Group #31909111 — subgroups

Subgroups — #31909111

Order 18176.

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

normal subgroups only

This group has 180 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	#22590155	C₂	1	2	yes	maximal
2	#22589938	C₂	1	1	yes	maximal, characteristic
3	#22586415	C₂	1	2	yes	maximal, complemented
4	#22586408	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1369	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556067	C₄	1	1	yes	characteristic
7	C₂×C₁₆×D₇₁	C₄	1	1	yes	characteristic
8	#17985953	C₂²	1	1	yes	characteristic
9	#17985820	C₂²	1	2	yes	—
10	#17985780	—	2	2	no	—
11	#17985776	—	2	2	no	—
12	#17985774	C₄	1	1	yes	characteristic
13	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₁₆×Dic₇₁	C₄	1	1	yes	characteristic
15	#364×C₇₁	C₂²	1	2	yes	—
16	#346×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×#11765848	D₄	1	1	yes	characteristic
18	C₂×C₈×D₇₁	C₂×C₄	1	1	yes	characteristic
19	#14556085	Q₈	1	1	yes	characteristic
20	#14556068	—	2	2	no	—
21	#14556067	—	2	2	no	—
22	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₁₆×D₇₁	—	2	2	no	—
25	#14556039	Q₈	1	1	yes	characteristic
26	#14556038	D₄	1	1	yes	characteristic
27	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₇₁	—	2	2	no	—
29	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₂×C₄×D₇₁	C₄⋊C₄	1	1	yes	characteristic
33	#11765848	—	2	2	no	—
34	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₈×D₇₁	—	2	1	no	—
37	C₈×D₇₁	—	2	1	no	—
38	Dic₇₁⋊C₄	—	2	1	no	—
39	(C₂×D₇₁)⋊C₄	—	2	1	no	—
40	C₄×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
41	C₇₁⋊C₁₆	—	2	1	no	—
42	C₇₁⋊C₁₆	—	4	2	no	—
43	C₇₁⋊C₁₆	M₁₆	1	2	yes	—
44	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
47	C₁₆×C₇₁	—	2	1	no	—
48	C₁₆×C₇₁	—	4	2	no	—
49	C₁₆×C₇₁	M₁₆	1	2	yes	—
50	C₂²×D₇₁	#108	1	1	yes	characteristic
51	C₂×C₄×C₇₁	—	2	1	no	—
52	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×Dic₇₁	—	2	1	no	—
54	C₂×Dic₇₁	#108	1	1	yes	characteristic
55	C₄×D₇₁	—	2	1	no	—
56	C₄×D₇₁	—	2	1	no	—
57	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
58	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
59	C₈×C₇₁	#135	1	2	yes	—
60	C₇₁⋊C₈	—	2	1	no	—
61	C₇₁⋊C₈	—	2	1	no	—
62	C₂²×C₇₁	C₂×#108	1	1	yes	characteristic
63	C₂×D₇₁	—	2	1	no	—
64	C₂×D₇₁	—	4	1	no	—
65	C₄×C₇₁	—	4	1	no	—
66	C₄×C₇₁	#423	1	1	yes	characteristic
67	C₄×C₇₁	#427	1	1	yes	characteristic
68	Dic₇₁	—	2	1	no	—
69	Dic₇₁	—	4	1	no	—
70	#12027	—	71	1	no	maximal, Sylow
71	C₂×C₇₁	—	2	1	no	—
72	C₂×C₇₁	#1568	1	1	yes	characteristic
73	D₇₁	—	4	1	no	—
74	#2166	—	71	2	no	—
75	#2108	—	71	1	no	—
76	#1373	—	71	2	no	—
77	#1369	—	71	1	no	—
78	#1369	D₇₁	1	1	yes	characteristic, complemented
79	C₇₁	#12027	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	—	71	1	no	—
81	C₂²×C₁₆	—	71	1	no	—
82	#405	—	71	1	no	—
83	#364	—	71	2	no	—
84	#364	C₂×D₇₁	1	2	yes	—
85	#349	—	142	2	no	—
86	#347	—	142	2	no	—
87	#346	—	71	1	no	—
88	#346	C₂×D₇₁	1	1	yes	characteristic
89	C₄×C₁₆	—	71	1	no	—
90	C₄×C₁₆	—	71	1	no	—
91	C₂×M₁₆	—	71	1	no	—
92	C₂²×C₈	—	71	1	no	—
93	#117	—	71	1	no	—
94	M₃₂	—	2	2	no	—
95	M₃₂	—	142	2	no	—
96	M₃₂	—	142	2	no	—
97	C₂×C₁₆	—	71	1	no	—
98	C₂×C₁₆	—	71	1	no	—
99	C₂×C₁₆	—	142	2	no	—
100	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
101	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
102	#97	—	71	1	no	—
103	#97	—	71	1	no	—
104	C₄×C₈	—	71	1	no	—
105	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
106	C₂²×C₄	—	71	1	no	—
107	M₁₆	—	142	2	no	—
108	C₂×C₈	—	71	1	no	—
109	C₂×C₈	—	71	1	no	—
110	C₂×C₈	—	142	1	no	—
111	C₂×C₈	—	142	1	no	—
112	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
113	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
114	C₄⋊C₄	—	142	1	no	—
115	C₂²⋊C₄	—	142	1	no	—
116	C₄²	—	71	1	no	—
117	C₄²	Q₈×D₇₁	1	1	yes	characteristic
118	C₁₆	—	2	1	no	—
119	C₁₆	—	4	2	no	—
120	C₁₆	—	284	2	no	—
121	C₁₆	—	71	2	no	—
122	C₁₆	—	142	1	no	—
123	C₁₆	#11765848	1	2	yes	—
124	C₂³	—	71	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	71	1	no	—
127	C₂×C₄	—	142	1	no	—
128	C₂×C₄	—	142	1	no	—
129	C₂×C₄	—	142	1	no	—
130	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
131	C₈	—	142	1	no	—
132	C₈	—	142	1	no	—
133	C₈	M₁₆×D₇₁	1	1	yes	characteristic
134	C₈	C₂×#11765848	1	1	yes	characteristic
135	C₈	#14556149	1	2	yes	—
136	C₂²	—	284	1	no	—
137	C₂²	—	142	1	no	—
138	C₂²	#108×D₇₁	1	1	yes	characteristic
139	C₄	—	4	1	no	—
140	C₄	—	284	1	no	—
141	C₄	—	142	1	no	—
142	C₄	#17986021	1	1	yes	characteristic
143	C₄	#17986034	1	1	yes	characteristic
144	C₂	—	2	1	no	—
145	C₂	—	284	1	no	—
146	C₂	#22587100	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590155

