Group #2924 — subgroups

Subgroups — #2924

Order 128.

201 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 134 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×#117	C₂	1	1	yes	maximal, characteristic
2	#433	C₂	1	4	yes	maximal, complemented
3	#432	C₂	1	2	yes	maximal, complemented
4	#431	C₂	1	2	yes	maximal, complemented
5	#424	C₂	1	2	yes	maximal, complemented
6	#407	C₂	1	1	yes	maximal, characteristic
7	C₂×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
8	#405	C₂	1	2	yes	maximal, complemented
9	C₂³×C₄	C₂²	1	1	yes	characteristic
10	C₂×M₁₆	C₂²	1	2	yes	—
11	C₂²×C₈	C₂²	1	2	yes	—
12	#117	C₄	1	1	yes	characteristic
13	#117	C₄	1	1	yes	characteristic
14	#117	C₄	1	2	yes	—
15	#117	C₂²	1	1	yes	characteristic
16	#117	C₂²	1	1	yes	characteristic
17	#117	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄	C₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄	C₄	1	1	yes	characteristic
20	C₂×C₂²⋊C₄	C₄	1	2	yes	—
21	C₂×C₄²	C₂²	1	2	yes	—
22	C₄⋊C₈	C₂²	1	4	yes	complemented
23	C₄⋊C₈	C₂²	1	4	yes	complemented
24	C₂²⋊C₈	C₂²	1	1	yes	characteristic
25	C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂²⋊C₈	C₂²	1	2	yes	—
27	C₂²⋊C₈	C₂²	1	2	yes	—
28	C₂²⋊C₈	C₂²	1	2	yes	—
29	#97	C₂²	1	4	yes	complemented
30	C₄×C₈	C₂²	1	4	yes	complemented
31	C₂⁴	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄	—	2	4	no	—
33	C₂²×C₄	C₂×C₄	1	2	yes	—
34	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄	C₂×C₄	1	2	yes	—
38	C₂²×C₄	C₂³	1	1	yes	characteristic
39	C₂²×C₄	C₂³	1	1	yes	characteristic
40	C₂²×C₄	C₂³	1	1	yes	characteristic
41	M₁₆	—	2	4	no	—
42	C₂×C₈	—	2	4	no	—
43	C₂×C₈	C₂³	1	2	yes	—
44	C₂×C₈	C₂³	1	4	yes	—
45	C₂×C₈	C₂³	1	2	yes	—
46	C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
52	C₂²⋊C₄	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄	C₂×C₄	1	2	yes	—
54	C₂²⋊C₄	C₂×C₄	1	2	yes	—
55	C₂²⋊C₄	C₂×C₄	1	2	yes	—
56	C₄²	C₂×C₄	1	2	yes	—
57	C₄²	C₂×C₄	1	2	yes	—
58	C₄²	C₂³	1	2	yes	—
59	C₄²	C₂³	1	2	yes	—
60	C₂³	—	2	2	no	—
61	C₂³	—	2	2	no	—
62	C₂³	—	4	1	no	—
63	C₂³	C₂²×C₄	1	1	yes	characteristic
64	C₂³	C₂²×C₄	1	1	yes	characteristic
65	C₂³	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	4	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄	C₂²×C₄	1	2	yes	—
75	C₂×C₄	C₂²×C₄	1	2	yes	—
76	C₂×C₄	C₂²×C₄	1	2	yes	—
77	C₂×C₄	C₂²×C₄	1	2	yes	—
78	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄	#41	1	4	yes	—
81	C₂×C₄	C₂⁴	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	—	2	2	no	—
84	C₈	—	2	4	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	2	1	no	—
87	C₂²	—	2	1	no	—
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	2	1	no	—
91	C₂²	—	2	1	no	—
92	C₂²	—	4	4	no	—
93	C₂²	#117	1	2	yes	—
94	C₂²	#131	1	2	yes	—
95	C₂²	C₂³×C₄	1	1	yes	characteristic
96	C₄	—	2	1	no	—
97	C₄	—	2	1	no	—
98	C₄	—	2	2	no	—
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	2	2	no	—
102	C₄	#117	1	2	yes	—
103	C₄	C₂×#41	1	2	yes	—
104	C₂	—	2	2	no	—
105	C₂	—	2	2	no	—
106	C₂	—	4	1	no	—
107	C₂	C₂×#117	1	1	yes	characteristic
108	C₂	C₂×#131	1	1	yes	characteristic
109	C₂	#568	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#117

