Group #31912172 — subgroups

Subgroups — #31912172

Order 18176.

330 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22590595	C₂	1	2	yes	maximal, complemented
2	#22590593	C₂	1	1	yes	maximal, characteristic, complemented
3	#22587945	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#2246	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₆₈	—	2	2	no	—
7	C₂×#14556072	—	2	4	no	—
8	#17986379	C₂²	1	2	yes	—
9	#17986345	—	2	4	no	—
10	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×D₁₆×C₇₁	—	2	2	no	—
13	#493×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	#14556072	—	4	8	no	—
20	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
21	#14556049	—	4	8	no	—
22	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
23	C₂×D₈×C₇₁	—	2	2	no	—
24	C₂×D₈×C₇₁	—	2	2	no	—
25	#127×C₇₁	—	2	2	no	—
26	D₁₆×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×D₄×C₇₁	—	4	2	no	—
36	C₂×D₄×C₇₁	—	4	4	no	—
37	D₈×C₇₁	—	4	4	no	—
38	D₈×C₇₁	—	4	4	no	—
39	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₁	D₈	1	4	yes	—
43	C₂³×C₇₁	—	8	2	no	—
44	D₄×C₇₁	—	8	2	no	—
45	D₄×C₇₁	—	8	2	no	—
46	D₄×C₇₁	—	8	8	no	—
47	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
49	C₂×Dic₇₁	—	8	2	no	—
50	Dic₁₄₂	—	8	2	no	—
51	Dic₁₄₂	—	8	2	no	—
52	C₈×C₇₁	C₂×D₈	1	2	yes	—
53	C₈×C₇₁	C₂×D₈	1	2	yes	—
54	C₇₁⋊C₈	—	2	4	no	—
55	C₂²×C₇₁	—	16	2	no	—
56	C₂²×C₇₁	—	16	4	no	—
57	C₂²×C₇₁	#493	1	1	yes	characteristic
58	C₄×C₇₁	#493	1	1	yes	characteristic
59	C₄×C₇₁	#493	1	1	yes	characteristic
60	C₄×C₇₁	#508	1	4	yes	—
61	Dic₇₁	—	16	2	no	—
62	#12518	—	71	1	no	maximal, Sylow
63	C₂×C₇₁	—	32	2	no	—
64	C₂×C₇₁	#1712	1	1	yes	characteristic
65	C₂×C₇₁	#2249	1	2	yes	—
66	#2248	—	71	2	no	—
67	#2247	—	71	1	no	—
68	#2246	D₇₁	1	1	yes	characteristic
69	#1709	—	71	2	no	—
70	C₈×C₁₆	—	71	1	no	—
71	C₇₁	#12518	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₃₂	—	142	2	no	—
73	C₂×SD₃₂	—	142	4	no	—
74	C₂×D₁₆	—	2	2	no	—
75	#494	—	71	2	no	—
76	#493	C₂×D₇₁	1	2	yes	—
77	#486	—	142	4	no	—
78	C₄×C₁₆	—	71	1	no	—
79	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
80	C₈²	—	71	1	no	—
81	C₂×Q₁₆	—	142	2	no	—
82	C₂×Q₁₆	—	142	2	no	—
83	C₂×D₈	—	2	2	no	—
84	C₂×D₈	—	2	2	no	—
85	C₄⋊Q₈	—	142	2	no	—
86	#127	—	2	2	no	—
87	Q₃₂	—	284	4	no	—
88	SD₃₂	—	284	8	no	—
89	D₁₆	—	4	4	no	—
90	C₂×C₁₆	—	71	2	no	—
91	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
92	#102	—	284	8	no	—
93	C₄×C₈	—	71	2	no	—
94	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
95	C₂×Q₈	—	284	2	no	—
96	C₂×D₄	—	4	2	no	—
97	C₂×D₄	—	4	4	no	—
98	Q₁₆	—	284	4	no	—
99	Q₁₆	—	284	4	no	—
100	D₈	—	4	4	no	—
101	D₈	—	4	4	no	—
102	C₂×C₈	—	71	4	no	—
103	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
104	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
105	C₄⋊C₄	—	284	4	no	—
106	C₄²	D₄×D₇₁	1	1	yes	characteristic
107	C₁₆	—	71	4	no	—
108	C₁₆	#11765850	1	4	yes	—
109	C₂³	—	8	2	no	—
110	Q₈	—	568	2	no	—
111	Q₈	—	568	2	no	—
112	D₄	—	8	2	no	—
113	D₄	—	8	2	no	—
114	D₄	—	8	8	no	—
115	C₂×C₄	—	568	2	no	—
116	C₂×C₄	#14556124	1	1	yes	characteristic
117	C₂×C₄	D₈×D₇₁	1	2	yes	—
118	C₈	—	142	4	no	—
119	C₈	D₈×D₇₁	1	2	yes	—
120	C₈	C₂×#11765850	1	2	yes	—
121	C₂²	—	16	2	no	—
122	C₂²	—	16	4	no	—
123	C₂²	#17986374	1	1	yes	characteristic
124	C₄	—	1136	2	no	—
125	C₄	#17986374	1	1	yes	characteristic
126	C₄	#17986375	1	1	yes	characteristic
127	C₄	#17986428	1	4	yes	—
128	C₂	—	32	2	no	—
129	C₂	#22587954	1	1	yes	characteristic
130	C₂	#22590598	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590595

