Group #31891228 — subgroups

Subgroups — #31891228

Order 18176.

422 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 203 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	#22588500	C₂	1	1	yes	maximal, characteristic
2	#22588467	C₂	1	4	yes	maximal
3	C₂×C₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556062	—	2	2	no	—
6	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
7	#17985869	C₂²	1	2	yes	—
8	#17985761	—	2	8	no	—
9	C₈×C₇₁⋊C₈	C₄	1	4	yes	complemented
10	C₂×#107×C₇₁	—	2	2	no	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	#384×C₇₁	C₂²	1	2	yes	—
13	C₂²×C₇₁⋊C₈	D₄	1	4	yes	—
14	C₂×C₄⋊Dic₇₁	—	2	4	no	—
15	C₂×C₄⋊Dic₇₁	—	2	2	no	—
16	#14556062	—	4	4	no	—
17	C₄×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
18	C₄×C₇₁⋊C₈	Q₈	1	4	yes	—
19	C₂²×C₈×C₇₁	D₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
21	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
22	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
23	#107×C₇₁	—	4	4	no	—
24	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₇₁	Q₈	1	2	yes	—
26	C₂²×Dic₇₁	—	4	2	no	—
27	C₂×C₇₁⋊C₈	—	2	4	no	—
28	C₂×C₇₁⋊C₈	—	2	4	no	—
29	C₂×C₇₁⋊C₈	C₄⋊C₄	1	8	yes	—
30	C₄⋊Dic₇₁	—	4	4	no	—
31	C₄⋊Dic₇₁	—	4	8	no	—
32	C₂²×C₄×C₇₁	—	4	2	no	—
33	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
34	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
36	C₂×C₈×C₇₁	D₈	1	4	yes	—
37	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
38	C₄⋊C₄×C₇₁	—	4	4	no	—
39	C₄⋊C₄×C₇₁	—	4	8	no	—
40	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
42	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
43	C₂³×C₇₁	#127	1	1	yes	characteristic
44	C₂×C₄×C₇₁	—	8	2	no	—
45	C₂×C₄×C₇₁	—	8	4	no	—
46	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
50	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
51	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
52	C₂×C₄×C₇₁	#135	1	4	yes	—
53	C₂×C₄×C₇₁	#135	1	4	yes	—
54	C₂×Dic₇₁	—	8	2	no	—
55	C₂×Dic₇₁	—	8	4	no	—
56	C₈×C₇₁	#107	1	8	yes	—
57	C₇₁⋊C₈	—	2	8	no	—
58	C₂²×C₇₁	#384	1	1	yes	characteristic
59	C₂²×C₇₁	#493	1	1	yes	characteristic
60	C₂²×C₇₁	#494	1	1	yes	characteristic
61	C₂²×C₇₁	#495	1	2	yes	—
62	C₂²×C₇₁	#495	1	2	yes	—
63	C₄×C₇₁	—	16	2	no	—
64	C₄×C₇₁	C₂×#107	1	2	yes	—
65	C₄×C₇₁	#427	1	4	yes	—
66	C₄×C₇₁	#499	1	4	yes	—
67	C₄×C₇₁	#500	1	2	yes	—
68	Dic₇₁	—	16	2	no	—
69	#8816	—	71	1	no	maximal, Sylow
70	C₂×C₇₁	#1709	1	1	yes	characteristic
71	C₂×C₇₁	#1709	1	1	yes	characteristic
72	C₂×C₇₁	#1710	1	2	yes	—
73	C₂×C₇₁	#1845	1	1	yes	characteristic
74	C₂×C₇₁	#1846	1	2	yes	—
75	#1845	—	71	1	no	—
76	#1845	D₇₁	1	1	yes	characteristic
77	#1834	—	71	4	no	—
78	C₂×C₈²	—	71	1	no	—
79	C₇₁	#8816	1	1	yes	characteristic, complemented, Sylow
80	C₂×#107	—	2	2	no	—
81	C₂×#107	—	142	2	no	—
82	C₂×C₄×C₈	—	71	2	no	—
83	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
84	#384	—	71	2	no	—
85	#384	C₂×D₇₁	1	2	yes	—
86	#340	—	142	8	no	—
87	C₈²	—	71	4	no	—
88	C₂²×C₈	—	71	4	no	—
89	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
90	C₂×C₄⋊C₄	—	2	4	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	—	142	4	no	—
93	C₂×C₄⋊C₄	—	142	2	no	—
94	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
95	#107	—	4	4	no	—
96	#107	—	284	4	no	—
97	C₄×C₈	—	71	4	no	—
98	C₄×C₈	—	71	4	no	—
99	C₄×C₈	Dic₁₄₂	1	2	yes	—
100	C₄×C₈	C₄×D₇₁	1	2	yes	—
101	C₂²×C₄	—	4	2	no	—
102	C₂²×C₄	—	284	2	no	—
103	C₂²×C₄	D₄×D₇₁	1	2	yes	—
104	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
105	C₂×C₈	—	71	8	no	—
106	C₂×C₈	—	142	4	no	—
107	C₂×C₈	—	142	4	no	—
108	C₂×C₈	Dic₇₁⋊C₄	1	4	yes	—
109	C₂×C₈	#11765850	1	4	yes	—
110	C₂×C₈	#11765855	1	4	yes	—
111	C₄⋊C₄	—	4	4	no	—
112	C₄⋊C₄	—	4	8	no	—
113	C₄⋊C₄	—	284	4	no	—
114	C₄⋊C₄	—	284	8	no	—
115	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
116	C₄²	Q₈×D₇₁	1	2	yes	—
117	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
118	C₂³	#14556124	1	1	yes	characteristic
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	—	8	4	no	—
121	C₂×C₄	—	568	2	no	—
122	C₂×C₄	—	568	4	no	—
123	C₂×C₄	C₄⋊C₄×D₇₁	1	2	yes	—
124	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
125	C₂×C₄	#14556126	1	1	yes	characteristic
126	C₂×C₄	#14556127	1	2	yes	—
127	C₂×C₄	C₂×#11765850	1	2	yes	—
128	C₂×C₄	C₂×#11765855	1	2	yes	—
129	C₂×C₄	#14556147	1	4	yes	—
130	C₂×C₄	#14556151	1	4	yes	—
131	C₈	—	142	8	no	—
132	C₈	#14556061	1	8	yes	—
133	C₂²	#17985868	1	1	yes	characteristic
134	C₂²	#17986375	1	1	yes	characteristic
135	C₂²	#17986378	1	1	yes	characteristic
136	C₂²	#17986382	1	2	yes	—
137	C₂²	#17986384	1	2	yes	—
138	C₄	—	16	2	no	—
139	C₄	—	1136	2	no	—
140	C₄	C₂×#14556061	1	2	yes	—
141	C₄	#17986036	1	4	yes	—
142	C₄	#17986400	1	4	yes	—
143	C₄	#17986402	1	2	yes	—
144	C₂	#22587945	1	1	yes	characteristic
145	C₂	#22587947	1	1	yes	characteristic
146	C₂	#22587949	1	2	yes	—
147	C₂	#22588499	1	1	yes	characteristic
148	C₂	#22588503	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588500

