Group #30637626 — subgroups

Subgroups — #30637626

Order 16256.

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

normal subgroups only

This group has 152 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	#22550003	C₂	1	2	yes	maximal
2	C₄×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic
3	#22549606	C₂	1	2	yes	maximal
4	C₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#335×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₀₈	C₄	1	1	yes	characteristic
7	C₄×Dic₂₅₄	C₂²	1	2	yes	—
8	#16861143	C₄	1	1	yes	characteristic
9	Dic₁₂₇⋊C₈	—	2	2	no	—
10	#16861134	C₄	1	2	yes	—
11	#16861119	—	2	2	no	—
12	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
15	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₅₄	C₂×C₄	1	2	yes	—
17	Dic₅₀₈	—	2	2	no	—
18	C₂×C₁₂₇⋊C₈	—	2	1	no	—
19	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
20	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
21	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
22	Dic₁₂₇⋊C₄	—	2	2	no	—
23	C₄×Dic₁₂₇	—	2	2	no	—
24	C₂×C₈×C₁₂₇	—	2	2	no	—
25	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
28	Dic₂₅₄	—	2	2	no	—
29	Dic₂₅₄	—	2	2	no	—
30	C₂×Dic₁₂₇	—	2	2	no	—
31	C₂×Dic₁₂₇	—	2	2	no	—
32	C₁₂₇⋊C₈	—	2	2	no	—
33	C₁₂₇⋊C₈	—	2	1	no	—
34	C₁₂₇⋊C₈	Q₁₆	1	2	yes	complemented
35	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
38	C₈×C₁₂₇	—	2	1	no	—
39	C₈×C₁₂₇	—	4	2	no	—
40	C₈×C₁₂₇	M₁₆	1	2	yes	—
41	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
42	C₄×C₁₂₇	—	2	1	no	—
43	C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₁₂₇	#131	1	1	yes	characteristic
45	C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₁₂₇	#135	1	1	yes	characteristic
47	Dic₁₂₇	—	4	2	no	—
48	Dic₁₂₇	—	4	2	no	—
49	C₂×C₁₂₇	#436	1	1	yes	characteristic
50	C₂×C₁₂₇	C₄×Q₁₆	1	1	yes	characteristic
51	C₂×C₁₂₇	#443	1	1	yes	characteristic
52	#1591	—	127	1	no	maximal, Sylow
53	C₁₂₇	#1591	1	1	yes	characteristic, complemented, Sylow
54	#446	—	127	2	no	—
55	C₄×Q₁₆	—	127	1	no	—
56	#335	D₁₂₇	1	1	yes	characteristic
57	Q₈⋊C₈	—	127	2	no	—
58	C₈²	—	127	1	no	—
59	C₂×Q₁₆	—	127	1	no	—
60	C₄×Q₈	—	127	2	no	—
61	#107	—	127	1	no	—
62	C₄⋊C₈	—	254	2	no	—
63	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
64	#103	—	127	2	no	—
65	#97	—	254	2	no	—
66	C₄×C₈	—	127	1	no	—
67	C₄×C₈	—	127	1	no	—
68	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
69	C₂×Q₈	—	127	2	no	—
70	Q₁₆	—	254	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	127	1	no	—
73	C₂×C₈	—	127	1	no	—
74	C₂×C₈	—	254	1	no	—
75	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
76	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
77	C₄⋊C₄	—	127	2	no	—
78	C₄⋊C₄	—	254	2	no	—
79	C₄²	—	254	2	no	—
80	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
81	Q₈	—	254	2	no	—
82	Q₈	—	254	2	no	—
83	C₂×C₄	—	254	2	no	—
84	C₂×C₄	—	254	2	no	—
85	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
86	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
87	C₂×C₄	#14523249	1	1	yes	characteristic
88	C₈	—	2	1	no	—
89	C₈	—	4	2	no	—
90	C₈	—	127	2	no	—
91	C₈	—	254	2	no	—
92	C₈	—	254	1	no	—
93	C₈	#14523231	1	2	yes	—
94	C₂²	#16861171	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	508	2	no	—
97	C₄	—	508	2	no	—
98	C₄	C₂×#14523231	1	1	yes	characteristic
99	C₄	#16861217	1	1	yes	characteristic
100	C₄	Q₁₆×D₁₂₇	1	1	yes	characteristic
101	C₄	#16861228	1	1	yes	characteristic
102	C₂	#22549953	1	1	yes	characteristic
103	C₂	#22549970	1	1	yes	characteristic
104	C₂	#22549992	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550003

