Group #30638358 — subgroups

Subgroups — #30638358

Order 16256.

220 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 69 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	#22550274	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550253	C₂	1	1	yes	maximal, characteristic
3	#22550249	C₂	1	2	yes	maximal, complemented
4	#22550223	C₂	1	2	yes	maximal, complemented
5	#322×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523237	—	2	4	no	—
7	C₂×D₁₀₁₆	—	2	2	no	—
8	#16861209	C₂²	1	2	yes	—
9	#16861206	C₂²	1	1	yes	characteristic
10	#16861206	C₂²	1	1	yes	characteristic
11	#16861141	—	2	2	no	—
12	#16861131	—	2	4	no	—
13	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
15	C₂×Dic₂₅₄	—	2	2	no	—
16	C₂×D₅₀₈	—	2	1	no	—
17	C₂×D₅₀₈	—	2	1	no	—
18	C₂×D₅₀₈	—	2	2	no	—
19	C₂×D₅₀₈	—	2	2	no	—
20	#14523237	—	4	8	no	—
21	D₁₀₁₆	—	4	4	no	—
22	C₄⋊Dic₁₂₇	—	2	2	no	—
23	C₄⋊Dic₁₂₇	—	2	2	no	—
24	C₂×C₈×C₁₂₇	D₄	1	4	yes	—
25	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
26	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₂²×D₁₂₇	—	4	1	no	—
28	C₂²×D₁₂₇	—	4	1	no	—
29	Dic₂₅₄	—	4	4	no	—
30	D₅₀₈	—	4	2	no	—
31	D₅₀₈	—	4	2	no	—
32	D₅₀₈	—	4	4	no	—
33	D₅₀₈	—	4	4	no	—
34	C₂×Dic₁₂₇	—	4	2	no	—
35	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
37	C₈×C₁₂₇	—	2	4	no	—
38	C₈×C₁₂₇	SD₁₆	1	4	yes	complemented
39	C₂²×C₁₂₇	#127	1	1	yes	characteristic
40	C₂×D₁₂₇	—	8	1	no	—
41	C₂×D₁₂₇	—	8	1	no	—
42	C₂×D₁₂₇	—	8	2	no	—
43	C₂×D₁₂₇	—	8	2	no	—
44	C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
45	C₄×C₁₂₇	AGL(1,ℤ/8)	1	4	yes	—
46	Dic₁₂₇	—	8	2	no	—
47	C₂×C₁₂₇	#492	1	1	yes	characteristic
48	C₂×C₁₂₇	#496	1	2	yes	—
49	D₁₂₇	—	16	1	no	—
50	D₁₂₇	—	16	1	no	—
51	#1715	—	127	1	no	maximal, Sylow
52	C₁₂₇	#1715	1	1	yes	characteristic, complemented, Sylow
53	#498	—	127	1	no	—
54	#493	—	127	1	no	—
55	#492	—	127	2	no	—
56	#486	—	127	2	no	—
57	#322	D₁₂₇	1	1	yes	characteristic, complemented
58	C₂×SD₁₆	—	254	4	no	—
59	C₂×D₈	—	254	2	no	—
60	C₄⋊Q₈	—	127	2	no	—
61	#127	—	127	1	no	—
62	#127	—	127	1	no	—
63	#106	—	254	2	no	—
64	#102	—	254	4	no	—
65	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
66	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
67	C₂×Q₈	—	254	2	no	—
68	C₂×D₄	—	254	2	no	—
69	C₂×D₄	—	254	1	no	—
70	C₂×D₄	—	254	1	no	—
71	C₂×D₄	—	254	2	no	—
72	SD₁₆	—	508	8	no	—
73	D₈	—	508	4	no	—
74	C₂×C₈	D₅₀₈	1	4	yes	—
75	C₂×C₈	D₅₀₈	1	2	yes	—
76	C₄⋊C₄	—	254	2	no	—
77	C₄⋊C₄	—	254	2	no	—
78	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
79	C₂³	—	508	1	no	—
80	C₂³	—	508	1	no	—
81	Q₈	—	508	4	no	—
82	D₄	—	508	2	no	—
83	D₄	—	508	2	no	—
84	D₄	—	508	4	no	—
85	D₄	—	508	4	no	—
86	C₂×C₄	—	508	2	no	—
87	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
88	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	#14523237	1	4	yes	complemented
91	C₂²	—	1016	1	no	—
92	C₂²	—	1016	1	no	—
93	C₂²	—	1016	2	no	—
94	C₂²	—	1016	2	no	—
95	C₂²	#16861206	1	1	yes	characteristic
96	C₄	—	1016	2	no	—
97	C₄	C₂×#14523237	1	2	yes	—
98	C₄	#16861237	1	4	yes	—
99	C₂	—	2032	1	no	—
100	C₂	—	2032	1	no	—
101	C₂	#22550249	1	1	yes	characteristic
102	C₂	#22550265	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550274

