Group #31913819 — subgroups

Subgroups — #31913819

Order 18176.

234 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 111 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	#22590106	C₂	1	1	yes	maximal, characteristic
2	#22590099	C₂	1	1	yes	maximal, characteristic
3	#22586429	C₂	1	2	yes	maximal
4	#22586426	C₂	1	2	yes	maximal
5	C₇₁×#1747	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	1	yes	characteristic
7	C₂×#14556068	C₂²	1	2	yes	—
8	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
9	#17985820	—	2	2	no	—
10	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
11	#405×C₇₁	C₂²	1	2	yes	—
12	#405×C₇₁	C₂²	1	1	yes	characteristic
13	#338×C₇₁	C₄	1	1	yes	characteristic
14	H(3,ℤ/4)×C₇₁	C₄	1	1	yes	characteristic
15	H(3,ℤ/4)×C₇₁	C₄	1	2	yes	—
16	#14556068	—	2	2	no	—
17	#14556068	—	2	2	no	—
18	#14556068	—	2	2	no	—
19	C₂×C₇₁⋊C₁₆	—	2	1	no	—
20	C₂×C₇₁⋊C₁₆	—	2	1	no	—
21	C₂×C₇₁⋊C₁₆	—	2	2	no	—
22	C₂×C₇₁⋊C₁₆	—	2	2	no	—
23	C₂×M₁₆×C₇₁	C₂×C₄	1	2	yes	—
24	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
26	#117×C₇₁	C₂×C₄	1	2	yes	—
27	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
28	#97×C₇₁	—	2	2	no	—
29	#97×C₇₁	Q₈	1	2	yes	—
30	C₄×C₈×C₇₁	—	2	2	no	—
31	C₄×C₈×C₇₁	D₄	1	2	yes	—
32	C₇₁⋊C₁₆	—	4	1	no	—
33	C₇₁⋊C₁₆	—	4	1	no	—
34	C₇₁⋊C₁₆	—	4	2	no	—
35	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	M₁₆×C₇₁	—	2	2	no	—
37	M₁₆×C₇₁	—	2	2	no	—
38	M₁₆×C₇₁	—	2	2	no	—
39	C₂×C₈×C₇₁	—	2	2	no	—
40	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₇₁	#41	1	2	yes	—
44	C₂×C₈×C₇₁	#41	1	2	yes	—
45	C₄⋊C₄×C₇₁	—	2	1	no	—
46	C₄⋊C₄×C₇₁	—	2	2	no	—
47	C₂²⋊C₄×C₇₁	—	2	1	no	—
48	C₂²⋊C₄×C₇₁	—	2	2	no	—
49	C₄²×C₇₁	—	2	2	no	—
50	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
51	C₂³×C₇₁	#131	1	1	yes	characteristic
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	—	4	2	no	—
54	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₁	#117	1	2	yes	—
56	C₂×C₄×C₇₁	#131	1	2	yes	—
57	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
58	C₈×C₇₁	—	2	2	no	—
59	C₈×C₇₁	—	4	2	no	—
60	C₈×C₇₁	#124	1	1	yes	characteristic
61	C₈×C₇₁	#124	1	1	yes	characteristic
62	C₈×C₇₁	#125	1	2	yes	—
63	C₂²×C₇₁	—	4	1	no	—
64	C₂²×C₇₁	#424	1	1	yes	characteristic
65	C₂²×C₇₁	#433	1	2	yes	—
66	C₄×C₇₁	—	4	2	no	—
67	C₄×C₇₁	—	8	2	no	—
68	C₄×C₇₁	#382	1	1	yes	characteristic
69	C₄×C₇₁	#424	1	1	yes	characteristic
70	C₄×C₇₁	#433	1	2	yes	—
71	#12885	—	71	1	no	maximal, Sylow
72	C₂×C₇₁	—	2	1	no	—
73	C₂×C₇₁	—	2	2	no	—
74	C₂×C₇₁	#1832	1	1	yes	characteristic
75	#2153	—	71	1	no	—
76	#2151	—	71	1	no	—
77	#1747	D₇₁	1	1	yes	characteristic
78	#1379	—	71	2	no	—
79	#1378	—	71	2	no	—
80	C₇₁	#12885	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₃₂	—	71	1	no	—
82	C₂×M₃₂	—	71	2	no	—
83	C₂²×C₁₆	—	71	1	no	—
84	#405	C₂×D₇₁	1	2	yes	—
85	#405	C₂×D₇₁	1	1	yes	characteristic
86	#364	—	142	2	no	—
87	C₄⋊C₁₆	—	142	2	no	—
88	#338	Dic₇₁	1	1	yes	characteristic
89	H(3,ℤ/4)	Dic₇₁	1	1	yes	characteristic
90	H(3,ℤ/4)	Dic₇₁	1	2	yes	—
91	C₂×M₁₆	C₂×Dic₇₁	1	2	yes	—
92	C₂×M₁₆	C₂×Dic₇₁	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
94	#117	C₂×Dic₇₁	1	2	yes	—
95	#117	C₂×Dic₇₁	1	1	yes	characteristic
96	M₃₂	—	142	2	no	—
97	M₃₂	—	142	2	no	—
98	M₃₂	—	142	2	no	—
99	C₂×C₁₆	—	142	1	no	—
100	C₂×C₁₆	—	142	1	no	—
101	C₂×C₁₆	—	142	2	no	—
102	C₂×C₁₆	—	142	2	no	—
103	#97	—	2	2	no	—
104	#97	Dic₁₄₂	1	2	yes	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	D₂₈₄	1	2	yes	—
107	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
113	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
114	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
115	C₂×C₈	#11765865	1	2	yes	—
116	C₂×C₈	#11765865	1	2	yes	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	C₄⋊Dic₇₁	1	2	yes	—
123	C₁₆	—	284	1	no	—
124	C₁₆	—	284	1	no	—
125	C₁₆	—	284	2	no	—
126	C₂³	#14556135	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
130	C₂×C₄	#14556087	1	2	yes	—
131	C₂×C₄	#14556135	1	1	yes	characteristic
132	C₂×C₄	#14556135	1	2	yes	—
133	C₈	—	2	2	no	—
134	C₈	—	4	2	no	—
135	C₈	#14556116	1	1	yes	characteristic
136	C₈	#14556116	1	1	yes	characteristic
137	C₈	#14556120	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	#17986026	1	1	yes	characteristic
140	C₂²	#17986060	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	8	2	no	—
143	C₄	#17985865	1	1	yes	characteristic
144	C₄	#17986026	1	1	yes	characteristic
145	C₄	#17986060	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	#22588460	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590106

