Group #31911808 — subgroups

Subgroups — #31911808

Order 18176.

204 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₁₆×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
2	#22587731	C₂	1	2	yes	maximal
3	#22586412	C₂	1	1	yes	maximal, characteristic
4	#22586395	C₂	1	2	yes	maximal
5	C₇₁×#2252	C₂	1	1	yes	maximal, characteristic
6	#17986206	—	2	2	no	—
7	C₈×Dic₁₄₂	C₂²	1	1	yes	characteristic
8	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
9	Dic₇₁⋊C₁₆	—	2	1	no	—
10	C₁₆×Dic₇₁	—	2	1	no	—
11	#17985752	C₂²	1	2	yes	—
12	#17985745	—	2	2	no	—
13	#500×C₇₁	C₂²	1	2	yes	—
14	#366×C₇₁	—	2	1	no	—
15	#366×C₇₁	—	2	1	no	—
16	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×#11765855	—	4	2	no	—
18	C₄×Dic₁₄₂	D₄	1	1	yes	characteristic
19	C₂×C₇₁⋊C₁₆	—	2	1	no	—
20	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
21	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
22	Dic₇₁⋊C₈	—	2	1	no	—
23	#14556052	—	4	2	no	—
24	C₈×Dic₇₁	—	2	1	no	—
25	C₄⋊Q₈×C₇₁	—	2	2	no	—
26	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
28	#107×C₇₁	—	2	2	no	—
29	#107×C₇₁	—	2	2	no	—
30	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₄₂	D₈	1	1	yes	characteristic
32	#11765855	—	8	4	no	—
33	C₂×C₇₁⋊C₈	—	2	1	no	—
34	C₂×C₇₁⋊C₈	—	4	2	no	—
35	C₄⋊Dic₇₁	D₈	1	1	yes	characteristic
36	Dic₇₁⋊C₄	—	2	1	no	—
37	C₄×Dic₇₁	—	2	1	no	—
38	C₇₁⋊C₁₆	—	4	1	no	—
39	C₂×Q₈×C₇₁	—	4	2	no	—
40	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
42	C₄⋊C₄×C₇₁	—	4	2	no	—
43	C₄⋊C₄×C₇₁	—	4	2	no	—
44	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₁₆×C₇₁	—	2	1	no	—
46	C₁₆×C₇₁	Q₁₆	1	2	yes	complemented
47	Q₈×C₇₁	—	8	2	no	—
48	Q₈×C₇₁	—	8	2	no	—
49	C₂×C₄×C₇₁	—	8	2	no	—
50	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
53	C₂×Dic₇₁	—	2	1	no	—
54	C₂×Dic₇₁	—	2	1	no	—
55	Dic₁₄₂	—	2	1	no	—
56	Dic₁₄₂	D₁₆	1	2	yes	—
57	C₈×C₇₁	—	2	1	no	—
58	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₇₁	#137	1	1	yes	characteristic
60	C₇₁⋊C₈	—	4	1	no	—
61	C₇₁⋊C₈	—	8	2	no	—
62	C₂²×C₇₁	#466	1	1	yes	characteristic
63	C₄×C₇₁	—	2	1	no	—
64	C₄×C₇₁	—	16	2	no	—
65	C₄×C₇₁	#462	1	1	yes	characteristic
66	C₄×C₇₁	#473	1	1	yes	characteristic
67	C₄×C₇₁	C₂×D₁₆	1	1	yes	characteristic
68	C₄×C₇₁	#508	1	1	yes	characteristic
69	Dic₇₁	—	2	2	no	—
70	Dic₇₁	—	4	1	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	D₇₁	1	1	yes	characteristic
76	Q₈×C₁₆	—	71	1	no	—
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	C₂×D₇₁	1	2	yes	—
82	#462	—	142	2	no	—
83	C₈×Q₈	—	71	1	no	—
84	#366	—	2	1	no	—
85	#366	—	2	1	no	—
86	C₄⋊C₁₆	—	71	1	no	—
87	C₄⋊C₁₆	—	142	1	no	—
88	C₄×C₁₆	—	142	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	—	2	2	no	—
94	C₄×Q₈	—	71	1	no	—
95	C₂×C₁₆	—	142	1	no	—
96	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
98	#107	—	2	2	no	—
99	#107	—	2	2	no	—
100	C₄⋊C₈	—	71	1	no	—
101	C₄⋊C₈	—	142	1	no	—
102	C₄⋊C₈	—	142	2	no	—
103	#103	—	284	2	no	—
104	C₄×C₈	—	142	1	no	—
105	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	71	1	no	—
108	Q₁₆	—	568	4	no	—
109	C₂×C₈	—	284	2	no	—
110	C₂×C₈	—	142	1	no	—
111	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
112	C₂×C₈	#11765864	1	1	yes	characteristic
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	71	1	no	—
116	C₄⋊C₄	—	142	1	no	—
117	C₄²	—	142	1	no	—
118	C₄²	D₄×D₇₁	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	284	1	no	—
121	C₁₆	#11765855	1	2	yes	complemented
122	Q₈	—	8	2	no	—
123	Q₈	—	8	2	no	—
124	Q₈	—	71	2	no	—
125	Q₈	—	142	1	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	—	142	1	no	—
128	C₂×C₄	—	142	1	no	—
129	C₂×C₄	#14556099	1	1	yes	characteristic
130	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
131	C₂×C₄	#14556147	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	284	1	no	—
134	C₈	—	568	2	no	—
135	C₈	C₂×#11765855	1	1	yes	characteristic
136	C₈	#14556159	1	1	yes	characteristic
137	C₂²	#17986232	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	16	2	no	—
140	C₄	—	284	1	no	—
141	C₄	—	142	2	no	—
142	C₄	#17986206	1	1	yes	characteristic
143	C₄	#17986271	1	1	yes	characteristic
144	C₄	D₁₆×D₇₁	1	1	yes	characteristic
145	C₄	#17986428	1	1	yes	characteristic
146	C₂	#22587731	1	1	yes	characteristic
147	C₂	#22590423	1	1	yes	characteristic
148	C₂	#22590462	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₄₂

