Group #17787 — subgroups

Subgroups — #17787

Order 256.

389 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 267 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	C₂×#431	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	#1565	C₂	1	8	yes	maximal, complemented
4	#1564	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆	C₄	1	1	yes	characteristic
6	C₂²×M₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₄²	C₂²	1	1	yes	characteristic
8	#431	—	2	2	no	—
9	#431	—	2	2	no	—
10	#431	C₂²	1	4	yes	—
11	#431	C₂²	1	4	yes	—
12	C₂×C₄⋊C₈	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₈	—	2	2	no	—
14	C₄×M₁₆	C₄	1	4	yes	—
15	C₄×M₁₆	C₂²	1	4	yes	—
16	C₂×#97	C₄	1	2	yes	—
17	C₂×C₄×C₈	—	2	2	no	—
18	C₂×C₄×C₈	C₂²	1	2	yes	—
19	#335	C₂²	1	8	yes	complemented
20	#334	C₂²	1	8	yes	complemented
21	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄	D₄	1	1	yes	characteristic
23	C₂³×C₄	Q₈	1	1	yes	characteristic
24	C₂×M₁₆	—	2	4	no	—
25	C₂×M₁₆	C₈	1	8	yes	complemented
26	C₂×M₁₆	C₂×C₄	1	4	yes	—
27	C₂×M₁₆	C₂×C₄	1	4	yes	—
28	C₂²×C₈	—	2	4	no	—
29	C₂²×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²	C₂×C₄	1	2	yes	—
33	C₂×C₄²	D₄	1	1	yes	characteristic
34	C₂×C₄²	D₄	1	2	yes	—
35	C₂×C₄²	Q₈	1	1	yes	characteristic
36	C₂×C₄²	Q₈	1	2	yes	—
37	C₂×C₄²	C₂³	1	2	yes	—
38	C₂×C₄²	C₂³	1	1	yes	characteristic
39	C₄⋊C₈	—	2	4	no	—
40	C₄⋊C₈	C₂³	1	8	yes	—
41	C₂²⋊C₈	—	2	4	no	—
42	C₂²⋊C₈	—	2	4	no	—
43	#97	C₂×C₄	1	8	yes	—
44	C₄×C₈	—	2	4	no	—
45	C₄×C₈	—	2	4	no	—
46	C₄×C₈	C₂×C₄	1	4	yes	—
47	C₄×C₈	C₂³	1	4	yes	—
48	C₂⁴	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄	—	4	2	no	—
50	C₂²×C₄	—	4	2	no	—
51	C₂²×C₄	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄	C₂×C₈	1	4	yes	—
58	C₂²×C₄	M₁₆	1	4	yes	—
59	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄	C₂²×C₄	1	2	yes	—
61	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄	C₂×D₄	1	2	yes	—
63	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄	C₂×Q₈	1	2	yes	—
65	M₁₆	—	2	8	no	—
66	M₁₆	C₂×C₈	1	16	yes	complemented
67	C₂×C₈	—	2	4	no	—
68	C₂×C₈	—	2	8	no	—
69	C₂×C₈	—	4	8	no	—
70	C₂×C₈	C₂×C₈	1	8	yes	complemented
71	C₂×C₈	C₂²×C₄	1	4	yes	—
72	C₂×C₈	C₂²×C₄	1	4	yes	—
73	C₄²	C₄⋊C₄	1	2	yes	—
74	C₄²	C₄⋊C₄	1	2	yes	—
75	C₄²	C₄⋊C₄	1	2	yes	—
76	C₄²	C₂²×C₄	1	1	yes	characteristic
77	C₄²	C₂²×C₄	1	2	yes	—
78	C₄²	C₂×D₄	1	2	yes	—
79	C₄²	C₂×D₄	1	1	yes	characteristic
80	C₄²	C₂×Q₈	1	2	yes	—
81	C₄²	C₂×Q₈	1	1	yes	characteristic
82	C₄²	C₂⁴	1	1	yes	characteristic
83	C₂³	—	4	1	no	—
84	C₂³	—	4	1	no	—
85	C₂³	C₄⋊C₈	1	4	yes	—
86	C₂³	C₂×C₄⋊C₄	1	2	yes	—
87	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	C₄⋊C₈	1	4	yes	—
97	C₂×C₄	C₄⋊C₈	1	8	yes	—
98	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄	C₂²×C₈	1	2	yes	—
105	C₂×C₄	C₂²×C₈	1	4	yes	—
106	C₂×C₄	C₂×M₁₆	1	2	yes	—
107	C₂×C₄	C₂×M₁₆	1	4	yes	—
108	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
111	C₈	—	2	4	no	—
112	C₈	—	4	8	no	—
113	C₈	C₂²×C₈	1	8	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	4	4	no	—
119	C₂²	C₂×C₄⋊C₈	1	2	yes	—
120	C₂²	C₂×C₄⋊C₈	1	4	yes	—
121	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	C₂×C₄⋊C₈	1	4	yes	—
127	C₄	C₂×C₄⋊C₈	1	2	yes	—
128	C₄	#430	1	2	yes	—
129	C₄	C₂³×C₈	1	1	yes	characteristic
130	C₄	C₂²×M₁₆	1	1	yes	characteristic
131	C₄	#575	1	2	yes	—
132	C₂	—	2	4	no	—
133	C₂	—	2	2	no	—
134	C₂	C₂²×C₄⋊C₈	1	1	yes	characteristic
135	C₂	#2911	1	1	yes	characteristic
136	C₂	C₂×#430	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#431

