Group #31912110 — subgroups

Subgroups — #31912110

Order 18176.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22590615	C₂	1	1	yes	maximal, characteristic
2	#22590593	C₂	1	1	yes	maximal, characteristic
3	#22590568	C₂	1	2	yes	maximal
4	#22587937	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₆₈	—	2	2	no	—
7	#17986404	C₂²	1	2	yes	—
8	#17986397	—	2	2	no	—
9	#17986379	C₂²	1	2	yes	—
10	#17986350	—	2	2	no	—
11	#17985825	—	2	2	no	—
12	#17985804	—	2	4	no	—
13	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₈²×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₈₄	—	2	2	no	—
17	C₂×Dic₂₈₄	—	2	2	no	—
18	#14556128	—	2	2	no	—
19	#14556128	—	2	2	no	—
20	Dic₅₆₈	—	4	4	no	—
21	#14556062	—	2	2	no	—
22	#14556062	—	2	2	no	—
23	#14556060	—	4	4	no	—
24	#14556053	—	4	4	no	—
25	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
26	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
27	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₄₂	—	4	2	no	—
31	C₂×Dic₁₄₂	—	4	2	no	—
32	Dic₂₈₄	—	4	4	no	—
33	Dic₂₈₄	—	4	4	no	—
34	C₄⋊Dic₇₁	—	4	2	no	—
35	C₄⋊Dic₇₁	—	4	2	no	—
36	C₄⋊Dic₇₁	—	4	2	no	—
37	C₄⋊Dic₇₁	—	4	2	no	—
38	C₂×C₈×C₇₁	D₈	1	2	yes	—
39	C₂×C₈×C₇₁	D₈	1	2	yes	—
40	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₇₁	—	2	2	no	—
44	C₁₆×C₇₁	SD₁₆	1	4	yes	—
45	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₇₁	—	8	2	no	—
49	C₂×Dic₇₁	—	8	2	no	—
50	Dic₁₄₂	—	8	2	no	—
51	Dic₁₄₂	—	8	2	no	—
52	Dic₁₄₂	—	8	4	no	—
53	C₈×C₇₁	—	2	2	no	—
54	C₈×C₇₁	Q₃₂	1	4	yes	complemented
55	C₈×C₇₁	C₂×SD₁₆	1	2	yes	—
56	C₈×C₇₁	#135	1	2	yes	—
57	C₂²×C₇₁	#493	1	1	yes	characteristic
58	C₄×C₇₁	#492	1	1	yes	characteristic
59	C₄×C₇₁	#495	1	1	yes	characteristic
60	C₄×C₇₁	C₂×Q₃₂	1	2	yes	—
61	C₄×C₇₁	#508	1	2	yes	—
62	Dic₇₁	—	16	2	no	—
63	Dic₇₁	—	16	2	no	—
64	#12504	—	71	1	no	maximal, Sylow
65	C₂×C₇₁	#1706	1	1	yes	characteristic
66	C₂×C₇₁	#2247	1	1	yes	characteristic
67	C₂×C₇₁	#2249	1	1	yes	characteristic
68	#2252	—	71	1	no	—
69	#2247	—	71	1	no	—
70	#2241	—	71	2	no	—
71	#1707	—	71	2	no	—
72	C₈×C₁₆	D₇₁	1	1	yes	characteristic
73	C₇₁	#12504	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₃₂	—	142	2	no	—
75	#500	—	71	2	no	—
76	#498	—	142	2	no	—
77	#494	—	71	2	no	—
78	#487	—	142	2	no	—
79	#366	—	142	2	no	—
80	#358	—	142	4	no	—
81	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
82	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
83	C₈²	C₂×D₇₁	1	1	yes	characteristic
84	C₂×Q₁₆	—	142	2	no	—
85	C₂×Q₁₆	—	142	2	no	—
86	C₄⋊Q₈	—	142	2	no	—
87	C₄⋊Q₈	—	142	2	no	—
88	Q₃₂	—	284	4	no	—
89	C₂×C₁₆	D₂₈₄	1	2	yes	—
90	C₂×C₁₆	D₂₈₄	1	2	yes	—
91	#107	—	142	2	no	—
92	#107	—	142	2	no	—
93	#106	—	284	4	no	—
94	#103	—	284	4	no	—
95	C₄×C₈	D₂₈₄	1	1	yes	characteristic
96	C₄×C₈	D₂₈₄	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
98	C₂×Q₈	—	284	2	no	—
99	C₂×Q₈	—	284	2	no	—
100	Q₁₆	—	284	4	no	—
101	Q₁₆	—	284	4	no	—
102	C₂×C₈	D₅₆₈	1	2	yes	—
103	C₂×C₈	D₅₆₈	1	2	yes	—
104	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
105	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
106	C₄⋊C₄	—	284	2	no	—
107	C₄⋊C₄	—	284	2	no	—
108	C₄⋊C₄	—	284	2	no	—
109	C₄⋊C₄	—	284	2	no	—
110	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	#11765854	1	4	yes	—
113	Q₈	—	568	2	no	—
114	Q₈	—	568	2	no	—
115	Q₈	—	568	4	no	—
116	C₂×C₄	—	568	2	no	—
117	C₂×C₄	—	568	2	no	—
118	C₂×C₄	#14556125	1	1	yes	characteristic
119	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
120	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	Dic₅₆₈	1	4	yes	complemented
123	C₈	C₂×#11765854	1	2	yes	—
124	C₈	#14556150	1	2	yes	—
125	C₂²	#17986376	1	1	yes	characteristic
126	C₄	—	1136	2	no	—
127	C₄	—	1136	2	no	—
128	C₄	#17986372	1	1	yes	characteristic
129	C₄	#17986383	1	1	yes	characteristic
130	C₄	C₂×Dic₅₆₈	1	2	yes	—
131	C₄	#17986431	1	2	yes	—
132	C₂	#22587932	1	1	yes	characteristic
133	C₂	#22590593	1	1	yes	characteristic
134	C₂	#22590601	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590615

