Group #2929 — subgroups

Subgroups — #2929

Order 128.

260 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 144 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×#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#525	C₂	1	1	yes	maximal, characteristic
4	#435	C₂	1	4	yes	maximal, complemented
5	C₈×D₄	C₂	1	4	yes	maximal, complemented
6	#424	C₂	1	1	yes	maximal, characteristic, complemented
7	#408	C₂	1	1	yes	maximal, characteristic, complemented
8	#407	C₂	1	1	yes	maximal, characteristic, complemented
9	#405	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41	C₂²	1	1	yes	characteristic
11	C₂³×C₄	C₂²	1	1	yes	characteristic
12	#131	—	2	4	no	—
13	C₂×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×M₁₆	C₂²	1	2	yes	complemented
16	C₂²×C₈	—	2	4	no	—
17	C₂²×C₈	C₂²	1	1	yes	characteristic
18	C₂²×C₈	C₂²	1	2	yes	—
19	C₂²×C₈	C₂²	1	2	yes	—
20	C₂²×C₈	C₂²	1	1	yes	characteristic
21	C₂²×C₈	C₂²	1	2	yes	complemented
22	#123	C₄	1	2	yes	—
23	C₂²⋊Q₈	C₄	1	2	yes	—
24	#121	C₄	1	2	yes	—
25	#120	C₄	1	2	yes	—
26	C₄×D₄	C₂²	1	4	yes	—
27	#117	C₂²	1	1	yes	characteristic
28	C₄⋊C₈	C₂²	1	4	yes	complemented
29	C₂²⋊C₈	C₂²	1	4	yes	—
30	C₂²⋊C₈	C₂²	1	4	yes	complemented
31	#97	C₂²	1	2	yes	complemented
32	C₄×C₈	C₂²	1	2	yes	complemented
33	C₂⁴	C₂×C₄	1	1	yes	characteristic
34	#41	—	2	2	no	—
35	#41	—	2	2	no	—
36	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄	C₂×C₄	1	4	yes	—
38	C₂×D₄	C₂×C₄	1	2	yes	—
39	C₂×D₄	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄	—	2	2	no	—
41	C₂²×C₄	—	2	2	no	—
42	C₂²×C₄	C₂×C₄	1	2	yes	—
43	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄	C₂³	1	2	yes	—
45	C₂²×C₄	C₂³	1	2	yes	—
46	C₂²×C₄	C₂³	1	1	yes	characteristic
47	M₁₆	—	2	2	no	—
48	M₁₆	—	2	2	no	—
49	M₁₆	—	2	4	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	—	2	4	no	—
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	2	4	no	—
54	C₂×C₈	D₄	1	4	yes	complemented
55	C₂×C₈	C₂³	1	2	yes	—
56	C₂×C₈	C₂³	1	2	yes	—
57	C₂×C₈	C₂³	1	2	yes	—
58	C₂×C₈	C₂³	1	1	yes	characteristic
59	C₂×C₈	C₂³	1	1	yes	characteristic
60	C₄⋊C₄	C₂×C₄	1	2	yes	—
61	C₄⋊C₄	C₂×C₄	1	4	yes	—
62	C₂²⋊C₄	C₂×C₄	1	2	yes	—
63	C₂²⋊C₄	C₂×C₄	1	4	yes	—
64	C₂²⋊C₄	C₂×C₄	1	4	yes	—
65	C₄²	C₂³	1	2	yes	—
66	C₂³	—	2	2	no	—
67	C₂³	—	2	4	no	—
68	C₂³	C₂²×C₄	1	2	yes	—
69	C₂³	C₂²×C₄	1	1	yes	characteristic
70	C₂³	C₂²×C₄	1	2	yes	—
71	Q₈	—	2	2	no	—
72	D₄	—	2	4	no	—
73	D₄	—	2	8	no	—
74	D₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	C₂²×C₄	1	4	yes	—
82	C₂×C₄	C₂²×C₄	1	2	yes	—
83	C₂×C₄	C₂²×C₄	1	2	yes	—
84	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄	C₂×D₄	1	2	yes	—
86	C₂×C₄	#41	1	2	yes	—
87	C₂×C₄	C₂⁴	1	1	yes	characteristic
88	C₈	—	2	2	no	—
89	C₈	—	2	2	no	—
90	C₈	—	2	2	no	—
91	C₈	C₂×D₄	1	2	yes	—
92	C₈	C₂×D₄	1	2	yes	—
93	C₂²	—	2	8	no	—
94	C₂²	—	2	2	no	—
95	C₂²	—	2	2	no	—
96	C₂²	—	2	2	no	—
97	C₂²	—	2	2	no	—
98	C₂²	—	2	4	no	—
99	C₂²	C₄×D₄	1	2	yes	—
100	C₂²	#131	1	4	yes	complemented
101	C₂²	C₂³×C₄	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	2	4	no	—
105	C₄	C₄×D₄	1	2	yes	—
106	C₄	C₂²×D₄	1	1	yes	characteristic
107	C₄	C₂×#41	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	—	2	4	no	—
110	C₂	—	4	2	no	—
111	C₂	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂	C₂×#131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#131

