Group #10636 — subgroups

Subgroups — #10636

Order 256.

139 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 139 conjugacy classes of subgroups — too many to draw. Only its 49 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	#1838	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	#423	C₂²	1	1	yes	characteristic
12	#423	C₂²	1	1	yes	characteristic
13	C₂×#97	C₂²	1	1	yes	characteristic
14	#382	C₂²	1	1	yes	characteristic
15	#341	—	2	1	no	—
16	#341	—	2	1	no	—
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	C₄⋊C₈	—	2	1	no	—
35	C₄⋊C₈	—	2	1	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	C₂²⋊C₈	—	2	1	no	—
41	C₂²⋊C₈	—	2	1	no	—
42	#97	—	2	1	no	—
43	#97	—	2	1	no	—
44	C₄×C₈	—	2	1	no	—
45	C₄×C₈	—	2	1	no	—
46	#95	—	2	1	no	—
47	#95	—	2	1	no	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×Q₈	—	4	1	no	—
50	C₂×Q₈	—	8	2	no	—
51	C₂²×C₄	—	4	1	no	—
52	C₂²×C₄	—	4	1	no	—
53	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄	#41	1	1	yes	characteristic
55	C₂²×C₄	#41	1	1	yes	characteristic
56	M₁₆	—	4	1	no	—
57	M₁₆	—	4	1	no	—
58	M₁₆	—	4	1	no	—
59	M₁₆	—	4	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	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	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	4	1	no	—
73	C₄⋊C₄	—	4	1	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄⋊C₄	—	4	1	no	—
76	C₄²	C₂×D₄	1	1	yes	characteristic
77	C₄²	#41	1	1	yes	characteristic
78	C₄²	#41	1	1	yes	characteristic
79	C₄²	#41	1	1	yes	characteristic
80	C₂³	#124	1	1	yes	characteristic
81	Q₈	—	8	2	no	—
82	Q₈	—	8	2	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	—	8	2	no	—
87	C₂×C₄	—	8	1	no	—
88	C₂×C₄	—	8	1	no	—
89	C₂×C₄	—	8	1	no	—
90	C₂×C₄	#121	1	1	yes	characteristic
91	C₂×C₄	#123	1	1	yes	characteristic
92	C₂×C₄	#123	1	1	yes	characteristic
93	C₂×C₄	#123	1	1	yes	characteristic
94	C₂×C₄	#126	1	1	yes	characteristic
95	C₂×C₄	#126	1	1	yes	characteristic
96	C₂×C₄	#135	1	1	yes	characteristic
97	C₂×C₄	#135	1	1	yes	characteristic
98	C₂×C₄	#137	1	1	yes	characteristic
99	C₂×C₄	#137	1	1	yes	characteristic
100	C₈	—	4	1	no	—
101	C₈	—	4	1	no	—
102	C₈	—	8	1	no	—
103	C₈	—	8	1	no	—
104	C₈	—	8	1	no	—
105	C₈	—	8	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#397	1	1	yes	characteristic
109	C₂²	#488	1	1	yes	characteristic
110	C₂²	#491	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	8	2	no	—
116	C₄	—	16	1	no	—
117	C₄	#464	1	1	yes	characteristic
118	C₄	#464	1	1	yes	characteristic
119	C₄	#485	1	1	yes	characteristic
120	C₄	#485	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	2	1	no	—
123	C₂	#2074	1	1	yes	characteristic
124	C₂	#2081	1	1	yes	characteristic
125	C₂	#2082	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1993

C₂

yes

maximal, characteristic

#1838

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

characteristic

#423

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

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

—

C₂×C₈

—

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

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₂²

—

107

C₂²

—

108

C₂²

#397

yes

characteristic

109

C₂²

#488

yes

characteristic

110

C₂²

#491

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

#464

yes

characteristic

118

C₄

#464

yes

characteristic

119

C₄

#485

yes

characteristic

120

C₄

#485

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

#2074

yes

characteristic

124

C₂

#2081

yes

characteristic

125

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
#1838	2
#1549	2
#1548	2
#1286	2
#1281	2

subgroup

index

Subgroups — #10636

Conjugacy classes of subgroups

Maximal subgroups