C₂

yes

maximal

C₂×C₄×M₁₆

C₂

yes

maximal, characteristic

#1565

C₂

yes

maximal, complemented

#1564

C₂

yes

maximal, complemented

C₂²×M₁₆

C₄

yes

characteristic

C₂²×M₁₆

C₄

yes

characteristic

C₂²×C₄²

C₂²

yes

—

—

C₂²

yes

—

#431

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₂²⋊C₈

—

C₄×M₁₆

C₄

yes

—

C₄×M₁₆

C₂²

yes

—

C₂×#97

C₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

—

#335

C₂²

yes

complemented

#334

C₂²

yes

complemented

C₂³×C₄

C₂×C₄

yes

characteristic

C₂³×C₄

D₄

yes

characteristic

C₂³×C₄

Q₈

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₈

yes

complemented

C₂×M₁₆

C₂×C₄

yes

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

—

C₂×C₄²

Q₈

yes

characteristic

C₂×C₄²

Q₈

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

C₂×C₄

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂³

yes

—

C₂⁴

C₄⋊C₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

M₁₆

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

—

M₁₆

—

M₁₆

C₂×C₈

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

complemented

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₄²

C₄⋊C₄

yes

—

C₄²

C₄⋊C₄

yes

—

C₄²

C₄⋊C₄

yes

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

—

C₄²

C₂×D₄

yes

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

—

C₄²

C₂×Q₈

yes

characteristic

C₄²

C₂⁴

yes

characteristic

C₂³

—

C₂³

—

C₂³

C₄⋊C₈

yes

—

C₂³

C₂×C₄⋊C₄

yes

—

C₂³

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

C₂×C₄⋊C₄

yes

—

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

100

C₂×C₄

C₂×C₄⋊C₄

yes

—

101

C₂×C₄

C₂×C₄⋊C₄

yes

—

102

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

103

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

104

C₂×C₄

C₂²×C₈

yes

—

105

C₂×C₄

C₂²×C₈

yes

—

106

C₂×C₄

C₂×M₁₆

yes

—

107

C₂×C₄

C₂×M₁₆

yes

—

108

C₂×C₄

C₂³×C₄

yes

characteristic

109

C₂×C₄

C₂²×D₄

yes

characteristic

110

C₂×C₄

C₂²×Q₈

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

C₂²×C₈

yes

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂×C₄⋊C₈

yes

—

120

C₂²

C₂×C₄⋊C₈

yes

—

121

C₂²

C₂²×C₄⋊C₄

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₂×C₄⋊C₈

yes

—

127

C₄

C₂×C₄⋊C₈

yes

—

128

C₄

#430

yes

—

129

C₄

C₂³×C₈

yes

characteristic

130

C₄

C₂²×M₁₆

yes

characteristic

131

C₄

#575

yes

—

132

C₂

—

133

C₂

—

134

C₂

C₂²×C₄⋊C₈

yes

characteristic

135

C₂

#2911

yes

characteristic

136

C₂

C₂×#430

yes

characteristic

	subgroup	G/H
center	C₄²	C₂×D₄
commutator	C₄	C₂³×C₈
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₄²

C₂×D₄

commutator

C₄

C₂³×C₈

Frattini, cosocle

C₄²

C₂⁴

socle

C₂²

C₂²×C₄⋊C₄

prime radical

C₂⁴

C₄⋊C₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
3. #1565	C₂	semidirect
4. #1564	C₂	semidirect
19. #335	C₂²	semidirect
20. #334	C₂²	semidirect
25. C₂×M₁₆	C₈	semidirect
66. M₁₆	C₂×C₈	semidirect
70. C₂×C₈	C₂×C₈	semidirect

H ≅ G/N

type

3. #1565

C₂

semidirect

4. #1564

C₂

semidirect

19. #335

C₂²

semidirect

20. #334

C₂²

semidirect

25. C₂×M₁₆

C₈

semidirect

66. M₁₆

C₂×C₈

semidirect

70. C₂×C₈

C₂×C₈

semidirect

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

subgroup	index
C₂×#431	2
C₂×C₄×M₁₆	2
#1565	2
#1564	2

subgroup

index

C₂×#431

C₂×C₄×M₁₆

#1565

#1564

Subgroups — #17787

Conjugacy classes of subgroups

Complements

Maximal subgroups