Group #30638397 — subgroups

Subgroups — #30638397

Order 16256.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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	#22550252	C₂	1	2	yes	maximal, complemented
2	#22550249	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550222	C₂	1	2	yes	maximal, complemented
4	#22549597	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523237	—	2	2	no	—
7	C₂×#14523233	—	2	4	no	—
8	#16861209	C₂²	1	1	yes	characteristic
9	#16861206	C₂²	1	1	yes	characteristic
10	#16861130	—	2	4	no	—
11	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
12	C₂×D₈×C₁₂₇	—	2	2	no	—
13	#127×C₁₂₇	C₂²	1	2	yes	—
14	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₅₄	—	2	1	no	—
16	C₂×D₅₀₈	—	2	2	no	—
17	C₂×D₅₀₈	—	2	1	no	—
18	#14523237	—	4	4	no	—
19	#14523233	—	4	8	no	—
20	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	—
21	C₄⋊Dic₁₂₇	—	2	2	no	—
22	C₂×D₄×C₁₂₇	—	2	2	no	—
23	C₂×D₄×C₁₂₇	—	2	2	no	—
24	C₂×D₄×C₁₂₇	—	2	2	no	—
25	D₈×C₁₂₇	—	4	4	no	—
26	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
27	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₂²×D₁₂₇	—	4	1	no	—
29	Dic₂₅₄	—	4	2	no	—
30	D₅₀₈	—	4	2	no	—
31	D₅₀₈	—	4	4	no	—
32	C₂×Dic₁₂₇	—	4	1	no	—
33	C₁₂₇⋊C₈	—	2	4	no	—
34	C₂³×C₁₂₇	—	4	2	no	—
35	D₄×C₁₂₇	—	4	4	no	—
36	D₄×C₁₂₇	—	4	4	no	—
37	D₄×C₁₂₇	—	4	4	no	—
38	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₈×C₁₂₇	D₈	1	4	yes	complemented
41	C₂²×C₁₂₇	—	8	2	no	—
42	C₂²×C₁₂₇	—	8	2	no	—
43	C₂²×C₁₂₇	—	8	2	no	—
44	C₂²×C₁₂₇	#127	1	1	yes	characteristic
45	C₂×D₁₂₇	—	8	1	no	—
46	C₂×D₁₂₇	—	8	2	no	—
47	C₄×C₁₂₇	C₂×D₈	1	2	yes	—
48	C₄×C₁₂₇	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₁₂₇	—	8	1	no	—
50	C₂×C₁₂₇	—	16	2	no	—
51	C₂×C₁₂₇	#493	1	1	yes	characteristic
52	C₂×C₁₂₇	#496	1	2	yes	—
53	D₁₂₇	—	16	1	no	—
54	#1721	—	127	1	no	maximal, Sylow
55	C₁₂₇	#1721	1	1	yes	characteristic, complemented, Sylow
56	#493	—	127	2	no	—
57	#493	D₁₂₇	1	1	yes	characteristic, complemented
58	#492	—	127	1	no	—
59	#486	—	127	2	no	—
60	#322	—	127	1	no	—
61	C₂×SD₁₆	—	254	2	no	—
62	C₂×D₈	—	2	2	no	—
63	C₂×D₈	—	254	4	no	—
64	C₄⋊Q₈	—	127	1	no	—
65	#127	—	127	1	no	—
66	#127	C₂×D₁₂₇	1	2	yes	—
67	#102	—	254	4	no	—
68	C₄×C₈	—	127	2	no	—
69	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
70	C₂×Q₈	—	254	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	2	no	—
73	C₂×D₄	—	2	2	no	—
74	C₂×D₄	—	254	2	no	—
75	C₂×D₄	—	254	1	no	—
76	SD₁₆	—	508	4	no	—
77	D₈	—	4	4	no	—
78	D₈	—	508	8	no	—
79	C₂×C₈	—	127	4	no	—
80	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
81	C₄⋊C₄	—	254	2	no	—
82	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
83	C₂³	—	4	2	no	—
84	C₂³	—	508	1	no	—
85	Q₈	—	508	2	no	—
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	4	no	—
89	D₄	—	508	2	no	—
90	D₄	—	508	4	no	—
91	C₂×C₄	—	508	1	no	—
92	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
94	C₈	—	254	4	no	—
95	C₈	#14523233	1	4	yes	complemented
96	C₂²	—	8	2	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	8	2	no	—
99	C₂²	—	1016	1	no	—
100	C₂²	—	1016	2	no	—
101	C₂²	#16861205	1	1	yes	characteristic
102	C₄	—	1016	1	no	—
103	C₄	C₂×#14523233	1	2	yes	—
104	C₄	#16861233	1	4	yes	—
105	C₂	—	16	2	no	—
106	C₂	—	2032	1	no	—
107	C₂	#22550252	1	1	yes	characteristic
108	C₂	#22550264	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550252

