Group #31911811 — subgroups

Subgroups — #31911811

Order 18176.

204 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 75 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	#22587734	C₂	1	2	yes	maximal
3	#22586412	C₂	1	1	yes	maximal, characteristic
4	#22586394	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#17986404	C₂²	1	2	yes	—
7	#17986209	—	2	2	no	—
8	#17985825	—	2	1	no	—
9	#17985825	—	2	1	no	—
10	#17985752	C₂²	1	2	yes	—
11	#17985744	—	2	2	no	—
12	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₇₁	—	2	1	no	—
14	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₇₁	—	2	1	no	—
16	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×#11765855	—	4	2	no	—
18	#14556128	—	2	2	no	—
19	#14556062	—	2	2	no	—
20	#14556062	—	2	2	no	—
21	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
22	#14556051	—	4	2	no	—
23	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₇₁	—	2	1	no	—
25	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₇₁	—	2	1	no	—
28	C₄⋊C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	—	2	1	no	—
30	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₄₂	—	4	2	no	—
32	#11765855	—	8	4	no	—
33	C₂×C₇₁⋊C₈	—	4	2	no	—
34	C₄⋊Dic₇₁	—	4	2	no	—
35	C₄⋊Dic₇₁	—	4	2	no	—
36	C₂×Q₈×C₇₁	D₈	1	1	yes	characteristic
37	C₂×C₈×C₇₁	—	2	1	no	—
38	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₇₁	—	2	1	no	—
41	C₄⋊C₄×C₇₁	D₈	1	1	yes	characteristic
42	C₄²×C₇₁	—	2	1	no	—
43	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₁	—	2	1	no	—
45	C₁₆×C₇₁	—	4	1	no	—
46	C₁₆×C₇₁	Q₁₆	1	2	yes	complemented
47	Q₈×C₇₁	—	2	1	no	—
48	Q₈×C₇₁	D₁₆	1	2	yes	—
49	C₂×C₄×C₇₁	—	2	1	no	—
50	C₂×C₄×C₇₁	—	2	1	no	—
51	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
54	C₂×Dic₇₁	—	8	2	no	—
55	Dic₁₄₂	—	8	2	no	—
56	Dic₁₄₂	—	8	2	no	—
57	C₈×C₇₁	—	2	1	no	—
58	C₈×C₇₁	—	4	1	no	—
59	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
60	C₈×C₇₁	#137	1	1	yes	characteristic
61	C₇₁⋊C₈	—	8	2	no	—
62	C₂²×C₇₁	#466	1	1	yes	characteristic
63	C₄×C₇₁	—	2	2	no	—
64	C₄×C₇₁	—	2	1	no	—
65	C₄×C₇₁	—	4	1	no	—
66	C₄×C₇₁	#462	1	1	yes	characteristic
67	C₄×C₇₁	#473	1	1	yes	characteristic
68	C₄×C₇₁	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₇₁	#508	1	1	yes	characteristic
70	Dic₇₁	—	16	2	no	—
71	#12457	—	71	1	no	maximal, Sylow
72	C₂×C₇₁	#1674	1	1	yes	characteristic
73	C₂×C₇₁	#2215	1	1	yes	characteristic
74	C₂×C₇₁	#2222	1	1	yes	characteristic
75	#2252	—	71	1	no	—
76	Q₈×C₁₆	D₇₁	1	1	yes	characteristic
77	#1674	—	71	2	no	—
78	#1371	—	71	1	no	—
79	#1364	—	71	2	no	—
80	C₇₁	#12457	1	1	yes	characteristic, complemented, Sylow
81	#500	—	71	2	no	—
82	#462	—	142	2	no	—
83	C₈×Q₈	C₂×D₇₁	1	1	yes	characteristic
84	#366	—	142	1	no	—
85	#366	—	142	1	no	—
86	C₄⋊C₁₆	—	2	1	no	—
87	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
88	C₄×C₁₆	—	2	1	no	—
89	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
90	#335	—	71	2	no	—
91	#332	—	142	2	no	—
92	C₂×Q₁₆	—	284	2	no	—
93	C₄⋊Q₈	—	142	2	no	—
94	C₄×Q₈	D₂₈₄	1	1	yes	characteristic
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
98	#107	—	142	2	no	—
99	#107	—	142	2	no	—
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	142	2	no	—
102	C₄⋊C₈	D₂₈₄	1	1	yes	characteristic
103	#103	—	284	2	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
106	C₂×Q₈	—	284	2	no	—
107	C₂×Q₈	D₅₆₈	1	1	yes	characteristic
108	Q₁₆	—	568	4	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	284	2	no	—
111	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
112	C₂×C₈	#11765865	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	284	2	no	—
115	C₄⋊C₄	—	284	2	no	—
116	C₄⋊C₄	D₅₆₈	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	4	1	no	—
121	C₁₆	#11765855	1	2	yes	complemented
122	Q₈	—	2	1	no	—
123	Q₈	—	568	2	no	—
124	Q₈	—	568	2	no	—
125	Q₈	D₁₁₃₆	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	568	2	no	—
129	C₂×C₄	#14556102	1	1	yes	characteristic
130	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
131	C₂×C₄	#14556150	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	568	2	no	—
135	C₈	C₂×#11765855	1	1	yes	characteristic
136	C₈	#14556158	1	1	yes	characteristic
137	C₂²	#17986235	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	2	no	—
140	C₄	—	4	1	no	—
141	C₄	—	1136	2	no	—
142	C₄	#17986209	1	1	yes	characteristic
143	C₄	#17986274	1	1	yes	characteristic
144	C₄	C₂×D₁₁₃₆	1	1	yes	characteristic
145	C₄	#17986431	1	1	yes	characteristic
146	C₂	#22587734	1	1	yes	characteristic
147	C₂	#22590426	1	1	yes	characteristic
148	C₂	#22590465	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590615

