Group #31909476 — subgroups

Subgroups — #31909476

Order 18176.

226 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 69 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	#22590217	C₂	1	1	yes	maximal, characteristic
2	#22587253	C₂	1	2	yes	maximal, complemented
3	#22586306	C₂	1	2	yes	maximal, complemented
4	#22586260	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1368	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	#17985827	C₄	1	1	yes	characteristic
10	#17985801	C₄	1	2	yes	—
11	#17985775	C₂²	1	1	yes	characteristic
12	#17985726	—	2	2	no	—
13	#17985720	C₂²	1	1	yes	characteristic
14	#346×C₇₁	C₂²	1	1	yes	characteristic
15	#334×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	2	yes	—
25	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₇₁	—	2	2	no	—
28	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×D₂₈₄	—	2	2	no	—
30	C₂×C₄×D₇₁	—	4	2	no	—
31	D₅₆₈	—	2	2	no	—
32	D₅₆₈	—	2	2	no	—
33	#11765848	—	8	4	no	—
34	C₂×C₇₁⋊C₈	—	2	2	no	—
35	C₄⋊Dic₇₁	—	2	2	no	—
36	(C₂×D₇₁)⋊C₄	—	4	2	no	—
37	C₇₁⋊C₁₆	—	4	1	no	—
38	C₂×C₈×C₇₁	—	4	2	no	—
39	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
41	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₁	—	2	1	no	—
43	C₁₆×C₇₁	M₁₆	1	2	yes	complemented
44	C₂²×D₇₁	—	4	2	no	—
45	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×Dic₇₁	—	4	2	no	—
48	D₂₈₄	—	4	2	no	—
49	D₂₈₄	—	4	2	no	—
50	C₄×D₇₁	—	8	2	no	—
51	C₄×D₇₁	—	8	2	no	—
52	C₈×C₇₁	—	2	1	no	—
53	C₈×C₇₁	—	8	2	no	—
54	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
55	C₈×C₇₁	#131	1	1	yes	characteristic
56	C₇₁⋊C₈	—	4	2	no	—
57	C₂²×C₇₁	#442	1	1	yes	characteristic
58	C₂×D₇₁	—	8	2	no	—
59	C₂×D₇₁	—	8	2	no	—
60	C₂×D₇₁	—	8	2	no	—
61	C₄×C₇₁	—	2	2	no	—
62	C₄×C₇₁	#436	1	1	yes	characteristic
63	C₄×C₇₁	#444	1	1	yes	characteristic
64	Dic₇₁	—	8	2	no	—
65	#12089	—	71	1	no	maximal, Sylow
66	C₂×C₇₁	#1594	1	1	yes	characteristic
67	C₂×C₇₁	AGL(1,ℤ/16)	1	2	yes	—
68	D₇₁	—	16	2	no	—
69	#2177	—	71	1	no	—
70	#1594	—	71	2	no	—
71	#1368	D₇₁	1	1	yes	characteristic, complemented
72	#1333	—	71	2	no	—
73	#1313	—	71	1	no	—
74	C₇₁	#12089	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₁₆	—	71	1	no	—
76	C₄×D₈	—	71	2	no	—
77	#436	—	142	2	no	—
78	#367	—	71	1	no	—
79	#357	—	71	2	no	—
80	#346	—	71	1	no	—
81	#346	C₂×D₇₁	1	1	yes	characteristic
82	#334	C₂×D₇₁	1	2	yes	—
83	D₄⋊C₈	—	142	2	no	—
84	#322	—	71	1	no	—
85	C₂×D₈	—	71	2	no	—
86	C₂×M₁₆	—	284	2	no	—
87	C₄×D₄	—	142	2	no	—
88	D₁₆	—	142	2	no	—
89	C₂×C₁₆	—	142	1	no	—
90	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
91	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
92	#107	—	71	2	no	—
93	C₄⋊C₈	—	2	2	no	—
94	#102	—	142	2	no	—
95	C₂²⋊C₈	—	284	2	no	—
96	C₄×C₈	—	71	2	no	—
97	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
98	C₂×D₄	—	142	2	no	—
99	C₂²×C₄	—	284	2	no	—
100	D₈	—	142	2	no	—
101	D₈	—	142	2	no	—
102	M₁₆	—	568	4	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	142	2	no	—
105	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
106	C₂×C₈	#11765866	1	1	yes	characteristic
107	C₄⋊C₄	—	142	2	no	—
108	C₂²⋊C₄	—	284	2	no	—
109	C₄²	D₄×D₇₁	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	284	1	no	—
112	C₁₆	#11765848	1	2	yes	complemented
113	C₂³	—	284	2	no	—
114	D₄	—	284	2	no	—
115	D₄	—	284	2	no	—
116	C₂×C₄	—	284	2	no	—
117	C₂×C₄	—	568	2	no	—
118	C₂×C₄	—	568	2	no	—
119	C₂×C₄	#14556090	1	1	yes	characteristic
120	C₂×C₄	#14556155	1	2	yes	—
121	C₈	—	2	1	no	—
122	C₈	—	8	2	no	—
123	C₈	—	284	2	no	—
124	C₈	C₂×#11765848	1	1	yes	characteristic
125	C₈	#14556136	1	1	yes	characteristic
126	C₂²	—	568	2	no	—
127	C₂²	—	568	2	no	—
128	C₂²	—	568	2	no	—
129	C₂²	#17986110	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	568	2	no	—
132	C₄	#17986076	1	1	yes	characteristic
133	C₄	#17986120	1	1	yes	characteristic
134	C₂	—	1136	2	no	—
135	C₂	#22587253	1	1	yes	characteristic
136	C₂	#22590237	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590217

