Group #31912165 — subgroups

Subgroups — #31912165

Order 18176.

310 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22590593	C₂	1	1	yes	maximal, characteristic
2	#22590592	C₂	1	2	yes	maximal
3	#22587956	C₂	1	2	yes	maximal
4	C₁₆×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#2247	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₆₈	—	2	2	no	—
7	C₂×#14556074	—	2	4	no	—
8	#17986402	—	2	4	no	—
9	#17986379	C₂²	1	2	yes	—
10	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂×C₇₁	—	2	2	no	—
13	#494×C₇₁	C₂²	1	2	yes	—
14	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₈₄	—	2	2	no	—
16	C₂×Dic₂₈₄	—	2	2	no	—
17	#14556128	—	2	2	no	—
18	Dic₅₆₈	—	4	4	no	—
19	#14556074	—	4	8	no	—
20	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
21	#14556061	—	4	8	no	—
22	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
23	C₂×Q₁₆×C₇₁	—	2	2	no	—
24	C₂×Q₁₆×C₇₁	—	2	2	no	—
25	C₄⋊Q₈×C₇₁	—	2	2	no	—
26	Q₃₂×C₇₁	—	4	4	no	—
27	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
28	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₄₂	—	4	2	no	—
30	Dic₂₈₄	—	4	4	no	—
31	Dic₂₈₄	—	4	4	no	—
32	C₂×C₇₁⋊C₈	D₈	1	4	yes	—
33	C₄⋊Dic₇₁	—	4	4	no	—
34	C₇₁⋊C₁₆	D₈	1	4	yes	—
35	C₂×Q₈×C₇₁	—	4	2	no	—
36	Q₁₆×C₇₁	—	4	4	no	—
37	Q₁₆×C₇₁	—	4	4	no	—
38	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₇₁	—	4	4	no	—
41	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₁	D₈	1	4	yes	—
43	Q₈×C₇₁	—	8	2	no	—
44	Q₈×C₇₁	—	8	2	no	—
45	C₂×C₄×C₇₁	—	8	2	no	—
46	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
48	C₂×Dic₇₁	—	8	2	no	—
49	Dic₁₄₂	—	8	2	no	—
50	Dic₁₄₂	—	8	2	no	—
51	C₈×C₇₁	C₂×D₈	1	2	yes	—
52	C₈×C₇₁	C₂×D₈	1	2	yes	—
53	C₇₁⋊C₈	Q₃₂	1	8	yes	complemented
54	C₂²×C₇₁	#493	1	1	yes	characteristic
55	C₄×C₇₁	—	16	2	no	—
56	C₄×C₇₁	#493	1	1	yes	characteristic
57	C₄×C₇₁	#493	1	1	yes	characteristic
58	C₄×C₇₁	C₂×Q₃₂	1	4	yes	—
59	Dic₇₁	—	16	2	no	—
60	#12516	—	71	1	no	maximal, Sylow
61	C₂×C₇₁	#1712	1	1	yes	characteristic
62	C₂×C₇₁	#2247	1	2	yes	—
63	#2247	—	71	2	no	—
64	#2247	—	71	1	no	—
65	#2247	D₇₁	1	1	yes	characteristic
66	#1713	—	71	2	no	—
67	C₈×C₁₆	—	71	1	no	—
68	C₇₁	#12516	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	—	2	2	no	—
70	C₂×Q₃₂	—	142	4	no	—
71	C₂×Q₃₂	—	142	2	no	—
72	#500	—	142	4	no	—
73	#494	—	71	2	no	—
74	#494	C₂×D₇₁	1	2	yes	—
75	C₄×C₁₆	—	71	1	no	—
76	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
77	C₈²	—	71	1	no	—
78	C₂×Q₁₆	—	2	2	no	—
79	C₂×Q₁₆	—	2	2	no	—
80	C₂×Q₁₆	—	142	2	no	—
81	C₂×Q₁₆	—	142	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	142	2	no	—
84	Q₃₂	—	4	4	no	—
85	Q₃₂	—	284	8	no	—
86	Q₃₂	—	284	4	no	—
87	C₂×C₁₆	—	71	2	no	—
88	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
89	#107	—	284	8	no	—
90	C₄×C₈	—	71	2	no	—
91	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
92	C₂×Q₈	—	4	2	no	—
93	C₂×Q₈	—	284	2	no	—
94	Q₁₆	—	4	4	no	—
95	Q₁₆	—	4	4	no	—
96	Q₁₆	—	284	4	no	—
97	Q₁₆	—	284	4	no	—
98	C₂×C₈	—	71	4	no	—
99	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	284	4	no	—
103	C₄²	D₄×D₇₁	1	1	yes	characteristic
104	C₁₆	—	71	4	no	—
105	C₁₆	#11765850	1	4	yes	—
106	Q₈	—	8	2	no	—
107	Q₈	—	8	2	no	—
108	Q₈	—	568	2	no	—
109	Q₈	—	568	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	568	2	no	—
112	C₂×C₄	#14556124	1	1	yes	characteristic
113	C₂×C₄	D₈×D₇₁	1	2	yes	—
114	C₈	—	71	8	no	—
115	C₈	D₈×D₇₁	1	2	yes	—
116	C₈	C₂×#11765850	1	2	yes	—
117	C₂²	#17986374	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	1136	2	no	—
120	C₄	#17986374	1	1	yes	characteristic
121	C₄	#17986375	1	1	yes	characteristic
122	C₄	Q₃₂×D₇₁	1	4	yes	—
123	C₂	#22587954	1	1	yes	characteristic
124	C₂	#22590591	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590593