C₂

yes

maximal, characteristic

#22587734

C₂

yes

maximal

#22586412

C₂

yes

maximal, characteristic

#22586394

C₂

yes

maximal

Q₈×C₁₆×C₇₁

C₂

yes

maximal, characteristic

#17986404

C₂²

yes

—

—

—

—

C₂²

yes

—

#17985744

—

C₈×Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₁

—

C₄⋊C₁₆×C₇₁

C₂²

yes

characteristic

C₄×C₁₆×C₇₁

—

C₄×C₁₆×C₇₁

C₂²

yes

—

—

—

—

C₄⋊(C₇₁⋊C₈)

—

#14556051

—

C₄×Q₈×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

#11765855

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₈

yes

characteristic

C₄²×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₁₆×C₇₁

Q₁₆

yes

complemented

Q₈×C₇₁

—

Q₈×C₇₁

D₁₆

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#121

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×Q₁₆

yes

characteristic

C₈×C₇₁

#137

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

#466

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×D₁₆

yes

characteristic

C₄×C₇₁

#508

yes

characteristic

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

Q₈×C₁₆

D₇₁

yes

—

—

—

C₇₁

#12457

yes

characteristic, complemented, Sylow

#500

—

#462

—

142

—

C₈×Q₈

C₂×D₇₁

yes

characteristic

#366

—

142

—

#366

—

142

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₇₁

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

—

#332

—

142

—

C₂×Q₁₆

—

284

—

C₄⋊Q₈

—

142

—

C₄×Q₈

D₂₈₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

#107

—

142

—

#107

—

142

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

142

—

102

C₄⋊C₈

D₂₈₄

yes

characteristic

103

#103

—

284

—

104

C₄×C₈

—

105

C₄×C₈

C₂²×D₇₁

yes

characteristic

106

C₂×Q₈

—

284

—

107

C₂×Q₈

D₅₆₈

yes

characteristic

108

Q₁₆

—

568

—

109

C₂×C₈

—

110

C₂×C₈

—

284

—

111

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

112

C₂×C₈

#11765865

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

284

—

115

C₄⋊C₄

—

284

—

116

C₄⋊C₄

D₅₆₈

yes

characteristic

117

C₄²

—

118

C₄²

C₂×D₂₈₄

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

#11765855

yes

complemented

122

Q₈

—

123

Q₈

—

568

—

124

Q₈

—

568

—

125

Q₈

D₁₁₃₆

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

568

—

129

C₂×C₄

#14556102

yes

characteristic

130

C₂×C₄

C₂×D₅₆₈

yes

characteristic

131

C₂×C₄

#14556150

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

568

—

135

C₈

C₂×#11765855

yes

characteristic

136

C₈

#14556158

yes

characteristic

137

C₂²

#17986235

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

1136

—

142

C₄

#17986209

yes

characteristic

143

C₄

#17986274

yes

characteristic

144

C₄

C₂×D₁₁₃₆

yes

characteristic

145

C₄

#17986431

yes

characteristic

146

C₂

#22587734

yes

characteristic

147

C₂

#22590426

yes

characteristic

148

C₂

#22590465

yes

characteristic

	subgroup	G/H
center	C₂²	#17986235
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	Q₈×C₁₆×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	#466
hypercenter	Q₈×C₁₆	D₇₁
hypocenter	C₇₁	#12457
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#17986235

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

Q₈×C₁₆×C₇₁

C₂

socle, prime radical

C₂²×C₇₁

#466

hypercenter

Q₈×C₁₆

D₇₁

hypocenter

C₇₁

#12457

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
46. C₁₆×C₇₁	Q₁₆	semidirect
80. C₇₁	#12457	semidirect
121. C₁₆	#11765855	semidirect

H ≅ G/N

type

46. C₁₆×C₇₁

Q₁₆

semidirect

80. C₇₁

#12457

semidirect

121. C₁₆

#11765855

semidirect

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

subgroup	index
#22590615	2
#22587734	2
#22586412	2
#22586394	2
Q₈×C₁₆×C₇₁	2
#12457	71

subgroup

index

Q₈×C₁₆×C₇₁

Subgroups — #31911811

Conjugacy classes of subgroups

Complements

Maximal subgroups