C₂

yes

maximal, characteristic

#433

C₂

yes

maximal, complemented

#432

C₂

yes

maximal, complemented

#431

C₂

yes

maximal, complemented

#424

C₂

yes

maximal, complemented

#407

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₈

C₂

yes

maximal, characteristic

#405

C₂

yes

maximal, complemented

C₂³×C₄

C₂²

yes

characteristic

C₂×M₁₆

C₂²

yes

—

C₂²×C₈

C₂²

yes

—

#117

C₄

yes

characteristic

#117

C₄

yes

characteristic

#117

C₄

yes

—

#117

C₂²

yes

characteristic

#117

C₂²

yes

characteristic

#117

C₂²

yes

—

C₂×C₄⋊C₄

C₄

yes

characteristic

C₂×C₄⋊C₄

C₄

yes

characteristic

C₂×C₂²⋊C₄

C₄

yes

—

C₂×C₄²

C₂²

yes

—

C₄⋊C₈

C₂²

yes

complemented

C₄⋊C₈

C₂²

yes

complemented

C₂²⋊C₈

C₂²

yes

characteristic

C₂²⋊C₈

C₂²

yes

characteristic

C₂²⋊C₈

C₂²

yes

—

C₂²⋊C₈

C₂²

yes

—

C₂²⋊C₈

C₂²

yes

—

#97

C₂²

yes

complemented

C₄×C₈

C₂²

yes

complemented

C₂⁴

C₂×C₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×C₄

yes

—

C₂²×C₄

C₂×C₄

yes

characteristic

C₂²×C₄

C₂×C₄

yes

characteristic

C₂²×C₄

C₂×C₄

yes

characteristic

C₂²×C₄

C₂×C₄

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

—

C₄⋊C₄

C₂×C₄

yes

—

C₄⋊C₄

C₂×C₄

yes

—

C₄⋊C₄

C₂×C₄

yes

characteristic

C₄⋊C₄

C₂×C₄

yes

characteristic

C₄⋊C₄

C₂×C₄

yes

characteristic

C₄⋊C₄

C₂×C₄

yes

characteristic

C₂²⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₄²

C₂×C₄

yes

—

C₄²

C₂×C₄

yes

—

C₄²

C₂³

yes

—

C₄²

C₂³

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²×C₄

yes

characteristic

C₂³

C₂²×C₄

yes

characteristic

C₂³

C₂²×C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄

yes

characteristic

C₂×C₄

C₂²×C₄

yes

—

C₂×C₄

C₂²×C₄

yes

—

C₂×C₄

C₂²×C₄

yes

—

C₂×C₄

C₂²×C₄

yes

—

C₂×C₄

C₂²×C₄

yes

characteristic

C₂×C₄

C₂²×C₄

yes

characteristic

C₂×C₄

#41

yes

—

C₂×C₄

C₂⁴

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#117

yes

—

C₂²

#131

yes

—

C₂²

C₂³×C₄

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

#117

yes

—

103

C₄

C₂×#41

yes

—

104

C₂

—

105

C₂

—

106

C₂

—

107

C₂

C₂×#117

yes

characteristic

108

C₂

C₂×#131

yes

characteristic

109

C₂

#568

yes

characteristic

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

subgroup

G/H

center, Frattini, cosocle

C₂×C₄

C₂⁴

commutator, socle

C₂²

C₂³×C₄

prime radical

C₂⁴

C₂×C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ G

LECS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #433	C₂	semidirect
3. #432	C₂	semidirect
4. #431	C₂	semidirect
5. #424	C₂	semidirect
8. #405	C₂	semidirect
22. C₄⋊C₈	C₂²	semidirect
23. C₄⋊C₈	C₂²	semidirect
29. #97	C₂²	semidirect
30. C₄×C₈	C₂²	semidirect

H ≅ G/N

type

2. #433

C₂

semidirect

3. #432

C₂

semidirect

4. #431

C₂

semidirect

5. #424

C₂

semidirect

8. #405

C₂

semidirect

22. C₄⋊C₈

C₂²

semidirect

23. C₄⋊C₈

C₂²

semidirect

29. #97

C₂²

semidirect

30. C₄×C₈

C₂²

semidirect

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

subgroup	index
C₂×#117	2
#433	2
#432	2
#431	2
#424	2
#407	2
C₂×C₂²⋊C₈	2
#405	2

subgroup

index

C₂×C₂²⋊C₈

Subgroups — #2924

Conjugacy classes of subgroups

Complements

Maximal subgroups