Group #3126 — subgroups

Subgroups — #3126

Order 128.

191 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 191 conjugacy classes of subgroups — too many to draw. Only its 88 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	#540	C₂	1	1	yes	maximal, characteristic
2	#519	C₂	1	1	yes	maximal, characteristic, complemented
3	#482	C₂	1	2	yes	maximal, complemented
4	#479	C₂	1	2	yes	maximal, complemented
5	#460	C₂	1	2	yes	maximal, complemented
6	#448	C₂	1	2	yes	maximal, complemented
7	#440	C₂	1	2	yes	maximal, complemented
8	C₄×SD₁₆	C₂	1	2	yes	maximal, complemented
9	#433	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆	C₂²	1	2	yes	—
12	C₂×SD₁₆	C₂²	1	2	yes	—
13	#127	C₂²	1	1	yes	characteristic
14	#125	C₂²	1	1	yes	characteristic
15	#123	—	2	2	no	—
16	#121	—	2	2	no	—
17	#121	C₂²	1	2	yes	—
18	C₄×Q₈	—	2	2	no	—
19	C₄×Q₈	C₂²	1	2	yes	—
20	C₄×Q₈	C₂²	1	2	yes	—
21	C₄×D₄	C₂²	1	2	yes	—
22	#117	—	2	2	no	—
23	#117	C₂²	1	1	yes	characteristic
24	#117	C₂²	1	1	yes	characteristic
25	#107	C₂²	1	2	yes	—
26	#106	C₂²	1	2	yes	—
27	C₄⋊C₈	C₂²	1	2	yes	—
28	#103	C₂²	1	2	yes	—
29	#103	C₂²	1	2	yes	—
30	#102	C₂²	1	2	yes	—
31	#102	C₂²	1	2	yes	—
32	C₂²⋊C₈	C₂²	1	2	yes	—
33	#97	C₂²	1	1	yes	characteristic
34	C₄×C₈	C₂²	1	1	yes	characteristic
35	C₂×Q₈	—	2	4	no	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×Q₈	C₂³	1	2	yes	—
38	C₂×D₄	—	2	2	no	—
39	C₂×D₄	—	2	2	no	—
40	C₂×D₄	C₂³	1	2	yes	—
41	C₂²×C₄	—	2	1	no	—
42	C₂²×C₄	—	2	2	no	—
43	C₂²×C₄	C₂³	1	1	yes	characteristic
44	SD₁₆	—	2	4	no	—
45	SD₁₆	—	2	4	no	—
46	C₂×C₈	C₂³	1	2	yes	—
47	C₂×C₈	C₂³	1	2	yes	—
48	C₄⋊C₄	—	2	2	no	—
49	C₄⋊C₄	—	2	2	no	—
50	C₄⋊C₄	—	2	2	no	—
51	C₄⋊C₄	D₄	1	2	yes	—
52	C₄⋊C₄	C₂³	1	2	yes	—
53	C₄⋊C₄	C₂³	1	2	yes	—
54	C₂²⋊C₄	—	2	1	no	—
55	C₂²⋊C₄	—	2	2	no	—
56	C₂²⋊C₄	—	2	2	no	—
57	C₂²⋊C₄	D₄	1	2	yes	—
58	C₄²	—	2	2	no	—
59	C₄²	—	2	2	no	—
60	C₄²	—	2	1	no	—
61	C₄²	C₂³	1	1	yes	characteristic
62	C₄²	C₂³	1	1	yes	characteristic
63	C₂³	—	2	2	no	—
64	C₂³	C₂×D₄	1	1	yes	characteristic
65	Q₈	—	2	4	no	—
66	Q₈	—	2	2	no	—
67	Q₈	#41	1	4	yes	—
68	D₄	—	2	2	no	—
69	D₄	—	2	2	no	—
70	D₄	—	8	2	no	—
71	D₄	—	8	2	no	—
72	C₂×C₄	—	2	4	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	8	2	no	—
78	C₂×C₄	C₂×D₄	1	2	yes	—
79	C₂×C₄	C₂×D₄	1	2	yes	—
80	C₂×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄	C₂⁴	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	—	2	2	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	2	no	—
86	C₂²	—	4	2	no	—
87	C₂²	—	4	2	no	—
88	C₂²	—	4	2	no	—
89	C₂²	C₂²×D₄	1	1	yes	characteristic
90	C₄	—	2	4	no	—
91	C₄	—	2	1	no	—
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	4	2	no	—
95	C₄	—	4	2	no	—
96	C₄	C₂×#41	1	2	yes	—
97	C₂	—	4	1	no	—
98	C₂	—	8	2	no	—
99	C₂	#525	1	1	yes	characteristic
100	C₂	#577	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#540

C₂

yes

maximal, characteristic

#519

C₂

yes

maximal, characteristic, complemented

#482

C₂

yes

maximal, complemented

#479

C₂

yes

maximal, complemented

#460

C₂

yes

maximal, complemented

#448

C₂

yes

maximal, complemented

#440

C₂

yes

maximal, complemented

C₄×SD₁₆

C₂

yes

maximal, complemented

#433

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈

C₂²

yes

characteristic

C₂×SD₁₆

C₂²

yes

—

C₂×SD₁₆

C₂²

yes

—

#127

C₂²

yes

characteristic

#125

C₂²

yes

—

—

C₂²

yes

—

C₄×Q₈

—

C₄×Q₈

C₂²

yes

—

C₄×Q₈

C₂²

yes

—

C₄×D₄

C₂²

yes

—

#117

—

#117

C₂²

yes

characteristic

#117

C₂²

yes

characteristic

#107

C₂²

yes

—

#106

C₂²

yes

—

C₄⋊C₈

C₂²

yes

—

#103

C₂²

yes

—

#103

C₂²

yes

—

#102

C₂²

yes

—

#102

C₂²

yes

—

C₂²⋊C₈

C₂²

yes

—

#97

C₂²

yes

characteristic

C₄×C₈

C₂²

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂³

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂³

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂³

yes

characteristic

SD₁₆

—

SD₁₆

—

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄

yes

—

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

D₄

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂³

yes

characteristic

C₄²

C₂³

yes

characteristic

C₂³

—

C₂³

C₂×D₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

#41

yes

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂⁴

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×D₄

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

C₂×#41

yes

—

C₂

—

C₂

—

C₂

#525

yes

characteristic

100

C₂

#577

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	#540	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴

C₂

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂² ◁ #117 ◁ G

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #519	C₂	semidirect
3. #482	C₂	semidirect
4. #479	C₂	semidirect
5. #460	C₂	semidirect
6. #448	C₂	semidirect
7. #440	C₂	semidirect
8. C₄×SD₁₆	C₂	semidirect
9. #433	C₂	semidirect

H ≅ G/N

type

2. #519

C₂

semidirect

3. #482

C₂

semidirect

4. #479

C₂

semidirect

5. #460

C₂

semidirect

6. #448

C₂

semidirect

7. #440

C₂

semidirect

8. C₄×SD₁₆

C₂

semidirect

9. #433

C₂

semidirect

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

subgroup	index
#540	2
#519	2
#482	2
#479	2
#460	2
#448	2
#440	2
C₄×SD₁₆	2
#433	2

subgroup

index

C₄×SD₁₆

Subgroups — #3126

Conjugacy classes of subgroups

Complements

Maximal subgroups