Group #31891224 — subgroups

Subgroups — #31891224

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	#22588497	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×#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556060	—	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×#106×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	#14556060	—	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	#106×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	8	no	—
28	C₂×C₇₁⋊C₈	C₄⋊C₄	1	8	yes	—
29	C₄⋊Dic₇₁	—	4	4	no	—
30	C₄⋊Dic₇₁	—	4	8	no	—
31	C₂²×C₄×C₇₁	—	4	2	no	—
32	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
33	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
36	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
37	C₄⋊C₄×C₇₁	—	4	4	no	—
38	C₄⋊C₄×C₇₁	—	4	8	no	—
39	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
41	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
42	C₂³×C₇₁	#127	1	1	yes	characteristic
43	C₂×C₄×C₇₁	—	8	2	no	—
44	C₂×C₄×C₇₁	—	8	4	no	—
45	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
49	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
50	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
51	C₂×C₄×C₇₁	#135	1	8	yes	—
52	C₂×Dic₇₁	—	8	2	no	—
53	C₂×Dic₇₁	—	8	4	no	—
54	C₈×C₇₁	#106	1	8	yes	—
55	C₇₁⋊C₈	—	2	8	no	—
56	C₂²×C₇₁	#384	1	1	yes	characteristic
57	C₂²×C₇₁	#492	1	1	yes	characteristic
58	C₂²×C₇₁	#492	1	1	yes	characteristic
59	C₂²×C₇₁	#495	1	4	yes	—
60	C₄×C₇₁	—	16	2	no	—
61	C₄×C₇₁	C₂×#106	1	2	yes	—
62	C₄×C₇₁	#427	1	4	yes	—
63	C₄×C₇₁	#498	1	2	yes	—
64	C₄×C₇₁	#499	1	4	yes	—
65	Dic₇₁	—	16	2	no	—
66	#8815	—	71	1	no	maximal, Sylow
67	C₂×C₇₁	#1708	1	2	yes	—
68	C₂×C₇₁	#1708	1	2	yes	—
69	C₂×C₇₁	#1844	1	1	yes	characteristic
70	C₂×C₇₁	#1846	1	2	yes	—
71	#1844	—	71	1	no	—
72	#1844	D₇₁	1	1	yes	characteristic
73	#1834	—	71	4	no	—
74	C₂×C₈²	—	71	1	no	—
75	C₇₁	#8815	1	1	yes	characteristic, complemented, Sylow
76	C₂×#106	—	2	2	no	—
77	C₂×#106	—	142	2	no	—
78	C₂×C₄×C₈	—	71	2	no	—
79	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
80	#384	—	71	2	no	—
81	#384	C₂×D₇₁	1	2	yes	—
82	#340	—	142	8	no	—
83	C₈²	—	71	4	no	—
84	C₂²×C₈	—	71	4	no	—
85	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
86	C₂×C₄⋊C₄	—	2	4	no	—
87	C₂×C₄⋊C₄	—	2	2	no	—
88	C₂×C₄⋊C₄	—	142	4	no	—
89	C₂×C₄⋊C₄	—	142	2	no	—
90	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
91	#106	—	4	4	no	—
92	#106	—	284	4	no	—
93	C₄×C₈	—	71	4	no	—
94	C₄×C₈	—	71	4	no	—
95	C₄×C₈	Dic₁₄₂	1	2	yes	—
96	C₄×C₈	C₄×D₇₁	1	2	yes	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	—	284	2	no	—
99	C₂²×C₄	D₄×D₇₁	1	2	yes	—
100	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
101	C₂×C₈	—	71	8	no	—
102	C₂×C₈	—	142	8	no	—
103	C₂×C₈	Dic₇₁⋊C₄	1	4	yes	—
104	C₂×C₈	#11765852	1	4	yes	—
105	C₂×C₈	#11765853	1	4	yes	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄⋊C₄	—	4	8	no	—
108	C₄⋊C₄	—	284	4	no	—
109	C₄⋊C₄	—	284	8	no	—
110	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
111	C₄²	Q₈×D₇₁	1	2	yes	—
112	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
113	C₂³	#14556124	1	1	yes	characteristic
114	C₂×C₄	—	8	2	no	—
115	C₂×C₄	—	8	4	no	—
116	C₂×C₄	—	568	2	no	—
117	C₂×C₄	—	568	4	no	—
118	C₂×C₄	C₄⋊C₄×D₇₁	1	2	yes	—
119	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
120	C₂×C₄	#14556126	1	1	yes	characteristic
121	C₂×C₄	#14556127	1	2	yes	—
122	C₂×C₄	C₂×#11765852	1	2	yes	—
123	C₂×C₄	C₂×#11765853	1	2	yes	—
124	C₂×C₄	#14556149	1	8	yes	—
125	C₈	—	142	8	no	—
126	C₈	#14556059	1	8	yes	—
127	C₂²	#17985868	1	1	yes	characteristic
128	C₂²	#17986371	1	1	yes	characteristic
129	C₂²	#17986373	1	1	yes	characteristic
130	C₂²	#17986380	1	4	yes	—
131	C₄	—	16	2	no	—
132	C₄	—	1136	2	no	—
133	C₄	C₂×#14556059	1	2	yes	—
134	C₄	#17986034	1	4	yes	—
135	C₄	#17986395	1	2	yes	—
136	C₄	#17986399	1	4	yes	—
137	C₂	#22587939	1	2	yes	—
138	C₂	#22587942	1	2	yes	—
139	C₂	#22588496	1	1	yes	characteristic
140	C₂	#22588501	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588497

