Group #22237 — subgroups

Subgroups — #22237

Order 256.

347 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 151 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	#3135	C₂	1	1	yes	maximal, characteristic
2	#2961	C₂	1	1	yes	maximal, characteristic, complemented
3	#2932	C₂	1	1	yes	maximal, characteristic
4	#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	#1962	C₂	1	1	yes	maximal, characteristic
6	#1929	C₂	1	4	yes	maximal, complemented
7	#1866	C₂	1	4	yes	maximal, complemented
8	#1849	C₂	1	1	yes	maximal, characteristic, complemented
9	#1745	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131	C₂²	1	2	yes	—
11	#533	C₂²	1	1	yes	characteristic
12	#532	C₂²	1	1	yes	characteristic
13	#491	C₄	1	2	yes	complemented
14	#490	C₄	1	2	yes	complemented
15	#489	C₄	1	4	yes	complemented
16	#446	—	2	4	no	—
17	#435	—	2	4	no	—
18	C₂×#108	C₂²	1	4	yes	—
19	#424	C₂²	1	2	yes	—
20	#423	C₂²	1	1	yes	characteristic
21	#423	C₂²	1	1	yes	characteristic
22	C₂×#104	C₂²	1	4	yes	—
23	#416	C₂²	1	4	yes	—
24	#408	C₂²	1	2	yes	—
25	#405	C₂²	1	2	yes	—
26	C₂×#97	C₂²	1	1	yes	characteristic
27	C₂×#97	C₂²	1	1	yes	characteristic
28	C₄³	C₂²	1	1	yes	characteristic
29	#339	C₂²	1	4	yes	—
30	#336	C₂²	1	2	yes	—
31	#336	C₂²	1	2	yes	—
32	C₂×#41	C₂³	1	2	yes	—
33	#131	—	4	4	no	—
34	C₂×M₁₆	—	2	2	no	—
35	C₂×M₁₆	C₂³	1	2	yes	—
36	C₂×M₁₆	C₂³	1	2	yes	—
37	C₂²×C₈	—	2	2	no	—
38	C₂²×C₈	C₂³	1	2	yes	—
39	C₂²×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₈	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
42	C₄⋊Q₈	C₂×C₄	1	2	yes	—
43	#127	C₂×C₄	1	1	yes	characteristic
44	#125	C₂×C₄	1	2	yes	—
45	#124	C₂×C₄	1	2	yes	—
46	C₂²⋊Q₈	—	2	2	no	—
47	#121	—	2	2	no	—
48	C₄×Q₈	—	2	2	no	—
49	C₄×D₄	—	2	2	no	—
50	#117	C₂³	1	2	yes	—
51	C₂×C₄²	—	2	2	no	—
52	C₂×C₄²	C₂³	1	2	yes	—
53	C₂×C₄²	C₂³	1	1	yes	characteristic
54	#108	—	4	4	no	—
55	C₄⋊C₈	—	2	4	no	—
56	C₄⋊C₈	—	2	2	no	—
57	C₄⋊C₈	—	2	2	no	—
58	#104	—	4	4	no	—
59	#103	C₂×C₄	1	8	yes	—
60	#102	C₂×C₄	1	8	yes	—
61	C₂²⋊C₈	—	2	4	no	—
62	#97	—	2	2	no	—
63	#97	C₂×C₄	1	2	yes	—
64	#97	C₂×C₄	1	2	yes	—
65	#97	D₄	1	4	yes	—
66	C₄×C₈	—	2	2	no	—
67	#41	—	2	4	no	—
68	#41	—	2	4	no	—
69	C₂×Q₈	—	2	2	no	—
70	C₂×Q₈	C₂²×C₄	1	2	yes	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	2	no	—
73	C₂×D₄	C₂²×C₄	1	2	yes	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	#41	1	2	yes	—
77	C₂²×C₄	C₂⁴	1	1	yes	characteristic
78	M₁₆	—	2	4	no	—
79	M₁₆	—	2	4	no	—
80	M₁₆	—	4	4	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	4	4	no	—
88	C₂×C₈	C₂²×C₄	1	4	yes	—
89	C₂×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₈	C₂×D₄	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	C₂²×C₄	1	4	yes	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₂²⋊C₄	—	2	4	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	2	2	no	—
100	C₄²	—	4	4	no	—
101	C₄²	C₂²×C₄	1	1	yes	characteristic
102	C₄²	C₂²×C₄	1	1	yes	characteristic
103	C₄²	C₂×D₄	1	2	yes	—
104	C₂³	—	2	2	no	—
105	C₂³	C₂×#41	1	1	yes	characteristic
106	Q₈	—	2	4	no	—
107	Q₈	—	8	2	no	—
108	D₄	—	2	4	no	—
109	D₄	—	4	4	no	—
110	D₄	—	8	2	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	C₄×D₄	1	2	yes	—
120	C₂×C₄	C₄×D₄	1	2	yes	—
121	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄	H(5,𝔽₂)	1	2	yes	—
124	C₂×C₄	2₋¹⁺⁴	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₈	—	4	2	no	—
129	C₈	—	4	2	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	4	2	no	—
132	C₂²	—	4	4	no	—
133	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²	#542	1	2	yes	—
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	4	4	no	—
138	C₄	—	4	2	no	—
139	C₄	—	4	4	no	—
140	C₄	#520	1	2	yes	—
141	C₄	#535	1	1	yes	characteristic
142	C₄	#536	1	1	yes	characteristic
143	C₂	—	2	2	no	—
144	C₂	—	8	2	no	—
145	C₂	#2326	1	1	yes	characteristic
146	C₂	#2956	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3135

