Group #30652878 — subgroups

Subgroups — #30652878

Order 16256.

930 conjugacy classes of subgroups; only the 451 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 451 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	#22550630	C₂	1	10	yes	maximal, complemented
2	#22550624	C₂	1	5	yes	maximal, complemented
3	#22550432	C₂	1	5	yes	maximal, complemented
4	#22550411	C₂	1	10	yes	maximal, complemented
5	#564×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861208	C₂²	1	10	yes	—
7	#16861204	C₂²	1	20	yes	—
8	#16861201	C₂²	1	5	yes	—
9	#16861197	C₂²	1	10	yes	—
10	#16861194	C₂²	1	20	yes	—
11	#16861189	C₂²	1	10	yes	—
12	#16861187	C₂²	1	20	yes	—
13	#16861184	C₂²	1	10	yes	—
14	#16861174	C₂²	1	10	yes	—
15	#16861171	C₂²	1	10	yes	—
16	#16861167	C₂²	1	10	yes	—
17	C₄²×D₁₂₇	C₂²	1	5	yes	—
18	#125×C₁₂₇	C₂²	1	5	yes	—
19	#125×C₁₂₇	C₂²	1	10	yes	—
20	C₂×Dic₂₅₄	C₂³	1	10	yes	—
21	C₂×D₅₀₈	C₂³	1	10	yes	—
22	C₂×C₄×D₁₂₇	C₂³	1	5	yes	—
23	C₂×C₄×D₁₂₇	C₂³	1	10	yes	—
24	C₄⋊Dic₁₂₇	C₂³	1	10	yes	—
25	Dic₁₂₇⋊C₄	C₂³	1	20	yes	—
26	(C₂×D₁₂₇)⋊C₄	C₂³	1	20	yes	—
27	(C₂×D₁₂₇)⋊C₄	C₂³	1	20	yes	—
28	C₄×Dic₁₂₇	C₂³	1	5	yes	—
29	C₄×Dic₁₂₇	C₂³	1	10	yes	—
30	C₄⋊C₄×C₁₂₇	C₂³	1	10	yes	—
31	C₄⋊C₄×C₁₂₇	C₂³	1	10	yes	—
32	C₄⋊C₄×C₁₂₇	C₂³	1	10	yes	—
33	C₄²×C₁₂₇	C₂³	1	5	yes	—
34	C₂²×D₁₂₇	C₂⁴	1	5	yes	—
35	Dic₂₅₄	—	4	10	no	—
36	D₅₀₈	—	4	10	no	—
37	C₂×Dic₁₂₇	C₂⁴	1	10	yes	—
38	C₂×Dic₁₂₇	C₂⁴	1	1	yes	characteristic
39	C₄×D₁₂₇	—	2	10	no	—
40	C₄×D₁₂₇	—	4	10	no	—
41	C₂×C₄×C₁₂₇	C₂⁴	1	5	yes	—
42	C₂×C₄×C₁₂₇	C₂⁴	1	10	yes	—
43	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
44	C₂×D₁₂₇	—	2	5	no	—
45	C₂×D₁₂₇	—	2	10	no	—
46	C₄×C₁₂₇	—	2	5	no	—
47	C₄×C₁₂₇	—	2	10	no	—
48	Dic₁₂₇	—	2	10	no	—
49	Dic₁₂₇	2₋¹⁺⁴	1	2	yes	—
50	C₂×C₁₂₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
51	C₂×C₁₂₇	#585	1	2	yes	—
52	D₁₂₇	—	4	5	no	—
53	#3531	—	127	1	no	maximal, Sylow
54	C₁₂₇	#3531	1	1	yes	characteristic, complemented, Sylow
55	#564	D₁₂₇	1	1	yes	characteristic, complemented
56	#563	—	127	10	no	—
57	#562	—	127	5	no	—
58	#533	—	127	5	no	—
59	#529	—	127	10	no	—
60	C₄⋊Q₈	—	127	10	no	—
61	#126	—	127	20	no	—
62	#125	—	127	5	no	—
63	#125	C₂×D₁₂₇	1	10	yes	—
64	#125	C₂×D₁₂₇	1	5	yes	—
65	#124	—	127	10	no	—
66	#123	—	127	20	no	—
67	C₂²⋊Q₈	—	127	20	no	—
68	C₂²⋊Q₈	—	127	10	no	—
69	#121	—	127	10	no	—
70	C₄×Q₈	—	127	10	no	—
71	C₄×D₄	—	127	10	no	—
72	#117	—	127	10	no	—
73	C₂×C₄²	—	127	5	no	—
74	C₂×Q₈	—	127	10	no	—
75	C₂×D₄	—	127	10	no	—
76	C₂²×C₄	—	127	10	no	—
77	C₂²×C₄	—	127	5	no	—
78	C₄⋊C₄	—	127	20	no	—
79	C₄⋊C₄	—	127	10	no	—
80	C₄⋊C₄	C₂²×D₁₂₇	1	10	yes	—
81	C₄⋊C₄	C₂²×D₁₂₇	1	10	yes	—
82	C₄⋊C₄	C₂²×D₁₂₇	1	10	yes	—
83	C₂²⋊C₄	—	127	20	no	—
84	C₂²⋊C₄	—	127	20	no	—
85	C₄²	—	127	10	no	—
86	C₄²	—	127	5	no	—
87	C₄²	C₂²×D₁₂₇	1	5	yes	—
88	C₂³	—	127	5	no	—
89	Q₈	—	508	10	no	—
90	D₄	—	508	10	no	—
91	C₂×C₄	—	127	1	no	—
92	C₂×C₄	—	127	10	no	—
93	C₂×C₄	—	508	10	no	—
94	C₂×C₄	—	254	10	no	—
95	C₂×C₄	C₂³×D₁₂₇	1	5	yes	—
96	C₂×C₄	C₂³×D₁₂₇	1	10	yes	—
97	C₂²	—	254	5	no	—
98	C₂²	—	254	10	no	—
99	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
100	C₄	—	2	5	no	—
101	C₄	—	2	10	no	—
102	C₄	—	127	2	no	—
103	C₄	—	254	10	no	—
104	C₂	—	508	5	no	—
105	C₂	2₋¹⁺⁴×D₁₂₇	1	1	yes	characteristic
106	C₂	#22550713	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550630

