Group #31909456 — subgroups

Subgroups — #31909456

Order 18176.

206 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₄×Dic₅₆₈	C₂	1	1	yes	maximal, characteristic
2	#22587238	C₂	1	2	yes	maximal
3	#22586414	C₂	1	1	yes	maximal, characteristic
4	#22586305	C₂	1	2	yes	maximal
5	C₇₁×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₆₈	C₄	1	1	yes	characteristic
7	#17986129	—	2	2	no	—
8	C₄×Dic₂₈₄	C₂²	1	2	yes	—
9	#17985825	C₄	1	1	yes	characteristic
10	#17985804	C₄	1	2	yes	—
11	#17985752	C₂²	1	2	yes	—
12	#17985731	—	2	2	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	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₄₂	—	2	2	no	—
18	Dic₅₆₈	—	2	2	no	—
19	#14556062	C₂×C₄	1	2	yes	—
20	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
21	Dic₇₁⋊C₈	—	4	2	no	—
22	#14556053	—	2	2	no	—
23	#14556038	—	4	2	no	—
24	C₂×C₁₆×C₇₁	—	2	1	no	—
25	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
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	—	2	2	no	—
31	Dic₂₈₄	—	2	2	no	—
32	Dic₂₈₄	—	2	2	no	—
33	C₂×C₇₁⋊C₈	—	4	2	no	—
34	C₄⋊Dic₇₁	—	2	2	no	—
35	Dic₇₁⋊C₄	—	4	2	no	—
36	C₄×Dic₇₁	—	4	2	no	—
37	C₂×C₈×C₇₁	—	2	1	no	—
38	C₂×C₈×C₇₁	D₈	1	2	yes	—
39	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
41	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₁	—	2	2	no	—
43	C₁₆×C₇₁	—	2	1	no	—
44	C₁₆×C₇₁	M₁₆	1	2	yes	—
45	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
48	C₂×Dic₇₁	—	4	2	no	—
49	C₂×Dic₇₁	—	4	2	no	—
50	Dic₁₄₂	—	4	2	no	—
51	Dic₁₄₂	—	4	2	no	—
52	C₈×C₇₁	—	2	1	no	—
53	C₈×C₇₁	—	2	2	no	—
54	C₈×C₇₁	—	4	1	no	—
55	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
56	C₈×C₇₁	#131	1	1	yes	characteristic
57	C₇₁⋊C₈	—	8	2	no	—
58	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
59	C₄×C₇₁	—	2	1	no	—
60	C₄×C₇₁	#436	1	1	yes	characteristic
61	C₄×C₇₁	#443	1	1	yes	characteristic
62	C₄×C₇₁	#510	1	2	yes	—
63	Dic₇₁	—	8	2	no	—
64	Dic₇₁	—	8	2	no	—
65	#12085	—	71	1	no	maximal, Sylow
66	C₂×C₇₁	#1589	1	1	yes	characteristic
67	C₂×C₇₁	#2176	1	1	yes	characteristic
68	C₂×C₇₁	#2179	1	1	yes	characteristic
69	C₄×Q₃₂	—	71	1	no	—
70	#1591	—	71	2	no	—
71	#1372	—	71	1	no	—
72	#1332	—	71	2	no	—
73	#1311	D₇₁	1	1	yes	characteristic
74	C₇₁	#12085	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₃₂	—	71	1	no	—
76	#446	—	142	2	no	—
77	C₄×Q₁₆	—	71	2	no	—
78	#366	—	71	1	no	—
79	#358	—	71	2	no	—
80	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
81	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
82	#335	—	71	2	no	—
83	Q₈⋊C₈	—	142	2	no	—
84	C₈²	C₂×D₇₁	1	1	yes	characteristic
85	C₂×Q₁₆	—	71	2	no	—
86	C₄×Q₈	—	142	2	no	—
87	Q₃₂	—	142	2	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
90	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
91	#107	—	71	2	no	—
92	C₄⋊C₈	—	284	2	no	—
93	C₄⋊C₈	—	142	2	no	—
94	#103	—	142	2	no	—
95	#97	—	284	2	no	—
96	C₄×C₈	D₂₈₄	1	1	yes	characteristic
97	C₄×C₈	D₂₈₄	1	1	yes	characteristic
98	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
99	C₂×Q₈	—	142	2	no	—
100	Q₁₆	—	142	2	no	—
101	Q₁₆	—	142	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	284	2	no	—
104	C₂×C₈	D₅₆₈	1	2	yes	—
105	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
106	C₂×C₈	#11765865	1	1	yes	characteristic
107	C₄⋊C₄	—	284	2	no	—
108	C₄⋊C₄	—	142	2	no	—
109	C₄²	—	284	2	no	—
110	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	#11765848	1	2	yes	—
114	Q₈	—	284	2	no	—
115	Q₈	—	284	2	no	—
116	C₂×C₄	—	284	2	no	—
117	C₂×C₄	—	284	2	no	—
118	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
119	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
120	C₂×C₄	#14556150	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	2	no	—
123	C₈	—	4	1	no	—
124	C₈	—	568	2	no	—
125	C₈	C₂×#11765848	1	1	yes	characteristic
126	C₈	#14556134	1	1	yes	characteristic
127	C₂²	C₄×D₅₆₈	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	568	2	no	—
130	C₄	—	568	2	no	—
131	C₄	#17986078	1	1	yes	characteristic
132	C₄	#17986117	1	1	yes	characteristic
133	C₄	#17986442	1	2	yes	—
134	C₂	#22587226	1	1	yes	characteristic
135	C₂	#22590212	1	1	yes	characteristic
136	C₂	#22590227	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₅₆₈

