Group #30637323 — subgroups

Subgroups — #30637323

Order 16256.

226 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 63 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	#22550432	C₂	1	1	yes	maximal, characteristic
2	#22549828	C₂	1	2	yes	maximal
3	#22549618	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549617	C₂	1	2	yes	maximal, complemented
5	#330×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523231	—	2	2	no	—
7	#16861208	C₄	1	2	yes	—
8	#16861201	C₂²	1	1	yes	characteristic
9	#16861187	—	2	2	no	—
10	#16861174	—	2	2	no	—
11	#16861167	C₄	1	2	yes	—
12	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
13	#16861120	C₂²	1	2	yes	—
14	C₈×Dic₁₂₇	—	2	2	no	—
15	#125×C₁₂₇	C₂²	1	1	yes	characteristic
16	#97×C₁₂₇	C₂²	1	2	yes	—
17	C₂×Dic₂₅₄	—	2	2	no	—
18	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₁₂₇	D₄	1	2	yes	—
20	#14523231	—	2	8	no	—
21	C₂×C₁₂₇⋊C₈	—	2	2	no	—
22	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
23	Dic₁₂₇⋊C₄	—	2	2	no	—
24	Dic₁₂₇⋊C₄	—	2	2	no	—
25	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
26	C₄×Dic₁₂₇	—	2	2	no	—
27	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₁₂₇	D₄	1	2	yes	—
29	C₂×C₈×C₁₂₇	—	2	2	no	—
30	C₄⋊C₄×C₁₂₇	—	2	2	no	—
31	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
32	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂²×D₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₂₅₄	—	4	4	no	—
35	C₂×Dic₁₂₇	—	2	2	no	—
36	C₂×Dic₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×Dic₁₂₇	C₂²⋊C₄	1	2	yes	—
38	C₄×D₁₂₇	—	2	2	no	—
39	C₄×D₁₂₇	—	2	4	no	—
40	C₁₂₇⋊C₈	—	2	4	no	—
41	C₂×C₄×C₁₂₇	—	2	2	no	—
42	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
44	C₈×C₁₂₇	—	2	4	no	—
45	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂×D₁₂₇	—	2	1	no	—
47	C₂×D₁₂₇	—	2	2	no	—
48	C₄×C₁₂₇	—	2	1	no	—
49	C₄×C₁₂₇	—	2	2	no	—
50	C₄×C₁₂₇	—	4	2	no	—
51	Dic₁₂₇	—	4	2	no	—
52	Dic₁₂₇	#101	1	2	yes	—
53	Dic₁₂₇	#104	1	4	yes	—
54	C₂×C₁₂₇	C₂×#101	1	1	yes	characteristic
55	C₂×C₁₂₇	C₂×#104	1	2	yes	—
56	D₁₂₇	—	4	1	no	—
57	#1526	—	127	1	no	maximal, Sylow
58	C₁₂₇	#1526	1	1	yes	characteristic, complemented, Sylow
59	#533	—	127	1	no	—
60	C₄×M₁₆	—	127	2	no	—
61	#330	—	127	1	no	—
62	#330	—	127	2	no	—
63	#330	D₁₂₇	1	1	yes	characteristic, complemented
64	C₂×M₁₆	—	254	2	no	—
65	C₄⋊Q₈	—	127	2	no	—
66	#125	—	127	1	no	—
67	#125	C₂×D₁₂₇	1	1	yes	characteristic
68	C₂²⋊Q₈	—	254	2	no	—
69	C₄×Q₈	—	254	2	no	—
70	#117	—	127	2	no	—
71	C₂×C₄²	—	127	1	no	—
72	#97	—	127	2	no	—
73	#97	C₂×D₁₂₇	1	2	yes	—
74	C₄×C₈	—	254	2	no	—
75	C₂×Q₈	—	254	2	no	—
76	C₂²×C₄	—	127	1	no	—
77	C₂²×C₄	—	127	2	no	—
78	M₁₆	—	254	8	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	254	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	127	2	no	—
83	C₄⋊C₄	—	254	2	no	—
84	C₄⋊C₄	—	254	2	no	—
85	C₄⋊C₄	C₄×D₁₂₇	1	2	yes	—
86	C₂²⋊C₄	—	254	2	no	—
87	C₄²	—	127	1	no	—
88	C₄²	—	127	2	no	—
89	C₄²	—	254	2	no	—
90	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
91	C₂³	—	127	1	no	—
92	Q₈	—	508	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	127	1	no	—
95	C₂×C₄	—	127	2	no	—
96	C₂×C₄	—	254	4	no	—
97	C₂×C₄	—	254	2	no	—
98	C₂×C₄	—	254	2	no	—
99	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
101	C₈	—	2	4	no	—
102	C₈	—	254	4	no	—
103	C₂²	—	254	1	no	—
104	C₂²	—	254	2	no	—
105	C₂²	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	2	no	—
108	C₄	—	4	2	no	—
109	C₄	—	127	2	no	—
110	C₄	—	127	4	no	—
111	C₄	—	508	2	no	—
112	C₂	—	508	1	no	—
113	C₂	#101×D₁₂₇	1	1	yes	characteristic
114	C₂	#104×D₁₂₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550432

