Group #30639551 — subgroups

Subgroups — #30639551

Order 16256.

308 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×#16861133	C₂	1	4	yes	maximal
2	C₂×#16861120	C₂	1	1	yes	maximal, characteristic
3	#22549742	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
7	#16861133	—	2	8	no	—
8	#16861120	—	2	2	no	—
9	C₁₂₇⋊#95	—	2	2	no	—
10	C₂²×Q₈×C₁₂₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
12	C₄⋊Q₈×C₁₂₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
14	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₂₇	—	2	2	no	—
16	C₂×C₁₂₇⋊C₈	—	2	2	no	—
17	C₂×C₁₂₇⋊C₈	—	2	2	no	—
18	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₁₂₇	—	2	4	no	—
20	C₂×Q₈×C₁₂₇	—	2	2	no	—
21	C₂×Q₈×C₁₂₇	—	2	4	no	—
22	C₂×Q₈×C₁₂₇	—	2	4	no	—
23	C₂×Q₈×C₁₂₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₂₇	—	2	2	no	—
25	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₂₇	—	2	8	no	—
28	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₄²×C₁₂₇	D₄	1	2	yes	—
30	C₂×Dic₁₂₇	—	4	2	no	—
31	C₂×Dic₁₂₇	—	4	4	no	—
32	C₁₂₇⋊C₈	—	2	4	no	—
33	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₁₂₇	—	2	8	no	—
35	Q₈×C₁₂₇	—	4	4	no	—
36	C₂×C₄×C₁₂₇	—	2	4	no	—
37	C₂×C₄×C₁₂₇	—	2	4	no	—
38	C₂×C₄×C₁₂₇	—	4	2	no	—
39	C₂×C₄×C₁₂₇	—	4	2	no	—
40	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	#41	1	2	yes	—
45	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₂₇	#124	1	1	yes	characteristic
47	C₂²×C₁₂₇	#127	1	1	yes	characteristic
48	C₂²×C₁₂₇	#137	1	4	yes	—
49	C₄×C₁₂₇	—	2	4	no	—
50	C₄×C₁₂₇	—	4	4	no	—
51	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
52	C₄×C₁₂₇	#121	1	2	yes	—
53	Dic₁₂₇	—	8	2	no	—
54	C₂×C₁₂₇	#390	1	1	yes	characteristic
55	C₂×C₁₂₇	#419	1	2	yes	—
56	C₂×C₁₂₇	#491	1	2	yes	—
57	C₂×C₁₂₇	#497	1	2	yes	—
58	#1960	—	127	1	no	maximal, Sylow
59	C₁₂₇	#1960	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic
61	C₂×#103	—	127	4	no	—
62	C₂×#97	—	127	1	no	—
63	#382	—	127	1	no	—
64	C₂²×Q₈	C₂×D₁₂₇	1	2	yes	—
65	C₂²×C₈	—	127	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	Dic₁₂₇	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	127	2	no	—
70	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
71	#103	—	254	8	no	—
72	#97	—	254	2	no	—
73	#95	—	254	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₁₂₇	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	254	2	no	—
80	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
82	C₂×C₈	—	127	4	no	—
83	C₂×C₈	—	254	2	no	—
84	C₂×C₈	—	254	2	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	254	4	no	—
87	C₄²	C₂×Dic₁₂₇	1	2	yes	—
88	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	508	2	no	—
97	C₂×C₄	—	508	4	no	—
98	C₂×C₄	C₂²⋊Dic₁₂₇	1	4	yes	—
99	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
102	C₂×C₄	#14523247	1	2	yes	—
103	C₈	—	254	4	no	—
104	C₂²	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
105	C₂²	#16861196	1	1	yes	characteristic
106	C₂²	#16861205	1	1	yes	characteristic
107	C₂²	#16861240	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	1016	2	no	—
111	C₄	D₄×Dic₁₂₇	1	2	yes	—
112	C₄	#16861180	1	2	yes	—
113	C₂	#22549779	1	1	yes	characteristic
114	C₂	#22549885	1	2	yes	—
115	C₂	#22550244	1	2	yes	—
116	C₂	#22550268	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861133

C₂

yes

maximal

C₂×#16861120

C₂

yes

maximal, characteristic

#22549742

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861133

—

#16861120

—

C₁₂₇⋊#95

—

C₂²×Q₈×C₁₂₇

C₂²

yes

—

C₄⋊Q₈×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₄⋊Dic₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄×D₄

yes

—

C₄×C₁₂₇

#121

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

#390

yes

characteristic

C₂×C₁₂₇

#419

yes

—

C₂×C₁₂₇

#491

yes

—

C₂×C₁₂₇

#497

yes

—

#1960

—

127

maximal, Sylow

C₁₂₇

#1960

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₁₂₇

yes

characteristic

C₂×#103

—

127

—

C₂×#97

—

127

—

#382

—

127

—

C₂²×Q₈

C₂×D₁₂₇

yes

—

C₂²×C₈

—

127

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₂₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#103

—

254

—

#97

—

254

—

#95

—

254

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₂₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

254

—

C₄²

C₂×Dic₁₂₇

yes

—

C₄²

C₂²⋊D₁₂₇

yes

—

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

—

508

—

C₂×C₄

C₂²⋊Dic₁₂₇

yes

—

C₂×C₄

C₂²×Dic₁₂₇

yes

characteristic

100

C₂×C₄

D₄×D₁₂₇

yes

—

101

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

102

C₂×C₄

#14523247

yes

—

103

C₈

—

254

—

104

C₂²

C₂×C₂²⋊Dic₁₂₇

yes

characteristic

105

C₂²

#16861196

yes

characteristic

106

C₂²

#16861205

yes

characteristic

107

C₂²

#16861240

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

1016

—

111

C₄

D₄×Dic₁₂₇

yes

—

112

C₄

#16861180

yes

—

113

C₂

#22549779

yes

characteristic

114

C₂

#22549885

yes

—

115

C₂

#22550244

yes

—

116

C₂

#22550268

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×C₄⋊Q₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	#1960
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×C₄⋊Q₈×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₂₇

hypocenter

C₁₂₇

#1960

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₁₂₇	C₄	semidirect
59. C₁₂₇	#1960	semidirect
67. C₄⋊Q₈	Dic₁₂₇	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₁₂₇

C₄

semidirect

59. C₁₂₇

#1960

semidirect

67. C₄⋊Q₈

Dic₁₂₇

semidirect

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

subgroup	index
C₂×#16861133	2
C₂×#16861120	2
#22549742	2
C₂×C₄⋊Q₈×C₁₂₇	2
#1960	127

subgroup

index

C₂×C₄⋊Q₈×C₁₂₇

127

Subgroups — #30639551

Conjugacy classes of subgroups

Complements

Maximal subgroups