C₂

yes

maximal

C₄×Dic₅₀₈

C₂

yes

maximal, characteristic

#22549606

C₂

yes

maximal

C₈×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic

#335×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₅₀₈

C₄

yes

characteristic

C₄×Dic₂₅₄

C₂²

yes

—

#16861143

C₄

yes

characteristic

Dic₁₂₇⋊C₈

—

#16861134

C₄

yes

—

#16861119

—

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

C₂×C₄

yes

—

Dic₅₀₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

D₄

yes

characteristic

C₂×C₁₂₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

Q₁₆

yes

complemented

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

M₁₆

yes

—

C₂²×C₁₂₇

C₄×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×M₁₆

yes

characteristic

C₄×C₁₂₇

#131

yes

characteristic

C₄×C₁₂₇

C₂×Q₁₆

yes

characteristic

C₄×C₁₂₇

#135

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#436

yes

characteristic

C₂×C₁₂₇

C₄×Q₁₆

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1591

yes

characteristic, complemented, Sylow

#446

—

127

—

C₄×Q₁₆

—

127

—

#335

D₁₂₇

yes

characteristic

Q₈⋊C₈

—

127

—

C₈²

—

127

—

C₂×Q₁₆

—

127

—

C₄×Q₈

—

127

—

#107

—

127

—

C₄⋊C₈

—

254

—

C₄⋊C₈

C₂×D₁₂₇

yes

—

#103

—

127

—

#97

—

254

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

127

—

Q₁₆

—

254

—

C₂×C₈

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄²

—

254

—

C₄²

C₂²×D₁₂₇

yes

characteristic

Q₈

—

254

—

Q₈

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

C₂×C₄

D₄×D₁₂₇

yes

characteristic

C₂×C₄

#14523249

yes

characteristic

C₈

—

C₈

—

C₈

—

127

—

C₈

—

254

—

C₈

—

254

—

C₈

#14523231

yes

—

C₂²

#16861171

yes

characteristic

C₄

—

C₄

—

508

—

C₄

—

508

—

C₄

yes

C₄

yes

100

C₄

Q₁₆×D₁₂₇

yes

characteristic

101

C₄

#16861228

yes

characteristic

102

C₂

#22549953

yes

characteristic

103

C₂

#22549970

yes

characteristic

104

C₂

#22549992

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#335×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₄×D₄
cosocle	C₄²×C₁₂₇	C₂³
hypercenter	#335	D₁₂₇
hypocenter	C₁₂₇	#1591
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#335×C₁₂₇

C₂

socle, prime radical

C₂²×C₁₂₇

C₄×D₄

cosocle

C₄²×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
34. C₁₂₇⋊C₈	Q₁₆	semidirect
53. C₁₂₇	#1591	semidirect

H ≅ G/N

type

34. C₁₂₇⋊C₈

Q₁₆

semidirect

53. C₁₂₇

#1591

semidirect

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

subgroup	index
#22550003	2
C₄×Dic₅₀₈	2
#22549606	2
C₈×C₁₂₇⋊C₈	2
#335×C₁₂₇	2
#1591	127

subgroup

index

#22550003

C₄×Dic₅₀₈

#22549606

C₈×C₁₂₇⋊C₈

#335×C₁₂₇

#1591

127

Subgroups — #30637626

Conjugacy classes of subgroups

Complements

Maximal subgroups