Group #21995 — subgroups

Subgroups — #21995

Order 256.

329 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 183 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	#2921	C₂	1	1	yes	maximal, characteristic
2	C₂×#433	C₂	1	1	yes	maximal, characteristic
3	#2905	C₂	1	1	yes	maximal, characteristic
4	#1950	C₂	1	4	yes	maximal, complemented
5	#1855	C₂	1	4	yes	maximal
6	#1849	C₂	1	1	yes	maximal, characteristic, complemented
7	#1848	C₂	1	1	yes	maximal, characteristic, complemented
8	#1838	C₂	1	1	yes	maximal, characteristic, complemented
9	#1837	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆	C₂²	1	2	yes	—
11	C₂×#117	C₂²	1	1	yes	characteristic
12	#433	—	2	4	no	—
13	#433	C₄	1	8	yes	—
14	C₂×#108	C₂²	1	4	yes	—
15	C₂×#108	C₂²	1	4	yes	—
16	#424	—	2	4	no	—
17	#423	C₂²	1	1	yes	characteristic
18	#423	C₂²	1	1	yes	characteristic
19	#423	C₂²	1	1	yes	characteristic
20	#423	C₂²	1	1	yes	characteristic
21	#423	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈	C₂²	1	2	yes	—
23	#407	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₄×M₁₆	C₂²	1	2	yes	—
26	C₂×#97	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
28	#341	C₂²	1	4	yes	—
29	#341	C₂²	1	4	yes	—
30	C₂³×C₄	C₂³	1	1	yes	characteristic
31	C₂×M₁₆	—	2	2	no	—
32	C₂×M₁₆	—	4	4	no	—
33	C₂×M₁₆	C₂³	1	2	yes	—
34	C₂×M₁₆	C₂³	1	2	yes	—
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	2	yes	—
40	#117	C₂×C₄	1	4	yes	—
41	#117	D₄	1	4	yes	—
42	C₂×C₄⋊C₄	Q₈	1	2	yes	—
43	C₂×C₂²⋊C₄	Q₈	1	2	yes	—
44	C₂×C₄²	C₂³	1	1	yes	characteristic
45	C₂×C₄²	C₂³	1	1	yes	characteristic
46	#108	—	4	4	no	—
47	#108	—	4	4	no	—
48	C₄⋊C₈	—	2	4	no	—
49	C₄⋊C₈	—	2	2	no	—
50	C₄⋊C₈	—	2	2	no	—
51	C₄⋊C₈	—	2	2	no	—
52	C₄⋊C₈	—	2	2	no	—
53	C₄⋊C₈	C₂×C₄	1	8	yes	—
54	C₂²⋊C₈	—	2	4	no	—
55	C₂²⋊C₈	C₂×C₄	1	8	yes	—
56	#97	—	2	2	no	—
57	#97	C₂×C₄	1	4	yes	—
58	C₄×C₈	—	2	2	no	—
59	C₄×C₈	C₂×C₄	1	4	yes	—
60	C₂⁴	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂²×C₄	1	2	yes	—
64	C₂²×C₄	C₂×D₄	1	2	yes	—
65	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄	C₂×Q₈	1	2	yes	—
67	C₂²×C₄	C₂×Q₈	1	2	yes	—
68	C₂²×C₄	C₂⁴	1	1	yes	characteristic
69	M₁₆	—	2	4	no	—
70	M₁₆	—	2	4	no	—
71	M₁₆	—	2	4	no	—
72	M₁₆	—	2	4	no	—
73	M₁₆	—	4	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	4	4	no	—
83	C₂×C₈	C₂²×C₄	1	4	yes	—
84	C₂×C₈	C₂²×C₄	1	4	yes	—
85	C₄⋊C₄	C₄⋊C₄	1	8	yes	—
86	C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
87	C₄²	C₂²×C₄	1	2	yes	—
88	C₄²	C₂²×C₄	1	2	yes	—
89	C₄²	C₂×D₄	1	2	yes	—
90	C₄²	C₂×D₄	1	2	yes	—
91	C₂³	—	2	2	no	—
92	C₂³	—	4	2	no	—
93	C₂³	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄	H(5,𝔽₂)	1	2	yes	—
107	C₂×C₄	2₋¹⁺⁴	1	2	yes	—
108	C₈	—	2	4	no	—
109	C₈	—	2	4	no	—
110	C₈	—	4	2	no	—
111	C₈	—	4	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	4	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	2	4	no	—
117	C₂²	—	8	2	no	—
118	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²	#544	1	2	yes	—
120	C₄	—	2	2	no	—
121	C₄	—	2	4	no	—
122	C₄	—	2	4	no	—
123	C₄	#520	1	2	yes	—
124	C₄	#535	1	1	yes	characteristic
125	C₄	#536	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	—	4	2	no	—
128	C₂	#2317	1	1	yes	characteristic
129	C₂	#2916	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#2921