C₂

yes

maximal, characteristic

#22588467

C₂

yes

maximal

C₂×C₈×C₇₁⋊C₈

C₂

yes

maximal, characteristic

C₇₁×#1845

C₂

yes

maximal, characteristic

C₂×#14556062

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

#17985869

C₂²

yes

—

#17985761

—

C₈×C₇₁⋊C₈

C₄

yes

complemented

C₂×#107×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#384×C₇₁

C₂²

yes

—

C₂²×C₇₁⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

#14556062

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×C₇₁⋊C₈

Q₈

yes

—

C₂²×C₈×C₇₁

D₄

yes

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#107×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

Q₁₆

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

—

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₂³×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×C₄×C₇₁

C₂×Q₁₆

yes

—

C₂×C₄×C₇₁

#135

yes

—

C₂×C₄×C₇₁

#135

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

#107

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₂²×C₇₁

#495

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#107

yes

—

C₄×C₇₁

#427

yes

—

C₄×C₇₁

#499

yes

—

C₄×C₇₁

#500

yes

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#1845

yes

characteristic

C₂×C₇₁

#1846

yes

—

#1845

—

#1845

D₇₁

yes

characteristic

#1834

—

C₂×C₈²

—

C₇₁

#8816

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#107

—

142

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#384

—

#384

C₂×D₇₁

yes

—

#340

—

142

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₇₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

#107

—

#107

—

284

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₄₂

yes

—

100

C₄×C₈

C₄×D₇₁

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

284

—

103

C₂²×C₄

D₄×D₇₁

yes

—

104

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

142

—

107

C₂×C₈

—

142

—

108

C₂×C₈

Dic₇₁⋊C₄

yes

—

109

C₂×C₈

#11765850

yes

—

110

C₂×C₈

#11765855

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

284

—

114

C₄⋊C₄

—

284

—

115

C₄²

C₂×C₄×D₇₁

yes

characteristic

116

C₄²

Q₈×D₇₁

yes

—

117

C₄²

C₂×Dic₁₄₂

yes

characteristic

118

C₂³

#14556124

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

568

—

122

C₂×C₄

—

568

—

123

C₂×C₄

C₄⋊C₄×D₇₁

yes

—

124

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

125

C₂×C₄

#14556126

yes

characteristic

126

C₂×C₄

#14556127

yes

—

127

C₂×C₄

C₂×#11765850

yes

—

128

C₂×C₄

C₂×#11765855

yes

—

129

C₂×C₄

#14556147

yes

—

130

C₂×C₄

#14556151

yes

—

131

C₈

—

142

—

132

C₈

#14556061

yes

—

133

C₂²

#17985868

yes

characteristic

134

C₂²

#17986375

yes

characteristic

135

C₂²

#17986378

yes

characteristic

136

C₂²

#17986382

yes

—

137

C₂²

#17986384

yes

—

138

C₄

—

139

C₄

—

1136

—

140

C₄

C₂×#14556061

yes

—

141

C₄

#17986036

yes

—

142

C₄

#17986400

yes

—

143

C₄

#17986402

yes

—

144

C₂

#22587945

yes

characteristic

145

C₂

#22587947

yes

characteristic

146

C₂

#22587949

yes

—

147

C₂

#22588499

yes

characteristic

148

C₂

#22588503

yes

—

	subgroup	G/H
center	C₂³	#14556124
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1845	C₂
socle, prime radical	C₂³×C₇₁	#127
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1845	D₇₁
hypocenter	C₇₁	#8816
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#14556124

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1845

C₂

socle, prime radical

C₂³×C₇₁

#127

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
9. C₈×C₇₁⋊C₈	C₄	semidirect
79. C₇₁	#8816	semidirect

H ≅ G/N

type

9. C₈×C₇₁⋊C₈

C₄

semidirect

79. C₇₁

#8816

semidirect

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

subgroup	index
#22588500	2
#22588467	2
C₂×C₈×C₇₁⋊C₈	2
C₇₁×#1845	2
#8816	71

subgroup

index

#22588500

#22588467

C₂×C₈×C₇₁⋊C₈

C₇₁×#1845

#8816

Subgroups — #31891228

Conjugacy classes of subgroups

Complements

Maximal subgroups