Group #30636823 — subgroups

Subgroups — #30636823

Order 16256.

128 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₂²×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861145	C₂	1	1	yes	maximal, characteristic
3	C₂×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523232	—	2	1	no	—
5	C₂×C₁₂₇⋊C₁₆	—	2	1	no	—
6	C₂×C₁₂₇⋊C₁₆	—	2	1	no	—
7	C₂×C₁₂₇⋊C₁₆	C₄	1	2	yes	—
8	#16861145	—	2	1	no	—
9	#16861145	C₄	1	2	yes	—
10	C₂×M₁₆×C₁₂₇	—	2	1	no	—
11	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	#108×C₁₂₇	—	2	1	no	—
13	#108×C₁₂₇	C₄	1	2	yes	—
14	#14523232	—	2	2	no	—
15	#14523232	—	4	1	no	—
16	C₂×C₁₂₇⋊C₈	—	4	1	no	—
17	C₁₂₇⋊C₁₆	—	2	2	no	—
18	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
19	M₁₆×C₁₂₇	—	2	2	no	—
20	M₁₆×C₁₂₇	—	4	1	no	—
21	C₂×C₈×C₁₂₇	—	4	1	no	—
22	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
27	C₂×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
28	C₁₂₇⋊C₈	—	4	2	no	—
29	C₂³×C₁₂₇	Q₁₆	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
36	C₈×C₁₂₇	—	4	2	no	—
37	C₈×C₁₂₇	C₄²	1	1	yes	characteristic
38	C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
40	C₈×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₁₂₇	—	2	2	no	—
42	C₂²×C₁₂₇	#103	1	1	yes	characteristic
43	C₂²×C₁₂₇	#103	1	1	yes	characteristic
44	C₂²×C₁₂₇	#107	1	1	yes	characteristic
45	C₄×C₁₂₇	#95	1	1	yes	characteristic
46	C₄×C₁₂₇	#102	1	1	yes	characteristic
47	C₄×C₁₂₇	#102	1	1	yes	characteristic
48	C₄×C₁₂₇	#106	1	1	yes	characteristic
49	C₂×C₁₂₇	—	2	1	no	—
50	C₂×C₁₂₇	—	2	1	no	—
51	C₂×C₁₂₇	#340	1	1	yes	characteristic
52	C₂×C₁₂₇	#368	1	2	yes	—
53	#1381	—	127	1	no	maximal, Sylow
54	C₁₂₇	#1381	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₁₆	—	127	1	no	—
56	C₂×#108	—	127	1	no	—
57	C₂×#108	D₁₂₇	1	1	yes	characteristic
58	C₂×M₁₆	—	2	1	no	—
59	C₂×M₁₆	—	254	1	no	—
60	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
61	C₂×C₁₆	—	127	2	no	—
62	C₂×C₁₆	—	254	1	no	—
63	C₂×C₁₆	—	254	1	no	—
64	#108	—	2	1	no	—
65	#108	—	127	2	no	—
66	#108	—	254	1	no	—
67	#108	Dic₁₂₇	1	2	yes	—
68	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	4	1	no	—
71	M₁₆	—	508	1	no	—
72	M₁₆	—	254	2	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	—	508	1	no	—
75	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
76	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
77	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
79	C₂×C₈	D₅₀₈	1	1	yes	characteristic
80	C₂×C₈	Dic₂₅₄	1	1	yes	characteristic
81	C₁₆	—	254	2	no	—
82	C₂³	#14523238	1	1	yes	characteristic
83	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
84	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
85	C₂×C₄	Dic₁₂₇⋊C₄	1	1	yes	characteristic
86	C₂×C₄	#14523233	1	1	yes	characteristic
87	C₂×C₄	#14523235	1	1	yes	characteristic
88	C₂×C₄	#14523236	1	1	yes	characteristic
89	C₈	—	4	2	no	—
90	C₈	—	508	2	no	—
91	C₈	C₄×Dic₁₂₇	1	1	yes	characteristic
92	C₈	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
93	C₈	Dic₁₂₇⋊C₄	1	1	yes	characteristic
94	C₈	C₄⋊Dic₁₂₇	1	1	yes	characteristic
95	C₂²	—	2	2	no	—
96	C₂²	#16861132	1	1	yes	characteristic
97	C₂²	#16861133	1	1	yes	characteristic
98	C₂²	#16861142	1	1	yes	characteristic
99	C₄	C₁₂₇⋊#95	1	1	yes	characteristic
100	C₄	#16861129	1	1	yes	characteristic
101	C₄	#16861130	1	1	yes	characteristic
102	C₄	#16861140	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	#22549638	1	1	yes	characteristic
106	C₂	#22549705	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₂₇⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#16861145

