Group #30637809 — subgroups

Subgroups — #30637809

Order 16256.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22550606	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550250	C₂	1	1	yes	maximal, characteristic
3	#22550077	C₂	1	2	yes	maximal
4	#22549606	C₂	1	2	yes	maximal, complemented
5	#333×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523237	—	2	2	no	—
7	C₂×#14523235	—	2	2	no	—
8	#16861208	—	2	2	no	—
9	#16861207	—	2	2	no	—
10	#16861206	C₂²	1	1	yes	characteristic
11	C₄×Dic₂₅₄	C₂²	1	2	yes	—
12	#16861174	—	2	2	no	—
13	#16861131	—	2	2	no	—
14	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
17	C₂×Dic₂₅₄	—	2	2	no	—
18	C₂×Dic₂₅₄	D₄	1	2	yes	—
19	C₂×D₅₀₈	—	2	1	no	—
20	C₂×D₅₀₈	—	2	2	no	—
21	#14523237	—	4	4	no	—
22	#14523235	—	4	4	no	—
23	C₂×C₁₂₇⋊C₈	—	2	1	no	—
24	C₄⋊Dic₁₂₇	D₄	1	2	yes	—
25	Dic₁₂₇⋊C₄	—	2	4	no	—
26	Dic₁₂₇⋊C₄	—	2	2	no	—
27	C₄×Dic₁₂₇	—	2	2	no	—
28	C₄×Dic₁₂₇	—	2	2	no	—
29	C₂×Q₈×C₁₂₇	D₄	1	2	yes	—
30	C₂×C₈×C₁₂₇	—	2	2	no	—
31	C₄⋊C₄×C₁₂₇	—	2	2	no	—
32	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂²×D₁₂₇	—	4	1	no	—
34	Dic₂₅₄	—	2	2	no	—
35	Dic₂₅₄	—	8	2	no	—
36	Dic₂₅₄	SD₁₆	1	4	yes	complemented
37	D₅₀₈	—	4	2	no	—
38	D₅₀₈	—	4	2	no	—
39	D₅₀₈	—	4	2	no	—
40	C₂×Dic₁₂₇	—	2	2	no	—
41	C₂×Dic₁₂₇	—	2	2	no	—
42	C₁₂₇⋊C₈	—	2	2	no	—
43	Q₈×C₁₂₇	—	4	2	no	—
44	C₂×C₄×C₁₂₇	—	2	2	no	—
45	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
47	C₈×C₁₂₇	—	4	2	no	—
48	C₂²×C₁₂₇	#120	1	1	yes	characteristic
49	C₂×D₁₂₇	—	8	2	no	—
50	C₂×D₁₂₇	—	8	1	no	—
51	C₄×C₁₂₇	—	2	1	no	—
52	C₄×C₁₂₇	—	4	2	no	—
53	C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
54	C₄×C₁₂₇	#137	1	2	yes	—
55	Dic₁₂₇	—	2	4	no	—
56	Dic₁₂₇	—	4	2	no	—
57	C₂×C₁₂₇	#448	1	2	yes	—
58	C₂×C₁₂₇	#453	1	1	yes	characteristic
59	D₁₂₇	—	16	1	no	—
60	#1623	—	127	1	no	maximal, Sylow
61	C₁₂₇	#1623	1	1	yes	characteristic, complemented, Sylow
62	Q₈²	—	127	1	no	—
63	#492	—	127	1	no	—
64	#460	—	127	2	no	—
65	#333	D₁₂₇	1	1	yes	characteristic, complemented
66	Q₈⋊C₈	—	127	2	no	—
67	C₂×SD₁₆	—	254	2	no	—
68	C₂×SD₁₆	—	254	2	no	—
69	C₄⋊Q₈	—	254	2	no	—
70	C₄⋊Q₈	—	254	2	no	—
71	C₄⋊Q₈	C₂×D₁₂₇	1	1	yes	characteristic
72	#127	—	127	1	no	—
73	C₄×Q₈	—	127	2	no	—
74	C₄×Q₈	—	254	2	no	—
75	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
76	#102	—	254	2	no	—
77	C₄×C₈	—	127	1	no	—
78	C₂×Q₈	—	127	2	no	—
79	C₂×Q₈	—	254	2	no	—
80	C₂×Q₈	D₅₀₈	1	2	yes	—
81	C₂×D₄	—	254	2	no	—
82	C₂×D₄	—	254	1	no	—
83	SD₁₆	—	508	4	no	—
84	SD₁₆	—	508	4	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	254	1	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	127	2	no	—
89	C₄⋊C₄	—	254	4	no	—
90	C₄⋊C₄	—	254	2	no	—
91	C₄²	—	254	2	no	—
92	C₄²	—	254	2	no	—
93	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
94	C₂³	—	508	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	127	4	no	—
97	Q₈	—	1016	2	no	—
98	Q₈	—	254	2	no	—
99	D₄	—	508	2	no	—
100	D₄	—	508	2	no	—
101	D₄	—	508	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	254	2	no	—
104	C₂×C₄	—	254	2	no	—
105	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
106	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
107	C₈	—	4	2	no	—
108	C₈	—	254	2	no	—
109	C₂²	—	1016	2	no	—
110	C₂²	—	1016	1	no	—
111	C₂²	#16861179	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	508	2	no	—
115	C₄	—	254	4	no	—
116	C₄	SD₁₆×D₁₂₇	1	2	yes	—
117	C₄	#16861241	1	2	yes	—
118	C₂	—	2032	1	no	—
119	C₂	#22550016	1	2	yes	—
120	C₂	#22550046	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550606

