Group #2968 — subgroups

Subgroups — #2968

Order 128.

186 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 130 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	#529	C₂	1	1	yes	maximal, characteristic
2	#446	C₂	1	2	yes	maximal, complemented
3	#435	C₂	1	2	yes	maximal, complemented
4	#435	C₂	1	2	yes	maximal, complemented
5	C₈×D₄	C₂	1	2	yes	maximal, complemented
6	#433	C₂	1	2	yes	maximal, complemented
7	#431	C₂	1	1	yes	maximal, characteristic, complemented
8	#405	C₂	1	2	yes	maximal, complemented
9	C₂×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆	C₂²	1	2	yes	—
11	C₂²×C₈	C₂²	1	2	yes	—
12	C₂²×C₈	C₂²	1	2	yes	—
13	#126	C₄	1	2	yes	—
14	#125	C₄	1	1	yes	characteristic
15	#124	C₄	1	1	yes	characteristic
16	#123	C₄	1	2	yes	—
17	C₂²⋊Q₈	C₄	1	1	yes	characteristic
18	#121	C₄	1	1	yes	characteristic
19	C₄×Q₈	C₂²	1	1	yes	characteristic
20	C₄×D₄	C₂²	1	2	yes	—
21	C₄×D₄	C₂²	1	1	yes	characteristic
22	#117	C₂²	1	2	yes	—
23	C₂×C₄²	C₂²	1	1	yes	characteristic
24	C₄⋊C₈	C₂²	1	2	yes	—
25	C₄⋊C₈	C₂²	1	2	yes	—
26	C₄⋊C₈	C₂²	1	2	yes	—
27	C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₂²⋊C₈	C₂²	1	2	yes	—
30	C₂²⋊C₈	C₂²	1	2	yes	—
31	#97	C₂²	1	1	yes	characteristic
32	#97	C₂²	1	1	yes	characteristic
33	#97	C₂²	1	2	yes	—
34	#97	C₂²	1	2	yes	—
35	C₄×C₈	C₂²	1	2	yes	—
36	C₄×C₈	C₂²	1	2	yes	—
37	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄	C₂×C₄	1	2	yes	—
39	C₂×D₄	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄	C₂³	1	1	yes	characteristic
43	C₂²×C₄	C₂³	1	2	yes	—
44	M₁₆	—	2	4	no	—
45	C₂×C₈	—	2	4	no	—
46	C₂×C₈	—	2	4	no	—
47	C₂×C₈	C₂³	1	2	yes	—
48	C₂×C₈	C₂³	1	2	yes	—
49	C₂×C₈	C₂³	1	2	yes	—
50	C₂×C₈	C₂³	1	2	yes	—
51	C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₄	C₂×C₄	1	2	yes	—
55	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₄	C₂×C₄	1	2	yes	—
57	C₂²⋊C₄	C₂×C₄	1	2	yes	—
58	C₂²⋊C₄	C₂×C₄	1	2	yes	—
59	C₂²⋊C₄	C₂×C₄	1	2	yes	—
60	C₂²⋊C₄	C₂×C₄	1	2	yes	—
61	C₂²⋊C₄	C₂×C₄	1	2	yes	—
62	C₄²	C₂×C₄	1	1	yes	characteristic
63	C₄²	C₂×C₄	1	1	yes	characteristic
64	C₄²	C₂³	1	1	yes	characteristic
65	C₄²	C₂³	1	2	yes	—
66	C₄²	C₂³	1	1	yes	characteristic
67	C₂³	C₂²×C₄	1	2	yes	—
68	C₂³	C₂²×C₄	1	1	yes	characteristic
69	Q₈	—	4	1	no	—
70	D₄	—	2	4	no	—
71	D₄	—	4	1	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄	C₂²×C₄	1	2	yes	—
83	C₂×C₄	C₂²×C₄	1	2	yes	—
84	C₂×C₄	C₂²×C₄	1	2	yes	—
85	C₂×C₄	C₂⁴	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	2	2	no	—
88	C₈	—	2	2	no	—
89	C₈	#41	1	4	yes	—
90	C₂²	—	2	1	no	—
91	C₂²	—	2	1	no	—
92	C₂²	—	2	2	no	—
93	C₂²	—	2	2	no	—
94	C₂²	—	2	2	no	—
95	C₂²	#131	1	2	yes	—
96	C₂²	C₂³×C₄	1	1	yes	characteristic
97	C₄	—	2	1	no	—
98	C₄	—	2	2	no	—
99	C₄	—	2	1	no	—
100	C₄	—	2	1	no	—
101	C₄	—	2	2	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	2	no	—
105	C₄	C₂×#41	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	4	2	no	—
108	C₂	C₄×#41	1	1	yes	characteristic
109	C₂	C₂×#131	1	1	yes	characteristic
110	C₂	#568	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#529

