Group #31911817 — subgroups

Subgroups — #31911817

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	#22590615	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
3	#22587745	C₂	1	2	yes	maximal
4	#22586399	C₂	1	2	yes	maximal
5	C₇₁×#1371	C₂	1	1	yes	maximal, characteristic
6	#17986404	C₂²	1	2	yes	—
7	#17986295	—	2	2	no	—
8	C₈×Dic₁₄₂	C₂²	1	1	yes	characteristic
9	#17985825	—	2	1	no	—
10	#17985825	—	2	1	no	—
11	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
12	Dic₇₁⋊C₁₆	—	2	1	no	—
13	C₁₆×Dic₇₁	—	2	1	no	—
14	#17985747	—	2	2	no	—
15	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
16	#335×C₇₁	C₂²	1	2	yes	—
17	#14556128	—	2	2	no	—
18	C₄×Dic₁₄₂	D₄	1	1	yes	characteristic
19	C₂×C₇₁⋊C₁₆	—	2	1	no	—
20	#14556062	—	2	2	no	—
21	#14556062	—	2	2	no	—
22	#14556062	—	4	2	no	—
23	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
24	Dic₇₁⋊C₈	—	2	1	no	—
25	#14556053	—	4	2	no	—
26	C₈×Dic₇₁	—	2	1	no	—
27	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₇₁	—	2	2	no	—
30	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₄₂	—	4	2	no	—
32	C₂×Dic₁₄₂	D₈	1	1	yes	characteristic
33	C₂×C₇₁⋊C₈	—	2	1	no	—
34	C₄⋊Dic₇₁	—	4	2	no	—
35	C₄⋊Dic₇₁	—	4	2	no	—
36	C₄⋊Dic₇₁	D₈	1	1	yes	characteristic
37	Dic₇₁⋊C₄	—	2	1	no	—
38	C₄×Dic₇₁	—	2	1	no	—
39	C₇₁⋊C₁₆	—	4	1	no	—
40	C₂×C₈×C₇₁	—	4	2	no	—
41	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
43	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₁	—	2	1	no	—
45	C₁₆×C₇₁	Q₁₆	1	2	yes	—
46	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
49	C₂×Dic₇₁	—	2	1	no	—
50	C₂×Dic₇₁	—	2	1	no	—
51	C₂×Dic₇₁	—	8	2	no	—
52	Dic₁₄₂	—	2	1	no	—
53	Dic₁₄₂	—	16	2	no	—
54	Dic₁₄₂	Q₃₂	1	2	yes	—
55	C₈×C₇₁	—	2	1	no	—
56	C₈×C₇₁	—	8	2	no	—
57	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₇₁	#137	1	1	yes	characteristic
59	C₇₁⋊C₈	—	4	1	no	—
60	C₂²×C₇₁	#466	1	1	yes	characteristic
61	C₄×C₇₁	—	2	1	no	—
62	C₄×C₇₁	#462	1	1	yes	characteristic
63	C₄×C₇₁	#473	1	1	yes	characteristic
64	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
65	C₄×C₇₁	#508	1	1	yes	characteristic
66	Dic₇₁	—	2	2	no	—
67	Dic₇₁	—	4	1	no	—
68	Dic₇₁	—	16	2	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	—	71	1	no	—
74	Q₈×C₁₆	—	71	1	no	—
75	#1676	—	71	2	no	—
76	#1371	D₇₁	1	1	yes	characteristic
77	#1365	—	71	2	no	—
78	C₇₁	#12458	1	1	yes	characteristic, complemented, Sylow
79	#500	—	71	2	no	—
80	#477	—	142	2	no	—
81	C₈×Q₈	—	71	1	no	—
82	#366	—	142	1	no	—
83	#366	—	142	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	C₂×D₇₁	1	2	yes	—
89	#333	—	142	2	no	—
90	C₄⋊Q₈	—	142	2	no	—
91	C₄×Q₈	—	71	1	no	—
92	C₂×C₁₆	—	142	1	no	—
93	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
94	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
95	#107	—	284	2	no	—
96	#107	—	142	2	no	—
97	#107	—	142	2	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	71	1	no	—
100	C₄⋊C₈	—	142	1	no	—
101	#103	—	284	2	no	—
102	C₄×C₈	—	142	1	no	—
103	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
104	C₂×Q₈	—	284	2	no	—
105	C₂×Q₈	—	71	1	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	142	1	no	—
108	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
109	C₂×C₈	#11765866	1	1	yes	characteristic
110	C₄⋊C₄	—	284	2	no	—
111	C₄⋊C₄	—	284	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	Dic₂₈₄	1	2	yes	—
119	Q₈	—	71	2	no	—
120	Q₈	—	1136	2	no	—
121	Q₈	—	142	1	no	—
122	C₂×C₄	—	568	2	no	—
123	C₂×C₄	—	142	1	no	—
124	C₂×C₄	—	142	1	no	—
125	C₂×C₄	#14556103	1	1	yes	characteristic
126	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
127	C₂×C₄	#14556151	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	8	2	no	—
130	C₈	—	284	1	no	—
131	C₈	C₂×Dic₂₈₄	1	1	yes	characteristic
132	C₈	#14556161	1	1	yes	characteristic
133	C₂²	#17986236	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	284	1	no	—
136	C₄	—	1136	2	no	—
137	C₄	—	142	2	no	—
138	C₄	#17986210	1	1	yes	characteristic
139	C₄	#17986275	1	1	yes	characteristic
140	C₄	Q₃₂×D₇₁	1	1	yes	characteristic
141	C₄	#17986428	1	1	yes	characteristic
142	C₂	#22587735	1	1	yes	characteristic
143	C₂	#22590432	1	1	yes	characteristic
144	C₂	#22590461	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590615