C₂

yes

maximal, characteristic

C₂×#433

C₂

yes

maximal, characteristic

#2905

C₂

yes

maximal, characteristic

#1950

C₂

yes

maximal, complemented

#1855

C₂

yes

maximal

#1849

C₂

yes

maximal, characteristic, complemented

#1848

C₂

yes

maximal, characteristic, complemented

#1838

C₂

yes

maximal, characteristic, complemented

#1837

C₂

yes

maximal, characteristic, complemented

C₂²×M₁₆

C₂²

yes

—

C₂×#117

C₂²

yes

characteristic

#433

—

#433

C₄

yes

—

C₂×#108

C₂²

yes

—

C₂×#108

C₂²

yes

—

#424

—

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

#423

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

#407

C₂²

yes

—

C₂×C₂²⋊C₈

C₂²

yes

—

C₄×M₁₆

C₂²

yes

—

C₂×#97

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#341

C₂²

yes

—

#341

C₂²

yes

—

C₂³×C₄

C₂³

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

#117

C₂×C₄

yes

—

#117

D₄

yes

—

C₂×C₄⋊C₄

Q₈

yes

—

C₂×C₂²⋊C₄

Q₈

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×C₄

yes

—

#97

—

#97

C₂×C₄

yes

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₂⁴

C₂×Q₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

—

C₂²×C₄

C₂×Q₈

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₄⋊C₄

yes

—

C₂²⋊C₄

C₄⋊C₄

yes

—

C₄²

C₂²×C₄

yes

—

C₄²

C₂²×C₄

yes

—

C₄²

C₂×D₄

yes

—

C₄²

C₂×D₄

yes

—

C₂³

—

C₂³

—

C₂³

C₂×C₄⋊C₄

yes

—

C₂³

C₂²×Q₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×C₄⋊C₄

yes

—

102

C₂×C₄

C₂×C₄⋊C₄

yes

—

103

C₂×C₄

C₂×C₄⋊C₄

yes

—

104

C₂×C₄

C₂³×C₄

yes

characteristic

105

C₂×C₄

C₂²×D₄

yes

characteristic

106

C₂×C₄

H(5,𝔽₂)

yes

—

107

C₂×C₄

2₋¹⁺⁴

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂²×C₄⋊C₄

yes

characteristic

119

C₂²

#544

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#520

yes

—

124

C₄

#535

yes

characteristic

125

C₄

#536

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#2317

yes

characteristic

129

C₂

#2916

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ 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

C₂²×C₄⋊C₄

prime radical

C₂⁴

C₂×Q₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
4. #1950	C₂	semidirect
6. #1849	C₂	semidirect
7. #1848	C₂	semidirect
8. #1838	C₂	semidirect
9. #1837	C₂	semidirect

H ≅ G/N

type

4. #1950

C₂

semidirect

6. #1849

C₂

semidirect

7. #1848

C₂

semidirect

8. #1838

C₂

semidirect

9. #1837

C₂

semidirect

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

subgroup	index
#2921	2
C₂×#433	2
#2905	2
#1950	2
#1855	2
#1849	2
#1848	2
#1838	2
#1837	2

subgroup

index

Subgroups — #21995

Conjugacy classes of subgroups

Complements

Maximal subgroups