Group #31912151 — subgroups

Subgroups — #31912151

Order 18176.

268 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 83 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	#22590568	C₂	1	2	yes	maximal, complemented
2	#22590564	C₂	1	2	yes	maximal
3	#22587958	C₂	1	1	yes	maximal, characteristic, complemented
4	#22587946	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986404	—	2	1	no	—
7	#17986404	C₂²	1	2	yes	—
8	#17986379	—	2	1	no	—
9	#17986379	C₂²	1	1	yes	characteristic
10	#17986376	C₂²	1	1	yes	characteristic
11	#17986346	—	2	1	no	—
12	#17986346	—	2	1	no	—
13	#17985804	—	2	4	no	—
14	#17985801	—	2	4	no	—
15	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
16	C₈²×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₈₄	—	2	2	no	—
18	C₂×Dic₂₈₄	—	4	2	no	—
19	C₂×D₅₆₈	—	2	2	no	—
20	#14556128	—	2	1	no	—
21	#14556128	—	2	2	no	—
22	#14556125	—	2	1	no	—
23	#14556062	—	2	2	no	—
24	#14556062	—	2	2	no	—
25	#14556062	—	4	2	no	—
26	#14556050	—	4	2	no	—
27	#14556050	—	4	2	no	—
28	C₂×C₁₆×C₇₁	D₄	1	4	yes	—
29	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₄₂	—	4	1	no	—
33	C₂×Dic₁₄₂	—	4	2	no	—
34	C₂×D₂₈₄	—	4	1	no	—
35	C₂×D₂₈₄	—	4	1	no	—
36	C₂×D₂₈₄	—	4	1	no	—
37	Dic₂₈₄	—	4	4	no	—
38	Dic₂₈₄	—	8	4	no	—
39	D₅₆₈	—	4	4	no	—
40	C₄⋊Dic₇₁	—	4	2	no	—
41	C₄⋊Dic₇₁	—	4	1	no	—
42	C₄⋊Dic₇₁	—	4	1	no	—
43	C₄⋊Dic₇₁	—	4	2	no	—
44	C₂×C₈×C₇₁	D₈	1	2	yes	—
45	C₂×C₈×C₇₁	D₈	1	2	yes	—
46	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
47	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₇₁	—	2	4	no	—
49	C₂²×D₇₁	—	8	1	no	—
50	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₇₁	—	8	1	no	—
54	C₂×Dic₇₁	—	8	2	no	—
55	D₂₈₄	—	8	2	no	—
56	D₂₈₄	—	8	2	no	—
57	D₂₈₄	—	8	2	no	—
58	Dic₁₄₂	—	8	2	no	—
59	Dic₁₄₂	—	8	4	no	—
60	C₈×C₇₁	—	2	2	no	—
61	C₈×C₇₁	—	2	2	no	—
62	C₈×C₇₁	#135	1	4	yes	—
63	C₂²×C₇₁	#493	1	1	yes	characteristic
64	C₂×D₇₁	—	16	1	no	—
65	C₂×D₇₁	—	16	1	no	—
66	C₂×D₇₁	—	16	1	no	—
67	C₄×C₇₁	#495	1	2	yes	—
68	C₄×C₇₁	#509	1	2	yes	—
69	C₄×C₇₁	#510	1	2	yes	—
70	Dic₇₁	—	16	1	no	—
71	Dic₇₁	—	16	2	no	—
72	#12512	—	71	1	no	maximal, Sylow
73	C₂×C₇₁	#1709	1	1	yes	characteristic
74	C₂×C₇₁	#2250	1	1	yes	characteristic
75	C₂×C₇₁	#2251	1	1	yes	characteristic
76	D₇₁	—	32	1	no	—
77	#2241	—	71	2	no	—
78	#2240	—	71	2	no	—
79	#1713	—	71	1	no	—
80	#1709	—	71	1	no	—
81	#1311	D₇₁	1	1	yes	characteristic, complemented
82	C₇₁	#12512	1	1	yes	characteristic, complemented, Sylow
83	#500	—	71	2	no	—
84	#500	—	142	1	no	—
85	#494	—	71	1	no	—
86	#494	—	142	1	no	—
87	#493	—	71	1	no	—
88	#486	—	142	1	no	—
89	#486	—	142	1	no	—
90	#358	—	142	4	no	—
91	#357	—	142	4	no	—
92	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
93	C₈²	C₂×D₇₁	1	1	yes	characteristic
94	C₂×Q₁₆	—	284	2	no	—
95	C₂×Q₁₆	—	142	2	no	—
96	C₂×D₈	—	142	2	no	—
97	C₄⋊Q₈	—	142	1	no	—
98	C₄⋊Q₈	—	142	2	no	—
99	#127	—	142	1	no	—
100	C₂×C₁₆	D₂₈₄	1	4	yes	—
101	#107	—	284	2	no	—
102	#107	—	142	2	no	—
103	#107	—	142	2	no	—
104	#102	—	284	2	no	—
105	#102	—	284	2	no	—
106	C₄×C₈	D₂₈₄	1	1	yes	characteristic
107	C₄×C₈	D₂₈₄	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
109	C₂×Q₈	—	284	1	no	—
110	C₂×Q₈	—	284	2	no	—
111	C₂×D₄	—	284	1	no	—
112	C₂×D₄	—	284	1	no	—
113	C₂×D₄	—	284	1	no	—
114	Q₁₆	—	284	4	no	—
115	Q₁₆	—	568	4	no	—
116	D₈	—	284	4	no	—
117	C₂×C₈	D₅₆₈	1	2	yes	—
118	C₂×C₈	D₅₆₈	1	2	yes	—
119	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
120	C₄⋊C₄	—	284	2	no	—
121	C₄⋊C₄	—	284	1	no	—
122	C₄⋊C₄	—	284	1	no	—
123	C₄⋊C₄	—	284	2	no	—
124	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
125	C₁₆	—	2	4	no	—
126	C₂³	—	568	1	no	—
127	Q₈	—	568	2	no	—
128	Q₈	—	568	4	no	—
129	D₄	—	568	2	no	—
130	D₄	—	568	2	no	—
131	D₄	—	568	2	no	—
132	C₂×C₄	—	568	1	no	—
133	C₂×C₄	—	568	2	no	—
134	C₂×C₄	#14556125	1	1	yes	characteristic
135	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
136	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	#14556150	1	4	yes	—
140	C₂²	—	1136	1	no	—
141	C₂²	—	1136	1	no	—
142	C₂²	—	1136	1	no	—
143	C₂²	#17986376	1	1	yes	characteristic
144	C₄	—	1136	1	no	—
145	C₄	—	1136	2	no	—
146	C₄	#17986383	1	2	yes	—
147	C₄	#17986437	1	2	yes	—
148	C₄	#17986442	1	2	yes	—
149	C₂	—	2272	1	no	—
150	C₂	#22587946	1	1	yes	characteristic
151	C₂	#22590606	1	1	yes	characteristic
152	C₂	#22590611	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590568

