Group #31911813 — subgroups

Subgroups — #31911813

Order 18176.

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

normal subgroups only

This group has 192 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	#22587741	C₂	1	2	yes	maximal
3	#22586412	C₂	1	1	yes	maximal, characteristic
4	#22586398	C₂	1	2	yes	maximal
5	C₇₁×#2252	C₂	1	1	yes	maximal, characteristic
6	#17986291	—	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	#17985748	—	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	C₄×Dic₁₄₂	D₄	1	1	yes	characteristic
18	C₂×C₇₁⋊C₁₆	—	2	1	no	—
19	#14556061	—	4	2	no	—
20	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
21	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
22	Dic₇₁⋊C₈	—	2	1	no	—
23	#14556051	—	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	C₂×C₇₁⋊C₈	—	2	1	no	—
33	C₂×C₇₁⋊C₈	—	4	2	no	—
34	C₄⋊Dic₇₁	D₈	1	1	yes	characteristic
35	Dic₇₁⋊C₄	—	2	1	no	—
36	C₄×Dic₇₁	—	2	1	no	—
37	C₇₁⋊C₁₆	—	4	1	no	—
38	C₂×Q₈×C₇₁	—	4	2	no	—
39	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
41	C₄⋊C₄×C₇₁	—	4	2	no	—
42	C₄⋊C₄×C₇₁	—	4	2	no	—
43	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₁	—	2	1	no	—
45	C₁₆×C₇₁	Q₁₆	1	2	yes	—
46	Q₈×C₇₁	—	16	2	no	—
47	C₂×C₄×C₇₁	—	8	2	no	—
48	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
51	C₂×Dic₇₁	—	2	1	no	—
52	C₂×Dic₇₁	—	2	1	no	—
53	Dic₁₄₂	—	2	1	no	—
54	Dic₁₄₂	Q₃₂	1	2	yes	—
55	C₈×C₇₁	—	2	1	no	—
56	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
57	C₈×C₇₁	#137	1	1	yes	characteristic
58	C₇₁⋊C₈	—	4	1	no	—
59	C₇₁⋊C₈	—	8	2	no	—
60	C₂²×C₇₁	#466	1	1	yes	characteristic
61	C₄×C₇₁	—	2	1	no	—
62	C₄×C₇₁	—	16	2	no	—
63	C₄×C₇₁	#462	1	1	yes	characteristic
64	C₄×C₇₁	#473	1	1	yes	characteristic
65	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
66	C₄×C₇₁	#508	1	1	yes	characteristic
67	Dic₇₁	—	2	2	no	—
68	Dic₇₁	—	4	1	no	—
69	#12458	—	71	1	no	maximal, Sylow
70	C₂×C₇₁	#1674	1	1	yes	characteristic
71	C₂×C₇₁	#2216	1	1	yes	characteristic
72	C₂×C₇₁	#2221	1	1	yes	characteristic
73	#2252	D₇₁	1	1	yes	characteristic
74	Q₈×C₁₆	—	71	1	no	—
75	#1676	—	71	2	no	—
76	#1371	—	71	1	no	—
77	#1365	—	71	2	no	—
78	C₇₁	#12458	1	1	yes	characteristic, complemented, Sylow
79	#500	C₂×D₇₁	1	2	yes	—
80	#477	—	142	2	no	—
81	C₈×Q₈	—	71	1	no	—
82	#366	—	2	1	no	—
83	#366	—	2	1	no	—
84	C₄⋊C₁₆	—	71	1	no	—
85	C₄⋊C₁₆	—	142	1	no	—
86	C₄×C₁₆	—	142	1	no	—
87	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
88	#335	—	71	2	no	—
89	#333	—	142	2	no	—
90	C₄⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	71	1	no	—
92	C₂×C₁₆	—	142	1	no	—
93	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
94	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
95	#107	—	2	2	no	—
96	#107	—	2	2	no	—
97	#107	—	284	2	no	—
98	C₄⋊C₈	—	71	1	no	—
99	C₄⋊C₈	—	142	1	no	—
100	C₄⋊C₈	—	142	2	no	—
101	#103	—	284	2	no	—
102	C₄×C₈	—	142	1	no	—
103	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	—	71	1	no	—
106	C₂×C₈	—	284	2	no	—
107	C₂×C₈	—	142	1	no	—
108	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
109	C₂×C₈	#11765864	1	1	yes	characteristic
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	71	1	no	—
113	C₄⋊C₄	—	142	1	no	—
114	C₄²	—	142	1	no	—
115	C₄²	D₄×D₇₁	1	1	yes	characteristic
116	C₁₆	—	2	1	no	—
117	C₁₆	—	284	1	no	—
118	C₁₆	#11765855	1	2	yes	—
119	Q₈	—	16	2	no	—
120	Q₈	—	71	2	no	—
121	Q₈	—	142	1	no	—
122	C₂×C₄	—	8	2	no	—
123	C₂×C₄	—	142	1	no	—
124	C₂×C₄	—	142	1	no	—
125	C₂×C₄	#14556099	1	1	yes	characteristic
126	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
127	C₂×C₄	#14556147	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	284	1	no	—
130	C₈	—	568	2	no	—
131	C₈	C₂×#11765855	1	1	yes	characteristic
132	C₈	#14556159	1	1	yes	characteristic
133	C₂²	#17986232	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	16	2	no	—
136	C₄	—	284	1	no	—
137	C₄	—	142	2	no	—
138	C₄	#17986206	1	1	yes	characteristic
139	C₄	#17986271	1	1	yes	characteristic
140	C₄	Q₃₂×D₇₁	1	1	yes	characteristic
141	C₄	#17986432	1	1	yes	characteristic
142	C₂	#22587731	1	1	yes	characteristic
143	C₂	#22590428	1	1	yes	characteristic
144	C₂	#22590457	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₄₂

