Group #31896953 — subgroups

Subgroups — #31896953

Order 18176.

454 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 127 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	#22588478	C₂	1	1	yes	maximal, characteristic
2	#22587012	C₂	1	4	yes	maximal
3	#22586632	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17986052	C₂²	1	2	yes	—
6	#17986023	C₂²	1	2	yes	—
7	#17985764	—	2	2	no	—
8	#17985745	—	2	8	no	—
9	#17985721	—	2	2	no	—
10	C₂²×Q₁₆×C₇₁	—	2	2	no	—
11	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
12	#494×C₇₁	—	2	2	no	—
13	#494×C₇₁	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#11765849	—	2	2	no	—
16	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
17	C₄⋊(C₇₁⋊C₈)	D₄	1	4	yes	—
18	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
19	C₄×C₇₁⋊C₈	—	2	2	no	—
20	C₂²×Q₈×C₇₁	—	2	2	no	—
21	C₂×Q₁₆×C₇₁	—	2	8	no	—
22	C₂×Q₁₆×C₇₁	—	4	8	no	—
23	C₂²×C₈×C₇₁	D₄	1	2	yes	—
24	C₄⋊Q₈×C₇₁	—	2	4	no	—
25	C₄⋊Q₈×C₇₁	—	2	2	no	—
26	C₄⋊Q₈×C₇₁	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
28	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₇₁	D₄	1	2	yes	—
31	#11765849	—	4	4	no	—
32	C₂×C₇₁⋊C₈	—	2	4	no	—
33	C₂×C₇₁⋊C₈	—	4	2	no	—
34	C₂×Q₈×C₇₁	—	2	4	no	—
35	C₂×Q₈×C₇₁	—	4	2	no	—
36	C₂×Q₈×C₇₁	—	4	8	no	—
37	C₂²×C₄×C₇₁	—	4	2	no	—
38	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₁	#41	1	2	yes	—
40	Q₁₆×C₇₁	—	4	16	no	—
41	C₂×C₈×C₇₁	—	2	4	no	—
42	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
43	C₄⋊C₄×C₇₁	—	2	8	no	—
44	C₄⋊C₄×C₇₁	—	4	4	no	—
45	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₇₁	C₂×D₄	1	2	yes	—
47	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂³×C₇₁	#124	1	1	yes	characteristic
49	Q₈×C₇₁	—	4	8	no	—
50	Q₈×C₇₁	—	8	4	no	—
51	C₂×C₄×C₇₁	—	2	4	no	—
52	C₂×C₄×C₇₁	—	4	4	no	—
53	C₂×C₄×C₇₁	—	8	4	no	—
54	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
56	C₂×C₄×C₇₁	#121	1	2	yes	—
57	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
58	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	4	yes	—
59	C₈×C₇₁	—	2	4	no	—
60	C₇₁⋊C₈	—	4	4	no	—
61	C₇₁⋊C₈	—	8	2	no	—
62	C₂²×C₇₁	—	2	2	no	—
63	C₂²×C₇₁	#390	1	1	yes	characteristic
64	C₂²×C₇₁	#490	1	2	yes	—
65	C₄×C₇₁	—	2	2	no	—
66	C₄×C₇₁	—	8	4	no	—
67	C₄×C₇₁	#418	1	2	yes	—
68	C₄×C₇₁	#473	1	4	yes	—
69	C₄×C₇₁	#496	1	2	yes	—
70	#9779	—	71	1	no	maximal, Sylow
71	C₂×C₇₁	—	2	2	no	—
72	C₂×C₇₁	#1961	1	1	yes	characteristic
73	C₂×C₇₁	#1979	1	2	yes	—
74	C₂×#494	D₇₁	1	1	yes	characteristic
75	#1837	—	71	1	no	—
76	#1546	—	71	4	no	—
77	#1452	—	71	1	no	—
78	C₇₁	#9779	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₁₆	—	2	2	no	—
80	C₂×C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
81	#494	—	2	2	no	—
82	#494	Dic₇₁	1	4	yes	—
83	#431	—	71	2	no	—
84	#423	—	71	2	no	—
85	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
86	#341	—	142	2	no	—
87	#332	—	142	8	no	—
88	#322	—	142	2	no	—
89	C₂²×Q₈	—	2	2	no	—
90	C₂×Q₁₆	—	2	8	no	—
91	C₂×Q₁₆	—	4	8	no	—
92	C₂×M₁₆	—	142	2	no	—
93	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
94	C₄⋊Q₈	—	2	4	no	—
95	C₄⋊Q₈	—	2	2	no	—
96	C₄⋊Q₈	C₂×Dic₇₁	1	4	yes	—
97	C₂×C₄⋊C₄	—	2	4	no	—
98	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
99	C₄⋊C₈	—	71	4	no	—
100	C₄⋊C₈	—	142	4	no	—
101	C₂²⋊C₈	—	142	2	no	—
102	C₄×C₈	—	142	2	no	—
103	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
104	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	4	8	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
110	C₂²×C₄	#11765864	1	2	yes	—
111	Q₁₆	—	4	16	no	—
112	M₁₆	—	284	4	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	284	2	no	—
115	C₂×C₈	—	142	4	no	—
116	C₂×C₈	C₂²⋊Dic₇₁	1	4	yes	—
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	—	4	4	no	—
119	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
120	C₄²	D₄×D₇₁	1	2	yes	—
121	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
122	C₂³	#14556115	1	1	yes	characteristic
123	Q₈	—	4	8	no	—
124	Q₈	—	8	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	—	8	4	no	—
128	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
129	C₂×C₄	D₄×Dic₇₁	1	2	yes	—
130	C₂×C₄	#14556099	1	2	yes	—
131	C₂×C₄	#14556124	1	1	yes	characteristic
132	C₂×C₄	#14556154	1	4	yes	—
133	C₈	—	2	4	no	—
134	C₈	—	284	4	no	—
135	C₈	—	568	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	#17985902	1	1	yes	characteristic
138	C₂²	#17986363	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	4	no	—
141	C₄	#17986004	1	2	yes	—
142	C₄	#17986271	1	4	yes	—
143	C₄	#17986386	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#22589165	1	1	yes	characteristic
146	C₂	#22589256	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588478