C₂

yes

maximal, complemented

#22590593

C₂

yes

maximal, characteristic, complemented

#22587945

C₂

yes

maximal, complemented

C₁₆×C₇₁⋊C₈

C₂

yes

maximal, characteristic, complemented

C₇₁×#2246

C₂

yes

maximal, characteristic

C₂×Dic₅₆₈

—

C₂×#14556072

—

#17986379

C₂²

yes

—

#17986345

—

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₈×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×D₁₆×C₇₁

—

#493×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

#14556128

—

Dic₅₆₈

—

#14556072

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

#14556049

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₂×D₈×C₇₁

—

C₂×D₈×C₇₁

—

#127×C₇₁

—

D₁₆×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×D₄×C₇₁

—

C₂×D₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

D₈

yes

—

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×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

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

#12518

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1712

yes

characteristic

C₂×C₇₁

#2249

yes

—

#2248

—

#2247

—

#2246

D₇₁

yes

characteristic

#1709

—

C₈×C₁₆

—

C₇₁

#12518

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

142

—

C₂×SD₃₂

—

142

—

C₂×D₁₆

—

#494

—

#493

C₂×D₇₁

yes

—

#486

—

142

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₈²

—

C₂×Q₁₆

—

142

—

C₂×Q₁₆

—

142

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

142

—

#127

—

Q₃₂

—

284

—

SD₃₂

—

284

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₁

yes

—

#102

—

284

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

284

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

284

—

Q₁₆

—

284

—

100

D₈

—

101

D₈

—

102

C₂×C₈

—

103

C₂×C₈

D₄×D₇₁

yes

characteristic

104

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

105

C₄⋊C₄

—

284

—

106

C₄²

D₄×D₇₁

yes

characteristic

107

C₁₆

—

108

C₁₆

#11765850

yes

—

109

C₂³

—

110

Q₈

—

568

—

111

Q₈

—

568

—

112

D₄

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

568

—

116

C₂×C₄

#14556124

yes

characteristic

117

C₂×C₄

D₈×D₇₁

yes

—

118

C₈

—

142

—

119

C₈

D₈×D₇₁

yes

—

120

C₈

C₂×#11765850

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

#17986374

yes

characteristic

124

C₄

—

1136

—

125

C₄

#17986374

yes

characteristic

126

C₄

#17986375

yes

characteristic

127

C₄

#17986428

yes

—

128

C₂

—

129

C₂

#22587954

yes

characteristic

130

C₂

#22590598

yes

—

	subgroup	G/H
center	C₂²	#17986374
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting, prime radical	C₇₁×#2246	C₂
socle	C₂²×C₇₁	#493
hypercenter	#2246	D₇₁
hypocenter	C₇₁	#12518
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#17986374

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting, prime radical

C₇₁×#2246

C₂

C₂²×C₇₁

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. #22590595	C₂	semidirect
2. #22590593	C₂	semidirect
3. #22587945	C₂	semidirect
4. C₁₆×C₇₁⋊C₈	C₂	semidirect
71. C₇₁	#12518	semidirect

H ≅ G/N

type

1. #22590595

C₂

semidirect

2. #22590593

C₂

semidirect

3. #22587945

C₂

semidirect

4. C₁₆×C₇₁⋊C₈

C₂

semidirect

71. C₇₁

#12518

semidirect

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

subgroup	index
#22590595	2
#22590593	2
#22587945	2
C₁₆×C₇₁⋊C₈	2
C₇₁×#2246	2
#12518	71

subgroup

index

#22590595

#22590593

#22587945

C₁₆×C₇₁⋊C₈

C₇₁×#2246

#12518

Subgroups — #31912172

Conjugacy classes of subgroups

Complements

Maximal subgroups