Group #30639893 — subgroups

Subgroups — #30639893

Order 16256.

466 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 71 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	#22549731	C₂	1	3	yes	maximal
2	#22549643	C₂	1	3	yes	maximal, complemented
3	#543×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₁₂₇	C₂²	1	1	yes	characteristic
5	C₂×C₂²⋊Dic₁₂₇	C₂²	1	3	yes	—
6	C₂×(C₂×D₁₂₇)⋊C₄	—	2	6	no	—
7	C₂²⋊Q₈×C₁₂₇	—	2	6	no	—
8	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	3	yes	—
9	C₂³×D₁₂₇	—	2	3	no	—
10	C₂³×D₁₂₇	—	2	3	no	—
11	C₂³×D₁₂₇	—	4	4	no	—
12	C₂³×D₁₂₇	Q₈	1	2	yes	complemented
13	C₂²×Dic₁₂₇	—	2	3	no	—
14	C₂²⋊Dic₁₂₇	—	2	6	no	—
15	(C₂×D₁₂₇)⋊C₄	—	4	12	no	—
16	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
17	C₂×Q₈×C₁₂₇	—	2	2	no	—
18	C₂²×C₄×C₁₂₇	D₄	1	6	yes	—
19	C₄⋊C₄×C₁₂₇	—	2	6	no	—
20	C₂²⋊C₄×C₁₂₇	—	2	6	no	—
21	C₂²×D₁₂₇	—	2	12	no	—
22	C₂²×D₁₂₇	—	2	6	no	—
23	C₂²×D₁₂₇	—	4	4	no	—
24	C₂²×D₁₂₇	—	4	12	no	—
25	C₂²×D₁₂₇	—	4	6	no	—
26	C₂²×D₁₂₇	—	4	1	no	—
27	C₂²×D₁₂₇	—	4	3	no	—
28	C₂×Dic₁₂₇	—	4	6	no	—
29	C₂×Dic₁₂₇	—	4	3	no	—
30	C₂³×C₁₂₇	—	4	2	no	—
31	C₂³×C₁₂₇	C₂×D₄	1	3	yes	—
32	C₂³×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
33	C₂³×C₁₂₇	#41	1	3	yes	—
34	Q₈×C₁₂₇	—	8	2	no	—
35	C₂×C₄×C₁₂₇	—	2	6	no	—
36	C₂×C₄×C₁₂₇	—	4	6	no	—
37	C₂²×C₁₂₇	—	2	6	no	—
38	C₂²×C₁₂₇	—	4	1	no	—
39	C₂²×C₁₂₇	—	4	3	no	—
40	C₂²×C₁₂₇	#120	1	1	yes	characteristic
41	C₂²×C₁₂₇	C₂²⋊Q₈	1	3	yes	—
42	C₂²×C₁₂₇	#124	1	3	yes	—
43	C₂×D₁₂₇	—	2	12	no	—
44	C₂×D₁₂₇	—	4	12	no	—
45	C₂×D₁₂₇	—	4	6	no	—
46	C₂×D₁₂₇	—	4	2	no	—
47	C₂×D₁₂₇	—	4	4	no	—
48	C₄×C₁₂₇	—	4	6	no	—
49	Dic₁₂₇	—	8	3	no	—
50	C₂×C₁₂₇	—	2	3	no	—
51	C₂×C₁₂₇	—	2	3	no	—
52	C₂×C₁₂₇	#393	1	1	yes	characteristic
53	C₂×C₁₂₇	B(4,𝔽₂)	1	2	yes	—
54	D₁₂₇	—	4	4	no	—
55	#2029	—	127	1	no	maximal, Sylow
56	C₁₂₇	#2029	1	1	yes	characteristic, complemented, Sylow
57	#543	D₁₂₇	1	1	yes	characteristic, complemented
58	#379	—	127	3	no	—
59	#342	—	127	3	no	—
60	C₂⁵	—	127	1	no	—
61	C₂²⋊Q₈	—	2	6	no	—
62	C₂×C₂²⋊C₄	—	127	3	no	—
63	C₂×C₂²⋊C₄	—	254	6	no	—
64	C₂×C₂²⋊C₄	C₂×D₁₂₇	1	3	yes	—
65	C₂⁴	—	127	2	no	—
66	C₂⁴	—	508	4	no	—
67	C₂⁴	—	254	3	no	—
68	C₂⁴	—	254	3	no	—
69	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
70	C₂×Q₈	—	2	2	no	—
71	C₂²×C₄	—	254	3	no	—
72	C₂²×C₄	C₂²⋊D₁₂₇	1	6	yes	—
73	C₄⋊C₄	—	2	6	no	—
74	C₂²⋊C₄	—	2	6	no	—
75	C₂²⋊C₄	—	508	12	no	—
76	C₂²⋊C₄	—	254	6	no	—
77	C₂³	—	4	2	no	—
78	C₂³	—	508	6	no	—
79	C₂³	—	508	12	no	—
80	C₂³	—	508	4	no	—
81	C₂³	—	508	3	no	—
82	C₂³	—	508	1	no	—
83	C₂³	—	254	12	no	—
84	C₂³	—	254	6	no	—
85	C₂³	C₂×C₂²⋊D₁₂₇	1	3	yes	—
86	C₂³	Q₈×D₁₂₇	1	1	yes	characteristic
87	C₂³	#14523249	1	3	yes	—
88	Q₈	—	8	2	no	—
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	4	6	no	—
91	C₂×C₄	—	508	6	no	—
92	C₂×C₄	—	508	3	no	—
93	C₂²	—	2	6	no	—
94	C₂²	—	4	1	no	—
95	C₂²	—	4	3	no	—
96	C₂²	—	508	12	no	—
97	C₂²	—	508	6	no	—
98	C₂²	—	508	2	no	—
99	C₂²	—	508	4	no	—
100	C₂²	—	254	12	no	—
101	C₂²	#16861178	1	1	yes	characteristic
102	C₂²	#16861185	1	3	yes	—
103	C₂²	#16861198	1	3	yes	—
104	C₄	—	4	6	no	—
105	C₄	—	1016	3	no	—
106	C₂	—	2	3	no	—
107	C₂	—	2	3	no	—
108	C₂	—	508	4	no	—
109	C₂	#22549787	1	1	yes	characteristic
110	C₂	#22550063	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22549731