C₂

yes

maximal, characteristic

#2961

C₂

yes

maximal, characteristic, complemented

#2932

C₂

yes

maximal, characteristic

#1994

C₂

yes

maximal, characteristic, complemented

#1962

C₂

yes

maximal, characteristic

#1929

C₂

yes

maximal, complemented

#1866

C₂

yes

maximal, complemented

#1849

C₂

yes

maximal, characteristic, complemented

#1745

C₂

yes

maximal, characteristic, complemented

C₂×#131

C₂²

yes

—

#533

C₂²

yes

characteristic

#532

C₂²

yes

characteristic

#491

C₄

yes

complemented

#490

C₄

yes

complemented

#489

C₄

yes

—

—

C₂²

yes

—

#424

C₂²

yes

—

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×#104

C₂²

yes

—

#416

C₂²

yes

—

#408

C₂²

yes

—

#405

C₂²

yes

—

C₂×#97

C₂²

yes

characteristic

C₂×#97

C₂²

yes

characteristic

C₄³

C₂²

yes

characteristic

#339

C₂²

yes

—

#336

C₂²

yes

—

#336

C₂²

yes

—

C₂×#41

C₂³

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

—

#127

C₂×C₄

yes

characteristic

#125

C₂×C₄

yes

—

#124

C₂×C₄

yes

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

C₂³

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#104

—

#103

C₂×C₄

yes

—

#102

C₂×C₄

yes

—

C₂²⋊C₈

—

#97

—

#97

C₂×C₄

yes

—

#97

C₂×C₄

yes

—

#97

D₄

yes

—

C₄×C₈

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

C₂²×C₄

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#41

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×C₄

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₄²

C₂²×C₄

yes

characteristic

102

C₄²

C₂²×C₄

yes

characteristic

103

C₄²

C₂×D₄

yes

—

104

C₂³

—

105

C₂³

C₂×#41

yes

characteristic

106

Q₈

—

107

Q₈

—

108

D₄

—

109

D₄

—

110

D₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₄×D₄

yes

—

120

C₂×C₄

C₄×D₄

yes

—

121

C₂×C₄

C₂³×C₄

yes

characteristic

122

C₂×C₄

C₂²×D₄

yes

characteristic

123

C₂×C₄

H(5,𝔽₂)

yes

—

124

C₂×C₄

2₋¹⁺⁴

yes

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

C₂×C₄×D₄

yes

characteristic

134

C₂²

#542

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#520

yes

—

141

C₄

#535

yes

characteristic

142

C₄

#536

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

#2326

yes

characteristic

146

C₂

#2956

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

C₂²×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₂²×C₄

C₂⁴

socle

C₂²

C₂×C₄×D₄

C₂²

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2961	C₂	semidirect
4. #1994	C₂	semidirect
6. #1929	C₂	semidirect
7. #1866	C₂	semidirect
8. #1849	C₂	semidirect
9. #1745	C₂	semidirect
13. #491	C₄	semidirect
14. #490	C₄	semidirect
15. #489	C₄	semidirect

H ≅ G/N

type

2. #2961

C₂

semidirect

4. #1994

C₂

semidirect

6. #1929

C₂

semidirect

7. #1866

C₂

semidirect

8. #1849

C₂

semidirect

9. #1745

C₂

semidirect

13. #491

C₄

semidirect

14. #490

C₄

semidirect

15. #489

C₄

semidirect

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

subgroup	index
#3135	2
#2961	2
#2932	2
#1994	2
#1962	2
#1929	2
#1866	2
#1849	2
#1745	2

subgroup

index

Subgroups — #22237

Conjugacy classes of subgroups

Complements

Maximal subgroups