Group #30640494 — subgroups

Subgroups — #30640494

Order 16256.

152 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 75 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	M₃₂×D₁₂₇	C₂	1	2	yes	maximal
2	#22549830	C₂	1	1	yes	maximal, characteristic
3	#22549697	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549696	C₂	1	2	yes	maximal, complemented
5	#364×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523231	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
8	#16861166	C₄	1	1	yes	characteristic
9	#16861149	C₂²	1	2	yes	—
10	#16861148	—	2	2	no	—
11	C₁₆×D₁₂₇	—	2	2	no	—
12	#16861120	C₄	1	1	yes	characteristic
13	#16861119	C₄	1	1	yes	characteristic
14	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
15	M₃₂×C₁₂₇	C₂²	1	2	yes	—
16	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
18	#14523231	C₈	1	4	yes	—
19	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₈×D₁₂₇	D₄	1	2	yes	—
22	C₈×D₁₂₇	Q₈	1	2	yes	—
23	Dic₁₂₇⋊C₄	—	2	1	no	—
24	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
25	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₁₂₇⋊C₁₆	—	2	2	no	—
27	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₁₆×C₁₂₇	—	2	2	no	—
31	C₂²×D₁₂₇	M₁₆	1	1	yes	characteristic
32	C₂×Dic₁₂₇	—	2	1	no	—
33	C₂×Dic₁₂₇	M₁₆	1	1	yes	characteristic
34	C₄×D₁₂₇	C₄⋊C₄	1	2	yes	—
35	C₄×D₁₂₇	C₂×C₈	1	2	yes	—
36	C₁₂₇⋊C₈	C₄⋊C₄	1	2	yes	—
37	C₁₂₇⋊C₈	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₁₂₇	—	2	1	no	—
39	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₁₂₇	C₂×C₈	1	2	yes	—
41	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
44	C₂×D₁₂₇	—	4	1	no	—
45	C₂×D₁₂₇	C₄⋊C₈	1	2	yes	—
46	C₄×C₁₂₇	—	4	1	no	—
47	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
49	Dic₁₂₇	—	4	1	no	—
50	Dic₁₂₇	C₄⋊C₈	1	2	yes	—
51	C₂×C₁₂₇	—	2	1	no	—
52	C₂×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
53	D₁₂₇	—	2	2	no	—
54	#2153	—	127	1	no	maximal, Sylow
55	C₁₂₇	#2153	1	1	yes	characteristic, complemented, Sylow
56	C₂×M₃₂	—	127	2	no	—
57	#405	—	127	1	no	—
58	#364	—	127	2	no	—
59	#364	—	127	1	no	—
60	#364	D₁₂₇	1	1	yes	characteristic, complemented
61	C₂×M₁₆	—	127	1	no	—
62	C₂²×C₈	—	127	1	no	—
63	#117	—	127	1	no	—
64	M₃₂	—	127	2	no	—
65	M₃₂	—	254	2	no	—
66	M₃₂	C₂×D₁₂₇	1	2	yes	—
67	C₂×C₁₆	—	254	2	no	—
68	#97	—	127	1	no	—
69	#97	—	127	1	no	—
70	C₄×C₈	—	127	1	no	—
71	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
72	C₂²×C₄	—	127	1	no	—
73	M₁₆	—	127	4	no	—
74	C₂×C₈	—	127	1	no	—
75	C₂×C₈	—	127	1	no	—
76	C₂×C₈	—	127	2	no	—
77	C₂×C₈	—	127	2	no	—
78	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
79	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
80	C₄⋊C₄	—	254	1	no	—
81	C₂²⋊C₄	—	254	1	no	—
82	C₄²	—	127	1	no	—
83	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
84	C₁₆	—	2	2	no	—
85	C₁₆	—	254	2	no	—
86	C₂³	—	127	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	127	1	no	—
89	C₂×C₄	—	127	2	no	—
90	C₂×C₄	—	127	2	no	—
91	C₂×C₄	—	254	1	no	—
92	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
93	C₈	—	127	2	no	—
94	C₈	—	127	2	no	—
95	C₈	C₈×D₁₂₇	1	2	yes	—
96	C₈	D₄×D₁₂₇	1	1	yes	characteristic
97	C₈	Q₈×D₁₂₇	1	1	yes	characteristic
98	C₂²	—	127	2	no	—
99	C₂²	—	508	1	no	—
100	C₂²	M₁₆×D₁₂₇	1	1	yes	characteristic
101	C₄	—	4	1	no	—
102	C₄	—	127	2	no	—
103	C₄	—	508	1	no	—
104	C₄	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
105	C₄	C₂×C₈×D₁₂₇	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	254	2	no	—
108	C₂	C₄⋊C₈×D₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₃₂×D₁₂₇