C₂

yes

maximal, characteristic

#22590592

C₂

yes

maximal

#22587956

C₂

yes

maximal

C₁₆×C₇₁⋊C₈

C₂

yes

maximal, characteristic

C₇₁×#2247

C₂

yes

maximal, characteristic

C₂×Dic₅₆₈

—

C₂×#14556074

—

#17986402

—

#17986379

C₂²

yes

—

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₈×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×Q₃₂×C₇₁

—

#494×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

#14556128

—

Dic₅₆₈

—

#14556074

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

#14556061

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

—

Q₃₂×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

C₂×C₇₁⋊C₈

D₈

yes

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

D₈

yes

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

D₈

yes

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

C₂×D₈

yes

—

C₈×C₇₁

C₂×D₈

yes

—

C₇₁⋊C₈

Q₃₂

yes

complemented

C₂²×C₇₁

#493

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×Q₃₂

yes

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

#2247

—

#2247

—

#2247

D₇₁

yes

characteristic

#1713

—

C₈×C₁₆

—

C₇₁

#12516

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

142

—

C₂×Q₃₂

—

142

—

#500

—

142

—

#494

—

#494

C₂×D₇₁

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₈²

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

142

—

C₂×Q₁₆

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

Q₃₂

—

Q₃₂

—

284

—

Q₃₂

—

284

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₁

yes

—

#107

—

284

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

284

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

284

—

Q₁₆

—

284

—

C₂×C₈

—

C₂×C₈

D₄×D₇₁

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

284

—

103

C₄²

D₄×D₇₁

yes

characteristic

104

C₁₆

—

105

C₁₆

#11765850

yes

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

568

—

109

Q₈

—

568

—

110

C₂×C₄

—

111

C₂×C₄

—

568

—

112

C₂×C₄

#14556124

yes

characteristic

113

C₂×C₄

D₈×D₇₁

yes

—

114

C₈

—

115

C₈

D₈×D₇₁

yes

—

116

C₈

C₂×#11765850

yes

—

117

C₂²

#17986374

yes

characteristic

118

C₄

—

119

C₄

—

1136

—

120

C₄

#17986374

yes

characteristic

121

C₄

#17986375

yes

characteristic

122

C₄

Q₃₂×D₇₁

yes

—

123

C₂

#22587954

yes

characteristic

124

C₂

#22590591

yes

—

	subgroup	G/H
center	C₂²	#17986374
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#2247	C₂
socle, prime radical	C₂²×C₇₁	#493
hypercenter	#2247	D₇₁
hypocenter	C₇₁	#12516
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#17986374

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#2247

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
53. C₇₁⋊C₈	Q₃₂	semidirect
68. C₇₁	#12516	semidirect

H ≅ G/N

type

53. C₇₁⋊C₈

Q₃₂

semidirect

68. C₇₁

#12516

semidirect

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

subgroup	index
#22590593	2
#22590592	2
#22587956	2
C₁₆×C₇₁⋊C₈	2
C₇₁×#2247	2
#12516	71

subgroup

index

#22590593

#22590592

#22587956

C₁₆×C₇₁⋊C₈

C₇₁×#2247

#12516

Subgroups — #31912165

Conjugacy classes of subgroups

Complements

Maximal subgroups