Group #30637510 — subgroups

Subgroups — #30637510

Order 16256.

150 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 81 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	#22549933	C₂	1	1	yes	maximal, characteristic
2	#22549929	C₂	1	1	yes	maximal, characteristic
3	#22549629	C₂	1	2	yes	maximal, complemented
4	#22549627	C₂	1	2	yes	maximal, complemented
5	#431×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
7	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
8	C₂²⋊(C₁₂₇⋊C₈)	—	2	1	no	—
9	C₂²⋊(C₁₂₇⋊C₈)	—	2	1	no	—
10	#16861120	—	2	1	no	—
11	C₄×C₁₂₇⋊C₈	—	2	1	no	—
12	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₂₇	C₄	1	2	yes	—
14	C₂²⋊C₈×C₁₂₇	C₄	1	2	yes	—
15	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₁₂₇⋊C₈	—	2	2	no	—
18	C₂×C₁₂₇⋊C₈	—	2	2	no	—
19	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
26	C₄²×C₁₂₇	Q₈	1	1	yes	characteristic
27	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₁₂₇⋊C₈	—	4	2	no	—
29	C₁₂₇⋊C₈	—	4	2	no	—
30	C₂³×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	—	2	2	no	—
32	C₂×C₄×C₁₂₇	—	2	2	no	—
33	C₂×C₄×C₁₂₇	—	4	1	no	—
34	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₁₂₇	—	2	2	no	—
41	C₈×C₁₂₇	—	2	2	no	—
42	C₂²×C₁₂₇	—	2	1	no	—
43	C₂²×C₁₂₇	—	2	1	no	—
44	C₂²×C₁₂₇	—	2	1	no	—
45	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₁₂₇	—	2	1	no	—
47	C₄×C₁₂₇	—	2	1	no	—
48	C₄×C₁₂₇	—	2	1	no	—
49	C₄×C₁₂₇	—	2	1	no	—
50	C₄×C₁₂₇	#131	1	2	yes	—
51	C₄×C₁₂₇	#135	1	2	yes	—
52	C₂×C₁₂₇	—	4	1	no	—
53	C₂×C₁₂₇	#424	1	1	yes	characteristic
54	C₂×C₁₂₇	#427	1	1	yes	characteristic
55	C₂×C₁₂₇	#430	1	1	yes	characteristic
56	#1572	—	127	1	no	maximal, Sylow
57	C₁₂₇	#1572	1	1	yes	characteristic, complemented, Sylow
58	#432	—	127	1	no	—
59	#431	—	127	1	no	—
60	#431	D₁₂₇	1	1	yes	characteristic
61	#335	—	127	2	no	—
62	#334	—	127	2	no	—
63	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
64	C₄⋊C₈	—	127	2	no	—
65	C₄⋊C₈	—	127	2	no	—
66	C₄⋊C₈	Dic₁₂₇	1	2	yes	—
67	C₂²⋊C₈	—	254	1	no	—
68	C₂²⋊C₈	—	254	1	no	—
69	C₂²⋊C₈	Dic₁₂₇	1	2	yes	—
70	#97	—	254	1	no	—
71	C₄×C₈	—	254	1	no	—
72	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
73	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
74	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
75	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
76	C₂²×C₄	Dic₂₅₄	1	1	yes	characteristic
77	C₂×C₈	—	254	2	no	—
78	C₂×C₈	—	254	2	no	—
79	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
80	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
81	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
82	C₄²	D₅₀₈	1	1	yes	characteristic
83	C₄²	Dic₂₅₄	1	1	yes	characteristic
84	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
85	C₂³	C₄⋊Dic₁₂₇	1	1	yes	characteristic
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
92	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
93	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
95	C₈	—	2	2	no	—
96	C₈	—	2	2	no	—
97	C₈	—	508	2	no	—
98	C₈	—	508	2	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	#16861216	1	2	yes	—
108	C₄	#16861231	1	2	yes	—
109	C₂	—	4	1	no	—
110	C₂	#22549903	1	1	yes	characteristic
111	C₂	#22549914	1	1	yes	characteristic
112	C₂	#22549925	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22549933

C₂

yes

maximal, characteristic

#22549929

C₂

yes

maximal, characteristic

#22549629

C₂

yes

maximal, complemented

#22549627

C₂

yes

maximal, complemented

#431×C₁₂₇

C₂

yes

maximal, characteristic

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₂₇⋊C₈)

—

C₂²⋊(C₁₂₇⋊C₈)

—

#16861120

—

C₄×C₁₂₇⋊C₈

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

C₄

yes

—

C₂²⋊C₈×C₁₂₇

C₄

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

Q₈

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

Q₈

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

characteristic

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#131

yes

—

C₄×C₁₂₇

#135

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1572

yes

characteristic, complemented, Sylow

#432

—

127

—

#431

—

127

—

#431

D₁₂₇

yes

characteristic

#335

—

127

—

#334

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

Dic₁₂₇

yes

—

C₂²⋊C₈

—

254

—

C₂²⋊C₈

—

254

—

C₂²⋊C₈

Dic₁₂₇

yes

—

#97

—

254

—

C₄×C₈

—

254

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

C₂²×C₄

D₅₀₈

yes

characteristic

C₂²×C₄

Dic₂₅₄

yes

characteristic

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₂×C₈

C₂×Dic₁₂₇

yes

—

C₂×C₈

C₂×Dic₁₂₇

yes

—

C₄²

C₂×Dic₁₂₇

yes

characteristic

C₄²

D₅₀₈

yes

characteristic

C₄²

Dic₂₅₄

yes

characteristic

C₄²

C₂²×D₁₂₇

yes

characteristic

C₂³

C₄⋊Dic₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₁₂₇

yes

characteristic

C₂×C₄

C₄⋊Dic₁₂₇

yes

characteristic

C₂×C₄

C₄⋊Dic₁₂₇

yes

characteristic

C₂×C₄

C₂²×Dic₁₂₇

yes

characteristic

C₂×C₄

C₂×D₅₀₈

yes

characteristic

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

C₈

—

C₈

—

C₈

—

508

—

C₈

—

508

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂×C₄⋊Dic₁₂₇

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

#16861216

yes

—

108

C₄

#16861231

yes

—

109

C₂

—

110

C₂

#22549903

yes

characteristic

111

C₂

#22549914

yes

characteristic

112

C₂

#22549925

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₀₈
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#431×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₂₇	C₄⋊C₄
cosocle	C₄²×C₁₂₇	C₂³
hypercenter	#431	D₁₂₇
hypocenter	C₁₂₇	#1572
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₅₀₈

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#431×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₂₇

C₄⋊C₄

cosocle

C₄²×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #22549629	C₂	semidirect
4. #22549627	C₂	semidirect
57. C₁₂₇	#1572	semidirect

H ≅ G/N

type

3. #22549629

C₂

semidirect

4. #22549627

C₂

semidirect

57. C₁₂₇

#1572

semidirect

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

subgroup	index
#22549933	2
#22549929	2
#22549629	2
#22549627	2
#431×C₁₂₇	2
#1572	127

subgroup

index

#431×C₁₂₇

127

Subgroups — #30637510

Conjugacy classes of subgroups

Complements

Maximal subgroups