Group #3402 — subgroups

Subgroups — #3402

Order 128.

178 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 86 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	#556	C₂	1	2	yes	maximal, complemented
2	#497	C₂	1	2	yes	maximal, complemented
3	#495	C₂	1	1	yes	maximal, characteristic
4	#464	C₂	1	2	yes	maximal, complemented
5	#463	C₂	1	2	yes	maximal
6	#462	C₂	1	2	yes	maximal, complemented
7	#446	C₂	1	1	yes	maximal, characteristic, complemented
8	#440	C₂	1	2	yes	maximal, complemented
9	C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆	C₂²	1	2	yes	—
13	C₂×SD₁₆	C₂²	1	2	yes	—
14	C₂×SD₁₆	C₂²	1	2	yes	—
15	C₂×D₈	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈	C₂²	1	2	yes	—
17	#126	—	2	2	no	—
18	#126	—	2	2	no	—
19	#124	C₂²	1	2	yes	—
20	#124	C₂²	1	2	yes	—
21	C₂²⋊Q₈	—	2	2	no	—
22	C₄×Q₈	C₂²	1	2	yes	—
23	C₄×Q₈	C₂²	1	2	yes	—
24	C₄×Q₈	C₂²	1	1	yes	characteristic
25	C₄×D₄	C₂²	1	2	yes	—
26	#117	—	2	2	no	—
27	#107	C₂²	1	1	yes	characteristic
28	C₄⋊C₈	C₂²	1	2	yes	—
29	C₄⋊C₈	C₂²	1	1	yes	characteristic
30	#103	C₂²	1	2	yes	—
31	#103	C₂²	1	2	yes	—
32	#102	C₂²	1	2	yes	—
33	#97	C₂²	1	2	yes	—
34	C₄×C₈	C₂²	1	1	yes	characteristic
35	C₂×Q₈	D₄	1	1	yes	characteristic
36	C₂×Q₈	C₂³	1	2	yes	—
37	C₂×Q₈	C₂³	1	2	yes	—
38	C₂×D₄	C₂³	1	2	yes	—
39	C₂²×C₄	—	2	2	no	—
40	Q₁₆	—	2	2	no	—
41	Q₁₆	—	4	1	no	—
42	Q₁₆	—	4	2	no	—
43	SD₁₆	—	2	4	no	—
44	SD₁₆	—	4	2	no	—
45	D₈	—	4	1	no	—
46	C₂×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₈	C₂³	1	2	yes	—
48	C₂×C₈	C₂³	1	1	yes	characteristic
49	C₄⋊C₄	—	2	2	no	—
50	C₄⋊C₄	—	2	2	no	—
51	C₄⋊C₄	—	2	2	no	—
52	C₄⋊C₄	—	2	2	no	—
53	C₄⋊C₄	D₄	1	2	yes	—
54	C₄⋊C₄	D₄	1	1	yes	characteristic
55	C₄⋊C₄	C₂³	1	2	yes	—
56	C₂²⋊C₄	—	2	2	no	—
57	C₂²⋊C₄	—	2	2	no	—
58	C₂²⋊C₄	—	2	2	no	—
59	C₂²⋊C₄	—	2	2	no	—
60	C₂²⋊C₄	—	2	2	no	—
61	C₄²	—	2	2	no	—
62	C₄²	—	2	2	no	—
63	C₄²	C₂³	1	1	yes	characteristic
64	C₄²	C₂³	1	2	yes	—
65	C₂³	—	2	2	no	—
66	Q₈	—	2	2	no	—
67	Q₈	—	2	2	no	—
68	Q₈	—	2	2	no	—
69	Q₈	—	2	2	no	—
70	Q₈	—	4	1	no	—
71	D₄	—	2	2	no	—
72	D₄	—	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	4	no	—
77	C₂×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄	C₂×D₄	1	2	yes	—
79	C₂×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄	C₂×D₄	1	2	yes	—
81	C₂×C₄	C₂⁴	1	1	yes	characteristic
82	C₈	—	2	1	no	—
83	C₈	—	2	2	no	—
84	C₈	#41	1	2	yes	—
85	C₂²	—	4	2	no	—
86	C₂²	—	4	2	no	—
87	C₂²	—	4	2	no	—
88	C₂²	C₂²×D₄	1	1	yes	characteristic
89	C₄	—	2	2	no	—
90	C₄	—	2	1	no	—
91	C₄	—	2	1	no	—
92	C₄	—	2	2	no	—
93	C₄	—	4	2	no	—
94	C₄	—	4	2	no	—
95	C₄	—	4	2	no	—
96	C₄	C₂×#41	1	1	yes	characteristic
97	C₄	H(5,𝔽₂)	1	1	yes	characteristic
98	C₂	—	8	2	no	—
99	C₂	#550	1	1	yes	characteristic
100	C₂	#575	1	1	yes	characteristic
101	C₂	#578	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#556