C₂

yes

maximal, characteristic

C₁₆×Dic₁₄₂

C₂

yes

maximal, characteristic

#22587745

C₂

yes

maximal

#22586399

C₂

yes

maximal

C₇₁×#1371

C₂

yes

maximal, characteristic

#17986404

C₂²

yes

—

#17986295

—

C₈×Dic₁₄₂

C₂²

yes

characteristic

#17985825

—

#17985825

—

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

characteristic

Dic₇₁⋊C₁₆

—

C₁₆×Dic₇₁

—

#17985747

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

#335×C₇₁

C₂²

yes

—

#14556128

—

C₄×Dic₁₄₂

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

—

#14556062

—

#14556062

—

#14556062

—

C₄⋊(C₇₁⋊C₈)

D₄

yes

characteristic

Dic₇₁⋊C₈

—

#14556053

—

C₈×Dic₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₈

yes

characteristic

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₈

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

Q₁₆

yes

—

C₂×C₄×C₇₁

#121

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

Q₃₂

yes

—

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

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×Q₃₂

yes

characteristic

C₄×C₇₁

#508

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

Q₈×C₁₆

—

#1676

—

#1371

D₇₁

yes

characteristic

#1365

—

C₇₁

#12458

yes

characteristic, complemented, Sylow

#500

—

#477

—

142

—

C₈×Q₈

—

#366

—

142

—

#366

—

142

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

142

—

C₄×C₁₆

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

C₂×D₇₁

yes

—

#333

—

142

—

C₄⋊Q₈

—

142

—

C₄×Q₈

—

C₂×C₁₆

—

142

—

C₂×C₁₆

D₂₈₄

yes

characteristic

C₂×C₁₆

D₂₈₄

yes

characteristic

#107

—

284

—

#107

—

142

—

#107

—

142

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

142

—

101

#103

—

284

—

102

C₄×C₈

—

142

—

103

C₄×C₈

C₂²×D₇₁

yes

characteristic

104

C₂×Q₈

—

284

—

105

C₂×Q₈

—

106

C₂×C₈

—

107

C₂×C₈

—

142

—

108

C₂×C₈

C₂×D₂₈₄

yes

characteristic

109

C₂×C₈

#11765866

yes

characteristic

110

C₄⋊C₄

—

284

—

111

C₄⋊C₄

—

284

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

142

—

114

C₄²

—

142

—

115

C₄²

D₄×D₇₁

yes

characteristic

116

C₁₆

—

117

C₁₆

—

284

—

118

C₁₆

Dic₂₈₄

yes

—

119

Q₈

—

120

Q₈

—

1136

—

121

Q₈

—

142

—

122

C₂×C₄

—

568

—

123

C₂×C₄

—

142

—

124

C₂×C₄

—

142

—

125

C₂×C₄

#14556103

yes

characteristic

126

C₂×C₄

D₈×D₇₁

yes

characteristic

127

C₂×C₄

#14556151

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

284

—

131

C₈

C₂×Dic₂₈₄

yes

characteristic

132

C₈

#14556161

yes

characteristic

133

C₂²

#17986236

yes

characteristic

134

C₄

—

135

C₄

—

284

—

136

C₄

—

1136

—

137

C₄

—

142

—

138

C₄

#17986210

yes

characteristic

139

C₄

#17986275

yes

characteristic

140

C₄

Q₃₂×D₇₁

yes

characteristic

141

C₄

#17986428

yes

characteristic

142

C₂

#22587735

yes

characteristic

143

C₂

#22590432

yes

characteristic

144

C₂

#22590461

yes

characteristic

	subgroup	G/H
center	C₂²	#17986236
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1371	C₂
socle, prime radical	C₂²×C₇₁	#466
hypercenter	#1371	D₇₁
hypocenter	C₇₁	#12458
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂²

#17986236

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1371

C₂

C₂²×C₇₁

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

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
#22590615	2
C₁₆×Dic₁₄₂	2
#22587745	2
#22586399	2
C₇₁×#1371	2
#12458	71

subgroup

index

C₁₆×Dic₁₄₂

C₇₁×#1371

Subgroups — #31911817

Conjugacy classes of subgroups

Complements

Maximal subgroups