Group #17804 — subgroups

Subgroups — C₂×#1573 #17804

Order 256.

363 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 207 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	C₂²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#1573	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	C₂³×C₈	—	2	1	no	—
8	C₂²×C₄²	C₂²	1	1	yes	characteristic
9	#432	—	2	2	no	—
10	#432	C₂²	1	4	yes	—
11	#431	C₄	1	4	yes	—
12	#431	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈	—	2	2	no	—
14	C₂×C₄⋊C₈	—	2	1	no	—
15	C₂×C₄⋊C₈	C₄	1	2	yes	—
16	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₈	—	2	1	no	—
21	C₂×C₂²⋊C₈	C₄	1	2	yes	—
22	C₂×#97	—	2	1	no	—
23	C₂×C₄×C₈	C₂²	1	2	yes	—
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	4	no	—
30	C₂²×C₈	—	2	2	no	—
31	C₂²×C₈	—	2	1	no	—
32	C₂²×C₈	—	2	1	no	—
33	C₂²×C₈	—	4	4	no	—
34	C₂²×C₈	C₂×C₄	1	4	yes	—
35	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²	C₂×C₄	1	2	yes	—
37	C₂×C₄²	D₄	1	2	yes	—
38	C₂×C₄²	D₄	1	1	yes	characteristic
39	C₂×C₄²	Q₈	1	1	yes	characteristic
40	C₂×C₄²	Q₈	1	2	yes	—
41	C₂×C₄²	C₂³	1	2	yes	—
42	C₂×C₄²	C₂³	1	1	yes	characteristic
43	C₄⋊C₈	—	2	4	no	—
44	C₄⋊C₈	—	2	2	no	—
45	C₄⋊C₈	—	2	1	no	—
46	C₄⋊C₈	—	2	1	no	—
47	C₄⋊C₈	C₂×C₄	1	8	yes	—
48	C₄⋊C₈	C₂³	1	2	yes	—
49	C₄⋊C₈	C₂³	1	4	yes	—
50	C₄⋊C₈	C₂³	1	2	yes	—
51	C₂²⋊C₈	—	2	4	no	—
52	C₂²⋊C₈	C₂×C₄	1	8	yes	—
53	#97	—	2	4	no	—
54	C₄×C₈	C₂×C₄	1	4	yes	—
55	C₄×C₈	C₂³	1	4	yes	—
56	C₂⁴	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄	—	2	4	no	—
58	C₂²×C₄	—	2	4	no	—
59	C₂²×C₄	—	2	2	no	—
60	C₂²×C₄	—	2	2	no	—
61	C₂²×C₄	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄	C₂²×C₄	1	2	yes	—
68	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄	C₂×D₄	1	2	yes	—
70	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄	C₂×Q₈	1	2	yes	—
72	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₈	—	2	8	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	4	8	no	—
78	C₂×C₈	—	4	2	no	—
79	C₂×C₈	—	4	4	no	—
80	C₂×C₈	—	4	2	no	—
81	C₂×C₈	C₂²×C₄	1	8	yes	—
82	C₄²	C₄⋊C₄	1	2	yes	—
83	C₄²	C₄⋊C₄	1	2	yes	—
84	C₄²	C₄⋊C₄	1	2	yes	—
85	C₄²	C₂²×C₄	1	1	yes	characteristic
86	C₄²	C₂²×C₄	1	2	yes	—
87	C₄²	C₂×D₄	1	2	yes	—
88	C₄²	C₂×D₄	1	1	yes	characteristic
89	C₄²	C₂×Q₈	1	2	yes	—
90	C₄²	C₂×Q₈	1	1	yes	characteristic
91	C₄²	C₂⁴	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	C₂³	—	2	2	no	—
94	C₂³	#108	1	4	yes	—
95	C₂³	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄	#131	1	4	yes	—
115	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄	#137	1	2	yes	—
117	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
120	C₈	—	2	8	no	—
121	C₈	—	4	2	no	—
122	C₈	—	4	4	no	—
123	C₈	—	4	2	no	—
124	C₂²	—	2	8	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	#424	1	2	yes	—
128	C₂²	#428	1	2	yes	—
129	C₂²	C₂×#108	1	4	yes	—
130	C₂²	C₂×#108	1	2	yes	—
131	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	2	no	—
136	C₄	—	2	1	no	—
137	C₄	#424	1	2	yes	—
138	C₄	#428	1	2	yes	—
139	C₄	C₂×#131	1	2	yes	—
140	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₄	C₂×#137	1	1	yes	characteristic
142	C₂	—	2	4	no	—
143	C₂	#1573	1	4	yes	direct product factor
144	C₂	C₂×#424	1	1	yes	characteristic
145	C₂	C₂×#428	1	1	yes	characteristic
146	C₂	C₂²×#108	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#432

C₂

yes

maximal, characteristic

C₂×#431

C₂

yes

maximal, characteristic

C₂²×C₄⋊C₈

C₂

yes

maximal, characteristic

#1573

C₂

yes

maximal, direct product factor

C₂×#335

C₂

yes

maximal, complemented

C₂×#334

C₂

yes

maximal, complemented

C₂³×C₈

—

C₂²×C₄²

C₂²

yes

characteristic

#432

—

#432

C₂²

yes

—

#431

C₄

yes

—

#431

C₂²

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₄

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

C₄

yes

—

C₂×#97

—

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

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

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

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×C₄

yes

—

#97

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂³

yes

—

C₂⁴

C₄⋊C₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

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

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

—

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

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

#108

yes

—

C₂³

C₂×C₄⋊C₄

yes

—

C₂³

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

109

C₂×C₄

C₂×C₄⋊C₄

yes

—

110

C₂×C₄

C₂×C₄⋊C₄

yes

—

111

C₂×C₄

C₂×C₄⋊C₄

yes

—

112

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

113

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

114

C₂×C₄

#131

yes

—

115

C₂×C₄

AGL(1,ℤ/8)

yes

—

116

C₂×C₄

#137

yes

—

117

C₂×C₄

C₂³×C₄

yes

characteristic

118

C₂×C₄

C₂²×D₄

yes

characteristic

119

C₂×C₄

C₂²×Q₈

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

#424

yes

—

128

C₂²

#428

yes

—

129

C₂²

C₂×#108

yes

—

130

C₂²

C₂×#108

yes

—

131

C₂²

C₂²×C₄⋊C₄

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#424

yes

—

138

C₄

#428

yes

—

139

C₄

C₂×#131

yes

—

140

C₄

C₂×AGL(1,ℤ/8)

yes

characteristic

141

C₄

C₂×#137

yes

characteristic

142

C₂

—

143

C₂

#1573

yes

direct product factor

144

C₂

C₂×#424

yes

characteristic

145

C₂

C₂×#428

yes

characteristic

146

C₂

C₂²×#108

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. #1573	C₂	direct (N × H)
5. C₂×#335	C₂	semidirect
6. C₂×#334	C₂	semidirect
24. #335	C₂²	semidirect
25. #334	C₂²	semidirect
143. C₂	#1573	direct (N × H)

H ≅ G/N

type

4. #1573

C₂

direct (N × H)

5. C₂×#335

C₂

semidirect

6. C₂×#334

C₂

semidirect

24. #335

C₂²

semidirect

25. #334

C₂²

semidirect

143. C₂

#1573

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
C₂²×C₄⋊C₈	2
#1573	2
C₂×#335	2
C₂×#334	2

subgroup

index

C₂²×C₄⋊C₈

Subgroups — C₂×#1573 #17804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#1573 #17804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1573 #17804