Group #3509 — subgroups

Subgroups — #3509

Order 128.

496 conjugacy classes of subgroups; only the 392 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 392 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	#557	C₂	1	4	yes	maximal, complemented
2	#556	C₂	1	4	yes	maximal, complemented
3	#552	C₂	1	8	yes	maximal, complemented
4	#541	C₂	1	4	yes	maximal, complemented
5	#536	C₂	1	2	yes	maximal
6	#533	C₂	1	1	yes	maximal, characteristic, complemented
7	#533	C₂	1	4	yes	maximal, complemented
8	C₂×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#519	C₂	1	2	yes	maximal, complemented
10	C₄×#41	C₂	1	1	yes	maximal, characteristic
11	C₂×#41	C₂²	1	1	yes	characteristic
12	C₂²×Q₈	C₂²	1	2	yes	—
13	C₄⋊Q₈	C₂²	1	2	yes	—
14	C₄⋊Q₈	C₂²	1	2	yes	—
15	C₄⋊Q₈	C₂²	1	4	yes	—
16	C₄⋊Q₈	C₂²	1	2	yes	—
17	C₄⋊Q₈	C₂²	1	4	yes	—
18	#126	C₂²	1	16	yes	—
19	#125	C₂²	1	2	yes	—
20	#125	C₂²	1	8	yes	—
21	#125	C₂²	1	16	yes	—
22	#124	C₂²	1	4	yes	—
23	#123	C₂²	1	8	yes	—
24	C₂²⋊Q₈	C₂²	1	8	yes	—
25	C₂²⋊Q₈	C₂²	1	4	yes	—
26	C₂²⋊Q₈	C₂²	1	8	yes	—
27	C₄×Q₈	C₂²	1	2	yes	—
28	C₄×Q₈	C₂²	1	8	yes	—
29	C₄×Q₈	C₂²	1	8	yes	—
30	C₄×Q₈	C₂²	1	8	yes	—
31	C₄×Q₈	C₂²	1	4	yes	—
32	C₄×D₄	C₂²	1	4	yes	—
33	C₄×D₄	C₂²	1	2	yes	—
34	#117	C₂²	1	1	yes	characteristic
35	#117	C₂²	1	2	yes	—
36	#117	C₂²	1	2	yes	—
37	#117	C₂²	1	8	yes	—
38	#117	C₂²	1	8	yes	—
39	C₂×C₄⋊C₄	C₂²	1	4	yes	—
40	C₂×C₄²	C₂²	1	1	yes	characteristic
41	C₂×C₄²	C₂²	1	2	yes	—
42	#41	—	2	4	no	—
43	C₂×Q₈	—	2	8	no	—
44	C₂×Q₈	C₂³	1	4	yes	—
45	C₂×Q₈	C₂³	1	4	yes	—
46	C₂×Q₈	C₂³	1	4	yes	—
47	C₂×Q₈	C₂³	1	1	yes	characteristic
48	C₂×D₄	C₂³	1	2	yes	—
49	C₂×D₄	C₂³	1	1	yes	characteristic
50	C₂²×C₄	C₂³	1	2	yes	—
51	C₂²×C₄	C₂³	1	4	yes	—
52	C₂²×C₄	C₂³	1	2	yes	—
53	C₂²×C₄	C₂³	1	4	yes	—
54	C₂²×C₄	C₂³	1	1	yes	characteristic
55	C₄⋊C₄	C₂³	1	8	yes	—
56	C₄⋊C₄	C₂³	1	8	yes	—
57	C₄⋊C₄	C₂³	1	4	yes	—
58	C₄⋊C₄	C₂³	1	2	yes	—
59	C₄⋊C₄	C₂³	1	4	yes	—
60	C₄⋊C₄	C₂³	1	4	yes	—
61	C₄⋊C₄	C₂³	1	16	yes	—
62	C₄⋊C₄	C₂³	1	16	yes	—
63	C₄⋊C₄	C₂³	1	8	yes	—
64	C₂²⋊C₄	C₂³	1	2	yes	—
65	C₂²⋊C₄	C₂³	1	4	yes	—
66	C₂²⋊C₄	C₂³	1	4	yes	—
67	C₂²⋊C₄	C₂³	1	16	yes	—
68	C₄²	C₂³	1	1	yes	characteristic
69	C₄²	C₂³	1	2	yes	—
70	C₄²	C₂³	1	1	yes	characteristic
71	C₄²	C₂³	1	4	yes	—
72	C₄²	C₂³	1	2	yes	—
73	C₄²	C₂³	1	16	yes	—
74	C₄²	C₂³	1	4	yes	—
75	C₂³	C₂⁴	1	2	yes	—
76	C₂³	C₂⁴	1	1	yes	characteristic
77	Q₈	—	2	8	no	—
78	Q₈	—	2	8	no	—
79	Q₈	—	2	2	no	—
80	Q₈	—	2	8	no	—
81	D₄	—	2	4	no	—
82	D₄	—	2	2	no	—
83	C₂×C₄	—	2	8	no	—
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	8	no	—
87	C₂×C₄	#41	1	4	yes	—
88	C₂×C₄	C₂⁴	1	4	yes	—
89	C₂×C₄	C₂⁴	1	8	yes	—
90	C₂×C₄	C₂⁴	1	2	yes	—
91	C₂×C₄	C₂⁴	1	8	yes	—
92	C₂×C₄	C₂⁴	1	2	yes	—
93	C₂×C₄	C₂⁴	1	1	yes	characteristic
94	C₂×C₄	C₂⁴	1	1	yes	characteristic
95	C₂×C₄	C₂⁴	1	2	yes	—
96	C₂²	—	2	2	no	—
97	C₂²	—	2	2	no	—
98	C₂²	—	2	4	no	—
99	C₂²	C₂×#41	1	2	yes	—
100	C₂²	C₂⁵	1	1	yes	characteristic
101	C₄	—	2	2	no	—
102	C₄	—	2	8	no	—
103	C₄	—	2	4	no	—
104	C₄	—	2	2	no	—
105	C₄	—	2	8	no	—
106	C₄	C₂×#41	1	2	yes	—
107	C₄	C₂×#41	1	2	yes	—
108	C₄	2₋¹⁺⁴	1	4	yes	—
109	C₂	—	2	2	no	—
110	C₂	—	4	2	no	—
111	C₂	C₂²×#41	1	1	yes	characteristic
112	C₂	C₂×2₋¹⁺⁴	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#557