C₂

yes

maximal, characteristic

#446

C₂

yes

maximal, complemented

#435

C₂

yes

maximal, complemented

#435

C₂

yes

maximal, complemented

C₈×D₄

C₂

yes

maximal, complemented

#433

C₂

yes

maximal, complemented

#431

C₂

yes

maximal, characteristic, complemented

#405

C₂

yes

maximal, complemented

C₂×#97

C₂

yes

maximal, characteristic, complemented

C₂×M₁₆

C₂²

yes

—

C₂²×C₈

C₂²

yes

—

C₂²×C₈

C₂²

yes

—

#126

C₄

yes

—

#125

C₄

yes

characteristic

#124

C₄

yes

characteristic

#123

C₄

yes

—

C₂²⋊Q₈

C₄

yes

characteristic

#121

C₄

yes

characteristic

C₄×Q₈

C₂²

yes

characteristic

C₄×D₄

C₂²

yes

—

C₄×D₄

C₂²

yes

characteristic

#117

C₂²

yes

—

C₂×C₄²

C₂²

yes

characteristic

C₄⋊C₈

C₂²

yes

—

C₄⋊C₈

C₂²

yes

—

C₄⋊C₈

C₂²

yes

—

C₂²⋊C₈

C₂²

yes

characteristic

C₂²⋊C₈

C₂²

yes

characteristic

C₂²⋊C₈

C₂²

yes

—

C₂²⋊C₈

C₂²

yes

—

#97

C₂²

yes

characteristic

#97

C₂²

yes

characteristic

#97

C₂²

yes

—

#97

C₂²

yes

—

C₄×C₈

C₂²

yes

—

C₄×C₈

C₂²

yes

—

C₂×Q₈

C₂×C₄

yes

characteristic

C₂×D₄

C₂×C₄

yes

—

C₂×D₄

C₂×C₄

yes

characteristic

C₂²×C₄

C₂×C₄

yes

characteristic

C₂²×C₄

C₂×C₄

yes

characteristic

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂³

yes

—

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

—

C₄⋊C₄

C₂×C₄

yes

characteristic

C₄⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₂²⋊C₄

C₂×C₄

yes

—

C₄²

C₂×C₄

yes

characteristic

C₄²

C₂×C₄

yes

characteristic

C₄²

C₂³

yes

characteristic

C₄²

C₂³

yes

—

C₄²

C₂³

yes

characteristic

C₂³

C₂²×C₄

yes

—

C₂³

C₂²×C₄

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄

yes

characteristic

C₂×C₄

C₂²×C₄

yes

characteristic

C₂×C₄

C₂²×C₄

yes

characteristic

C₂×C₄

C₂²×C₄

yes

characteristic

C₂×C₄

C₂²×C₄

yes

characteristic

C₂×C₄

C₂²×C₄

yes

—

C₂×C₄

C₂²×C₄

yes

—

C₂×C₄

C₂²×C₄

yes

—

C₂×C₄

C₂⁴

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

#41

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#131

yes

—

C₂²

C₂³×C₄

yes

characteristic

C₄

—

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₂×#41

yes

—

106

C₂

—

107

C₂

—

108

C₂

C₄×#41

yes

characteristic

109

C₂

C₂×#131

yes

characteristic

110

C₂

#568

yes

characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂⁴
commutator, socle	C₂²	C₂³×C₄
prime radical	#121	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

C₄

G ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ G

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #446	C₂	semidirect
3. #435	C₂	semidirect
4. #435	C₂	semidirect
5. C₈×D₄	C₂	semidirect
6. #433	C₂	semidirect
7. #431	C₂	semidirect
8. #405	C₂	semidirect
9. C₂×#97	C₂	semidirect

H ≅ G/N

type

2. #446

C₂

semidirect

3. #435

C₂

semidirect

4. #435

C₂

semidirect

5. C₈×D₄

C₂

semidirect

6. #433

C₂

semidirect

7. #431

C₂

semidirect

8. #405

C₂

semidirect

9. C₂×#97

C₂

semidirect

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

subgroup	index
#529	2
#446	2
#435	2
C₈×D₄	2
#433	2
#431	2
#405	2
C₂×#97	2

subgroup

index

C₈×D₄

C₂×#97

Subgroups — #2968

Conjugacy classes of subgroups

Complements

Maximal subgroups