C₂

yes

maximal, characteristic

#22590099

C₂

yes

maximal, characteristic

#22586429

C₂

yes

maximal

#22586426

C₂

yes

maximal

C₇₁×#1747

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

characteristic

C₂×#14556068

C₂²

yes

—

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

#17985820

—

C₄⋊(C₇₁⋊C₁₆)

—

#405×C₇₁

C₂²

yes

—

#405×C₇₁

C₂²

yes

characteristic

#338×C₇₁

C₄

yes

characteristic

H(3,ℤ/4)×C₇₁

C₄

yes

characteristic

H(3,ℤ/4)×C₇₁

C₄

yes

—

#14556068

—

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×M₁₆×C₇₁

C₂×C₄

yes

—

C₂×M₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

#117×C₇₁

C₂×C₄

yes

—

#117×C₇₁

C₂×C₄

yes

characteristic

#97×C₇₁

—

#97×C₇₁

Q₈

yes

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₄⋊C₄

yes

—

C₂³×C₇₁

#131

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

#131

yes

—

C₂×C₄×C₇₁

#131

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₈×C₇₁

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

#424

yes

characteristic

C₂²×C₇₁

#433

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

#12885

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

—

D₇₁

yes

characteristic

#1379

—

#1378

—

C₇₁

#12885

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

C₂×D₇₁

yes

—

#405

C₂×D₇₁

yes

characteristic

#364

—

142

—

C₄⋊C₁₆

—

142

—

#338

Dic₇₁

yes

characteristic

H(3,ℤ/4)

Dic₇₁

yes

characteristic

H(3,ℤ/4)

Dic₇₁

yes

—

C₂×M₁₆

C₂×Dic₇₁

yes

—

C₂×M₁₆

C₂×Dic₇₁

yes

characteristic

C₂²×C₈

C₂²×D₇₁

yes

characteristic

#117

C₂×Dic₇₁

yes

—

#117

C₂×Dic₇₁

yes

characteristic

M₃₂

—

142

—

M₃₂

—

142

—

M₃₂

—

142

—

C₂×C₁₆

—

142

—

100

C₂×C₁₆

—

142

—

101

C₂×C₁₆

—

142

—

102

C₂×C₁₆

—

142

—

103

#97

—

104

#97

Dic₁₄₂

yes

—

105

C₄×C₈

—

106

C₄×C₈

D₂₈₄

yes

—

107

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

C₄⋊Dic₇₁

yes

—

113

C₂×C₈

C₂×D₂₈₄

yes

characteristic

114

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

115

C₂×C₈

#11765865

yes

—

116

C₂×C₈

#11765865

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

C₄⋊Dic₇₁

yes

—

123

C₁₆

—

284

—

124

C₁₆

—

284

—

125

C₁₆

—

284

—

126

C₂³

#14556135

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄⋊Dic₇₁

yes

characteristic

130

C₂×C₄

#14556087

yes

—

131

C₂×C₄

#14556135

yes

characteristic

132

C₂×C₄

#14556135

yes

—

133

C₈

—

134

C₈

—

135

C₈

#14556116

yes

characteristic

136

C₈

#14556116

yes

characteristic

137

C₈

#14556120

yes

—

138

C₂²

—

139

C₂²

#17986026

yes

characteristic

140

C₂²

#17986060

yes

—

141

C₄

—

142

C₄

—

143

C₄

#17985865

yes

characteristic

144

C₄

#17986026

yes

characteristic

145

C₄

#17986060

yes

—

146

C₂

—

147

C₂

—

148

C₂

#22588460

yes

characteristic

	subgroup	G/H
center	C₈	#14556116
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1747	C₂
socle	C₂×C₇₁	#1832
prime radical	C₂³×C₇₁	#131
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1747	D₇₁
hypocenter	C₇₁	#12885
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1747

subgroup

G/H

center

C₈

#14556116

commutator

C₂²×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1747

C₂

C₂×C₇₁

C₂³×C₇₁

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1747

N	H ≅ G/N	type
80. C₇₁	#12885	semidirect

H ≅ G/N

type

80. C₇₁

#12885

semidirect

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

subgroup	index
#22590106	2
#22590099	2
#22586429	2
#22586426	2
C₇₁×#1747	2
#12885	71

subgroup

index

C₇₁×#1747

Subgroups — #31913819

Conjugacy classes of subgroups

Complements

Maximal subgroups