C₂

yes

maximal

#22549643

C₂

yes

maximal, complemented

#543×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₁₂₇

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₂₇

C₂²

yes

—

C₂×(C₂×D₁₂₇)⋊C₄

—

C₂²⋊Q₈×C₁₂₇

—

C₂×C₂²⋊C₄×C₁₂₇

C₂²

yes

—

C₂³×D₁₂₇

—

C₂³×D₁₂₇

—

C₂³×D₁₂₇

—

C₂³×D₁₂₇

Q₈

yes

complemented

C₂²×Dic₁₂₇

—

C₂²⋊Dic₁₂₇

—

(C₂×D₁₂₇)⋊C₄

—

C₂⁴×C₁₂₇

C₂³

yes

characteristic

C₂×Q₈×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₄⋊C₄×C₁₂₇

—

C₂²⋊C₄×C₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

—

C₂³×C₁₂₇

C₂×Q₈

yes

characteristic

C₂³×C₁₂₇

#41

yes

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#120

yes

characteristic

C₂²×C₁₂₇

C₂²⋊Q₈

yes

—

C₂²×C₁₂₇

#124

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#393

yes

characteristic

C₂×C₁₂₇

B(4,𝔽₂)

yes

—

D₁₂₇

—

#2029

—

127

maximal, Sylow

C₁₂₇

#2029

yes

characteristic, complemented, Sylow

#543

D₁₂₇

yes

characteristic, complemented

#379

—

127

—

#342

—

127

—

C₂⁵

—

127

—

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

—

127

—

C₂×C₂²⋊C₄

—

254

—

C₂×C₂²⋊C₄

C₂×D₁₂₇

yes

—

C₂⁴

—

127

—

C₂⁴

—

508

—

C₂⁴

—

254

—

C₂⁴

—

254

—

C₂⁴

C₂²×D₁₂₇

yes

characteristic

C₂×Q₈

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

508

—

C₂²⋊C₄

—

254

—

C₂³

—

C₂³

—

508

—

C₂³

—

508

—

C₂³

—

508

—

C₂³

—

508

—

C₂³

—

508

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

C₂×C₂²⋊D₁₂₇

yes

—

C₂³

Q₈×D₁₂₇

yes

characteristic

C₂³

#14523249

yes

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

—

508

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

508

—

100

C₂²

—

254

—

101

C₂²

#16861178

yes

characteristic

102

C₂²

#16861185

yes

—

103

C₂²

#16861198

yes

—

104

C₄

—

105

C₄

—

1016

—

106

C₂

—

107

C₂

—

108

C₂

—

508

—

109

C₂

#22549787

yes

characteristic

110

C₂

#22550063

yes

—

	subgroup	G/H
center	C₂²	#16861178
commutator, cosocle	C₂⁴×C₁₂₇	C₂³
Frattini	C₂⁴	C₂²×D₁₂₇
Fitting	#543×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#120
prime radical	C₂⁴×D₁₂₇	C₂²
hypercenter	#543	D₁₂₇
hypocenter	C₁₂₇	#2029
LCS	G ▷ C₂⁴×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

subgroup

G/H

center

C₂²

#16861178

commutator, cosocle

C₂⁴×C₁₂₇

C₂³

Frattini

C₂⁴

C₂²×D₁₂₇

Fitting

#543×C₁₂₇

C₂

socle

C₂²×C₁₂₇

#120

prime radical

C₂⁴×D₁₂₇

C₂²

D₁₂₇

C₁₂₇

G ▷ C₂⁴×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

N	H ≅ G/N	type
2. #22549643	C₂	semidirect
3. #543×C₁₂₇	C₂	semidirect
12. C₂³×D₁₂₇	Q₈	semidirect
56. C₁₂₇	#2029	semidirect
57. #543	D₁₂₇	semidirect

H ≅ G/N

type

2. #22549643

C₂

semidirect

3. #543×C₁₂₇

C₂

semidirect

12. C₂³×D₁₂₇

Q₈

semidirect

56. C₁₂₇

#2029

semidirect

57. #543

D₁₂₇

semidirect

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

subgroup	index
#22549731	2
#22549643	2
#543×C₁₂₇	2
#2029	127

subgroup

index

#22549731

#22549643

#543×C₁₂₇

#2029

127

Subgroups — #30639893

Conjugacy classes of subgroups

Complements

Maximal subgroups