C₂

yes

maximal, complemented

#497

C₂

yes

maximal, complemented

#495

C₂

yes

maximal, characteristic

#464

C₂

yes

maximal, complemented

#463

C₂

yes

maximal

#462

C₂

yes

maximal, complemented

#446

C₂

yes

maximal, characteristic, complemented

#440

C₂

yes

maximal, complemented

C₄×Q₁₆

C₂

yes

maximal, characteristic, complemented

C₂×Q₁₆

C₂²

yes

characteristic

C₂×Q₁₆

C₂²

yes

characteristic

C₂×Q₁₆

C₂²

yes

—

C₂×SD₁₆

C₂²

yes

—

C₂×SD₁₆

C₂²

yes

—

C₂×D₈

C₂²

yes

characteristic

C₄⋊Q₈

C₂²

yes

—

#126

—

#126

—

#124

C₂²

yes

—

#124

C₂²

yes

—

C₂²⋊Q₈

—

C₄×Q₈

C₂²

yes

—

C₄×Q₈

C₂²

yes

—

C₄×Q₈

C₂²

yes

characteristic

C₄×D₄

C₂²

yes

—

#117

—

#107

C₂²

yes

characteristic

C₄⋊C₈

C₂²

yes

—

C₄⋊C₈

C₂²

yes

characteristic

#103

C₂²

yes

—

#103

C₂²

yes

—

#102

C₂²

yes

—

#97

C₂²

yes

—

C₄×C₈

C₂²

yes

characteristic

C₂×Q₈

D₄

yes

characteristic

C₂×Q₈

C₂³

yes

—

C₂×Q₈

C₂³

yes

—

C₂×D₄

C₂³

yes

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄

yes

—

C₄⋊C₄

D₄

yes

characteristic

C₄⋊C₄

C₂³

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂³

yes

characteristic

C₄²

C₂³

yes

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂⁴

yes

characteristic

C₈

—

C₈

—

C₈

#41

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×D₄

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

C₂×#41

yes

characteristic

C₄

H(5,𝔽₂)

yes

characteristic

C₂

—

C₂

#550

yes

characteristic

100

C₂

#575

yes

characteristic

101

C₂

#578

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴

prime radical

C₂×D₈

C₂²

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ G

LECS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #556	C₂	semidirect
2. #497	C₂	semidirect
4. #464	C₂	semidirect
6. #462	C₂	semidirect
7. #446	C₂	semidirect
8. #440	C₂	semidirect
9. C₄×Q₁₆	C₂	semidirect

H ≅ G/N

type

1. #556

C₂

semidirect

2. #497

C₂

semidirect

4. #464

C₂

semidirect

6. #462

C₂

semidirect

7. #446

C₂

semidirect

8. #440

C₂

semidirect

9. C₄×Q₁₆

C₂

semidirect

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

subgroup	index
#556	2
#497	2
#495	2
#464	2
#463	2
#462	2
#446	2
#440	2
C₄×Q₁₆	2

subgroup

index

C₄×Q₁₆

Subgroups — #3402

Conjugacy classes of subgroups

Complements

Maximal subgroups