Group #93613 — subgroups

Subgroups — #93613

Order 512.

159 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 159 conjugacy classes of subgroups — too many to draw. Only its 56 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	#12896	C₂	1	1	yes	maximal, characteristic
2	#12882	C₂	1	1	yes	maximal, characteristic
3	#8269	C₂	1	1	yes	maximal, characteristic
4	C₂×#364	—	2	1	no	—
5	C₄×#97	C₂²	1	1	yes	characteristic
6	#1564	—	2	1	no	—
7	#1452	—	2	1	no	—
8	#1377	—	2	1	no	—
9	#1377	C₄	1	2	yes	—
10	#1376	—	2	1	no	—
11	#1375	—	2	1	no	—
12	#1375	C₄	1	1	yes	characteristic
13	#1375	C₄	1	1	yes	characteristic
14	#1276	C₄	1	2	yes	—
15	C₂×M₃₂	—	2	2	no	—
16	C₂×M₃₂	—	2	1	no	—
17	C₂×M₃₂	—	2	1	no	—
18	#431	—	2	2	no	—
19	C₂×#97	C₂×C₄	1	1	yes	characteristic
20	C₂×#97	C₂×C₄	1	1	yes	characteristic
21	C₂×#97	D₄	1	1	yes	characteristic
22	C₂×#97	Q₈	1	1	yes	characteristic
23	C₂×C₄×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	C₄³	C₂×C₄	1	1	yes	characteristic
26	#364	—	4	1	no	—
27	#364	—	4	1	no	—
28	#335	—	2	2	no	—
29	#334	—	2	2	no	—
30	#322	—	4	1	no	—
31	#322	—	4	1	no	—
32	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄²	—	4	1	no	—
37	C₂×C₄²	C₄²	1	1	yes	characteristic
38	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
40	M₃₂	—	4	2	no	—
41	M₃₂	—	4	1	no	—
42	M₃₂	—	4	1	no	—
43	M₃₂	—	4	2	no	—
44	M₃₂	—	4	1	no	—
45	M₃₂	—	4	1	no	—
46	C₂×C₁₆	—	4	1	no	—
47	C₂×C₁₆	—	4	1	no	—
48	C₂×C₁₆	—	4	2	no	—
49	C₄⋊C₈	—	2	4	no	—
50	C₂²⋊C₈	—	4	2	no	—
51	#97	—	2	2	no	—
52	#97	—	4	1	no	—
53	#97	—	4	1	no	—
54	#97	—	4	1	no	—
55	C₄×C₈	—	2	1	no	—
56	C₄×C₈	—	2	1	no	—
57	C₄×C₈	—	4	2	no	—
58	C₄×C₈	D₈	1	1	yes	characteristic
59	C₄×C₈	SD₁₆	1	1	yes	characteristic
60	C₄×C₈	SD₁₆	1	1	yes	characteristic
61	C₄×C₈	Q₁₆	1	1	yes	characteristic
62	C₂²×C₄	—	2	1	no	—
63	C₂²×C₄	—	2	1	no	—
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	#95	1	1	yes	characteristic
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	4	4	no	—
75	C₂×C₈	#102	1	1	yes	characteristic
76	C₂×C₈	#103	1	1	yes	characteristic
77	C₂×C₈	#104	1	1	yes	characteristic
78	C₂×C₈	#104	1	1	yes	characteristic
79	C₂×C₈	#106	1	1	yes	characteristic
80	C₂×C₈	#107	1	1	yes	characteristic
81	C₂×C₈	#108	1	2	yes	—
82	C₄²	—	2	1	no	—
83	C₄²	—	2	1	no	—
84	C₄²	—	2	1	no	—
85	C₄²	—	2	1	no	—
86	C₄²	—	2	1	no	—
87	C₄²	—	8	2	no	—
88	C₄²	#102	1	1	yes	characteristic
89	C₄²	#103	1	1	yes	characteristic
90	C₁₆	—	8	2	no	—
91	C₁₆	—	8	1	no	—
92	C₁₆	—	8	1	no	—
93	C₂³	#344	1	1	yes	characteristic
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	8	1	no	—
100	C₂×C₄	—	8	1	no	—
101	C₂×C₄	H(3,ℤ/4)	1	1	yes	characteristic
102	C₂×C₄	#339	1	1	yes	characteristic
103	C₂×C₄	#340	1	1	yes	characteristic
104	C₂×C₄	#341	1	1	yes	characteristic
105	C₂×C₄	#344	1	1	yes	characteristic
106	C₂×C₄	#344	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	2	1	no	—
111	C₈	—	2	1	no	—
112	C₈	—	2	1	no	—
113	C₈	—	2	2	no	—
114	C₈	—	8	4	no	—
115	C₂²	—	4	1	no	—
116	C₂²	#1287	1	1	yes	characteristic
117	C₂²	#1303	1	1	yes	characteristic
118	C₂²	#1308	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	4	1	no	—
121	C₄	—	8	1	no	—
122	C₄	—	8	1	no	—
123	C₄	#1286	1	1	yes	characteristic
124	C₄	#1288	1	1	yes	characteristic
125	C₄	#1301	1	1	yes	characteristic
126	C₄	#1302	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#7041	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12896

C₂

yes

maximal, characteristic

#12882

C₂

yes

maximal, characteristic

#8269

C₂

yes

maximal, characteristic

C₂×#364

—

C₄×#97

C₂²

yes

—

—

—

C₄

yes

—

#1376

—

#1375

—

#1375

C₄

yes

characteristic

#1375

C₄

yes

characteristic

#1276

C₄

yes

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

#431

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

Q₈

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₄³

C₂×C₄

yes

—

—

—

—

—

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄²

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#95

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#344

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

H(3,ℤ/4)

yes

characteristic

102

C₂×C₄

#339

yes

characteristic

103

C₂×C₄

#340

yes

characteristic

104

C₂×C₄

#341

yes

characteristic

105

C₂×C₄

#344

yes

characteristic

106

C₂×C₄

#344

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

#1287

yes

characteristic

117

C₂²

#1303

yes

characteristic

118

C₂²

#1308

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#1286

yes

characteristic

124

C₄

#1288

yes

characteristic

125

C₄

#1301

yes

characteristic

126

C₄

#1302

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#7041

yes

characteristic

	subgroup	G/H
center	C₄	#1286
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₄×#97	C₂²
socle	C₂	#7041
prime radical	C₂³	#344
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₄×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#1286

commutator

C₂×C₄²

C₄²

Frattini, cosocle

C₄×#97

C₂²

C₂

C₂³

G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G

LECS

G ▷ C₄×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#12896	2
#12882	2
#8269	2

subgroup

index

#12896

#12882

#8269

Subgroups — #93613

Conjugacy classes of subgroups

Maximal subgroups