Group #30649244 — subgroups

Subgroups — #30649244

Order 16256.

356 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 173 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	#22550433	C₂	1	2	yes	maximal
2	#22550281	C₂	1	2	yes	maximal, complemented
3	#22550278	C₂	1	4	yes	maximal, complemented
4	#22550274	C₂	1	2	yes	maximal, complemented
5	#22549914	C₂	1	4	yes	maximal
6	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	#16861209	C₂²	1	4	yes	—
8	#16861201	C₂²	1	4	yes	—
9	#16861188	—	2	4	no	—
10	C₄×Dic₂₅₄	—	2	4	no	—
11	#16861168	C₂²	1	4	yes	—
12	#16861143	C₂²	1	8	yes	—
13	#16861141	C₂²	1	8	yes	—
14	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
18	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₅₄	—	2	4	no	—
20	C₄⋊Dic₁₂₇	—	2	4	no	—
21	C₄⋊Dic₁₂₇	C₂³	1	8	yes	—
22	Dic₁₂₇⋊C₄	—	2	8	no	—
23	C₂²⋊Dic₁₂₇	—	2	4	no	—
24	C₄×Dic₁₂₇	—	2	4	no	—
25	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₂₇	Q₈	1	8	yes	complemented
28	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
29	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
30	C₄²×C₁₂₇	D₄	1	2	yes	—
31	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
33	Dic₂₅₄	—	4	8	no	—
34	C₂×Dic₁₂₇	—	2	8	no	—
35	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	—	2	4	no	—
37	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
41	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
42	C₈×C₁₂₇	C₂×Q₈	1	4	yes	—
43	C₈×C₁₂₇	C₂×Q₈	1	4	yes	—
44	C₂²×C₁₂₇	—	2	2	no	—
45	C₂²×C₁₂₇	C₄⋊Q₈	1	2	yes	—
46	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
47	C₄×C₁₂₇	—	2	2	no	—
48	C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
49	C₄×C₁₂₇	#135	1	4	yes	—
50	C₄×C₁₂₇	C₂²×Q₈	1	2	yes	—
51	Dic₁₂₇	—	4	8	no	—
52	C₂×C₁₂₇	—	2	2	no	—
53	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₁₂₇	C₂×#135	1	2	yes	—
55	#3160	—	127	1	no	maximal, Sylow
56	C₁₂₇	#3160	1	1	yes	characteristic, complemented, Sylow
57	#533	—	127	2	no	—
58	#500	—	127	2	no	—
59	#499	—	127	4	no	—
60	#498	—	127	2	no	—
61	#427	—	127	4	no	—
62	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
64	C₄⋊Q₈	—	127	4	no	—
65	#125	—	127	4	no	—
66	C₂²⋊Q₈	—	254	4	no	—
67	C₄×Q₈	—	254	4	no	—
68	#117	—	127	4	no	—
69	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
70	#107	—	127	8	no	—
71	#106	—	127	8	no	—
72	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
73	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
74	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
75	C₂×Q₈	—	254	4	no	—
76	C₂²×C₄	D₅₀₈	1	2	yes	—
77	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
78	C₂×C₈	Dic₂₅₄	1	8	yes	complemented
79	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
80	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
81	C₄⋊C₄	—	127	8	no	—
82	C₄⋊C₄	—	254	8	no	—
83	C₄⋊C₄	—	254	4	no	—
84	C₂²⋊C₄	—	254	4	no	—
85	C₄²	—	254	4	no	—
86	C₄²	D₅₀₈	1	2	yes	—
87	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
88	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
89	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
90	Q₈	—	508	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	254	8	no	—
93	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
94	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
95	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
96	C₂×C₄	C₂×Dic₂₅₄	1	4	yes	—
97	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
98	C₈	C₂×Dic₂₅₄	1	4	yes	—
99	C₈	C₂×Dic₂₅₄	1	4	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	#16861209	1	2	yes	—
102	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	508	8	no	—
105	C₄	#16861209	1	2	yes	—
106	C₄	#16861231	1	4	yes	—
107	C₄	C₂²×Dic₂₅₄	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₂×#16861209	1	1	yes	characteristic
110	C₂	C₂×#16861231	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550433

