Group #30640871 — subgroups

Subgroups — #30640871

Order 16256.

130 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 49 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	#22550280	C₂	1	1	yes	maximal, characteristic
2	#22549970	C₂	1	1	yes	maximal, characteristic
3	#22549703	C₂	1	1	yes	maximal, characteristic
4	Dic₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#22549681	C₂	1	1	yes	maximal, characteristic
6	#22549679	C₂	1	1	yes	maximal, characteristic
7	#367×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic
9	#16861208	—	2	1	no	—
10	#16861174	—	2	1	no	—
11	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#16861143	C₂²	1	1	yes	characteristic
13	#16861142	—	2	1	no	—
14	#16861132	—	2	1	no	—
15	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
17	#107×C₁₂₇	C₂²	1	1	yes	characteristic
18	#107×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₅₄	—	2	1	no	—
20	C₂×Dic₂₅₄	—	4	1	no	—
21	Dic₅₀₈	—	2	1	no	—
22	Dic₅₀₈	Q₈	1	2	yes	complemented
23	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₁₂₇	—	2	1	no	—
25	Dic₁₂₇⋊C₄	—	4	1	no	—
26	Dic₁₂₇⋊C₄	—	4	1	no	—
27	C₄×Dic₁₂₇	—	4	1	no	—
28	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
29	C₁₂₇⋊C₁₆	—	2	1	no	—
30	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₁₂₇	—	2	1	no	—
32	C₄⋊C₄×C₁₂₇	—	2	1	no	—
33	C₁₆×C₁₂₇	—	2	1	no	—
34	Dic₂₅₄	—	2	2	no	—
35	Dic₂₅₄	—	4	1	no	—
36	Dic₂₅₄	—	8	2	no	—
37	C₂×Dic₁₂₇	—	4	1	no	—
38	C₂×Dic₁₂₇	—	4	1	no	—
39	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
40	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
41	C₁₂₇⋊C₈	—	2	1	no	—
42	C₂×C₄×C₁₂₇	—	4	1	no	—
43	C₂×C₄×C₁₂₇	—	4	1	no	—
44	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₁₂₇	#41	1	1	yes	characteristic
47	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
48	C₄×C₁₂₇	—	8	1	no	—
49	C₄×C₁₂₇	—	8	1	no	—
50	C₄×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
51	C₄×C₁₂₇	#137	1	1	yes	characteristic
52	Dic₁₂₇	—	2	1	no	—
53	Dic₁₂₇	—	4	2	no	—
54	Dic₁₂₇	—	8	1	no	—
55	Dic₁₂₇	SD₃₂	1	2	yes	—
56	C₂×C₁₂₇	#474	1	1	yes	characteristic
57	C₂×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
58	C₂×C₁₂₇	#510	1	1	yes	characteristic
59	#2225	—	127	1	no	maximal, Sylow
60	C₁₂₇	#2225	1	1	yes	characteristic, complemented, Sylow
61	#500	—	127	1	no	—
62	C₄×Q₁₆	—	127	1	no	—
63	#367	—	127	1	no	—
64	#367	D₁₂₇	1	1	yes	characteristic
65	C₄⋊C₁₆	—	127	1	no	—
66	#358	—	127	1	no	—
67	#358	—	127	1	no	—
68	C₂×Q₁₆	—	127	1	no	—
69	C₄⋊Q₈	—	254	1	no	—
70	C₄×Q₈	—	254	1	no	—
71	C₂×C₁₆	—	127	1	no	—
72	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
73	#107	—	127	1	no	—
74	#107	—	254	1	no	—
75	#107	C₂×D₁₂₇	1	1	yes	characteristic
76	#107	C₂×D₁₂₇	1	1	yes	characteristic
77	#103	—	254	1	no	—
78	C₄×C₈	—	127	1	no	—
79	C₂×Q₈	—	508	1	no	—
80	C₂×Q₈	—	254	1	no	—
81	Q₁₆	—	127	2	no	—
82	Q₁₆	—	254	1	no	—
83	C₂×C₈	—	127	1	no	—
84	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	2	1	no	—
87	C₄⋊C₄	—	508	1	no	—
88	C₄⋊C₄	—	508	1	no	—
89	C₄⋊C₄	—	254	1	no	—
90	C₄²	—	127	1	no	—
91	C₄²	—	508	1	no	—
92	C₁₆	—	2	1	no	—
93	C₁₆	—	254	1	no	—
94	Q₈	—	1016	2	no	—
95	Q₈	—	508	1	no	—
96	Q₈	—	254	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	127	1	no	—
100	C₂×C₄	—	127	1	no	—
101	C₂×C₄	—	508	1	no	—
102	C₂×C₄	—	508	1	no	—
103	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
104	C₈	—	254	1	no	—
105	C₈	Q₈×D₁₂₇	1	1	yes	characteristic
106	C₈	#14523248	1	1	yes	characteristic
107	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
108	C₄	—	8	1	no	—
109	C₄	—	8	1	no	—
110	C₄	—	127	2	no	—
111	C₄	—	1016	1	no	—
112	C₄	—	508	2	no	—
113	C₄	—	254	1	no	—
114	C₄	#16861187	1	1	yes	characteristic
115	C₄	#16861241	1	1	yes	characteristic
116	C₂	#22550155	1	1	yes	characteristic
117	C₂	SD₃₂×D₁₂₇	1	1	yes	characteristic
118	C₂	#22550315	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550280