C₂

yes

maximal, characteristic

C₂×#108×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

—

C₂×C₁₂₇⋊C₁₆

—

C₂×C₁₂₇⋊C₁₆

—

C₂×C₁₂₇⋊C₁₆

C₄

yes

—

#16861145

—

#16861145

C₄

yes

—

C₂×M₁₆×C₁₂₇

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

#108×C₁₂₇

—

#108×C₁₂₇

C₄

yes

—

#14523232

—

#14523232

—

C₂×C₁₂₇⋊C₈

—

C₁₂₇⋊C₁₆

—

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

D₄

yes

characteristic

C₂×C₈×C₁₂₇

Q₈

yes

characteristic

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

Q₁₆

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

D₈

yes

characteristic

C₂×C₄×C₁₂₇

SD₁₆

yes

characteristic

C₂×C₄×C₁₂₇

SD₁₆

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

C₄²

yes

characteristic

C₈×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₈×C₁₂₇

C₄⋊C₄

yes

characteristic

C₈×C₁₂₇

C₄⋊C₄

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#340

yes

characteristic

C₂×C₁₂₇

#368

yes

—

#1381

—

127

maximal, Sylow

C₁₂₇

#1381

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

127

—

C₂×#108

—

127

—

C₂×#108

D₁₂₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

254

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂×C₁₆

—

127

—

C₂×C₁₆

—

254

—

C₂×C₁₆

—

254

—

#108

—

#108

—

127

—

#108

—

254

—

#108

Dic₁₂₇

yes

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

508

—

M₁₆

—

254

—

C₂×C₈

—

C₂×C₈

—

508

—

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₄×D₁₂₇

yes

characteristic

C₂×C₈

C₂×Dic₁₂₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₈

D₅₀₈

yes

characteristic

C₂×C₈

Dic₂₅₄

yes

characteristic

C₁₆

—

254

—

C₂³

#14523238

yes

characteristic

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₁₂₇

yes

characteristic

C₂×C₄

Dic₁₂₇⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

508

—

C₈

C₄×Dic₁₂₇

yes

characteristic

C₈

(C₂×D₁₂₇)⋊C₄

yes

characteristic

C₈

Dic₁₂₇⋊C₄

yes

characteristic

C₈

C₄⋊Dic₁₂₇

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

C₁₂₇⋊#95

yes

characteristic

100

C₄

#16861129

yes

characteristic

101

C₄

#16861130

yes

characteristic

102

C₄

#16861140

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

#22549638

yes

characteristic

106

C₂

#22549705

yes

—

	subgroup	G/H
center	C₂×C₄	#14523233
commutator	C₈×C₁₂₇	C₄²
Frattini	C₂²×C₈	C₂×D₁₂₇
Fitting	C₂×#108×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#107
prime radical	C₂³×C₁₂₇	Q₁₆
cosocle	C₂²×C₈×C₁₂₇	C₂²
hypercenter	C₂×#108	D₁₂₇
hypocenter	C₁₂₇	#1381
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

#14523233

commutator

C₈×C₁₂₇

C₄²

Frattini

C₂²×C₈

C₂×D₁₂₇

Fitting

C₂×#108×C₁₂₇

C₂

socle

C₂²×C₁₂₇

#107

prime radical

C₂³×C₁₂₇

Q₁₆

cosocle

C₂²×C₈×C₁₂₇

C₂²

D₁₂₇

C₁₂₇

G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

N	H ≅ G/N	type
54. C₁₂₇	#1381	semidirect

H ≅ G/N

type

54. C₁₂₇

#1381

semidirect

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

subgroup	index
C₂²×C₁₂₇⋊C₁₆	2
C₂×#16861145	2
C₂×#108×C₁₂₇	2
#1381	127

subgroup

index

C₂²×C₁₂₇⋊C₁₆

C₂×#16861145

C₂×#108×C₁₂₇

#1381

127

Subgroups — #30636823

Conjugacy classes of subgroups

Complements

Maximal subgroups