Subgroups — C₇₁×#13641 #31834513

Order 18176.

180 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#2108	C₂	1	1	yes	maximal, characteristic
2	C₇₁×C₄⋊C₃₂	C₂	1	2	yes	maximal, complemented
3	C₇₁×C₂²⋊C₃₂	C₂	1	2	yes	maximal
4	C₇₁×#1397	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×M₃₂×C₇₁	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
8	#405×C₇₁	C₄	1	1	yes	characteristic
9	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
10	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
11	#346×C₇₁	C₄	1	1	yes	characteristic
12	#346×C₇₁	C₄	1	1	yes	characteristic
13	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₇₁	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
17	#117×C₇₁	C₈	1	1	yes	characteristic
18	M₃₂×C₇₁	—	2	2	no	—
19	C₂×C₁₆×C₇₁	—	2	2	no	—
20	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
24	#97×C₇₁	C₈	1	1	yes	characteristic
25	#97×C₇₁	C₈	1	1	yes	characteristic
26	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₃₂×C₇₁	—	2	2	no	—
29	C₃₂×C₇₁	—	2	2	no	—
30	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
31	M₁₆×C₇₁	—	2	2	no	—
32	C₂×C₈×C₇₁	—	2	1	no	—
33	C₂×C₈×C₇₁	—	2	1	no	—
34	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
38	C₄⋊C₄×C₇₁	C₁₆	1	2	yes	—
39	C₂²⋊C₄×C₇₁	C₁₆	1	2	yes	—
40	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
42	C₁₆×C₇₁	—	2	1	no	—
43	C₁₆×C₇₁	—	2	1	no	—
44	C₁₆×C₇₁	M₁₆	1	2	yes	—
45	C₁₆×C₇₁	#41	1	2	yes	—
46	C₂³×C₇₁	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇₁	—	2	1	no	—
48	C₂×C₄×C₇₁	—	2	1	no	—
49	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
50	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
51	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
53	C₈×C₇₁	—	2	1	no	—
54	C₈×C₇₁	—	2	1	no	—
55	C₈×C₇₁	M₃₂	1	2	yes	—
56	C₈×C₇₁	#117	1	1	yes	characteristic
57	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
58	C₂²×C₇₁	—	2	1	no	—
59	C₂²×C₇₁	—	2	2	no	—
60	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
61	C₄×C₇₁	—	2	1	no	—
62	C₄×C₇₁	—	2	2	no	—
63	C₄×C₇₁	—	2	2	no	—
64	C₄×C₇₁	#431	1	1	yes	characteristic
65	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
66	#13641	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂×C₇₁	—	4	1	no	—
68	C₂×C₇₁	#2162	1	1	yes	characteristic
69	C₂×C₇₁	#2258	1	2	yes	—
70	#2108	C₂×C₇₁	1	1	yes	characteristic
71	C₄⋊C₃₂	C₂×C₇₁	1	2	yes	complemented
72	C₂²⋊C₃₂	C₂×C₇₁	1	2	yes	—
73	#1397	C₂×C₇₁	1	1	yes	characteristic, complemented
74	C₄×C₃₂	C₂×C₇₁	1	1	yes	characteristic, complemented
75	C₇₁	#13641	1	1	yes	characteristic, direct product factor, Sylow
76	C₂×M₃₂	C₄×C₇₁	1	1	yes	characteristic
77	C₂²×C₁₆	C₂²×C₇₁	1	1	yes	characteristic
78	#405	C₄×C₇₁	1	1	yes	characteristic
