Group #30636567 — subgroups

Subgroups — #30636567

Order 16256.

172 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 85 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	#22549929	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523232	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523232	C₄	1	2	yes	—
5	C₄⋊(C₁₂₇⋊C₈)	C₄	1	2	yes	—
6	C₂²⋊(C₁₂₇⋊C₈)	—	2	1	no	—
7	#16861120	—	2	1	no	—
8	C₄×C₁₂₇⋊C₈	—	2	1	no	—
9	C₄×C₁₂₇⋊C₈	—	2	1	no	—
10	C₂²×C₈×C₁₂₇	—	2	1	no	—
11	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₁₂₇	—	2	1	no	—
13	C₄×C₈×C₁₂₇	C₄	1	2	yes	—
14	#14523232	—	2	2	no	—
15	#14523232	C₈	1	4	yes	complemented
16	C₂×C₁₂₇⋊C₈	—	2	2	no	—
17	C₂×C₁₂₇⋊C₈	—	2	2	no	—
18	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₂₇	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₁₂₇	—	2	4	no	—
22	C₂×C₈×C₁₂₇	—	2	2	no	—
23	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
26	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
27	C₁₂₇⋊C₈	—	2	4	no	—
28	C₁₂₇⋊C₈	—	4	2	no	—
29	C₂³×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	—	2	2	no	—
31	C₂×C₄×C₁₂₇	—	2	2	no	—
32	C₂×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
39	C₂×C₄×C₁₂₇	M₁₆	1	2	yes	—
40	C₈×C₁₂₇	—	2	4	no	—
41	C₂²×C₁₂₇	—	2	1	no	—
42	C₂²×C₁₂₇	—	2	1	no	—
43	C₂²×C₁₂₇	#95	1	1	yes	characteristic
44	C₂²×C₁₂₇	C₄⋊C₈	1	2	yes	—
45	C₄×C₁₂₇	—	2	1	no	—
46	C₄×C₁₂₇	—	2	1	no	—
47	C₄×C₁₂₇	C₄×C₈	1	1	yes	characteristic
48	C₄×C₁₂₇	#97	1	1	yes	characteristic
49	C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
50	C₄×C₁₂₇	#104	1	2	yes	—
51	C₄×C₁₂₇	#108	1	2	yes	—
52	C₂×C₁₂₇	—	2	2	no	—
53	C₂×C₁₂₇	#336	1	1	yes	characteristic
54	C₂×C₁₂₇	#339	1	1	yes	characteristic
55	C₂×C₁₂₇	#341	1	1	yes	characteristic
56	#1278	—	127	1	no	maximal, Sylow
57	C₁₂₇	#1278	1	1	yes	characteristic, complemented, Sylow
58	#431	—	127	1	no	—
59	C₄×M₁₆	—	127	1	no	—
60	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
61	C₂×M₁₆	—	127	2	no	—
62	C₂²×C₈	—	2	1	no	—
63	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
64	C₄⋊C₈	—	127	2	no	—
65	C₂²⋊C₈	—	254	1	no	—
66	#97	—	254	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	254	1	no	—
69	C₄×C₈	—	254	1	no	—
70	C₄×C₈	Dic₁₂₇	1	2	yes	—
71	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
72	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
73	C₂²×C₄	Dic₂₅₄	1	1	yes	characteristic
74	M₁₆	—	127	4	no	—
75	M₁₆	—	254	2	no	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	254	2	no	—
79	C₂×C₈	—	254	2	no	—
80	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
81	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
82	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
83	C₄²	D₅₀₈	1	1	yes	characteristic
84	C₂³	Dic₁₂₇⋊C₄	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
88	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
89	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
91	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
92	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
93	C₂×C₄	C₈×D₁₂₇	1	2	yes	—
94	C₂×C₄	#14523231	1	2	yes	—
95	C₈	—	2	4	no	—
96	C₈	—	508	2	no	—
97	C₈	—	254	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	C₁₂₇⋊#95	1	1	yes	characteristic
101	C₂²	Dic₁₂₇⋊C₈	1	2	yes	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₁₂₇	1	1	yes	characteristic
105	C₄	#16861119	1	1	yes	characteristic
106	C₄	(C₂×D₁₂₇)⋊C₈	1	2	yes	—
107	C₄	#16861137	1	2	yes	—
108	C₄	#16861145	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#22549630	1	1	yes	characteristic
111	C₂	#22549636	1	1	yes	characteristic
112	C₂	#22549641	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22549929

