Group #31963016 — subgroups

Subgroups — C₇₁⋊#17309 #31963016

Order 18176.

1466 conjugacy classes of subgroups; only the 283 normal subgroups are shown.

normal subgroups only

This group has 1466 conjugacy classes of subgroups — too many to draw. Only its 283 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22586611	C₂	1	2	yes	maximal
2	#22586610	C₂	1	12	yes	maximal
3	C₇₁×#1446	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₄₂	C₂²	1	4	yes	—
5	C₂×C₇₁⋊#95	C₂²	1	12	yes	—
6	C₂×C₇₁⋊#95	C₂²	1	12	yes	—
7	C₂³×Q₈×C₇₁	C₂²	1	1	yes	characteristic
8	C₂×#95×C₇₁	C₂²	1	6	yes	—
9	C₂²×Dic₁₄₂	—	4	12	no	—
10	C₂²×Dic₁₄₂	—	4	12	no	—
11	C₂²×Dic₁₄₂	Q₈	1	16	yes	complemented
12	C₂³×Dic₇₁	C₂³	1	8	yes	—
13	C₇₁⋊#95	—	4	12	no	—
14	C₇₁⋊#95	—	4	12	no	—
15	C₂²×Q₈×C₇₁	—	4	6	no	—
16	C₂²×Q₈×C₇₁	Q₈	1	4	yes	complemented
17	C₂³×C₄×C₇₁	C₂³	1	4	yes	—
18	C₂³×C₄×C₇₁	C₂³	1	3	yes	—
19	#95×C₇₁	—	4	6	no	—
20	C₂×Dic₁₄₂	—	4	48	no	—
21	C₂×Dic₁₄₂	—	4	48	no	—
22	C₂×Dic₁₄₂	—	4	16	no	—
23	C₂²×Dic₇₁	—	2	24	no	—
24	C₂²×Dic₇₁	—	2	24	no	—
25	C₂²×Dic₇₁	C₂×Q₈	1	16	yes	—
26	C₂⁴×C₇₁	C₂⁴	1	1	yes	characteristic
27	C₂×Q₈×C₇₁	—	4	12	no	—
28	C₂×Q₈×C₇₁	—	4	16	no	—
29	C₂²×C₄×C₇₁	—	2	12	no	—
30	C₂²×C₄×C₇₁	—	2	12	no	—
31	C₂²×C₄×C₇₁	—	2	6	no	—
32	C₂²×C₄×C₇₁	—	2	12	no	—
33	C₂²×C₄×C₇₁	C₂×Q₈	1	8	yes	—
34	C₂²×C₄×C₇₁	C₂×Q₈	1	6	yes	—
35	C₂³×C₇₁	C₂²×Q₈	1	4	yes	—
36	C₂³×C₇₁	C₂²×Q₈	1	1	yes	characteristic
37	C₂³×C₇₁	H(5,𝔽₂)	1	6	yes	—
38	C₂³×C₇₁	H(5,𝔽₂)	1	4	yes	—
39	Q₈×C₇₁	—	4	16	no	—
40	C₂×C₄×C₇₁	—	2	6	no	—
41	C₂×C₄×C₇₁	—	2	12	no	—
42	C₂×C₄×C₇₁	—	2	24	no	—
43	C₂×C₄×C₇₁	—	4	12	no	—
44	C₂×C₄×C₇₁	—	4	12	no	—
45	C₂×C₄×C₇₁	—	4	4	no	—
46	C₂×Dic₇₁	—	2	48	no	—
47	C₂×Dic₇₁	—	4	24	no	—
48	C₂×Dic₇₁	—	4	8	no	—
49	Dic₁₄₂	—	4	64	no	—
50	C₂²×C₇₁	#543	1	12	yes	—
51	C₂²×C₇₁	#543	1	3	yes	—
52	C₂²×C₇₁	Q₈²	1	6	yes	—
53	C₂²×C₇₁	Q₈²	1	4	yes	—
54	C₂²×C₇₁	H(3,𝔽₄)	1	6	yes	—
55	C₂²×C₇₁	H(3,𝔽₄)	1	4	yes	—
56	C₄×C₇₁	—	4	8	no	—
57	C₄×C₇₁	—	4	6	no	—
58	Dic₇₁	—	4	16	no	—
59	#17309	—	71	1	no	maximal, Sylow
60	C₂×C₇₁	#2811	1	4	yes	—
61	C₂×C₇₁	#2811	1	6	yes	—
62	C₂×C₇₁	#2840	1	4	yes	—
63	C₂×C₇₁	#2840	1	1	yes	characteristic
64	#1446	—	71	12	no	—
65	#1446	—	71	2	no	—
66	#1446	D₇₁	1	1	yes	characteristic
67	C₇₁	#17309	1	1	yes	characteristic, complemented, Sylow
68	C₂³×Q₈	—	71	4	no	—
69	C₂³×Q₈	C₂×D₇₁	1	1	yes	characteristic
70	C₂×#95	—	71	12	no	—
71	C₂×#95	—	71	12	no	—
72	C₂×#95	C₂×D₇₁	1	6	yes	—
73	C₂²×Q₈	—	4	6	no	—
74	C₂²×Q₈	—	284	12	no	—
75	C₂²×Q₈	—	284	12	no	—
76	C₂²×Q₈	—	71	16	no	—
77	C₂²×Q₈	Dic₁₄₂	1	4	yes	complemented
78	C₂³×C₄	—	71	8	no	—
79	C₂³×C₄	C₂²×D₇₁	1	4	yes	—
80	C₂³×C₄	C₂²×D₇₁	1	3	yes	—
81	#95	—	4	6	no	—
82	#95	—	284	12	no	—
83	#95	—	284	12	no	—
84	C₂⁴	C₂³×D₇₁	1	1	yes	characteristic
85	C₂×Q₈	—	4	12	no	—
86	C₂×Q₈	—	4	16	no	—
87	C₂×Q₈	—	284	48	no	—
88	C₂×Q₈	—	284	48	no	—
89	C₂×Q₈	—	284	16	no	—
90	C₂²×C₄	—	2	6	no	—
91	C₂²×C₄	—	2	12	no	—
92	C₂²×C₄	—	2	12	no	—
93	C₂²×C₄	—	2	12	no	—
94	C₂²×C₄	—	71	16	no	—
95	C₂²×C₄	—	142	24	no	—
96	C₂²×C₄	—	142	24	no	—
97	C₂²×C₄	Q₈×D₇₁	1	8	yes	—
98	C₂²×C₄	C₂×Dic₁₄₂	1	6	yes	—
99	C₂³	C₂×Q₈×D₇₁	1	4	yes	—
100	C₂³	C₂²×Dic₁₄₂	1	1	yes	characteristic
101	C₂³	#14556173	1	6	yes	—
102	C₂³	#14556174	1	4	yes	—
103	Q₈	—	4	16	no	—
104	Q₈	—	284	64	no	—
105	C₂×C₄	—	2	24	no	—
106	C₂×C₄	—	2	6	no	—
107	C₂×C₄	—	2	12	no	—
108	C₂×C₄	—	4	12	no	—
109	C₂×C₄	—	4	12	no	—
110	C₂×C₄	—	4	4	no	—
111	C₂×C₄	—	284	24	no	—
112	C₂×C₄	—	284	8	no	—
113	C₂×C₄	—	142	48	no	—
114	C₂²	#17986624	1	12	yes	—
115	C₂²	#17986625	1	3	yes	—
116	C₂²	#17986729	1	6	yes	—
117	C₂²	Q₈×Dic₁₄₂	1	4	yes	—
118	C₂²	#17986741	1	6	yes	—
119	C₂²	#17986742	1	4	yes	—
120	C₄	—	4	8	no	—
121	C₄	—	4	6	no	—
122	C₄	—	284	16	no	—
123	C₂	#22596332	1	4	yes	—
124	C₂	#22596333	1	6	yes	—
125	C₂	#22596592	1	4	yes	—
126	C₂	#22596593	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22586611

