Group #31909459 — subgroups

Subgroups — #31909459

Order 18176.

234 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₄×D₁₁₃₆	C₂	1	1	yes	maximal, characteristic
2	#22587224	C₂	1	2	yes	maximal, complemented
3	#22586300	C₂	1	2	yes	maximal, complemented
4	#22586256	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1372	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₁₃₆	C₄	1	1	yes	characteristic
7	C₄×D₅₆₈	C₂²	1	2	yes	—
8	#17986076	—	2	2	no	—
9	#17985825	C₄	1	1	yes	characteristic
10	#17985801	C₄	1	2	yes	—
11	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	#17985726	—	2	2	no	—
13	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	#335×C₇₁	C₂²	1	2	yes	—
16	C₂×D₅₆₈	C₂×C₄	1	2	yes	—
17	C₂×#11765848	—	4	2	no	—
18	C₄×D₂₈₄	—	2	2	no	—
19	D₁₁₃₆	—	2	2	no	—
20	C₂×C₇₁⋊C₁₆	—	2	1	no	—
21	#14556062	C₂×C₄	1	2	yes	—
22	#14556050	—	2	2	no	—
23	(C₂×D₇₁)⋊C₈	—	4	2	no	—
24	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
25	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₈×C₇₁	—	2	2	no	—
29	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
30	C₂×D₂₈₄	—	2	2	no	—
31	C₂×C₄×D₇₁	—	4	2	no	—
32	D₅₆₈	—	2	2	no	—
33	D₅₆₈	—	2	2	no	—
34	#11765848	—	8	4	no	—
35	C₂×C₇₁⋊C₈	—	2	1	no	—
36	C₂×C₇₁⋊C₈	D₈	1	2	yes	—
37	C₄⋊Dic₇₁	—	2	2	no	—
38	(C₂×D₇₁)⋊C₄	—	4	2	no	—
39	C₇₁⋊C₁₆	—	2	2	no	—
40	C₂×C₈×C₇₁	—	4	2	no	—
41	C₂×C₈×C₇₁	C₂²×C₄	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	M₁₆	1	2	yes	complemented
46	C₂²×D₇₁	—	4	2	no	—
47	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
50	C₂×Dic₇₁	—	4	2	no	—
51	D₂₈₄	—	4	2	no	—
52	D₂₈₄	—	4	2	no	—
53	C₄×D₇₁	—	8	2	no	—
54	C₄×D₇₁	—	8	2	no	—
55	C₈×C₇₁	—	2	1	no	—
56	C₈×C₇₁	—	8	2	no	—
57	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
58	C₈×C₇₁	#131	1	1	yes	characteristic
59	C₇₁⋊C₈	—	2	2	no	—
60	C₇₁⋊C₈	—	4	1	no	—
61	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
62	C₂×D₇₁	—	8	2	no	—
63	C₂×D₇₁	—	8	2	no	—
64	C₂×D₇₁	—	8	2	no	—
65	C₄×C₇₁	—	2	1	no	—
66	C₄×C₇₁	#436	1	1	yes	characteristic
67	C₄×C₇₁	#443	1	1	yes	characteristic
68	C₄×C₇₁	#509	1	2	yes	—
69	Dic₇₁	—	8	2	no	—
70	#12086	—	71	1	no	maximal, Sylow
71	C₂×C₇₁	#1589	1	1	yes	characteristic
72	C₂×C₇₁	#2177	1	1	yes	characteristic
73	C₂×C₇₁	#2179	1	1	yes	characteristic
