Group #3550 — subgroups

Subgroups — #3550

Order 128.

501 conjugacy classes of subgroups; only the 384 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 384 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	#563	C₂	1	4	yes	maximal, complemented
2	#559	C₂	1	2	yes	maximal, complemented
3	Q₈²	C₂	1	2	yes	maximal, complemented
4	#556	C₂	1	4	yes	maximal, complemented
5	D₄×Q₈	C₂	1	4	yes	maximal, complemented
6	#542	C₂	1	8	yes	maximal, complemented
7	#536	C₂	1	4	yes	maximal, complemented
8	#533	C₂	1	2	yes	maximal, complemented
9	#532	C₂	1	1	yes	maximal, characteristic
10	C₂×#41	C₂²	1	2	yes	—
11	C₂²×Q₈	C₂²	1	4	yes	—
12	C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈	C₂²	1	2	yes	—
14	C₄⋊Q₈	C₂²	1	2	yes	—
15	C₄⋊Q₈	C₂²	1	8	yes	—
16	C₄⋊Q₈	C₂²	1	8	yes	—
17	#127	C₂²	1	1	yes	characteristic
18	#126	C₂²	1	16	yes	—
19	#125	C₂²	1	4	yes	—
20	#124	C₂²	1	2	yes	—
21	#124	C₂²	1	8	yes	—
22	#124	C₂²	1	8	yes	—
23	#123	C₂²	1	8	yes	—
24	#123	C₂²	1	8	yes	—
25	C₂²⋊Q₈	C₂²	1	4	yes	—
26	C₂²⋊Q₈	C₂²	1	4	yes	—
27	C₂²⋊Q₈	C₂²	1	8	yes	—
28	C₂²⋊Q₈	C₂²	1	16	yes	—
29	#121	C₂²	1	4	yes	—
30	C₄×Q₈	C₂²	1	4	yes	—
31	C₄×Q₈	C₂²	1	4	yes	—
32	C₄×Q₈	C₂²	1	8	yes	—
33	C₄×D₄	C₂²	1	4	yes	—
34	C₄×D₄	C₂²	1	8	yes	—
35	#117	C₂²	1	4	yes	—
36	#117	C₂²	1	4	yes	—
37	C₂×C₄²	C₂²	1	1	yes	characteristic
38	#41	—	4	4	no	—
39	C₂×Q₈	—	4	8	no	—
40	C₂×Q₈	C₂³	1	4	yes	—
41	C₂×Q₈	C₂³	1	4	yes	—
42	C₂×Q₈	C₂³	1	8	yes	—
43	C₂×Q₈	C₂³	1	2	yes	—
44	C₂×D₄	C₂³	1	4	yes	—
45	C₂×D₄	C₂³	1	4	yes	—
46	C₂×D₄	C₂³	1	2	yes	—
47	C₂²×C₄	C₂³	1	2	yes	—
48	C₂²×C₄	C₂³	1	1	yes	characteristic
49	C₂²×C₄	C₂³	1	4	yes	—
50	C₂²×C₄	C₂³	1	8	yes	—
51	C₄⋊C₄	C₂³	1	4	yes	—
52	C₄⋊C₄	C₂³	1	4	yes	—
53	C₄⋊C₄	C₂³	1	4	yes	—
54	C₄⋊C₄	C₂³	1	4	yes	—
55	C₄⋊C₄	C₂³	1	4	yes	—
56	C₄⋊C₄	C₂³	1	8	yes	—
57	C₄⋊C₄	C₂³	1	16	yes	—
58	C₄⋊C₄	C₂³	1	8	yes	—
59	C₂²⋊C₄	C₂³	1	8	yes	—
60	C₂²⋊C₄	C₂³	1	8	yes	—
61	C₂²⋊C₄	C₂³	1	8	yes	—
62	C₂²⋊C₄	C₂³	1	16	yes	—
63	C₄²	C₂³	1	2	yes	—
64	C₄²	C₂³	1	1	yes	characteristic
65	C₄²	C₂³	1	1	yes	characteristic
66	C₄²	C₂³	1	8	yes	—
67	C₄²	C₂³	1	8	yes	—
68	C₂³	C₂⁴	1	4	yes	—
69	C₂³	C₂⁴	1	1	yes	characteristic
70	Q₈	—	2	8	no	—
71	Q₈	—	2	8	no	—
72	Q₈	—	4	8	no	—
73	Q₈	—	4	2	no	—
74	D₄	—	2	8	no	—
75	D₄	—	4	2	no	—
76	D₄	—	4	4	no	—
77	C₂×C₄	—	2	8	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	—	4	8	no	—
81	C₂×C₄	C₂⁴	1	8	yes	—
82	C₂×C₄	C₂⁴	1	2	yes	—
83	C₂×C₄	C₂⁴	1	4	yes	—
84	C₂×C₄	C₂⁴	1	1	yes	characteristic
85	C₂×C₄	C₂⁴	1	8	yes	—
86	C₂×C₄	C₂⁴	1	1	yes	characteristic
87	C₂×C₄	C₂⁴	1	2	yes	—
88	C₂²	—	2	4	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	2	2	no	—
91	C₂²	—	2	8	no	—
92	C₂²	C₂⁵	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	4	no	—
95	C₄	—	2	8	no	—
96	C₄	—	2	2	no	—
97	C₄	—	2	8	no	—
98	C₄	H(5,𝔽₂)	1	2	yes	—
99	C₄	2₋¹⁺⁴	1	4	yes	—
100	C₂	—	4	4	no	—
101	C₂	—	4	1	no	—
102	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
103	C₂	C₂×2₋¹⁺⁴	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#563