C₂

yes

maximal, complemented

#22550249

C₂

yes

maximal, characteristic, complemented

#22550222

C₂

yes

maximal, complemented

#22549597

C₂

yes

maximal, characteristic, complemented

#493×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523237

—

C₂×#14523233

—

#16861209

C₂²

yes

characteristic

#16861206

C₂²

yes

characteristic

#16861130

—

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₂×D₈×C₁₂₇

—

#127×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

#14523237

—

#14523233

—

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₄⋊Dic₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

D₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₂²×D₁₂₇

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

D₈

yes

complemented

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#127

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

C₂×D₈

yes

—

C₄×C₁₂₇

AGL(1,ℤ/8)

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#493

yes

characteristic

C₂×C₁₂₇

#496

yes

—

D₁₂₇

—

#1721

—

127

maximal, Sylow

C₁₂₇

#1721

yes

characteristic, complemented, Sylow

#493

—

127

—

#493

D₁₂₇

yes

characteristic, complemented

#492

—

127

—

#486

—

127

—

#322

—

127

—

C₂×SD₁₆

—

254

—

C₂×D₈

—

C₂×D₈

—

254

—

C₄⋊Q₈

—

127

—

#127

—

127

—

#127

C₂×D₁₂₇

yes

—

#102

—

254

—

C₄×C₈

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

254

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

SD₁₆

—

508

—

D₈

—

D₈

—

508

—

C₂×C₈

—

127

—

C₂×C₈

C₂²⋊D₁₂₇

yes

—

C₄⋊C₄

—

254

—

C₄²

C₂²×D₁₂₇

yes

characteristic

C₂³

—

C₂³

—

508

—

Q₈

—

508

—

D₄

—

D₄

—

D₄

—

D₄

—

508

—

D₄

—

508

—

C₂×C₄

—

508

—

C₂×C₄

D₄×D₁₂₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₈

—

254

—

C₈

#14523233

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

1016

—

100

C₂²

—

1016

—

101

C₂²

#16861205

yes

characteristic

102

C₄

—

1016

—

103

C₄

C₂×#14523233

yes

—

104

C₄

#16861233

yes

—

105

C₂

—

106

C₂

—

2032

—

107

C₂

#22550252

yes

characteristic

108

C₂

#22550264

yes

—

	subgroup	G/H
center	C₂²	#16861205
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#493×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#127
hypercenter	#493	D₁₂₇
hypocenter	C₁₂₇	#1721
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#16861205

commutator, cosocle

C₄²×C₁₂₇

C₂³

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#493×C₁₂₇

C₂

C₂²×C₁₂₇

D₁₂₇

C₁₂₇

G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #22550252	C₂	semidirect
2. #22550249	C₂	semidirect
3. #22550222	C₂	semidirect
4. #22549597	C₂	semidirect
5. #493×C₁₂₇	C₂	semidirect
40. C₈×C₁₂₇	D₈	semidirect
55. C₁₂₇	#1721	semidirect
57. #493	D₁₂₇	semidirect
95. C₈	#14523233	semidirect

H ≅ G/N

type

1. #22550252

C₂

semidirect

2. #22550249

C₂

semidirect

3. #22550222

C₂

semidirect

4. #22549597

C₂

semidirect

5. #493×C₁₂₇

C₂

semidirect

40. C₈×C₁₂₇

D₈

semidirect

55. C₁₂₇

#1721

semidirect

57. #493

D₁₂₇

semidirect

95. C₈

#14523233

semidirect

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

subgroup	index
#22550252	2
#22550249	2
#22550222	2
#22549597	2
#493×C₁₂₇	2
#1721	127

subgroup

index

#493×C₁₂₇

127

Subgroups — #30638397

Conjugacy classes of subgroups

Complements

Maximal subgroups