C₂

yes

maximal

#22586610

C₂

yes

maximal

C₇₁×#1446

C₂

yes

maximal, characteristic

C₂³×Dic₁₄₂

C₂²

yes

—

C₂×C₇₁⋊#95

C₂²

yes

—

C₂×C₇₁⋊#95

C₂²

yes

—

C₂³×Q₈×C₇₁

C₂²

yes

characteristic

C₂×#95×C₇₁

C₂²

yes

—

C₂²×Dic₁₄₂

—

C₂²×Dic₁₄₂

—

C₂²×Dic₁₄₂

Q₈

yes

complemented

C₂³×Dic₇₁

C₂³

yes

—

C₇₁⋊#95

—

C₇₁⋊#95

—

C₂²×Q₈×C₇₁

—

C₂²×Q₈×C₇₁

Q₈

yes

complemented

C₂³×C₄×C₇₁

C₂³

yes

—

C₂³×C₄×C₇₁

C₂³

yes

—

#95×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

C₂×Q₈

yes

—

C₂⁴×C₇₁

C₂⁴

yes

characteristic

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×Q₈

yes

—

C₂²×C₄×C₇₁

C₂×Q₈

yes

—

C₂³×C₇₁

C₂²×Q₈

yes

—

C₂³×C₇₁

C₂²×Q₈

yes

characteristic

C₂³×C₇₁

H(5,𝔽₂)

