Group #17802 — subgroups

Subgroups — C₂×#1572 #17802

Order 256.

335 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 199 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×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×#431	C₂	1	1	yes	maximal, characteristic
4	#1572	C₂	1	8	yes	maximal, direct product factor
5	C₂×#335	C₂	1	2	yes	maximal, complemented
6	C₂×#334	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄²	C₂²	1	1	yes	characteristic
8	#432	—	2	2	no	—
9	#432	C₂²	1	4	yes	—
10	#431	—	2	2	no	—
11	#431	C₄	1	4	yes	—
12	#431	C₂²	1	4	yes	—
13	#431	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈	C₄	1	2	yes	—
15	C₂×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈	—	2	1	no	—
18	C₂×C₂²⋊C₈	—	2	1	no	—
19	C₂×C₂²⋊C₈	C₄	1	2	yes	—
20	C₂×#97	—	2	1	no	—
21	C₂×C₄×C₈	—	2	1	no	—
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	#335	C₂²	1	8	yes	complemented
25	#334	C₂²	1	8	yes	complemented
26	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄	D₄	1	1	yes	characteristic
28	C₂³×C₄	Q₈	1	1	yes	characteristic
29	C₂²×C₈	—	2	2	no	—
30	C₂²×C₈	—	2	2	no	—
31	C₂²×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²	C₂×C₄	1	2	yes	—
35	C₂×C₄²	D₄	1	2	yes	—
36	C₂×C₄²	D₄	1	1	yes	characteristic
37	C₂×C₄²	Q₈	1	1	yes	characteristic
38	C₂×C₄²	Q₈	1	2	yes	—
39	C₂×C₄²	C₂³	1	2	yes	—
40	C₂×C₄²	C₂³	1	1	yes	characteristic
41	C₄⋊C₈	—	2	2	no	—
42	C₄⋊C₈	—	2	2	no	—
43	C₄⋊C₈	C₂×C₄	1	8	yes	—
44	C₄⋊C₈	C₂³	1	4	yes	—
45	C₄⋊C₈	C₂³	1	4	yes	—
46	C₂²⋊C₈	—	2	4	no	—
47	C₂²⋊C₈	—	2	4	no	—
48	C₂²⋊C₈	C₂×C₄	1	8	yes	—
49	#97	—	2	4	no	—
50	C₄×C₈	—	2	4	no	—
51	C₄×C₈	C₂×C₄	1	2	yes	—
52	C₄×C₈	C₂×C₄	1	2	yes	—
53	C₄×C₈	C₂³	1	2	yes	—
54	C₄×C₈	C₂³	1	2	yes	—
55	C₂⁴	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄	—	2	4	no	—
57	C₂²×C₄	—	2	4	no	—
58	C₂²×C₄	—	4	2	no	—
59	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄	C₂²×C₄	1	2	yes	—
66	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄	C₂×D₄	1	2	yes	—
68	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄	C₂×Q₈	1	2	yes	—
71	C₂×C₈	—	2	4	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	4	no	—
75	C₂×C₈	—	4	4	no	—
76	C₂×C₈	—	4	4	no	—
77	C₂×C₈	C₂²×C₄	1	4	yes	—
78	C₂×C₈	C₂²×C₄	1	4	yes	—
79	C₄²	C₄⋊C₄	1	2	yes	—
80	C₄²	C₄⋊C₄	1	2	yes	—
81	C₄²	C₄⋊C₄	1	2	yes	—
82	C₄²	C₂²×C₄	1	1	yes	characteristic
83	C₄²	C₂²×C₄	1	2	yes	—
84	C₄²	C₂×D₄	1	2	yes	—
85	C₄²	C₂×D₄	1	1	yes	characteristic
86	C₄²	C₂×Q₈	1	2	yes	—
87	C₄²	C₂×Q₈	1	1	yes	characteristic
88	C₄²	C₂⁴	1	1	yes	characteristic
89	C₂³	—	2	2	no	—
90	C₂³	—	2	2	no	—
91	C₂³	—	2	2	no	—
92	C₂³	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄	#131	1	4	yes	—
111	C₂×C₄	#135	1	4	yes	—
112	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
115	C₈	—	2	4	no	—
116	C₈	—	2	4	no	—
117	C₈	—	4	4	no	—
118	C₈	—	4	4	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	4	4	no	—
123	C₂²	#424	1	2	yes	—
124	C₂²	#427	1	2	yes	—
125	C₂²	#430	1	2	yes	—
126	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	2	no	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	#424	1	2	yes	—
133	C₄	#427	1	2	yes	—
134	C₄	C₂×#131	1	2	yes	—
135	C₄	C₂×#135	1	2	yes	—
136	C₂	—	4	2	no	—
137	C₂	#1572	1	4	yes	direct product factor
138	C₂	C₂×#424	1	1	yes	characteristic
139	C₂	C₂×#427	1	1	yes	characteristic
140	C₂	C₂×#430	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#432

