Group #12557 — subgroups

Subgroups — #12557

Order 256.

315 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 167 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	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#427	C₂	1	1	yes	maximal, characteristic
3	C₂²×#107	C₂	1	1	yes	maximal, characteristic
4	#1380	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈	C₂²	1	1	yes	characteristic
6	C₂×#117	—	2	1	no	—
7	C₂²×C₄⋊C₄	—	2	1	no	—
8	C₂×M₃₂	—	2	2	no	—
9	C₂×M₃₂	—	2	2	no	—
10	C₂×M₃₂	C₄	1	4	yes	complemented
11	C₂²×C₁₆	C₂²	1	2	yes	—
12	#427	—	2	2	no	—
13	#427	C₄	1	4	yes	complemented
14	C₂×#107	—	2	1	no	—
15	C₂×#107	—	2	2	no	—
16	C₂×#107	C₄	1	4	yes	complemented
17	C₂×#107	C₂²	1	2	yes	—
18	C₂×#107	C₂²	1	2	yes	—
19	C₂×#106	—	2	1	no	—
20	C₂³×C₄	—	4	1	no	—
21	C₂³×C₄	D₄	1	1	yes	characteristic
22	C₂²×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₈	D₄	1	2	yes	—
27	C₂²×C₈	D₄	1	2	yes	—
28	C₂²×C₈	Q₈	1	2	yes	—
29	C₂²×C₈	C₂³	1	1	yes	characteristic
30	#117	—	2	4	no	—
31	#117	—	4	2	no	—
32	C₂×C₄⋊C₄	—	2	4	no	—
33	C₂×C₄⋊C₄	—	2	2	no	—
34	C₂×C₄⋊C₄	—	2	2	no	—
35	C₂×C₄⋊C₄	—	4	1	no	—
36	C₂×C₄⋊C₄	—	4	2	no	—
37	C₂×C₂²⋊C₄	—	4	1	no	—
38	C₂×C₄²	—	4	1	no	—
39	M₃₂	—	2	8	no	—
40	C₂×C₁₆	—	2	2	no	—
41	C₂×C₁₆	—	2	2	no	—
42	C₂×C₁₆	C₂×C₄	1	4	yes	complemented
43	#107	—	2	4	no	—
44	#107	—	2	2	no	—
45	#107	—	2	2	no	—
46	#107	C₂×C₄	1	4	yes	—
47	#107	C₂×C₄	1	4	yes	complemented
48	#106	—	2	4	no	—
49	C₂⁴	D₈	1	1	yes	characteristic
50	C₂²×C₄	—	4	4	no	—
51	C₂²×C₄	—	4	2	no	—
52	C₂²×C₄	—	4	2	no	—
53	C₂²×C₄	—	8	4	no	—
54	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄	D₈	1	1	yes	characteristic
58	C₂²×C₄	SD₁₆	1	2	yes	—
59	C₂²×C₄	SD₁₆	1	2	yes	—
60	C₂²×C₄	Q₁₆	1	2	yes	—
61	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈	C₄²	1	4	yes	—
63	C₂×C₈	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈	C₂²⋊C₄	1	4	yes	—
66	C₂×C₈	C₄⋊C₄	1	2	yes	—
67	C₂×C₈	C₄⋊C₄	1	4	yes	—
68	C₂×C₈	C₄⋊C₄	1	4	yes	—
69	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	—	2	4	no	—
77	C₄⋊C₄	—	4	4	no	—
78	C₄⋊C₄	—	4	2	no	—
79	C₄⋊C₄	—	4	2	no	—
80	C₂²⋊C₄	—	4	4	no	—
81	C₄²	—	4	4	no	—
82	C₁₆	—	2	4	no	—
83	C₂³	—	2	2	no	—
84	C₂³	—	2	2	no	—
85	C₂³	#102	1	2	yes	—
86	C₂³	#102	1	2	yes	—
87	C₂³	#107	1	2	yes	—
88	C₂³	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄	—	4	4	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	—	8	8	no	—
92	C₂×C₄	—	8	2	no	—
93	C₂×C₄	—	8	4	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	#95	1	4	yes	—
96	C₂×C₄	#102	1	2	yes	—
97	C₂×C₄	#102	1	2	yes	—
98	C₂×C₄	#103	1	4	yes	—
99	C₂×C₄	#103	1	4	yes	—
100	C₂×C₄	#106	1	4	yes	—
101	C₂×C₄	#107	1	2	yes	—
102	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
108	C₈	#95	1	4	yes	—
109	C₈	C₂×C₄²	1	1	yes	characteristic
110	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂²	—	2	4	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	2	4	no	—
117	C₂²	#340	1	4	yes	—
118	C₂²	C₂×#102	1	1	yes	characteristic
119	C₂²	C₂×#102	1	1	yes	characteristic
120	C₂²	C₂×#107	1	1	yes	characteristic
121	C₂²	#509	1	2	yes	—
122	C₂²	#510	1	2	yes	—
123	C₄	—	8	4	no	—
124	C₄	—	8	4	no	—
125	C₄	#340	1	4	yes	—
126	C₄	C₂×#95	1	1	yes	characteristic
127	C₄	C₂×#103	1	1	yes	characteristic
128	C₄	C₂×#103	1	1	yes	characteristic
129	C₄	C₂×#106	1	1	yes	characteristic
130	C₂	—	2	4	no	—
131	C₂	C₂×#340	1	1	yes	characteristic
132	C₂	#2139	1	2	yes	—
133	C₂	#2140	1	1	yes	characteristic
134	C₂	#2141	1	1	yes	characteristic
135	C₂	#2159	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×M₃₂