yes

—

C₂³×C₇₁

H(5,𝔽₂)

yes

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₂²×C₇₁

#543

yes

—

C₂²×C₇₁

#543

yes

—

C₂²×C₇₁

Q₈²

yes

—

C₂²×C₇₁

Q₈²

yes

—

C₂²×C₇₁

H(3,𝔽₄)

yes

—

C₂²×C₇₁

H(3,𝔽₄)

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

Dic₇₁

—

#17309

—

maximal, Sylow

C₂×C₇₁

#2811

yes

—

C₂×C₇₁

#2811

yes

—

C₂×C₇₁

#2840

yes

—

C₂×C₇₁

yes

—

—

D₇₁

yes

characteristic

C₇₁

#17309

yes

characteristic, complemented, Sylow

C₂³×Q₈

—

C₂³×Q₈

C₂×D₇₁

yes

characteristic

C₂×#95

—

C₂×#95

—

C₂×#95

C₂×D₇₁

yes

—

C₂²×Q₈

—

C₂²×Q₈

—

284

—

C₂²×Q₈

—

284

—

C₂²×Q₈

—

C₂²×Q₈

Dic₁₄₂

yes

complemented

C₂³×C₄

—

C₂³×C₄

C₂²×D₇₁

yes

—

C₂³×C₄

C₂²×D₇₁

yes

—

#95

—

#95

—

284

—

#95

—

284

—

C₂⁴

C₂³×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

284

—

C₂×Q₈

—

284

—

C₂×Q₈

—

284

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

Q₈×D₇₁

yes

—

C₂²×C₄

C₂×Dic₁₄₂

yes

—

C₂³

C₂×Q₈×D₇₁

yes

—

100

C₂³

C₂²×Dic₁₄₂

yes

characteristic

101

C₂³

#14556173

yes

—

102

C₂³

#14556174

yes

—

103

Q₈

—

104

Q₈

—

284

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

284

—

112

C₂×C₄

—

284

—

113

C₂×C₄

—

142

—

114

C₂²

#17986624

yes

—

115

C₂²

#17986625

yes

—

116

C₂²

#17986729

yes

—

117

C₂²

Q₈×Dic₁₄₂

yes

—

118

C₂²

#17986741

yes

—

119

C₂²

#17986742

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

284

—

123

C₂

#22596332

yes

—

124

C₂

#22596333

yes

—

125

C₂

#22596592

yes

—

126

C₂

#22596593

yes

characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₇₁
commutator, socle, prime radical, cosocle	C₂⁴×C₇₁	C₂⁴
Fitting	C₇₁×#1446	C₂
hypercenter	#1446	D₇₁
hypocenter	C₇₁	#17309
LCS	G ▷ C₂⁴×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁴ ◁ #1446

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₇₁

commutator, socle, prime radical, cosocle

C₂⁴×C₇₁

C₂⁴

Fitting

C₇₁×#1446

C₂

D₇₁

C₇₁

G ▷ C₂⁴×C₇₁ ▷ C₇₁

C₁ ◁ C₂⁴ ◁ #1446

N	H ≅ G/N	type
11. C₂²×Dic₁₄₂	Q₈	semidirect
16. C₂²×Q₈×C₇₁	Q₈	semidirect
67. C₇₁	#17309	semidirect
77. C₂²×Q₈	Dic₁₄₂	semidirect

H ≅ G/N

type

11. C₂²×Dic₁₄₂

Q₈

semidirect

16. C₂²×Q₈×C₇₁

Q₈

semidirect

67. C₇₁

#17309

semidirect

77. C₂²×Q₈

Dic₁₄₂

semidirect

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

subgroup	index
#22586611	2
#22586610	2
C₇₁×#1446	2
#17309	71

subgroup

index

#22586611

#22586610

C₇₁×#1446

#17309

Subgroups — C₇₁⋊#17309 #31963016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71⋊#17309 #31963016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁⋊#17309 #31963016