C₂

yes

maximal, characteristic, complemented

#22550250

C₂

yes

maximal, characteristic

#22550077

C₂

yes

maximal

#22549606

C₂

yes

maximal, complemented

#333×C₁₂₇

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₂₅₄

C₂²

yes

—

#16861174

—

#16861131

—

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

D₄

yes

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

#14523237

—

#14523235

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

D₄

yes

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂×Q₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₂²×D₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

SD₁₆

yes

complemented

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

#120

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×SD₁₆

yes

—

C₄×C₁₂₇

#137

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#448

yes

—

C₂×C₁₂₇

#453

yes

characteristic

D₁₂₇

—

#1623

—

127

maximal, Sylow

C₁₂₇

#1623

yes

characteristic, complemented, Sylow

Q₈²

—

127

—

#492

—

127

—

#460

—

127

—

#333

D₁₂₇

yes

characteristic, complemented

Q₈⋊C₈

—

127

—

C₂×SD₁₆

—

254

—

C₂×SD₁₆

—

254

—

C₄⋊Q₈

—

254

—

C₄⋊Q₈

—

254

—

C₄⋊Q₈

C₂×D₁₂₇

yes

characteristic

#127

—

127

—

C₄×Q₈

—

127

—

C₄×Q₈

—

254

—

C₄⋊C₈

C₂×D₁₂₇

yes

—

#102

—

254

—

C₄×C₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

D₅₀₈

yes

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

SD₁₆

—

508

—

SD₁₆

—

508

—

C₂×C₈

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₄²

—

254

—

C₄²

—

254

—

C₄²

C₂²×D₁₂₇

yes

characteristic

C₂³

—

508

—

Q₈

—

Q₈

—

127

—

Q₈

—

1016

—

Q₈

—

254

—

D₄

—

508

—

100

D₄

—

508

—

101

D₄

—

508

—

102

C₂×C₄

—

103

C₂×C₄

—

254

—

104

C₂×C₄

—

254

—

105

C₂×C₄

D₄×D₁₂₇

yes

—

106

C₂×C₄

C₂×D₅₀₈

yes

characteristic

107

C₈

—

108

C₈

—

254

—

109

C₂²

—

1016

—

110

C₂²

—

1016

—

111

C₂²

#16861179

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

508

—

115

C₄

—

254

—

116

C₄

SD₁₆×D₁₂₇

yes

—

117

C₄

#16861241

yes

—

118

C₂

—

2032

—

119

C₂

#22550016

yes

—

120

C₂

#22550046

yes

characteristic

	subgroup	G/H
center	C₂²	#16861179
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#333×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#120
prime radical	#16861206	C₂²
hypercenter	#333	D₁₂₇
hypocenter	C₁₂₇	#1623
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#16861179

commutator, cosocle

C₄²×C₁₂₇

C₂³

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#333×C₁₂₇

C₂

C₂²×C₁₂₇

C₂²

D₁₂₇

C₁₂₇

G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
1. #22550606	C₂	semidirect
4. #22549606	C₂	semidirect
5. #333×C₁₂₇	C₂	semidirect
36. Dic₂₅₄	SD₁₆	semidirect
61. C₁₂₇	#1623	semidirect
65. #333	D₁₂₇	semidirect

H ≅ G/N

type

1. #22550606

C₂

semidirect

4. #22549606

C₂

semidirect

5. #333×C₁₂₇

C₂

semidirect

36. Dic₂₅₄

SD₁₆

semidirect

61. C₁₂₇

#1623

semidirect

65. #333

D₁₂₇

semidirect

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

subgroup	index
#22550606	2
#22550250	2
#22550077	2
#22549606	2
#333×C₁₂₇	2
#1623	127

subgroup

index

#333×C₁₂₇

127

Subgroups — #30637809

Conjugacy classes of subgroups

Complements

Maximal subgroups