Group #10633 — subgroups

Subgroups — #10633

Order 256.

147 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 147 conjugacy classes of subgroups — too many to draw. Only its 53 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	#1993	C₂	1	1	yes	maximal, characteristic
2	#1849	C₂	1	1	yes	maximal, characteristic
3	#1549	C₂	1	1	yes	maximal, characteristic
4	#1548	C₂	1	1	yes	maximal, characteristic
5	#1286	C₂	1	1	yes	maximal, characteristic
6	#1286	C₂	1	1	yes	maximal, characteristic
7	#1281	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#431	C₂²	1	1	yes	characteristic
10	#431	C₂²	1	1	yes	characteristic
11	C₂×#108	—	2	1	no	—
12	C₂×#108	—	2	1	no	—
13	#423	C₂²	1	1	yes	characteristic
14	#423	C₂²	1	1	yes	characteristic
15	C₂×#97	C₂²	1	1	yes	characteristic
16	#382	C₂²	1	1	yes	characteristic
17	#340	—	2	1	no	—
18	#340	—	2	1	no	—
19	#333	—	2	2	no	—
20	#332	—	2	2	no	—
21	C₂²×Q₈	—	2	1	no	—
22	C₂×M₁₆	—	2	1	no	—
23	C₂×M₁₆	—	2	1	no	—
24	C₂²×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₈	D₄	1	1	yes	characteristic
26	C₄⋊Q₈	—	2	1	no	—
27	C₄⋊Q₈	—	2	1	no	—
28	C₄⋊Q₈	—	2	1	no	—
29	C₄⋊Q₈	D₄	1	2	yes	—
30	C₂×C₄⋊C₄	—	2	1	no	—
31	C₂×C₄⋊C₄	—	2	1	no	—
32	C₂×C₄⋊C₄	—	2	1	no	—
33	C₂×C₄²	C₂³	1	1	yes	characteristic
34	#108	—	4	2	no	—
35	#108	—	4	2	no	—
36	C₄⋊C₈	—	2	1	no	—
37	C₄⋊C₈	—	2	1	no	—
38	C₄⋊C₈	—	2	1	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₄⋊C₈	—	2	1	no	—
41	C₄⋊C₈	—	2	1	no	—
42	C₂²⋊C₈	—	2	1	no	—
43	C₂²⋊C₈	—	2	1	no	—
44	#97	—	2	1	no	—
45	#97	—	2	1	no	—
46	C₄×C₈	—	2	1	no	—
47	C₄×C₈	—	2	1	no	—
48	#95	—	2	1	no	—
49	#95	—	2	1	no	—
50	C₂×Q₈	—	2	2	no	—
51	C₂×Q₈	—	4	1	no	—
52	C₂×Q₈	—	8	2	no	—
53	C₂²×C₄	—	4	1	no	—
54	C₂²×C₄	—	4	1	no	—
55	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄	#41	1	1	yes	characteristic
57	C₂²×C₄	#41	1	1	yes	characteristic
58	M₁₆	—	4	1	no	—
59	M₁₆	—	4	1	no	—
60	M₁₆	—	4	1	no	—
61	M₁₆	—	4	1	no	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	—	4	1	no	—
67	C₂×C₈	—	4	1	no	—
68	C₂×C₈	—	4	1	no	—
69	C₂×C₈	—	4	1	no	—
70	C₂×C₈	SD₁₆	1	2	yes	—
71	C₂×C₈	Q₁₆	1	2	yes	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄⋊C₄	—	4	1	no	—
76	C₄⋊C₄	—	4	1	no	—
77	C₄⋊C₄	—	4	1	no	—
78	C₄²	C₂×D₄	1	1	yes	characteristic
79	C₄²	#41	1	1	yes	characteristic
80	C₄²	#41	1	1	yes	characteristic
81	C₄²	#41	1	1	yes	characteristic
82	C₂³	#124	1	1	yes	characteristic
83	Q₈	—	8	2	no	—
84	Q₈	—	8	2	no	—
85	C₂×C₄	—	4	1	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	8	2	no	—
89	C₂×C₄	—	8	1	no	—
90	C₂×C₄	—	8	1	no	—
91	C₂×C₄	—	8	1	no	—
92	C₂×C₄	#121	1	1	yes	characteristic
93	C₂×C₄	#123	1	1	yes	characteristic
94	C₂×C₄	#123	1	1	yes	characteristic
95	C₂×C₄	#123	1	1	yes	characteristic
96	C₂×C₄	#126	1	1	yes	characteristic
97	C₂×C₄	#126	1	1	yes	characteristic
98	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄	#137	1	1	yes	characteristic
102	C₈	—	2	2	no	—
103	C₈	—	2	2	no	—
104	C₈	—	8	1	no	—
105	C₈	—	8	1	no	—
106	C₈	—	8	1	no	—
107	C₈	—	8	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	—	2	1	no	—
110	C₂²	#397	1	1	yes	characteristic
111	C₂²	#487	1	1	yes	characteristic
112	C₂²	#489	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	1	no	—
116	C₄	—	2	1	no	—
117	C₄	—	8	2	no	—
118	C₄	—	16	1	no	—
119	C₄	#460	1	1	yes	characteristic
120	C₄	#462	1	1	yes	characteristic
121	C₄	#483	1	1	yes	characteristic
122	C₄	#484	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#2072	1	1	yes	characteristic
126	C₂	#2081	1	1	yes	characteristic
127	C₂	#2082	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1993

C₂

yes

maximal, characteristic

#1849

C₂

yes

maximal, characteristic

#1549

C₂

yes

maximal, characteristic

#1548

C₂

yes

maximal, characteristic

#1286

C₂

yes

maximal, characteristic

#1286

C₂

yes

maximal, characteristic

#1281

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#431

C₂²

yes

—

—

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×#97

C₂²

yes

characteristic

#382

C₂²

yes

—

—

—

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

characteristic

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₂³

yes

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

characteristic

100

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

101

C₂×C₄

#137

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₂²

—

109

C₂²

—

110

C₂²

#397

yes

characteristic

111

C₂²

#487

yes

characteristic

112

C₂²

#489

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#460

yes

characteristic

120

C₄

#462

yes

characteristic

121

C₄

#483

yes

characteristic

122

C₄

#484

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#2072

yes

characteristic

126

C₂

#2081

yes

characteristic

127

C₂

#2082

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#397

commutator, Frattini, cosocle

C₂×C₄²

C₂³

prime radical

C₂³

#124

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ 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
#1993	2
#1849	2
#1549	2
#1548	2
#1286	2
#1281	2

subgroup

index

Subgroups — #10633

Conjugacy classes of subgroups

Maximal subgroups