74	D₇₁	—	16	2	no	—
75	C₄×D₁₆	—	71	1	no	—
76	#1589	—	71	2	no	—
77	#1372	D₇₁	1	1	yes	characteristic, complemented
78	#1331	—	71	2	no	—
79	#1311	—	71	1	no	—
80	C₇₁	#12086	1	1	yes	characteristic, complemented, Sylow
81	C₂×D₁₆	—	71	1	no	—
82	C₄×D₈	—	71	2	no	—
83	#436	—	142	2	no	—
84	#366	—	71	1	no	—
85	#357	—	71	2	no	—
86	C₄×C₁₆	—	71	1	no	—
87	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
88	#335	C₂×D₇₁	1	2	yes	—
89	D₄⋊C₈	—	142	2	no	—
90	C₈²	—	71	1	no	—
91	C₂×D₈	—	71	2	no	—
92	C₂×M₁₆	—	284	2	no	—
93	C₄×D₄	—	142	2	no	—
94	D₁₆	—	142	2	no	—
95	C₂×C₁₆	—	142	1	no	—
96	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
97	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
98	#107	—	71	2	no	—
99	C₄⋊C₈	—	2	2	no	—
100	#102	—	142	2	no	—
101	C₂²⋊C₈	—	284	2	no	—
102	C₄×C₈	—	71	1	no	—
103	C₄×C₈	—	71	1	no	—
104	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
105	C₂×D₄	—	142	2	no	—
106	C₂²×C₄	—	284	2	no	—
107	D₈	—	142	2	no	—
108	D₈	—	142	2	no	—
109	M₁₆	—	568	4	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	71	2	no	—
112	C₂×C₈	—	142	1	no	—
113	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
114	C₂×C₈	#11765866	1	1	yes	characteristic
115	C₄⋊C₄	—	142	2	no	—
116	C₂²⋊C₄	—	284	2	no	—
117	C₄²	D₄×D₇₁	1	1	yes	characteristic
118	C₁₆	—	2	1	no	—
119	C₁₆	—	142	2	no	—
120	C₁₆	#11765848	1	2	yes	complemented
121	C₂³	—	284	2	no	—
122	D₄	—	284	2	no	—
123	D₄	—	284	2	no	—
124	C₂×C₄	—	284	2	no	—
125	C₂×C₄	—	568	2	no	—
126	C₂×C₄	—	568	2	no	—
127	C₂×C₄	#14556090	1	1	yes	characteristic
128	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
129	C₂×C₄	#14556151	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	8	2	no	—
132	C₈	—	284	1	no	—
133	C₈	—	142	2	no	—
134	C₈	C₂×#11765848	1	1	yes	characteristic
135	C₈	#14556136	1	1	yes	characteristic
136	C₂²	—	568	2	no	—
137	C₂²	—	568	2	no	—
138	C₂²	—	568	2	no	—
139	C₂²	#17986081	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	568	2	no	—
142	C₄	#17986076	1	1	yes	characteristic
143	C₄	#17986115	1	1	yes	characteristic
144	C₄	#17986436	1	2	yes	—
145	C₂	—	1136	2	no	—
146	C₂	#22587224	1	1	yes	characteristic
147	C₂	#22590215	1	1	yes	characteristic
148	C₂	#22590225	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₁₁₃₆