C₂

yes

maximal, characteristic

#22590593

C₂

yes

maximal, characteristic

#22590568

C₂

yes

maximal

#22587937

C₂

yes

maximal

C₈×C₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₅₆₈

—

#17986404

C₂²

yes

—

#17986397

—

#17986379

C₂²

yes

—

#17986350

—

#17985825

—

#17985804

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₈²×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

#14556128

—

#14556128

—

Dic₅₆₈

—

—

—

—

—

C₂×C₁₆×C₇₁

D₄

yes

—

C₂×C₁₆×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

Q₃₂

yes

complemented

C₈×C₇₁

C₂×SD₁₆

yes

—

C₈×C₇₁

#135

yes

—

C₂²×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×Q₃₂

yes

—

C₄×C₇₁

#508

yes

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

—

—

—

C₈×C₁₆

D₇₁

yes

characteristic

C₇₁

#12504

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

142

—

#500

—

#498

—

142

—

#494

—

#487

—

142

—

#366

—

142

—

#358

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₈²

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

—

142

—

C₂×Q₁₆

—

142

—

C₄⋊Q₈

—

142

—

C₄⋊Q₈

—

142

—

Q₃₂

—

284

—

C₂×C₁₆

D₂₈₄

yes

—

C₂×C₁₆

D₂₈₄

yes

—

#107

—

142

—

#107

—

142

—

#106

—

284

—

#103

—

284

—

C₄×C₈

D₂₈₄

yes

characteristic

C₄×C₈

D₂₈₄

yes

characteristic

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

284

—

C₂×Q₈

—

284

—

100

Q₁₆

—

284

—

101

Q₁₆

—

284

—

102

C₂×C₈

D₅₆₈

yes

—

103

C₂×C₈

D₅₆₈

yes

—

104

C₂×C₈

C₂×D₂₈₄

yes

characteristic

105

C₂×C₈

C₂×D₂₈₄

yes

characteristic

106

C₄⋊C₄

—

284

—

107

C₄⋊C₄

—

284

—

108

C₄⋊C₄

—

284

—

109

C₄⋊C₄

—

284

—

110

C₄²

C₂×D₂₈₄

yes

characteristic

111

C₁₆

—

112

C₁₆

#11765854

yes

—

113

Q₈

—

568

—

114

Q₈

—

568

—

115

Q₈

—

568

—

116

C₂×C₄

—

568

—

117

C₂×C₄

—

568

—

118

C₂×C₄

#14556125

yes

characteristic

119

C₂×C₄

C₂×D₅₆₈

yes

characteristic

120

C₂×C₄

C₂×D₅₆₈

yes

characteristic

121

C₈

—

122

C₈

Dic₅₆₈

yes

complemented

123

C₈

C₂×#11765854

yes

—

124

C₈

#14556150

yes

—

125

C₂²

#17986376

yes

characteristic

126

C₄

—

1136

—

127

C₄

—

1136

—

128

C₄

#17986372

yes

characteristic

129

C₄

#17986383

yes

characteristic

130

C₄

C₂×Dic₅₆₈

yes

—

131

C₄

#17986431

yes

—

132

C₂

#22587932

yes

characteristic

133

C₂

#22590593

yes

characteristic

134

C₂

#22590601

yes

characteristic

	subgroup	G/H
center	C₂²	#17986376
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₈×C₁₆×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	#493
hypercenter	C₈×C₁₆	D₇₁
hypocenter	C₇₁	#12504
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#17986376

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₈×C₁₆×C₇₁

C₂

socle, prime radical

C₂²×C₇₁

#493

hypercenter

C₈×C₁₆

D₇₁

hypocenter

C₇₁

#12504

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
54. C₈×C₇₁	Q₃₂	semidirect
73. C₇₁	#12504	semidirect
122. C₈	Dic₅₆₈	semidirect

H ≅ G/N

type

54. C₈×C₇₁

Q₃₂

semidirect

73. C₇₁

#12504

semidirect

122. C₈

Dic₅₆₈

semidirect

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

subgroup	index
#22590615	2
#22590593	2
#22590568	2
#22587937	2
C₈×C₁₆×C₇₁	2
#12504	71

subgroup

index

C₈×C₁₆×C₇₁

Subgroups — #31912110

Conjugacy classes of subgroups

Complements

Maximal subgroups