C₂

yes

maximal, characteristic

#22587238

C₂

yes

maximal

#22586414

C₂

yes

maximal, characteristic

#22586305

C₂

yes

maximal

C₇₁×#1311

C₂

yes

maximal, characteristic

C₂×Dic₅₆₈

C₄

yes

characteristic

#17986129

—

C₄×Dic₂₈₄

C₂²

yes

—

#17985825

C₄

yes

characteristic

#17985804

C₄

yes

—

#17985752

C₂²

yes

—

#17985731

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₈²×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

C₂×C₄

yes

—

C₄×Dic₁₄₂

—

Dic₅₆₈

—

#14556062

C₂×C₄

yes

—

C₄⋊(C₇₁⋊C₈)

—

Dic₇₁⋊C₈

—

#14556053

—

#14556038

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₁₆×C₇₁

M₁₆

yes

—

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₈×C₇₁

#131

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₄×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₄×Q₃₂

—

—

—

—

D₇₁

yes

characteristic

C₇₁

#12085

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

#446

—

142

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

—

Q₈⋊C₈

—

142

—

C₈²

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

142

—

Q₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₁

yes

characteristic

C₂×C₁₆

C₄×D₇₁

yes

characteristic

#107

—

C₄⋊C₈

—

284

—

C₄⋊C₈

—

142

—

#103

—

142

—

#97

—

284

—

C₄×C₈

D₂₈₄

yes

characteristic

C₄×C₈

D₂₈₄

yes

characteristic

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

142

—

100

Q₁₆

—

142

—

101

Q₁₆

—

142

—

102

C₂×C₈

—

103

C₂×C₈

—

284

—

104

C₂×C₈

D₅₆₈

yes

—

105

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

106

C₂×C₈

#11765865

yes

characteristic

107

C₄⋊C₄

—

284

—

108

C₄⋊C₄

—

142

—

109

C₄²

—

284

—

110

C₄²

C₂×D₂₈₄

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

#11765848

yes

—

114

Q₈

—

284

—

115

Q₈

—

284

—

116

C₂×C₄

—

284

—

117

C₂×C₄

—

284

—

118

C₂×C₄

C₄×D₂₈₄

yes

characteristic

119

C₂×C₄

C₂×D₅₆₈

yes

characteristic

120

C₂×C₄

#14556150

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

568

—

125

C₈

C₂×#11765848

yes

characteristic

126

C₈

#14556134

yes

characteristic

127

C₂²

C₄×D₅₆₈

yes

characteristic

128

C₄

—

129

C₄

—

568

—

130

C₄

—

568

—

131

C₄

#17986078

yes

characteristic

132

C₄

#17986117

yes

characteristic

133

C₄

#17986442

yes

—

134

C₂

#22587226

yes

characteristic

135

C₂

#22590212

yes

characteristic

136

C₂

#22590227

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1311	C₂
socle, prime radical	C₂²×C₇₁	C₄×D₈
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1311	D₇₁
hypocenter	C₇₁	#12085
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂×C₄

C₂×D₅₆₈

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1311

C₂

socle, prime radical

C₂²×C₇₁

C₄×D₈

cosocle

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

N	H ≅ G/N	type
74. C₇₁	#12085	semidirect

H ≅ G/N

type

74. C₇₁

#12085

semidirect

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

subgroup	index
C₄×Dic₅₆₈	2
#22587238	2
#22586414	2
#22586305	2
C₇₁×#1311	2
#12085	71

subgroup

index

C₄×Dic₅₆₈

C₇₁×#1311

Subgroups — #31909456

Conjugacy classes of subgroups

Complements

Maximal subgroups