C₂

yes

maximal, characteristic

#22587741

C₂

yes

maximal

#22586412

C₂

yes

maximal, characteristic

#22586398

C₂

yes

maximal

C₇₁×#2252

C₂

yes

maximal, characteristic

#17986291

—

C₈×Dic₁₄₂

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

characteristic

Dic₇₁⋊C₁₆

—

C₁₆×Dic₇₁

—

#17985752

C₂²

yes

—

#17985748

—

#500×C₇₁

C₂²

yes

—

#366×C₇₁

—

#366×C₇₁

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₄×Dic₁₄₂

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

—

#14556061

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

D₄

yes

characteristic

Dic₇₁⋊C₈

—

#14556051

—

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

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

—

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

Q₃₂

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×Q₃₂

yes

characteristic

C₄×C₇₁

#508

yes

characteristic

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

yes

characteristic

Q₈×C₁₆

—

#1676

—

#1371

—

#1365

—

C₇₁

#12458

yes

characteristic, complemented, Sylow

#500

C₂×D₇₁

yes

—

#477

—

142

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

142

—

C₄×C₁₆

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

—

#333

—

142

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

142

—

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₁

yes

—

—

—

284

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

100

C₄⋊C₈

—

142

—

101

#103

—

284

—

102

C₄×C₈

—

142

—

103

C₄×C₈

C₂²×D₇₁

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×C₈

—

284

—

107

C₂×C₈

—

142

—

108

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

109

C₂×C₈

#11765864

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

142

—

114

C₄²

—

142

—

115

C₄²

D₄×D₇₁

yes

characteristic

116

C₁₆

—

117

C₁₆

—

284

—

118

C₁₆

#11765855

yes

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

142

—

122

C₂×C₄

—

123

C₂×C₄

—

142

—

124

C₂×C₄

—

142

—

125

C₂×C₄

#14556099

yes

characteristic

126

C₂×C₄

D₈×D₇₁

yes

characteristic

127

C₂×C₄

#14556147

yes

characteristic

128

C₈

—

129

C₈

—

284

—

130

C₈

—

568

—

131

C₈

C₂×#11765855

yes

characteristic

132

C₈

#14556159

yes

characteristic

133

C₂²

#17986232

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

284

—

137

C₄

—

142

—

138

C₄

#17986206

yes

characteristic

139

C₄

#17986271

yes

characteristic

140

C₄

Q₃₂×D₇₁

yes

characteristic

141

C₄

#17986432

yes

characteristic

142

C₂

#22587731

yes

characteristic

143

C₂

#22590428

yes

characteristic

144

C₂

#22590457

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	#12458
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
78. C₇₁	#12458	semidirect

H ≅ G/N

type

78. C₇₁

#12458

semidirect

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

subgroup	index
C₁₆×Dic₁₄₂	2
#22587741	2
#22586412	2
#22586398	2
C₇₁×#2252	2
#12458	71

subgroup

index

C₁₆×Dic₁₄₂

C₇₁×#2252

Subgroups — #31911813

Conjugacy classes of subgroups

Complements

Maximal subgroups