C₂

yes

maximal, characteristic

C₄×#14523232

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₄

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₄

yes

—

C₂²⋊(C₁₂₇⋊C₈)

—

#16861120

—

C₄×C₁₂₇⋊C₈

—

C₄×C₁₂₇⋊C₈

—

C₂²×C₈×C₁₂₇

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

—

C₄×C₈×C₁₂₇

C₄

yes

—

#14523232

—

#14523232

C₈

yes

complemented

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×C₈×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₄²×C₁₂₇

D₄

yes

characteristic

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄²

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₈

yes

—

C₂×C₄×C₁₂₇

M₁₆

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#95

yes

characteristic

C₂²×C₁₂₇

C₄⋊C₈

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄×C₈

yes

characteristic

C₄×C₁₂₇

#97

yes

characteristic

C₄×C₁₂₇

C₂²⋊C₈

yes

—

C₄×C₁₂₇

#104

yes

—

C₄×C₁₂₇

#108

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1278

yes

characteristic, complemented, Sylow

#431

—

127

—

C₄×M₁₆

—

127

—

C₂×C₄×C₈

D₁₂₇

yes

characteristic

C₂×M₁₆

—

127

—

C₂²×C₈

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

C₂²⋊C₈

—

254

—

#97

—

254

—

C₄×C₈

—

C₄×C₈

—

254

—

C₄×C₈

—

254

—

C₄×C₈

Dic₁₂₇

yes

—

C₂²×C₄

C₄×D₁₂₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂²×C₄

Dic₂₅₄

yes

characteristic

M₁₆

—

127

—

M₁₆

—

254

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₄²

C₄×D₁₂₇

yes

characteristic

C₄²

C₂×Dic₁₂₇

yes

characteristic

C₄²

C₂²⋊D₁₂₇

yes

characteristic

C₄²

D₅₀₈

yes

characteristic

C₂³

Dic₁₂₇⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₂₇

yes

characteristic

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₁₂₇

yes

characteristic

C₂×C₄

Dic₁₂₇⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₁₂₇

yes

characteristic

C₂×C₄

C₈×D₁₂₇

yes

—

C₂×C₄

#14523231

yes

—

C₈

—

C₈

—

508

—

C₈

—

254

—

C₂²

—

C₂²

—

100

C₂²

C₁₂₇⋊#95

yes

characteristic

101

C₂²

Dic₁₂₇⋊C₈

yes

—

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₁₂₇

yes

characteristic

105

C₄

#16861119

yes

characteristic

106

C₄

(C₂×D₁₂₇)⋊C₈

yes

—

107

C₄

#16861137

yes

—

108

C₄

#16861145

yes

—

109

C₂

—

110

C₂

#22549630

yes

characteristic

111

C₂

#22549636

yes

characteristic

112

C₂

#22549641

yes

characteristic

	subgroup	G/H
center	C₄²	D₅₀₈
commutator	C₄×C₁₂₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#95
prime radical	C₂³×C₁₂₇	C₄⋊C₄
cosocle	C₂×C₄²×C₁₂₇	C₂²
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#1278
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄²

D₅₀₈

commutator

C₄×C₁₂₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₁₂₇

Fitting

C₂×C₄×C₈×C₁₂₇

C₂

socle

C₂²×C₁₂₇

#95

prime radical

C₂³×C₁₂₇

C₄⋊C₄

cosocle

C₂×C₄²×C₁₂₇

C₂²

hypercenter

C₂×C₄×C₈

D₁₂₇

hypocenter

C₁₂₇

#1278

LCS

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
15. #14523232	C₈	semidirect
57. C₁₂₇	#1278	semidirect

H ≅ G/N

type

15. #14523232

C₈

semidirect

57. C₁₂₇

#1278

semidirect

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

subgroup	index
#22549929	2
C₄×#14523232	2
C₂×C₄×C₈×C₁₂₇	2
#1278	127

subgroup

index

#22549929

C₄×#14523232

C₂×C₄×C₈×C₁₂₇

#1278

127

Subgroups — #30636567

Conjugacy classes of subgroups

Complements

Maximal subgroups