C₂

yes

maximal, complemented

#22590564

C₂

yes

maximal

#22587958

C₂

yes

maximal, characteristic, complemented

#22587946

C₂

yes

maximal, characteristic

C₇₁×#1311

C₂

yes

maximal, characteristic, complemented

#17986404

—

#17986404

C₂²

yes

—

#17986379

—

#17986379

C₂²

yes

characteristic

#17986376

C₂²

yes

—

—

—

—

C₄×C₁₆×C₇₁

C₂²

yes

—

C₈²×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

C₂×D₅₆₈

—

—

—

—

—

—

—

—

—

C₂×C₁₆×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

D₅₆₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₂²×D₇₁

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#135

yes

—

C₂²×C₇₁

#493

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

#495

yes

—

C₄×C₇₁

#509

yes

—

C₄×C₇₁

#510

yes

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₇₁

#12512

yes

characteristic, complemented, Sylow

#500

—

#500

—

142

—

#494

—

#494

—

142

—

#493

—

#486

—

142

—

#486

—

142

—

#358

—

142

—

#357

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

—

C₈²

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

—

284

—

C₂×Q₁₆

—

142

—

C₂×D₈

—

142

—

C₄⋊Q₈

—

142

—

C₄⋊Q₈

—

142

—

#127

—

142

—

100

C₂×C₁₆

D₂₈₄

yes

—

101

#107

—

284

—

102

#107

—

142

—

103

#107

—

142

—

104

#102

—

284

—

105

#102

—

284

—

106

C₄×C₈

D₂₈₄

yes

characteristic

107

C₄×C₈

D₂₈₄

yes

characteristic

108

C₄×C₈

C₂²×D₇₁

yes

characteristic

109

C₂×Q₈

—

284

—

110

C₂×Q₈

—

284

—

111

C₂×D₄

—

284

—

112

C₂×D₄

—

284

—

113

C₂×D₄

—

284

—

114

Q₁₆

—

284

—

115

Q₁₆

—

568

—

116

D₈

—

284

—

117

C₂×C₈

D₅₆₈

yes

—

118

C₂×C₈

D₅₆₈

yes

—

119

C₂×C₈

C₂×D₂₈₄

yes

—

120

C₄⋊C₄

—

284

—

121

C₄⋊C₄

—

284

—

122

C₄⋊C₄

—

284

—

123

C₄⋊C₄

—

284

—

124

C₄²

C₂×D₂₈₄

yes

characteristic

125

C₁₆

—

126

C₂³

—

568

—

127

Q₈

—

568

—

128

Q₈

—

568

—

129

D₄

—

568

—

130

D₄

—

568

—

131

D₄

—

568

—

132

C₂×C₄

—

568

—

133

C₂×C₄

—

568

—

134

C₂×C₄

#14556125

yes

characteristic

135

C₂×C₄

C₂×D₅₆₈

yes

characteristic

136

C₂×C₄

C₂×D₅₆₈

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#14556150

yes

—

140

C₂²

—

1136

—

141

C₂²

—

1136

—

142

C₂²

—

1136

—

143

C₂²

#17986376

yes

characteristic

144

C₄

—

1136

—

145

C₄

—

1136

—

146

C₄

#17986383

yes

—

147

C₄

#17986437

yes

—

148

C₄

#17986442

yes

—

149

C₂

—

2272

—

150

C₂

#22587946

yes

characteristic

151

C₂

#22590606

yes

characteristic

152

C₂

#22590611

yes

characteristic

	subgroup	G/H
center	C₂²	#17986376
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1311	C₂
socle	C₂²×C₇₁	#493
prime radical	#17986376	C₂²
hypercenter	#1311	D₇₁
hypocenter	C₇₁	#12512
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#17986376

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1311

C₂

C₂²×C₇₁

C₂²

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #22590568	C₂	semidirect
3. #22587958	C₂	semidirect
5. C₇₁×#1311	C₂	semidirect
81. #1311	D₇₁	semidirect
82. C₇₁	#12512	semidirect

H ≅ G/N

type

1. #22590568

C₂

semidirect

3. #22587958

C₂

semidirect

5. C₇₁×#1311

C₂

semidirect

81. #1311

D₇₁

semidirect

82. C₇₁

#12512

semidirect

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

subgroup	index
#22590568	2
#22590564	2
#22587958	2
#22587946	2
C₇₁×#1311	2
#12512	71

subgroup

index

C₇₁×#1311

Subgroups — #31912151

Conjugacy classes of subgroups

Complements

Maximal subgroups