C₂

yes

maximal, complemented

#556

C₂

yes

maximal, complemented

#552

C₂

yes

maximal, complemented

#541

C₂

yes

maximal, complemented

#536

C₂

yes

maximal

#533

C₂

yes

maximal, characteristic, complemented

#533

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

#519

C₂

yes

maximal, complemented

C₄×#41

C₂

yes

maximal, characteristic

C₂×#41

C₂²

yes

characteristic

C₂²×Q₈

C₂²

yes

—

C₄⋊Q₈

C₂²

yes

—

C₄⋊Q₈

C₂²

yes

—

C₄⋊Q₈

C₂²

yes

—

C₄⋊Q₈

C₂²

yes

—

C₄⋊Q₈

C₂²

yes

—

#126

C₂²

yes

—

#125

C₂²

yes

—

#125

C₂²

yes

—

#125

C₂²

yes

—

#124

C₂²

yes

—

#123

C₂²

yes

—

C₂²⋊Q₈

C₂²

yes

—

C₂²⋊Q₈

C₂²

yes

—

C₂²⋊Q₈

C₂²

yes

—

C₄×Q₈

C₂²

yes

—

C₄×Q₈

C₂²

yes

—

C₄×Q₈

C₂²

yes

—

C₄×Q₈

C₂²

yes

—

C₄×Q₈

C₂²

yes

—

C₄×D₄

C₂²

yes

—

C₄×D₄

C₂²

yes

—

#117

C₂²

yes

characteristic

#117

C₂²

yes

—

#117

C₂²

yes

—

#117

C₂²

yes

—

#117

C₂²

yes

—

C₂×C₄⋊C₄

C₂²

yes

—

C₂×C₄²

C₂²

yes

characteristic

C₂×C₄²

C₂²

yes

—

#41

—

C₂×Q₈

—

C₂×Q₈

C₂³

yes

—

C₂×Q₈

C₂³

yes

—

C₂×Q₈

C₂³

yes

—

C₂×Q₈

C₂³

yes

characteristic

C₂×D₄

C₂³

yes

—

C₂×D₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

—

C₂²⋊C₄

C₂³

yes

—

C₂²⋊C₄

C₂³

yes

—

C₂²⋊C₄

C₂³

yes

—

C₂²⋊C₄

C₂³

yes

—

C₄²

C₂³

yes

characteristic

C₄²

C₂³

yes

—

C₄²

C₂³

yes

characteristic

C₄²

C₂³

yes

—

C₄²

C₂³

yes

—

C₄²

C₂³

yes

—

C₄²

C₂³

yes

—

C₂³

C₂⁴

yes

—

C₂³

C₂⁴

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#41

yes

—

C₂×C₄

C₂⁴

yes

—

C₂×C₄

C₂⁴

yes

—

C₂×C₄

C₂⁴

yes

—

C₂×C₄

C₂⁴

yes

—

C₂×C₄

C₂⁴

yes

—

C₂×C₄

C₂⁴

yes

characteristic

C₂×C₄

C₂⁴

yes

characteristic

C₂×C₄

C₂⁴

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×#41

yes

—

100

C₂²

C₂⁵

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

C₂×#41

yes

—

107

C₄

C₂×#41

yes

—

108

C₄

2₋¹⁺⁴

yes

—

109

C₂

—

110

C₂

—

111

C₂

C₂²×#41

yes

characteristic

112

C₂

C₂×2₋¹⁺⁴

yes

—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

subgroup

G/H

center, commutator, Frattini, socle, cosocle

C₂²

C₂⁵

prime radical

C₂×#41

C₂²

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ G

LECS

G ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #557	C₂	semidirect
2. #556	C₂	semidirect
3. #552	C₂	semidirect
4. #541	C₂	semidirect
6. #533	C₂	semidirect
7. #533	C₂	semidirect
8. C₂×C₄⋊Q₈	C₂	semidirect
9. #519	C₂	semidirect

H ≅ G/N

type

1. #557

C₂

semidirect

2. #556

C₂

semidirect

3. #552

C₂

semidirect

4. #541

C₂

semidirect

6. #533

C₂

semidirect

7. #533

C₂

semidirect

8. C₂×C₄⋊Q₈

C₂

semidirect

9. #519

C₂

semidirect

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

subgroup	index
#557	2
#556	2
#552	2
#541	2
#536	2
#533	2
C₂×C₄⋊Q₈	2
#519	2
C₄×#41	2

subgroup

index

C₂×C₄⋊Q₈

C₄×#41

Subgroups — #3509

Conjugacy classes of subgroups

Complements

Maximal subgroups