C₂

yes

maximal, complemented

#22550624

C₂

yes

maximal, complemented

#22550432

C₂

yes

maximal, complemented

#22550411

C₂

yes

maximal, complemented

#564×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#16861208

C₂²

yes

—

#16861204

C₂²

yes

—

#16861201

C₂²

yes

—

#16861197

C₂²

yes

—

#16861194

C₂²

yes

—

#16861189

C₂²

yes

—

#16861187

C₂²

yes

—

#16861184

C₂²

yes

—

#16861174

C₂²

yes

—

#16861171

C₂²

yes

—

#16861167

C₂²

yes

—

C₄²×D₁₂₇

C₂²

yes

—

#125×C₁₂₇

C₂²

yes

—

#125×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

C₂³

yes

—

C₂×D₅₀₈

C₂³

yes

—

C₂×C₄×D₁₂₇

C₂³

yes

—

C₂×C₄×D₁₂₇

C₂³

yes

—

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

C₂³

yes

—

(C₂×D₁₂₇)⋊C₄

C₂³

yes

—

(C₂×D₁₂₇)⋊C₄

C₂³

yes

—

C₄×Dic₁₂₇

C₂³

yes

—

C₄×Dic₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₄²×C₁₂₇

C₂³

yes

—

C₂²×D₁₂₇

C₂⁴

yes

—

Dic₂₅₄

—

D₅₀₈

—

C₂×Dic₁₂₇

C₂⁴

yes

—

C₂×Dic₁₂₇

C₂⁴

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₂×C₄×C₁₂₇

C₂⁴

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

—

C₂²×C₁₂₇

C₂⁵

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

2₋¹⁺⁴

yes

—

C₂×C₁₂₇

C₂×2₋¹⁺⁴

yes

characteristic

C₂×C₁₂₇

#585

yes

—

D₁₂₇

—

#3531

—

127

maximal, Sylow

C₁₂₇

#3531

yes

characteristic, complemented, Sylow

#564

D₁₂₇

yes

characteristic, complemented

#563

—

127

—

#562

—

127

—

#533

—

127

—

#529

—

127

—

C₄⋊Q₈

—

127

—

#126

—

127

—

#125

—

127

—

#125

C₂×D₁₂₇

yes

—

#125

C₂×D₁₂₇

yes

—

#124

—

127

—

#123

—

127

—

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

—

127

—

#121

—

127

—

C₄×Q₈

—

127

—

C₄×D₄

—

127

—

#117

—

127

—

C₂×C₄²

—

127

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂²×D₁₂₇

yes

—

C₄⋊C₄

C₂²×D₁₂₇

yes

—

C₄⋊C₄

C₂²×D₁₂₇

yes

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₄²

—

127

—

C₄²

—

127

—

C₄²

C₂²×D₁₂₇

yes

—

C₂³

—

127

—

Q₈

—

508

—

D₄

—

508

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

C₂³×D₁₂₇

yes

—

C₂×C₄

C₂³×D₁₂₇

yes

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

C₂⁴×D₁₂₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

127

—

103

C₄

—

254

—

104

C₂

—

508

—

105

C₂

2₋¹⁺⁴×D₁₂₇

yes

characteristic

106

C₂

#22550713

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₂₇
commutator, socle, cosocle	C₂²×C₁₂₇	C₂⁵
Fitting	#564×C₁₂₇	C₂
hypercenter	#564	D₁₂₇
hypocenter	C₁₂₇	#3531
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₂₇

commutator, socle, cosocle

C₂²×C₁₂₇

C₂⁵

Fitting

#564×C₁₂₇

C₂

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
1. #22550630	C₂	semidirect
2. #22550624	C₂	semidirect
3. #22550432	C₂	semidirect
4. #22550411	C₂	semidirect
5. #564×C₁₂₇	C₂	semidirect
54. C₁₂₇	#3531	semidirect
55. #564	D₁₂₇	semidirect

H ≅ G/N

type

1. #22550630

C₂

semidirect

2. #22550624

C₂

semidirect

3. #22550432

C₂

semidirect

4. #22550411

C₂

semidirect

5. #564×C₁₂₇

C₂

semidirect

54. C₁₂₇

#3531

semidirect

55. #564

D₁₂₇

semidirect

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

subgroup	index
#22550630	2
#22550624	2
#22550432	2
#22550411	2
#564×C₁₂₇	2
#3531	127

subgroup

index

#564×C₁₂₇

127

Subgroups — #30652878

Conjugacy classes of subgroups

Complements

Maximal subgroups