C₂

yes

maximal, characteristic

#22549828

C₂

yes

maximal

#22549618

C₂

yes

maximal, characteristic, complemented

#22549617

C₂

yes

maximal, complemented

#330×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523231

—

#16861208

C₄

yes

—

#16861201

C₂²

yes

—

—

C₄

yes

—

C₄²×D₁₂₇

C₂²

yes

characteristic

#16861120

C₂²

yes

—

C₈×Dic₁₂₇

—

#125×C₁₂₇

C₂²

yes

characteristic

#97×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

—

C₂×C₄×D₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₂₇

D₄

yes

—

#14523231

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

(C₂×D₁₂₇)⋊C₄

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

C₂×C₄

yes

characteristic

C₄×Dic₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₂²×D₁₂₇

C₂²⋊C₄

yes

characteristic

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₂₇

C₂²⋊C₄

yes

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

#101

yes

—

Dic₁₂₇

#104

yes

—

C₂×C₁₂₇

C₂×#101

yes

characteristic

C₂×C₁₂₇

C₂×#104

yes

—

D₁₂₇

—

#1526

—

127

maximal, Sylow

C₁₂₇

#1526

yes

characteristic, complemented, Sylow

#533

—

127

—

C₄×M₁₆

—

127

—

#330

—

127

—

#330

—

127

—

#330

D₁₂₇

yes

characteristic, complemented

C₂×M₁₆

—

254

—

C₄⋊Q₈

—

127

—

#125

—

127

—

#125

C₂×D₁₂₇

yes

characteristic

C₂²⋊Q₈

—

254

—

C₄×Q₈

—

254

—

#117

—

127

—

C₂×C₄²

—

127

—

#97

—

127

—

#97

C₂×D₁₂₇

yes

—

C₄×C₈

—

254

—

C₂×Q₈

—

254

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

M₁₆

—

254

—

C₂×C₈

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄⋊C₄

C₄×D₁₂₇

yes

—

C₂²⋊C₄

—

254

—

C₄²

—

127

—

C₄²

—

127

—

C₄²

—

254

—

C₄²

C₂²×D₁₂₇

yes

characteristic

C₂³

—

127

—

Q₈

—

508

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

100

C₂×C₄

D₄×D₁₂₇

yes

—

101

C₈

—

102

C₈

—

254

—

103

C₂²

—

254

—

104

C₂²

—

254

—

105

C₂²

C₂²⋊C₄×D₁₂₇

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

127

—

110

C₄

—

127

—

111

C₄

—

508

—

112

C₂

—

508

—

113

C₂

#101×D₁₂₇

yes

characteristic

114

C₂

#104×D₁₂₇

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#330×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₂₇	C₂²⋊C₄
cosocle	C₄²×C₁₂₇	C₂³
hypercenter	#330	D₁₂₇
hypocenter	C₁₂₇	#1526
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#330×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₂₇

C₂²⋊C₄

cosocle

C₄²×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
3. #22549618	C₂	semidirect
4. #22549617	C₂	semidirect
5. #330×C₁₂₇	C₂	semidirect
58. C₁₂₇	#1526	semidirect
63. #330	D₁₂₇	semidirect

H ≅ G/N

type

3. #22549618

C₂

semidirect

4. #22549617

C₂

semidirect

5. #330×C₁₂₇

C₂

semidirect

58. C₁₂₇

#1526

semidirect

63. #330

D₁₂₇

semidirect

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

subgroup	index
#22550432	2
#22549828	2
#22549618	2
#22549617	2
#330×C₁₂₇	2
#1526	127

subgroup

index

#330×C₁₂₇

127

Subgroups — #30637323

Conjugacy classes of subgroups

Complements

Maximal subgroups