C₂

yes

maximal, characteristic

C₂×#431

C₂

yes

maximal, characteristic

C₂×#431

C₂

yes

maximal, characteristic

#1572

C₂

yes

maximal, direct product factor

C₂×#335

C₂

yes

maximal, complemented

C₂×#334

C₂

yes

maximal, complemented

C₂²×C₄²

C₂²

yes

characteristic

#432

—

#432

C₂²

yes

—

#431

—

#431

C₄

yes

—

#431

C₂²

yes

—

#431

C₂²

yes

—

C₂×C₄⋊C₈

C₄

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

C₄

yes

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#335

C₂²

yes

complemented

#334

C₂²

yes

complemented

C₂³×C₄

C₂×C₄

yes

characteristic

C₂³×C₄

D₄

yes

characteristic

C₂³×C₄

Q₈

yes

characteristic

C₂²×C₈

—

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

—

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

Q₈

yes

characteristic

C₂×C₄²

Q₈

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×C₄

yes

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₂⁴

C₄⋊C₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

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

C₄⋊C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

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

—

C₂³

C₂×C₄⋊C₄

yes

—

C₂³

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

105

C₂×C₄

C₂×C₄⋊C₄

yes

—

106

C₂×C₄

C₂×C₄⋊C₄

yes

—

107

C₂×C₄

C₂×C₄⋊C₄

yes

—

108

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

109

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

110

C₂×C₄

#131

yes

—

111

C₂×C₄

#135

yes

—

112

C₂×C₄

C₂³×C₄

yes

characteristic

113

C₂×C₄

C₂²×D₄

yes

characteristic

114

C₂×C₄

C₂²×Q₈

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

#424

yes

—

124

C₂²

#427

yes

—

125

C₂²

#430

yes

—

126

C₂²

C₂²×C₄⋊C₄

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#424

yes

—

133

C₄

#427

yes

—

134

C₄

C₂×#131

yes

—

135

C₄

C₂×#135

yes

—

136

C₂

—

137

C₂

#1572

yes

direct product factor

138

C₂

C₂×#424

yes

characteristic

139

C₂

C₂×#427

yes

characteristic

140

C₂

C₂×#430

yes

characteristic

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

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₄²

C₂⁴

socle

C₂³

C₂×C₄⋊C₄

prime radical

C₂⁴

C₄⋊C₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
4. #1572	C₂	direct (N × H)
5. C₂×#335	C₂	semidirect
6. C₂×#334	C₂	semidirect
24. #335	C₂²	semidirect
25. #334	C₂²	semidirect
137. C₂	#1572	direct (N × H)

H ≅ G/N

type

4. #1572

C₂

direct (N × H)

5. C₂×#335

C₂

semidirect

6. C₂×#334

C₂

semidirect

24. #335

C₂²

semidirect

25. #334

C₂²

semidirect

137. C₂

#1572

direct (N × H)

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

subgroup	index
C₂×#432	2
C₂×#431	2
#1572	2
C₂×#335	2
C₂×#334	2

subgroup

index

Subgroups — C₂×#1572 #17802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#1572 #17802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1572 #17802