Group #10548 — subgroups

Subgroups — #10548

Order 256.

182 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 57 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	#3162	C₂	1	1	yes	maximal, characteristic
2	#1940	C₂	1	1	yes	maximal, characteristic
3	#1542	C₂	1	2	yes	maximal
4	#1503	C₂	1	2	yes	maximal
5	#1275	C₂	1	1	yes	maximal, characteristic
6	#533	C₂²	1	2	yes	—
7	#501	—	2	2	no	—
8	#501	—	2	2	no	—
9	#431	C₂²	1	2	yes	—
10	#427	—	2	1	no	—
11	#427	—	2	1	no	—
12	C₂×C₄⋊C₈	—	2	1	no	—
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×#97	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	#336	—	2	1	no	—
17	#333	—	2	2	no	—
18	#332	—	2	2	no	—
19	#328	—	2	2	no	—
20	#328	—	2	2	no	—
21	C₂²×C₈	—	2	1	no	—
22	C₂²×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₈	D₄	1	1	yes	characteristic
24	C₄⋊Q₈	D₄	1	2	yes	—
25	C₄⋊Q₈	D₄	1	2	yes	—
26	#125	—	2	2	no	—
27	C₂²⋊Q₈	—	2	2	no	—
28	C₂²⋊Q₈	—	2	2	no	—
29	C₄×Q₈	—	2	2	no	—
30	C₄×Q₈	—	2	2	no	—
31	#117	—	2	2	no	—
32	C₂×C₄⋊C₄	—	2	1	no	—
33	C₂×C₄²	—	2	1	no	—
34	C₂×C₄²	C₂³	1	1	yes	characteristic
35	#107	—	2	1	no	—
36	#107	—	2	1	no	—
37	#107	—	2	1	no	—
38	#107	—	2	1	no	—
39	#106	—	2	2	no	—
40	#106	—	2	2	no	—
41	C₄⋊C₈	—	2	2	no	—
42	C₄⋊C₈	—	2	2	no	—
43	C₄⋊C₈	—	4	2	no	—
44	C₂²⋊C₈	—	2	2	no	—
45	#97	—	2	2	no	—
46	C₄×C₈	—	2	2	no	—
47	#95	D₄	1	1	yes	characteristic
48	#95	D₄	1	1	yes	characteristic
49	C₂×Q₈	—	2	2	no	—
50	C₂×Q₈	—	2	2	no	—
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	—	2	1	no	—
53	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄	#41	1	1	yes	characteristic
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	4	1	no	—
61	C₂×C₈	—	4	2	no	—
62	C₂×C₈	—	4	1	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	SD₁₆	1	2	yes	—
65	C₂×C₈	Q₁₆	1	2	yes	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄⋊C₄	—	4	2	no	—
72	C₄⋊C₄	—	4	2	no	—
73	C₂²⋊C₄	—	4	2	no	—
74	C₄²	—	4	2	no	—
75	C₄²	—	4	1	no	—
76	C₄²	—	4	1	no	—
77	C₄²	C₂×D₄	1	2	yes	—
78	C₄²	#41	1	2	yes	—
79	C₂³	#121	1	1	yes	characteristic
80	Q₈	—	8	2	no	—
81	Q₈	—	8	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	—	4	1	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	#120	1	1	yes	characteristic
93	C₂×C₄	#121	1	2	yes	—
94	C₂×C₄	#123	1	2	yes	—
95	C₂×C₄	#124	1	1	yes	characteristic
96	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄	#137	1	1	yes	characteristic
99	C₂×C₄	#137	1	1	yes	characteristic
100	C₈	—	2	2	no	—
101	C₈	—	2	2	no	—
102	C₈	—	8	2	no	—
103	C₈	—	8	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	#394	1	1	yes	characteristic
107	C₂²	#461	1	1	yes	characteristic
108	C₂²	#462	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	2	2	no	—
111	C₄	—	4	2	no	—
112	C₄	—	8	1	no	—
113	C₄	—	8	2	no	—
114	C₄	—	8	2	no	—
115	C₄	#448	1	1	yes	characteristic
116	C₄	#451	1	1	yes	characteristic
117	C₄	#487	1	1	yes	characteristic
118	C₄	#491	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#2043	1	1	yes	characteristic
122	C₂	#2050	1	1	yes	characteristic
123	C₂	#2053	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3162

C₂

yes

maximal, characteristic

#1940

C₂

yes

maximal, characteristic

#1542

C₂

yes

maximal

#1503

C₂

yes

maximal

#1275

C₂

yes

maximal, characteristic

#533

C₂²

yes

—

#501

—

#501

—

#431

C₂²

yes

—

#427

—

#427

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#97

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

—

—

—

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

D₄

yes

—

C₄⋊Q₈

D₄

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

#95

D₄

yes

characteristic

#95

D₄

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

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

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₄²

#41

yes

—

C₂³

#121

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#120

yes

characteristic

C₂×C₄

#121

yes

—

C₂×C₄

#123

yes

—

C₂×C₄

#124

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₄

C₂×Q₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

104

C₂²

—

105

C₂²

—

106

C₂²

#394

yes

characteristic

107

C₂²

#461

yes

characteristic

108

C₂²

#462

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#448

yes

characteristic

116

C₄

#451

yes

characteristic

117

C₄

#487

yes

characteristic

118

C₄

#491

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#2043

yes

characteristic

122

C₂

#2050

yes

characteristic

123

C₂

#2053

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#394
prime radical	C₂³	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#124

commutator, Frattini, cosocle

C₂×C₄²

C₂³

C₂²

C₂³

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

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
#3162	2
#1940	2
#1542	2
#1503	2
#1275	2

subgroup

index

Subgroups — #10548

Conjugacy classes of subgroups

Maximal subgroups