C₂

yes

maximal

#22589938

C₂

yes

maximal, characteristic

#22586415

C₂

yes

maximal, complemented

#22586408

C₂

yes

maximal, characteristic, complemented

C₇₁×#1369

C₂

yes

maximal, characteristic, complemented

C₂×#14556067

C₄

yes

characteristic

C₂×C₁₆×D₇₁

C₄

yes

characteristic

#17985953

C₂²

yes

characteristic

#17985820

C₂²

yes

—

#17985780

—

#17985776

—

#17985774

C₄

yes

characteristic

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₁₆×Dic₇₁

C₄

yes

characteristic

#364×C₇₁

C₂²

yes

—

#346×C₇₁

C₂²

yes

characteristic

C₂×#11765848

D₄

yes

characteristic

C₂×C₈×D₇₁

C₂×C₄

yes

characteristic

#14556085

Q₈

yes

characteristic

#14556068

—

#14556067

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₁₆×D₇₁

—

#14556039

Q₈

yes

characteristic

#14556038

D₄

yes

characteristic

C₈×Dic₇₁

C₂×C₄

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄×D₇₁

C₄⋊C₄

yes

characteristic

#11765848

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₈×D₇₁

—

C₈×D₇₁

—

Dic₇₁⋊C₄

—

(C₂×D₇₁)⋊C₄

—

C₄×Dic₇₁

C₄⋊C₄

yes

characteristic

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

M₁₆

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₁₆×C₇₁

M₁₆

yes

—

C₂²×D₇₁

#108

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

#108

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₈×C₇₁

#135

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×#108

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

#12027

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1568

yes

characteristic

D₇₁

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₇₁

#12027

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#364

—

#364

C₂×D₇₁

yes

—

#349

—

142

—

#347

—

142

—

#346

—

#346

C₂×D₇₁

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

—

142

—

M₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

142

—

100

C₂×C₁₆

C₄×D₇₁

yes

characteristic

101

C₂×C₁₆

C₄×D₇₁

yes

characteristic

102

#97

—

103

#97

—

104

C₄×C₈

—

105

C₄×C₈

C₂²×D₇₁

yes

characteristic

106

C₂²×C₄

—

107

M₁₆

—

142

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

142

—

111

C₂×C₈

—

142

—

112

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

113

C₂×C₈

D₄×D₇₁

yes

characteristic

114

C₄⋊C₄

—

142

—

115

C₂²⋊C₄

—

142

—

116

C₄²

—

117

C₄²

Q₈×D₇₁

yes

characteristic

118

C₁₆

—

119

C₁₆

—

120

C₁₆

—

284

—

121

C₁₆

—

122

C₁₆

—

142

—

123

C₁₆

#11765848

yes

—

124

C₂³

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

142

—

128

C₂×C₄

—

142

—

129

C₂×C₄

—

142

—

130

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

131

C₈

—

142

—

132

C₈

—

142

—

133

C₈

M₁₆×D₇₁

yes

characteristic

134

C₈

C₂×#11765848

yes

characteristic

135

C₈

#14556149

yes

—

136

C₂²

—

284

—

137

C₂²

—

142

—

138

C₂²

#108×D₇₁

yes

characteristic

139

C₄

—

140

C₄

—

284

—

141

C₄

—

142

—

142

C₄

#17986021

yes

characteristic

143

C₄

#17986034

yes

characteristic

144

C₂

—

145

C₂

—

284

—

146

C₂

#22587100

yes

characteristic

	subgroup	G/H
center	C₄	#17986034
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1369	C₂
socle	C₂×C₇₁	#1568
prime radical	C₂²×D₇₁	#108
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1369	D₇₁
hypocenter	C₇₁	#12027
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

subgroup

G/H

center

C₄

#17986034

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1369

C₂

C₂×C₇₁

C₂²×D₇₁

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

N	H ≅ G/N	type
3. #22586415	C₂	semidirect
4. #22586408	C₂	semidirect
5. C₇₁×#1369	C₂	semidirect
78. #1369	D₇₁	semidirect
79. C₇₁	#12027	semidirect

H ≅ G/N

type

3. #22586415

C₂

semidirect

4. #22586408

C₂

semidirect

5. C₇₁×#1369

C₂

semidirect

78. #1369

D₇₁

semidirect

79. C₇₁

#12027

semidirect

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

subgroup	index
#22590155	2
#22589938	2
#22586415	2
#22586408	2
C₇₁×#1369	2
#12027	71

subgroup

index

C₇₁×#1369

Subgroups — #31909111

Conjugacy classes of subgroups

Complements

Maximal subgroups