C₂

yes

maximal, characteristic, complemented

C₂³×C₈

C₂

yes

maximal, characteristic, complemented

#525

C₂

yes

maximal, characteristic

#435

C₂

yes

maximal, complemented

C₈×D₄

C₂

yes

maximal, complemented

#424

C₂

yes

maximal, characteristic, complemented

#408

C₂

yes

maximal, characteristic, complemented

#407

C₂

yes

maximal, characteristic, complemented

#405

C₂

yes

maximal, characteristic, complemented

C₂×#41

C₂²

yes

characteristic

C₂³×C₄

C₂²

yes

characteristic

#131

—

C₂×M₁₆

C₂²

yes

characteristic

C₂×M₁₆

C₂²

yes

characteristic

C₂×M₁₆

C₂²

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₂²

yes

characteristic

C₂²×C₈

C₂²

yes

—

C₂²×C₈

C₂²

yes

—

C₂²×C₈

C₂²

yes

characteristic

C₂²×C₈

C₂²

yes

complemented

#123

C₄

yes

—

C₂²⋊Q₈

C₄

yes

—

#121

C₄

yes

—

#120

C₄

yes

—

C₄×D₄

C₂²

yes

—

#117

C₂²

yes

characteristic

C₄⋊C₈

C₂²

yes

complemented

C₂²⋊C₈

C₂²

yes

—

C₂²⋊C₈

C₂²

yes

complemented

#97

C₂²

yes

complemented

C₄×C₈

C₂²

yes

complemented

C₂⁴

C₂×C₄

yes

characteristic

#41

—

#41

—

C₂×Q₈

C₂×C₄

yes

characteristic

C₂×D₄

C₂×C₄

yes

—

C₂×D₄

C₂×C₄

yes

—

C₂×D₄

C₂×C₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄

yes

—

C₂²×C₄

C₂×C₄

yes

characteristic

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄

yes

complemented

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

characteristic

C₄⋊C₄

C₂×C₄

yes

—

C₄⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₄²

C₂³

yes

—

C₂³

—

C₂³

—

C₂³

C₂²×C₄

yes

—

C₂³

C₂²×C₄

yes

characteristic

C₂³

C₂²×C₄

yes

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄

yes

—

C₂×C₄

C₂²×C₄

yes

—

C₂×C₄

C₂²×C₄

yes

—

C₂×C₄

C₂²×C₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

#41

yes

—

C₂×C₄

C₂⁴

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

C₂×D₄

yes

—

C₈

C₂×D₄

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₄×D₄

yes

—

100

C₂²

#131

yes

complemented

101

C₂²

C₂³×C₄

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄×D₄

yes

—

106

C₄

C₂²×D₄

yes

characteristic

107

C₄

C₂×#41

yes

characteristic

108

C₂

—

109

C₂

—

110

C₂

—

111

C₂

C₂×C₄×D₄

yes

characteristic

112

C₂

C₂×#131

yes

—

	subgroup	G/H
center	C₂×C₈	C₂³
commutator, socle	C₂²	C₂³×C₄
Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#525	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

C₂³

commutator, socle

C₂²

C₂³×C₄

Frattini, cosocle

C₂×C₄

C₂⁴

C₂

G ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₈ ◁ G

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂×#131	C₂	semidirect
2. C₂³×C₈	C₂	semidirect
4. #435	C₂	semidirect
5. C₈×D₄	C₂	semidirect
6. #424	C₂	semidirect
7. #408	C₂	semidirect
8. #407	C₂	semidirect
9. #405	C₂	semidirect
15. C₂×M₁₆	C₂²	semidirect
21. C₂²×C₈	C₂²	semidirect
28. C₄⋊C₈	C₂²	semidirect
30. C₂²⋊C₈	C₂²	semidirect
31. #97	C₂²	semidirect
32. C₄×C₈	C₂²	semidirect
54. C₂×C₈	D₄	semidirect
100. C₂²	#131	semidirect

H ≅ G/N

type

1. C₂×#131

C₂

semidirect

2. C₂³×C₈

C₂

semidirect

4. #435

C₂

semidirect

5. C₈×D₄

C₂

semidirect

6. #424

C₂

semidirect

7. #408

C₂

semidirect

8. #407

C₂

semidirect

9. #405

C₂

semidirect

15. C₂×M₁₆

C₂²

semidirect

21. C₂²×C₈

C₂²

semidirect

28. C₄⋊C₈

C₂²

semidirect

30. C₂²⋊C₈

C₂²

semidirect

31. #97

C₂²

semidirect

32. C₄×C₈

C₂²

semidirect

54. C₂×C₈

D₄

semidirect

100. C₂²

#131

semidirect

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

subgroup	index
C₂×#131	2
C₂³×C₈	2
#525	2
#435	2
C₈×D₄	2
#424	2
#408	2
#407	2
#405	2

subgroup

index

C₂³×C₈

C₈×D₄

Subgroups — #2929

Conjugacy classes of subgroups

Complements

Maximal subgroups