C₂

yes

maximal

#22550281

C₂

yes

maximal, complemented

#22550278

C₂

yes

maximal, complemented

#22550274

C₂

yes

maximal, complemented

#22549914

C₂

yes

maximal

C₂×C₄×C₈×C₁₂₇

C₂

yes

maximal, characteristic

#16861209

C₂²

yes

—

#16861201

C₂²

yes

—

#16861188

—

C₄×Dic₂₅₄

—

#16861168

C₂²

yes

—

#16861143

C₂²

yes

—

#16861141

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

Q₈

yes

complemented

C₂×C₈×C₁₂₇

C₂³

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

C₂×Q₈

yes

—

C₈×C₁₂₇

C₂×Q₈

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄⋊Q₈

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄⋊Q₈

yes

—

C₄×C₁₂₇

#135

yes

—

C₄×C₁₂₇

C₂²×Q₈

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₂₇

C₂×#135

yes

—

#3160

—

127

maximal, Sylow

C₁₂₇

#3160

yes

characteristic, complemented, Sylow

#533

—

127

—

#500

—

127

—

#499

—

127

—

#498

—

127

—

#427

—

127

—

C₂×C₄×C₈

D₁₂₇

yes

characteristic

C₂²×C₈

C₂×D₁₂₇

yes

—

C₄⋊Q₈

—

127

—

#125

—

127

—

C₂²⋊Q₈

—

254

—

C₄×Q₈

—

254

—

#117

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#107

—

127

—

#106

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₄×C₈

C₂×D₁₂₇

yes

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

254

—

C₂²×C₄

D₅₀₈

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

Dic₂₅₄

yes

complemented

C₂×C₈

C₂²×D₁₂₇

yes

—

C₂×C₈

C₂²×D₁₂₇

yes

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₄²

—

254

—

C₄²

D₅₀₈

yes

—

C₄²

C₂²×D₁₂₇

yes

characteristic

C₄²

C₂²×D₁₂₇

yes

characteristic

C₂³

C₂×D₅₀₈

yes

characteristic

Q₈

—

508

—

C₂×C₄

—

C₂×C₄

—

254

—

C₂×C₄

C₂×D₅₀₈

yes

—

C₂×C₄

C₂×D₅₀₈

yes

—

C₂×C₄

C₂×D₅₀₈

yes

characteristic

C₂×C₄

C₂×Dic₂₅₄

yes

—

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

C₈

C₂×Dic₂₅₄

yes

—

C₈

C₂×Dic₂₅₄

yes

—

100

C₂²

—

101

C₂²

#16861209

yes

—

102

C₂²

C₂²×D₅₀₈

yes

characteristic

103

C₄

—

104

C₄

—

508

—

105

C₄

#16861209

yes

—

106

C₄

#16861231

yes

—

107

C₄

C₂²×Dic₂₅₄

yes

—

108

C₂

—

109

C₂

C₂×#16861209

yes

characteristic

110

C₂

C₂×#16861231

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₀₈
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#3160
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₅₀₈

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

C₂×C₄×C₈×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×D₄

prime radical

C₂³×C₁₂₇

C₂×D₄

hypercenter

C₂×C₄×C₈

D₁₂₇

hypocenter

C₁₂₇

#3160

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #22550281	C₂	semidirect
3. #22550278	C₂	semidirect
4. #22550274	C₂	semidirect
27. C₂×C₈×C₁₂₇	Q₈	semidirect
56. C₁₂₇	#3160	semidirect
78. C₂×C₈	Dic₂₅₄	semidirect

H ≅ G/N

type

2. #22550281

C₂

semidirect

3. #22550278

C₂

semidirect

4. #22550274

C₂

semidirect

27. C₂×C₈×C₁₂₇

Q₈

semidirect

56. C₁₂₇

#3160

semidirect

78. C₂×C₈

Dic₂₅₄

semidirect

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

subgroup	index
#22550433	2
#22550281	2
#22550278	2
#22550274	2
#22549914	2
C₂×C₄×C₈×C₁₂₇	2
#3160	127

subgroup

index

C₂×C₄×C₈×C₁₂₇

127

Subgroups — #30649244

Conjugacy classes of subgroups

Complements

Maximal subgroups