Group #9727 — subgroups

Subgroups — #9727

Order 256.

196 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 65 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	#3170	C₂	1	1	yes	maximal, characteristic
2	C₄×#108	C₂	1	1	yes	maximal, characteristic
3	#1526	C₂	1	2	yes	maximal
4	#1485	C₂	1	2	yes	maximal
5	#1451	C₂	1	1	yes	maximal, characteristic
6	#533	C₂²	1	2	yes	—
7	#500	C₄	1	1	yes	characteristic
8	#494	C₄	1	1	yes	characteristic
9	#487	C₄	1	2	yes	—
10	#467	—	2	1	no	—
11	#467	—	2	1	no	—
12	C₄×Q₁₆	—	2	1	no	—
13	C₄×Q₁₆	—	2	1	no	—
14	#431	C₂²	1	1	yes	characteristic
15	C₂×#108	—	2	1	no	—
16	C₄×M₁₆	C₂²	1	2	yes	—
17	C₄×M₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	#341	—	2	1	no	—
20	#330	—	2	4	no	—
21	Q₈⋊C₈	—	2	4	no	—
22	#322	—	2	1	no	—
23	#322	—	2	1	no	—
24	C₂×Q₁₆	—	2	1	no	—
25	C₂×Q₁₆	—	2	1	no	—
26	C₂×M₁₆	—	2	1	no	—
27	C₂×M₁₆	—	2	2	no	—
28	C₂²×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₈	D₄	1	1	yes	characteristic
30	C₄⋊Q₈	C₂×C₄	1	2	yes	—
31	C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	#125	—	2	2	no	—
33	C₂²⋊Q₈	—	2	2	no	—
34	C₂²⋊Q₈	—	2	2	no	—
35	C₄×Q₈	—	2	2	no	—
36	C₄×Q₈	—	2	2	no	—
37	#117	—	2	2	no	—
38	C₂×C₄²	C₂³	1	1	yes	characteristic
39	#108	—	2	4	no	—
40	#107	—	2	1	no	—
41	#107	—	2	1	no	—
42	C₄⋊C₈	D₄	1	2	yes	—
43	#103	—	2	2	no	—
44	#103	—	2	2	no	—
45	C₂²⋊C₈	—	2	1	no	—
46	#97	—	2	2	no	—
47	#97	D₄	1	2	yes	—
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	—	2	1	no	—
50	C₄×C₈	—	2	2	no	—
51	C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈	D₄	1	1	yes	characteristic
54	C₄×C₈	D₄	1	1	yes	characteristic
55	C₂×Q₈	—	2	2	no	—
56	C₂×Q₈	—	2	2	no	—
57	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄	#41	1	1	yes	characteristic
59	C₂²×C₄	#41	1	1	yes	characteristic
60	Q₁₆	—	4	2	no	—
61	Q₁₆	—	4	2	no	—
62	M₁₆	—	2	4	no	—
63	M₁₆	—	4	2	no	—
64	M₁₆	—	4	2	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	4	1	no	—
69	C₂×C₈	—	4	2	no	—
70	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	4	2	no	—
79	C₄⋊C₄	—	4	2	no	—
80	C₂²⋊C₄	—	4	2	no	—
81	C₄²	—	4	2	no	—
82	C₄²	C₂²×C₄	1	1	yes	characteristic
83	C₄²	C₂×D₄	1	1	yes	characteristic
84	C₄²	C₂×D₄	1	1	yes	characteristic
85	C₄²	C₂×D₄	1	1	yes	characteristic
86	C₂³	#124	1	1	yes	characteristic
87	Q₈	—	4	2	no	—
88	Q₈	—	4	2	no	—
89	Q₈	—	4	4	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄	#121	1	1	yes	characteristic
100	C₂×C₄	#121	1	1	yes	characteristic
101	C₂×C₄	#127	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	2	1	no	—
104	C₈	—	2	1	no	—
105	C₈	—	4	2	no	—
106	C₈	—	4	4	no	—
107	C₈	—	4	2	no	—
108	C₈	#104	1	2	yes	—
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	#390	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	1	no	—
116	C₄	—	8	2	no	—
117	C₄	—	8	2	no	—
118	C₄	C₂×#104	1	1	yes	characteristic
119	C₄	#421	1	1	yes	characteristic
120	C₄	#444	1	2	yes	—
121	C₄	#473	1	2	yes	—
122	C₂	—	4	1	no	—
123	C₂	#1975	1	1	yes	characteristic
124	C₂	#1979	1	1	yes	characteristic
125	C₂	#1980	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3170

C₂

yes

maximal, characteristic

C₄×#108

C₂

yes

maximal, characteristic

#1526

C₂

yes

maximal

#1485

C₂

yes

maximal

#1451

C₂

yes

maximal, characteristic

#533

C₂²

yes

—

#500

C₄

yes

characteristic

#494

C₄

yes

characteristic

#487

C₄

yes

—

#467

—

#467

—

C₄×Q₁₆

—

C₄×Q₁₆

—

#431

C₂²

yes

characteristic

C₂×#108

—

C₄×M₁₆

C₂²

yes

—

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#341

—

#330

—

Q₈⋊C₈

—

#322

—

#322

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂³

yes

—

—

—

C₄⋊C₈

D₄

yes

—

#103

—

#103

—

C₂²⋊C₈

—

#97

—

#97

D₄

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×D₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₂³

#124

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

#121

yes

characteristic

100

C₂×C₄

#121

yes

characteristic

101

C₂×C₄

#127

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

#104

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

#390

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₂×#104

yes

characteristic

119

C₄

#421

yes

characteristic

120

C₄

#444

yes

—

121

C₄

#473

yes

—

122

C₂

—

123

C₂

#1975

yes

characteristic

124

C₂

#1979

yes

characteristic

125

C₂

#1980

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#390
prime radical	C₂³	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#121

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#3170	2
C₄×#108	2
#1526	2
#1485	2
#1451	2

subgroup

index

Subgroups — #9727

Conjugacy classes of subgroups

Maximal subgroups