C₂

yes

maximal, complemented

#559

C₂

yes

maximal, complemented

Q₈²

C₂

yes

maximal, complemented

#556

C₂

yes

maximal, complemented

D₄×Q₈

C₂

yes

maximal, complemented

#542

C₂

yes

maximal, complemented

#536

C₂

yes

maximal, complemented

#533

C₂

yes

maximal, complemented

#532

C₂

yes

maximal, characteristic

C₂×#41

C₂²

yes

—

C₂²×Q₈

C₂²

yes

—

C₄⋊Q₈

C₂²

yes

characteristic

C₄⋊Q₈

C₂²

yes

—

C₄⋊Q₈

C₂²

yes

—

C₄⋊Q₈

C₂²

yes

—

C₄⋊Q₈

C₂²

yes

—

#127

C₂²

yes

characteristic

#126

C₂²

yes

—

#125

C₂²

yes

—

#124

C₂²

yes

—

#124

C₂²

yes

—

#124

C₂²

yes

—

#123

C₂²

yes

—

#123

C₂²

yes

—

C₂²⋊Q₈

C₂²

yes

—

C₂²⋊Q₈

C₂²

yes

—

C₂²⋊Q₈

C₂²

yes

—

C₂²⋊Q₈

C₂²

yes

—

#121

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

—

#117

C₂²

yes

—

C₂×C₄²

C₂²

yes

characteristic

#41

—

C₂×Q₈

—

C₂×Q₈

C₂³

yes

—

C₂×Q₈

C₂³

yes

—

C₂×Q₈

C₂³

yes

—

C₂×Q₈

C₂³

yes

—

C₂×D₄

C₂³

yes

—

C₂×D₄

C₂³

yes

—

C₂×D₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

—

C₂²×C₄

C₂³

yes

characteristic

C₂²×C₄

C₂³

yes

—

C₂²×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

—

C₄²

C₂³

yes

characteristic

C₄²

C₂³

yes

characteristic

C₄²

C₂³

yes

—

C₄²

C₂³

yes

—

C₂³

C₂⁴

yes

—

C₂³

C₂⁴

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂⁴

yes

—

C₂×C₄

C₂⁴

yes

—

C₂×C₄

C₂⁴

yes

—

C₂×C₄

C₂⁴

yes

characteristic

C₂×C₄

C₂⁴

yes

—

C₂×C₄

C₂⁴

yes

characteristic

C₂×C₄

C₂⁴

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂⁵

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

H(5,𝔽₂)

yes

—

C₄

2₋¹⁺⁴

yes

—

100

C₂

—

101

C₂

—

102

C₂

C₂×H(5,𝔽₂)

yes

characteristic

103

C₂

C₂×2₋¹⁺⁴

yes

—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
prime radical	#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

subgroup

G/H

center, commutator, Frattini, socle, cosocle

C₂²

C₂⁵

C₂

G ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ G

G ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #563	C₂	semidirect
2. #559	C₂	semidirect
3. Q₈²	C₂	semidirect
4. #556	C₂	semidirect
5. D₄×Q₈	C₂	semidirect
6. #542	C₂	semidirect
7. #536	C₂	semidirect
8. #533	C₂	semidirect

H ≅ G/N

type

1. #563

C₂

semidirect

2. #559

C₂

semidirect

3. Q₈²

C₂

semidirect

4. #556

C₂

semidirect

5. D₄×Q₈

C₂

semidirect

6. #542

C₂

semidirect

7. #536

C₂

semidirect

8. #533

C₂

semidirect

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

subgroup	index
#563	2
#559	2
Q₈²	2
#556	2
D₄×Q₈	2
#542	2
#536	2
#533	2
#532	2

subgroup

index

Q₈²

D₄×Q₈

Subgroups — #3550

Conjugacy classes of subgroups

Complements

Maximal subgroups