C₂

yes

maximal, characteristic

#22588467

C₂

yes

maximal

C₂×C₈×C₇₁⋊C₈

C₂

yes

maximal, characteristic

C₇₁×#1844

C₂

yes

maximal, characteristic

C₂×#14556060

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

#17985869

C₂²

yes

—

#17985761

—

C₈×C₇₁⋊C₈

C₄

yes

complemented

C₂×#106×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#384×C₇₁

C₂²

yes

—

C₂²×C₇₁⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

#14556060

—

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

#106×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

C₂²×Dic₇₁

—

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

SD₁₆

yes

—

C₂×C₈×C₇₁

SD₁₆

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×SD₁₆

yes

—

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₂×C₄×C₇₁

#135

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

#106

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#106

yes

—

C₄×C₇₁

#427

yes

—

C₄×C₇₁

#498

yes

—

C₄×C₇₁

#499

yes

—

Dic₇₁

—

#8815

—

maximal, Sylow

C₂×C₇₁

#1708

yes

—

C₂×C₇₁

#1708

yes

—

C₂×C₇₁

#1844

yes

characteristic

C₂×C₇₁

#1846

yes

—

#1844

—

#1844

D₇₁

yes

characteristic

#1834

—

C₂×C₈²

—

C₇₁

#8815

yes

characteristic, complemented, Sylow

C₂×#106

—

C₂×#106

—

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

#106

—

#106

—

284

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₄₂

yes

—

C₄×C₈

C₄×D₇₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

284

—

C₂²×C₄

D₄×D₇₁

yes

—

100

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

142

—

103

C₂×C₈

Dic₇₁⋊C₄

yes

—

104

C₂×C₈

#11765852

yes

—

105

C₂×C₈

#11765853

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

284

—

109

C₄⋊C₄

—

284

—

110

C₄²

C₂×C₄×D₇₁

yes

characteristic

111

C₄²

Q₈×D₇₁

yes

—

112

C₄²

C₂×Dic₁₄₂

yes

characteristic

113

C₂³

#14556124

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

568

—

117

C₂×C₄

—

568

—

118

C₂×C₄

C₄⋊C₄×D₇₁

yes

—

119

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

120

C₂×C₄

#14556126

yes

characteristic

121

C₂×C₄

#14556127

yes

—

122

C₂×C₄

C₂×#11765852

yes

—

123

C₂×C₄

C₂×#11765853

yes

—

124

C₂×C₄

#14556149

yes

—

125

C₈

—

142

—

126

C₈

#14556059

yes

—

127

C₂²

#17985868

yes

characteristic

128

C₂²

#17986371

yes

characteristic

129

C₂²

#17986373

yes

characteristic

130

C₂²

#17986380

yes

—

131

C₄

—

132

C₄

—

1136

—

133

C₄

C₂×#14556059

yes

—

134

C₄

#17986034

yes

—

135

C₄

#17986395

yes

—

136

C₄

#17986399

yes

—

137

C₂

#22587939

yes

—

138

C₂

#22587942

yes

—

139

C₂

#22588496

yes

characteristic

140

C₂

#22588501

yes

—

	subgroup	G/H
center	C₂³	#14556124
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1844	C₂
socle, prime radical	C₂³×C₇₁	#127
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1844	D₇₁
hypocenter	C₇₁	#8815
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#14556124

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1844

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 ◁ #1844

N	H ≅ G/N	type
9. C₈×C₇₁⋊C₈	C₄	semidirect
75. C₇₁	#8815	semidirect

H ≅ G/N

type

9. C₈×C₇₁⋊C₈

C₄

semidirect

75. C₇₁

#8815

semidirect

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

subgroup	index
#22588497	2
#22588467	2
C₂×C₈×C₇₁⋊C₈	2
C₇₁×#1844	2
#8815	71

subgroup

index

#22588497

#22588467

C₂×C₈×C₇₁⋊C₈

C₇₁×#1844

#8815

Subgroups — #31891224

Conjugacy classes of subgroups

Complements

Maximal subgroups