C₂

yes

maximal, characteristic

C₂×#427

C₂

yes

maximal, characteristic

C₂²×#107

C₂

yes

maximal, characteristic

#1380

C₂

yes

maximal, complemented

C₂³×C₈

C₂²

yes

characteristic

C₂×#117

—

C₂²×C₄⋊C₄

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

C₄

yes

complemented

C₂²×C₁₆

C₂²

yes

—

#427

—

#427

C₄

yes

—

—

C₄

yes

complemented

C₂×#107

C₂²

yes

—

C₂×#107

C₂²

yes

—

C₂×#106

—

C₂³×C₄

—

C₂³×C₄

D₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

Q₈

yes

—

C₂²×C₈

C₂³

yes

characteristic

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄

yes

—

—

—

C₂×C₄

yes

—

#107

C₂×C₄

yes

complemented

#106

—

C₂⁴

D₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄

yes

—

C₂²×C₄

C₂²⋊C₄

yes

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

D₈

yes

characteristic

C₂²×C₄

SD₁₆

yes

—

C₂²×C₄

SD₁₆

yes

—

C₂²×C₄

Q₁₆

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

C₄²

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

#102

yes

—

C₂³

#102

yes

—

C₂³

#107

yes

—

C₂³

C₂×D₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#95

yes

—

C₂×C₄

#102

yes

—

C₂×C₄

#102

yes

—

C₂×C₄

#103

yes

—

C₂×C₄

#103

yes

—

100

C₂×C₄

#106

yes

—

101

C₂×C₄

#107

yes

—

102

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

103

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

104

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

105

C₂×C₄

C₂×SD₁₆

yes

characteristic

106

C₂×C₄

C₂×SD₁₆

yes

characteristic

107

C₂×C₄

C₂×Q₁₆

yes

characteristic

108

C₈

#95

yes

—

109

C₈

C₂×C₄²

yes

characteristic

110

C₈

C₂×C₂²⋊C₄

yes

characteristic

111

C₈

C₂×C₄⋊C₄

yes

characteristic

112

C₈

C₂×C₄⋊C₄

yes

characteristic

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

#340

yes

—

118

C₂²

C₂×#102

yes

characteristic

119

C₂²

C₂×#102

yes

characteristic

120

C₂²

C₂×#107

yes

characteristic

121

C₂²

#509

yes

—

122

C₂²

#510

yes

—

123

C₄

—

124

C₄

—

125

C₄

#340

yes

—

126

C₄

C₂×#95

yes

characteristic

127

C₄

C₂×#103

yes

characteristic

128

C₄

C₂×#103

yes

characteristic

129

C₄

C₂×#106

yes

characteristic

130

C₂

—

131

C₂

C₂×#340

yes

characteristic

132

C₂

#2139

yes

—

133

C₂

#2140

yes

characteristic

134

C₂

#2141

yes

characteristic

135

C₂

#2159

yes

—

	subgroup	G/H
center, socle	C₂³	C₂×D₈
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂⁴	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂³

C₂×D₈

commutator

C₈

C₂×C₄²

Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂⁴

D₈

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
4. #1380	C₂	semidirect
10. C₂×M₃₂	C₄	semidirect
13. #427	C₄	semidirect
16. C₂×#107	C₄	semidirect
42. C₂×C₁₆	C₂×C₄	semidirect
47. #107	C₂×C₄	semidirect

H ≅ G/N

type

4. #1380

C₂

semidirect

10. C₂×M₃₂

C₄

semidirect

13. #427

C₄

semidirect

16. C₂×#107

C₄

semidirect

42. C₂×C₁₆

C₂×C₄

semidirect

47. #107

C₂×C₄

semidirect

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

subgroup	index
C₂²×M₃₂	2
C₂×#427	2
C₂²×#107	2
#1380	2

subgroup

index

C₂²×M₃₂

C₂×#427

C₂²×#107

#1380

Subgroups — #12557

Conjugacy classes of subgroups

Complements

Maximal subgroups