C₂

yes

maximal, characteristic

#22587012

C₂

yes

maximal

#22586632

C₂

yes

maximal, characteristic

C₂×#494×C₇₁

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

#17986023

C₂²

yes

—

#17985764

—

#17985745

—

#17985721

—

C₂²×Q₁₆×C₇₁

—

C₂×C₄⋊Q₈×C₇₁

C₂²

yes

—

#494×C₇₁

—

#494×C₇₁

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×#11765849

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

D₄

yes

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₂²×Q₈×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₂³×C₇₁

#124

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

—

C₂×C₄×C₇₁

#121

yes

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

#390

yes

characteristic

C₂²×C₇₁

#490

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#418

yes

—

C₄×C₇₁

#473

yes

—

C₄×C₇₁

#496

yes

—

#9779

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1961

yes

characteristic

C₂×C₇₁

#1979

yes

—

C₂×#494

D₇₁

yes

—

—

—

C₇₁

#9779

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₇₁

yes

—

#494

—

#494

Dic₇₁

yes

—

#431

—

#423

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#341

—

142

—

#332

—

142

—

#322

—

142

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

142

—

C₂²×C₈

C₂²⋊D₇₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×Dic₇₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

142

—

101

C₂²⋊C₈

—

142

—

102

C₄×C₈

—

142

—

103

C₄×C₈

C₂×Dic₇₁

yes

—

104

C₄×C₈

C₂²⋊D₇₁

yes

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

110

C₂²×C₄

#11765864

yes

—

111

Q₁₆

—

112

M₁₆

—

284

—

113

C₂×C₈

—

114

C₂×C₈

—

284

—

115

C₂×C₈

—

142

—

116

C₂×C₈

C₂²⋊Dic₇₁

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

C₂²×Dic₇₁

yes

characteristic

120

C₄²

D₄×D₇₁

yes

—

121

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

122

C₂³

#14556115

yes

characteristic

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

129

C₂×C₄

D₄×Dic₇₁

yes

—

130

C₂×C₄

#14556099

yes

—

131

C₂×C₄

#14556124

yes

characteristic

132

C₂×C₄

#14556154

yes

—

133

C₈

—

134

C₈

—

284

—

135

C₈

—

568

—

136

C₂²

—

137

C₂²

#17985902

yes

characteristic

138

C₂²

#17986363

yes

—

139

C₄

—

140

C₄

—

141

C₄

#17986004

yes

—

142

C₄

#17986271

yes

—

143

C₄

#17986386

yes

—

144

C₂

—

145

C₂

#22589165

yes

characteristic

146

C₂

#22589256

yes

—

	subgroup	G/H
center	C₂²	#17985902
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#494×C₇₁	C₂
socle	C₂²×C₇₁	#390
prime radical	C₂³×C₇₁	#124
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#494	D₇₁
hypocenter	C₇₁	#9779
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂²

#17985902

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₂×#494×C₇₁

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
78. C₇₁	#9779	semidirect

H ≅ G/N

type

78. C₇₁

#9779

semidirect

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

subgroup	index
#22588478	2
#22587012	2
#22586632	2
C₂×#494×C₇₁	2
#9779	71

subgroup

index

C₂×#494×C₇₁

Subgroups — #31896953

Conjugacy classes of subgroups

Complements

Maximal subgroups