C₂

yes

maximal

#22549830

C₂

yes

maximal, characteristic

#22549697

C₂

yes

maximal, characteristic, complemented

#22549696

C₂

yes

maximal, complemented

#364×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523231

C₄

yes

characteristic

C₂×C₈×D₁₂₇

C₂²

yes

characteristic

#16861166

C₄

yes

characteristic

#16861149

C₂²

yes

—

#16861148

—

C₁₆×D₁₂₇

—

#16861120

C₄

yes

characteristic

#16861119

C₄

yes

characteristic

C₈×Dic₁₂₇

C₂²

yes

characteristic

M₃₂×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂×C₄

yes

characteristic

#14523231

C₈

yes

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₈×D₁₂₇

D₄

yes

—

C₈×D₁₂₇

Q₈

yes

—

Dic₁₂₇⋊C₄

—

(C₂×D₁₂₇)⋊C₄

—

C₄×Dic₁₂₇

C₂×C₄

yes

characteristic

C₁₂₇⋊C₁₆

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₁₆×C₁₂₇

—

C₂²×D₁₂₇

M₁₆

yes

characteristic

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

M₁₆

yes

characteristic

C₄×D₁₂₇

C₄⋊C₄

yes

—

C₄×D₁₂₇

C₂×C₈

yes

—

C₁₂₇⋊C₈

C₄⋊C₄

yes

—

C₁₂₇⋊C₈

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₈×C₁₂₇

C₂×C₈

yes

—

C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

C₂×Q₈

yes

characteristic

C₂²×C₁₂₇

C₂×M₁₆

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

C₄⋊C₈

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₂₇

C₂²×C₈

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

C₄⋊C₈

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₄⋊C₈

yes

characteristic

D₁₂₇

—

#2153

—

127

maximal, Sylow

C₁₂₇

#2153

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

127

—

#405

—

127

—

#364

—

127

—

#364

—

127

—

#364

D₁₂₇

yes

characteristic, complemented

C₂×M₁₆

—

127

—

C₂²×C₈

—

127

—

#117

—

127

—

M₃₂

—

127

—

M₃₂

—

254

—

M₃₂

C₂×D₁₂₇

yes

—

C₂×C₁₆

—

254

—

#97

—

127

—

#97

—

127

—

C₄×C₈

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

M₁₆

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₄²

—

127

—

C₄²

C₄×D₁₂₇

yes

characteristic

C₁₆

—

C₁₆

—

254

—

C₂³

—

127

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

C₈

—

127

—

C₈

—

127

—

C₈

C₈×D₁₂₇

yes

—

C₈

D₄×D₁₂₇

yes

characteristic

C₈

Q₈×D₁₂₇

yes

characteristic

C₂²

—

127

—

C₂²

—

508

—

100

C₂²

M₁₆×D₁₂₇

yes

characteristic

101

C₄

—

102

C₄

—

127

—

103

C₄

—

508

—

104

C₄

C₄⋊C₄×D₁₂₇

yes

characteristic

105

C₄

C₂×C₈×D₁₂₇

yes

characteristic

106

C₂

—

107

C₂

—

254

—

108

C₂

C₄⋊C₈×D₁₂₇

yes

characteristic

	subgroup	G/H
center	C₈	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#364×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×C₄⋊C₈
prime radical	C₂²×D₁₂₇	M₁₆
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#364	D₁₂₇
hypocenter	C₁₂₇	#2153
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

subgroup

G/H

center

C₈

D₄×D₁₂₇

commutator

C₄×C₁₂₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

#364×C₁₂₇

C₂

socle

C₂×C₁₂₇

C₂×C₄⋊C₈

prime radical

C₂²×D₁₂₇

M₁₆

cosocle

C₂×C₈×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

N	H ≅ G/N	type
3. #22549697	C₂	semidirect
4. #22549696	C₂	semidirect
5. #364×C₁₂₇	C₂	semidirect
55. C₁₂₇	#2153	semidirect
60. #364	D₁₂₇	semidirect

H ≅ G/N

type

3. #22549697

C₂

semidirect

4. #22549696

C₂

semidirect

5. #364×C₁₂₇

C₂

semidirect

55. C₁₂₇

#2153

semidirect

60. #364

D₁₂₇

semidirect

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

subgroup	index
M₃₂×D₁₂₇	2
#22549830	2
#22549697	2
#22549696	2
#364×C₁₂₇	2
#2153	127

subgroup

index

M₃₂×D₁₂₇

#364×C₁₂₇

127

Subgroups — #30640494

Conjugacy classes of subgroups

Complements

Maximal subgroups