C₂

yes

maximal, characteristic

#22549970

C₂

yes

maximal, characteristic

#22549703

C₂

yes

maximal, characteristic

Dic₁₂₇⋊C₁₆

C₂

yes

maximal, characteristic

#22549681

C₂

yes

maximal, characteristic

#22549679

C₂

yes

maximal, characteristic

#367×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₅₀₈

C₂²

yes

characteristic

#16861208

—

#16861174

—

C₂×C₁₂₇⋊C₁₆

C₂²

yes

characteristic

#16861143

C₂²

yes

characteristic

#16861142

—

#16861132

—

C₈×Dic₁₂₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₂₇

C₂²

yes

characteristic

#107×C₁₂₇

C₂²

yes

characteristic

#107×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

Dic₅₀₈

—

Dic₅₀₈

Q₈

yes

complemented

C₂×C₁₂₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₂₇

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

D₄

yes

characteristic

C₁₂₇⋊C₁₆

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₁₆×C₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₈

yes

characteristic

C₂×Dic₁₂₇

D₈

yes

characteristic

C₁₂₇⋊C₈

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

C₂×Q₈

yes

characteristic

C₈×C₁₂₇

#41

yes

characteristic

C₂²×C₁₂₇

C₂×D₈

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₂₇

#137

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

SD₃₂

yes

—

C₂×C₁₂₇

#474

yes

characteristic

C₂×C₁₂₇

C₂×SD₃₂

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#2225

yes

characteristic, complemented, Sylow

#500

—

127

—

C₄×Q₁₆

—

127

—

#367

—

127

—

#367

D₁₂₇

yes

characteristic

C₄⋊C₁₆

—

127

—

#358

—

127

—

#358

—

127

—

C₂×Q₁₆

—

127

—

C₄⋊Q₈

—

254

—

C₄×Q₈

—

254

—

C₂×C₁₆

—

127

—

C₂×C₁₆

C₂×D₁₂₇

yes

characteristic

#107

—

127

—

#107

—

254

—

#107

C₂×D₁₂₇

yes

characteristic

#107

C₂×D₁₂₇

yes

characteristic

#103

—

254

—

C₄×C₈

—

127

—

C₂×Q₈

—

508

—

C₂×Q₈

—

254

—

Q₁₆

—

127

—

Q₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

508

—

C₄⋊C₄

—

508

—

C₄⋊C₄

—

254

—

C₄²

—

127

—

C₄²

—

508

—

C₁₆

—

C₁₆

—

254

—

Q₈

—

1016

—

Q₈

—

508

—

Q₈

—

254

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

127

—

100

C₂×C₄

—

127

—

101

C₂×C₄

—

508

—

102

C₂×C₄

—

508

—

103

C₂×C₄

D₄×D₁₂₇

yes

characteristic

104

C₈

—

254

—

105

C₈

Q₈×D₁₂₇

yes

characteristic

106

C₈

#14523248

yes

characteristic

107

C₂²

D₈×D₁₂₇

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

127

—

111

C₄

—

1016

—

112

C₄

—

508

—

113

C₄

—

254

—

114

C₄

#16861187

yes

characteristic

115

C₄

#16861241

yes

characteristic

116

C₂

#22550155

yes

characteristic

117

C₂

SD₃₂×D₁₂₇

yes

characteristic

118

C₂

#22550315

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₂₇
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#367×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂×D₈
hypercenter	#367	D₁₂₇
hypocenter	C₁₂₇	#2225
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

D₈×D₁₂₇

commutator, cosocle

C₂×C₈×C₁₂₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

#367×C₁₂₇

C₂

socle, prime radical

C₂²×C₁₂₇

C₂×D₈

D₁₂₇

C₁₂₇

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

N	H ≅ G/N	type
22. Dic₅₀₈	Q₈	semidirect
60. C₁₂₇	#2225	semidirect

H ≅ G/N

type

22. Dic₅₀₈

Q₈

semidirect

60. C₁₂₇

#2225

semidirect

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

subgroup	index
#22550280	2
#22549970	2
#22549703	2
Dic₁₂₇⋊C₁₆	2
#22549681	2
#22549679	2
#367×C₁₂₇	2
#2225	127

subgroup

index

Dic₁₂₇⋊C₁₆

#367×C₁₂₇

127

Subgroups — #30640871

Conjugacy classes of subgroups

Complements

Maximal subgroups