C₂

yes

maximal, characteristic

#22587253

C₂

yes

maximal, complemented

#22586306

C₂

yes

maximal, complemented

#22586260

C₂

yes

maximal, characteristic, complemented

C₇₁×#1368

C₂

yes

maximal, characteristic, complemented

C₂×D₁₁₃₆

C₄

yes

characteristic

C₄×D₅₆₈

C₂²

yes

—

#17986076

—

#17985827

C₄

yes

characteristic

#17985801

C₄

yes

—

#17985775

C₂²

yes

characteristic

#17985726

—

#17985720

C₂²

yes

characteristic

#346×C₇₁

C₂²

yes

characteristic

#334×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

—

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

—

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

AGL(1,ℤ/8)

yes

—

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

—

C₂²×C₇₁

#442

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

—

#2177

—

#1594

—

#1368

D₇₁

yes

characteristic, complemented

#1333

—

#1313

—

C₇₁

#12089

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₄×D₈

—

#436

—

142

—

—

—

—

C₂×D₇₁

yes

characteristic

#334

C₂×D₇₁

yes

—

D₄⋊C₈

—

142

—

#322

—

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

—

#102

—

142

—

C₂²⋊C₈

—

284

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×D₄

—

142

—

C₂²×C₄

—

284

—

100

D₈

—

142

—

101

D₈

—

142

—

102

M₁₆

—

568

—

103

C₂×C₈

—

104

C₂×C₈

—

142

—

105

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

106

C₂×C₈

#11765866

yes

characteristic

107

C₄⋊C₄

—

142

—

108

C₂²⋊C₄

—

284

—

109

C₄²

D₄×D₇₁

yes

characteristic

110

C₁₆

—

111

C₁₆

—

284

—

112

C₁₆

#11765848

yes

complemented

113

C₂³

—

284

—

114

D₄

—

284

—

115

D₄

—

284

—

116

C₂×C₄

—

284

—

117

C₂×C₄

—

568

—

118

C₂×C₄

—

568

—

119

C₂×C₄

#14556090

yes

characteristic

120

C₂×C₄

#14556155

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

284

—

124

C₈

C₂×#11765848

yes

characteristic

125

C₈

#14556136

yes

characteristic

126

C₂²

—

568

—

127

C₂²

—

568

—

128

C₂²

—

568

—

129

C₂²

#17986110

yes

characteristic

130

C₄

—

131

C₄

—

568

—

132

C₄

#17986076

yes

characteristic

133

C₄

#17986120

yes

characteristic

134

C₂

—

1136

—

135

C₂

#22587253

yes

characteristic

136

C₂

#22590237

yes

—

	subgroup	G/H
center	C₂²	#17986110
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1368	C₂
socle	C₂²×C₇₁	#442
prime radical	C₂×D₁₁₃₆	C₄
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1368	D₇₁
hypocenter	C₇₁	#12089
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#17986110

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1368

C₂

socle

C₂²×C₇₁

#442

prime radical

C₂×D₁₁₃₆

C₄

cosocle

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
2. #22587253	C₂	semidirect
3. #22586306	C₂	semidirect
4. #22586260	C₂	semidirect
5. C₇₁×#1368	C₂	semidirect
43. C₁₆×C₇₁	M₁₆	semidirect
71. #1368	D₇₁	semidirect
74. C₇₁	#12089	semidirect
112. C₁₆	#11765848	semidirect

H ≅ G/N

type

2. #22587253

C₂

semidirect

3. #22586306

C₂

semidirect

4. #22586260

C₂

semidirect

5. C₇₁×#1368

C₂

semidirect

43. C₁₆×C₇₁

M₁₆

semidirect

71. #1368

D₇₁

semidirect

74. C₇₁

#12089

semidirect

112. C₁₆

#11765848

semidirect

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

subgroup	index
#22590217	2
#22587253	2
#22586306	2
#22586260	2
C₇₁×#1368	2
#12089	71

subgroup

index

C₇₁×#1368

Subgroups — #31909476

Conjugacy classes of subgroups

Complements

Maximal subgroups