79	C₂×C₃₂	C₂²×C₇₁	1	2	yes	—
80	C₂×C₃₂	C₂²×C₇₁	1	2	yes	—
81	#346	C₄×C₇₁	1	1	yes	characteristic
82	#346	C₄×C₇₁	1	1	yes	characteristic
83	C₄×C₁₆	C₂²×C₇₁	1	1	yes	characteristic
84	C₄×C₁₆	C₂²×C₇₁	1	1	yes	characteristic
85	C₂×M₁₆	C₈×C₇₁	1	1	yes	characteristic
86	C₂²×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
87	#117	C₈×C₇₁	1	1	yes	characteristic
88	M₃₂	—	2	2	no	—
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	C₂×C₄×C₇₁	1	1	yes	characteristic
91	C₂×C₁₆	C₂×C₄×C₇₁	1	1	yes	characteristic
92	C₂×C₁₆	C₂×C₄×C₇₁	1	1	yes	characteristic
93	C₂×C₁₆	C₂³×C₇₁	1	1	yes	characteristic
94	#97	C₈×C₇₁	1	1	yes	characteristic
95	#97	C₈×C₇₁	1	1	yes	characteristic
96	C₄×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
97	C₄×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
98	C₃₂	—	2	2	no	—
99	C₃₂	—	2	2	no	—
100	C₂²×C₄	C₂×C₈×C₇₁	1	1	yes	characteristic
101	M₁₆	—	2	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	C₂×C₈×C₇₁	1	1	yes	characteristic
105	C₂×C₈	C₂×C₈×C₇₁	1	1	yes	characteristic
106	C₂×C₈	C₂×C₈×C₇₁	1	1	yes	characteristic
107	C₂×C₈	C₂²×C₄×C₇₁	1	1	yes	characteristic
108	C₄⋊C₄	C₁₆×C₇₁	1	2	yes	—
109	C₂²⋊C₄	C₁₆×C₇₁	1	2	yes	—
110	C₄²	C₂×C₈×C₇₁	1	1	yes	characteristic
111	C₄²	C₂×C₈×C₇₁	1	1	yes	characteristic
112	C₁₆	—	2	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	M₁₆×C₇₁	1	2	yes	—
115	C₁₆	#41×C₇₁	1	2	yes	—
116	C₂³	C₂×C₁₆×C₇₁	1	1	yes	characteristic
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	C₂×C₁₆×C₇₁	1	2	yes	—
120	C₂×C₄	C₂×C₁₆×C₇₁	1	2	yes	—
121	C₂×C₄	C₂×C₁₆×C₇₁	1	1	yes	characteristic
122	C₂×C₄	C₂²×C₈×C₇₁	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	M₃₂×C₇₁	1	2	yes	—
126	C₈	#117×C₇₁	1	1	yes	characteristic
127	C₈	C₂×M₁₆×C₇₁	1	1	yes	characteristic
128	C₂²	—	2	1	no	—
129	C₂²	—	2	2	no	—
130	C₂²	C₂²×C₁₆×C₇₁	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	#431×C₇₁	1	1	yes	characteristic
135	C₄	C₂×M₃₂×C₇₁	1	1	yes	characteristic
136	C₂	—	4	1	no	—
137	C₂	C₇₁×#2162	1	1	yes	characteristic
138	C₂	C₇₁×#2258	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆×C₇₁	C₂³
commutator	C₂²	C₂²×C₁₆×C₇₁
Frattini, cosocle	C₂×C₁₆	C₂³×C₇₁
socle	C₂²×C₇₁	C₂²×C₁₆
prime radical	C₂³×C₇₁	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₁×C₄⋊C₃₂	C₂	semidirect
4. C₇₁×#1397	C₂	semidirect
5. C₄×C₃₂×C₇₁	C₂	semidirect
66. #13641	C₇₁	direct (N × H)
71. C₄⋊C₃₂	C₂×C₇₁	semidirect
73. #1397	C₂×C₇₁	semidirect
74. C₄×C₃₂	C₂×C₇₁	semidirect
75. C₇₁	#13641	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇₁×#2108	2
C₇₁×C₄⋊C₃₂	2
C₇₁×C₂²⋊C₃₂	2
C₇₁×#1397	2
C₄×C₃₂×C₇₁	2
#13641	71

Subgroups — C71×#13641 #31834513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#13641 #31834513