C₂

yes

maximal, characteristic

#22587731

C₂

yes

maximal

#22586412

C₂

yes

maximal, characteristic

#22586395

C₂

yes

maximal

C₇₁×#2252

C₂

yes

maximal, characteristic

#17986206

—

C₈×Dic₁₄₂

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

characteristic

Dic₇₁⋊C₁₆

—

C₁₆×Dic₇₁

—

#17985752

C₂²

yes

—

#17985745

—

#500×C₇₁

C₂²

yes

—

#366×C₇₁

—

#366×C₇₁

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#11765855

—

C₄×Dic₁₄₂

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

D₄

yes

characteristic

Dic₇₁⋊C₈

—

#14556052

—

C₈×Dic₇₁

—

C₄⋊Q₈×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

#107×C₇₁

—

#107×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

D₈

yes

characteristic

#11765855

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

D₈

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

Q₁₆

yes

complemented

Q₈×C₇₁

—

Q₈×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

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

D₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₂×Q₁₆

yes

characteristic

C₈×C₇₁

#137

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

#466

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×D₁₆

yes

characteristic

C₄×C₇₁

#508

yes

characteristic

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

yes

characteristic

Q₈×C₁₆

—

#1674

—

#1371

—

#1364

—

C₇₁

#12457

yes

characteristic, complemented, Sylow

#500

C₂×D₇₁

yes

—

#462

—

142

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

142

—

C₄×C₁₆

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

—

#332

—

142

—

C₂×Q₁₆

—

284

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

142

—

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

#107

—

#107

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

142

—

102

C₄⋊C₈

—

142

—

103

#103

—

284

—

104

C₄×C₈

—

142

—

105

C₄×C₈

C₂²×D₇₁

yes

characteristic

106

C₂×Q₈

—

107

C₂×Q₈

—

108

Q₁₆

—

568

—

109

C₂×C₈

—

284

—

110

C₂×C₈

—

142

—

111

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

112

C₂×C₈

#11765864

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

142

—

117

C₄²

—

142

—

118

C₄²

D₄×D₇₁

yes

characteristic

119

C₁₆

—

120

C₁₆

—

284

—

121

C₁₆

#11765855

yes

complemented

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

142

—

126

C₂×C₄

—

127

C₂×C₄

—

142

—

128

C₂×C₄

—

142

—

129

C₂×C₄

#14556099

yes

characteristic

130

C₂×C₄

D₈×D₇₁

yes

characteristic

131

C₂×C₄

#14556147

yes

characteristic

132

C₈

—

133

C₈

—

284

—

134

C₈

—

568

—

135

C₈

C₂×#11765855

yes

characteristic

136

C₈

#14556159

yes

characteristic

137

C₂²

#17986232

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

284

—

141

C₄

—

142

—

142

C₄

#17986206

yes

characteristic

143

C₄

#17986271

yes

characteristic

144

C₄

D₁₆×D₇₁

yes

characteristic

145

C₄

#17986428

yes

characteristic

146

C₂

#22587731

yes

characteristic

147

C₂

#22590423

yes

characteristic

148

C₂

#22590462

yes

characteristic

	subgroup	G/H
center	C₂²	#17986232
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#2252	C₂
socle, prime radical	C₂²×C₇₁	#466
hypercenter	#2252	D₇₁
hypocenter	C₇₁	#12457
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#17986232

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#2252

C₂

C₂²×C₇₁

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

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
C₁₆×Dic₁₄₂	2
#22587731	2
#22586412	2
#22586395	2
C₇₁×#2252	2
#12457	71

subgroup

index

C₁₆×Dic₁₄₂

C₇₁×#2252

Subgroups — #31911808

Conjugacy classes of subgroups

Complements

Maximal subgroups