C₂

yes

maximal, characteristic, complemented

#22550253

C₂

yes

maximal, characteristic

#22550249

C₂

yes

maximal, complemented

#22550223

C₂

yes

maximal, complemented

#322×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523237

—

C₂×D₁₀₁₆

—

#16861209

C₂²

yes

—

#16861206

C₂²

yes

characteristic

#16861206

C₂²

yes

characteristic

#16861141

—

#16861131

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

#14523237

—

D₁₀₁₆

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

C₂×Dic₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

SD₁₆

yes

complemented

C₂²×C₁₂₇

#127

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

C₂×SD₁₆

yes

—

C₄×C₁₂₇

AGL(1,ℤ/8)

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

#492

yes

characteristic

C₂×C₁₂₇

#496

yes

—

D₁₂₇

—

D₁₂₇

—

#1715

—

127

maximal, Sylow

C₁₂₇

#1715

yes

characteristic, complemented, Sylow

#498

—

127

—

#493

—

127

—

#492

—

127

—

#486

—

127

—

#322

D₁₂₇

yes

characteristic, complemented

C₂×SD₁₆

—

254

—

C₂×D₈

—

254

—

C₄⋊Q₈

—

127

—

#127

—

127

—

#127

—

127

—

#106

—

254

—

#102

—

254

—

C₄×C₈

C₂×D₁₂₇

yes

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

254

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

SD₁₆

—

508

—

D₈

—

508

—

C₂×C₈

D₅₀₈

yes

—

C₂×C₈

D₅₀₈

yes

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

C₂²×D₁₂₇

yes

characteristic

C₂³

—

508

—

C₂³

—

508

—

Q₈

—

508

—

D₄

—

508

—

D₄

—

508

—

D₄

—

508

—

D₄

—

508

—

C₂×C₄

—

508

—

C₂×C₄

C₂×D₅₀₈

yes

characteristic

C₂×C₄

C₂×D₅₀₈

yes

—

C₈

—

C₈

#14523237

yes

complemented

C₂²

—

1016

—

C₂²

—

1016

—

C₂²

—

1016

—

C₂²

—

1016

—

C₂²

#16861206

yes

characteristic

C₄

—

1016

—

C₄

C₂×#14523237

yes

—

C₄

#16861237

yes

—

C₂

—

2032

—

100

C₂

—

2032

—

101

C₂

#22550249

yes

characteristic

102

C₂

#22550265

yes

—

	subgroup	G/H
center	C₂²	#16861206
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#322×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#127
prime radical	#22550253	C₂
hypercenter	#322	D₁₂₇
hypocenter	C₁₂₇	#1715
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#16861206

commutator, cosocle

C₄²×C₁₂₇

C₂³

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#322×C₁₂₇

C₂

C₂²×C₁₂₇

C₂

D₁₂₇

C₁₂₇

G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #22550274	C₂	semidirect
3. #22550249	C₂	semidirect
4. #22550223	C₂	semidirect
5. #322×C₁₂₇	C₂	semidirect
38. C₈×C₁₂₇	SD₁₆	semidirect
52. C₁₂₇	#1715	semidirect
57. #322	D₁₂₇	semidirect
90. C₈	#14523237	semidirect

H ≅ G/N

type

1. #22550274

C₂

semidirect

3. #22550249

C₂

semidirect

4. #22550223

C₂

semidirect

5. #322×C₁₂₇

C₂

semidirect

38. C₈×C₁₂₇

SD₁₆

semidirect

52. C₁₂₇

#1715

semidirect

57. #322

D₁₂₇

semidirect

90. C₈

#14523237

semidirect

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

subgroup	index
#22550274	2
#22550253	2
#22550249	2
#22550223	2
#322×C₁₂₇	2
#1715	127

subgroup

index

#322×C₁₂₇

127

Subgroups — #30638358

Conjugacy classes of subgroups

Complements

Maximal subgroups