C₂

yes

maximal, characteristic

#22587224

C₂

yes

maximal, complemented

#22586300

C₂

yes

maximal, complemented

#22586256

C₂

yes

maximal, characteristic, complemented

C₇₁×#1372

C₂

yes

maximal, characteristic, complemented

C₂×D₁₁₃₆

C₄

yes

characteristic

C₄×D₅₆₈

C₂²

yes

—

#17986076

—

#17985825

C₄

yes

characteristic

#17985801

C₄

yes

—

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

#17985726

—

C₈×C₇₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

#335×C₇₁

C₂²

yes

—

C₂×D₅₆₈

C₂×C₄

yes

—

C₂×#11765848

—

C₄×D₂₈₄

—

D₁₁₃₆

—

C₂×C₇₁⋊C₁₆

—

#14556062

C₂×C₄

yes

—

#14556050

—

(C₂×D₇₁)⋊C₈

—

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×D₂₈₄

—

C₂×C₄×D₇₁

—

D₅₆₈

—

D₅₆₈

—

#11765848

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₈

yes

—

C₄⋊Dic₇₁

—

(C₂×D₇₁)⋊C₄

—

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

M₁₆

yes

complemented

C₂²×D₇₁

—

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₈×C₇₁

#131

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₄×D₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

C₄×D₁₆

—

#1589

—

#1372

D₇₁

yes

characteristic, complemented

#1331

—

#1311

—

C₇₁

#12086

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₄×D₈

—

#436

—

142

—

#366

—

#357

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

C₂×D₇₁

yes

—

D₄⋊C₈

—

142

—

C₈²

—

C₂×D₈

—

C₂×M₁₆

—

284

—

C₄×D₄

—

142

—

D₁₆

—

142

—

C₂×C₁₆

—

142

—

C₂×C₁₆

C₄×D₇₁

yes

characteristic

C₂×C₁₆

C₄×D₇₁

yes

characteristic

#107

—

C₄⋊C₈

—

100

#102

—

142

—

101

C₂²⋊C₈

—

284

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

C₂²×D₇₁

yes

characteristic

105

C₂×D₄

—

142

—

106

C₂²×C₄

—

284

—

107

D₈

—

142

—

108

D₈

—

142

—

109

M₁₆

—

568

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

142

—

113

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

114

C₂×C₈

#11765866

yes

characteristic

115

C₄⋊C₄

—

142

—

116

C₂²⋊C₄

—

284

—

117

C₄²

D₄×D₇₁

yes

characteristic

118

C₁₆

—

119

C₁₆

—

142

—

120

C₁₆

#11765848

yes

complemented

121

C₂³

—

284

—

122

D₄

—

284

—

123

D₄

—

284

—

124

C₂×C₄

—

284

—

125

C₂×C₄

—

568

—

126

C₂×C₄

—

568

—

127

C₂×C₄

#14556090

yes

characteristic

128

C₂×C₄

D₈×D₇₁

yes

characteristic

129

C₂×C₄

#14556151

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

284

—

133

C₈

—

142

—

134

C₈

C₂×#11765848

yes

characteristic

135

C₈

#14556136

yes

characteristic

136

C₂²

—

568

—

137

C₂²

—

568

—

138

C₂²

—

568

—

139

C₂²

#17986081

yes

characteristic

140

C₄

—

141

C₄

—

568

—

142

C₄

#17986076

yes

characteristic

143

C₄

#17986115

yes

characteristic

144

C₄

#17986436

yes

—

145

C₂

—

1136

—

146

C₂

#22587224

yes

characteristic

147

C₂

#22590215

yes

characteristic

148

C₂

#22590225

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1372	C₂
socle	C₂²×C₇₁	C₄×D₈
prime radical	C₂×D₁₁₃₆	C₄
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1372	D₇₁
hypocenter	C₇₁	#12086
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1372

subgroup

G/H

center

C₂×C₄

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1372

C₂

socle

C₂²×C₇₁

C₄×D₈

prime radical

C₂×D₁₁₃₆

C₄

cosocle

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1372

N	H ≅ G/N	type
2. #22587224	C₂	semidirect
3. #22586300	C₂	semidirect
4. #22586256	C₂	semidirect
5. C₇₁×#1372	C₂	semidirect
45. C₁₆×C₇₁	M₁₆	semidirect
77. #1372	D₇₁	semidirect
80. C₇₁	#12086	semidirect
120. C₁₆	#11765848	semidirect

H ≅ G/N

type

2. #22587224

C₂

semidirect

3. #22586300

C₂

semidirect

4. #22586256

C₂

semidirect

5. C₇₁×#1372

C₂

semidirect

45. C₁₆×C₇₁

M₁₆

semidirect

77. #1372

D₇₁

semidirect

80. C₇₁

#12086

semidirect

120. C₁₆

#11765848

semidirect

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

subgroup	index
C₄×D₁₁₃₆	2
#22587224	2
#22586300	2
#22586256	2
C₇₁×#1372	2
#12086	71

subgroup

index

C₄×D₁₁₃₆

C₇₁×#1372

Subgroups — #31909459

Conjugacy classes of subgroups

Complements

Maximal subgroups