C₇₁	1	1	yes	characteristic
202	C₄	#510×C₇₁	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	C₂×#332×C₇₁	1	1	yes	characteristic
205	C₂	C₇₁×#2141	1	1	yes	characteristic
206	C₂	C₂×#361×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇₁
Frattini, cosocle	C₄×C₈	C₂³×C₇₁
prime radical	C₂³×C₇₁	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇₁×#1365	C₂	semidirect
99. #11941	C₇₁	direct (N × H)
107. #1365	C₂×C₇₁	semidirect
108. C₇₁	#11941	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#500×C₇₁	2
C₇₁×#2150	2
C₇₁×#1564	2
C₇₁×#1365	2
#11941	71

Subgroups — C71×#11941 #31832813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#11941 #31832813