Group #30649239 — subgroups

Subgroups — #30649239

Order 16256.

356 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 145 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	#22550431	C₂	1	2	yes	maximal
2	#22550279	C₂	1	2	yes	maximal, complemented
3	#22550276	C₂	1	4	yes	maximal, complemented
4	#22550274	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549911	C₂	1	4	yes	maximal
6	C₄×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic
7	#498×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
9	#16861209	C₂²	1	2	yes	—
10	#16861200	C₂²	1	4	yes	—
11	#16861189	—	2	2	no	—
12	#16861185	—	2	2	no	—
13	Q₈×Dic₁₂₇	—	2	2	no	—
14	#16861174	—	2	2	no	—
15	#16861167	C₂²	1	4	yes	—
16	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
17	#16861142	C₂²	1	8	yes	—
18	#16861141	C₂²	1	4	yes	—
19	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₂₇	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
22	#106×C₁₂₇	C₂²	1	4	yes	—
23	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×Dic₂₅₄	—	2	2	no	—
25	C₂×C₄×D₁₂₇	D₄	1	2	yes	—
26	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
27	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
28	C₈×D₁₂₇	Q₈	1	8	yes	complemented
29	C₄⋊Dic₁₂₇	—	2	4	no	—
30	C₄⋊Dic₁₂₇	—	2	2	no	—
31	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
32	Dic₁₂₇⋊C₄	—	2	4	no	—
33	(C₂×D₁₂₇)⋊C₄	—	2	4	no	—
34	C₄×Dic₁₂₇	—	2	4	no	—
35	C₄×Dic₁₂₇	D₄	1	2	yes	—
36	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₁₂₇	—	2	2	no	—
38	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
39	C₄⋊C₄×C₁₂₇	—	2	2	no	—
40	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
41	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₂²×D₁₂₇	C₂×D₄	1	1	yes	characteristic
43	Dic₂₅₄	—	4	4	no	—
44	C₂×Dic₁₂₇	—	2	4	no	—
45	C₂×Dic₁₂₇	C₂×D₄	1	2	yes	—
46	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
47	C₄×D₁₂₇	—	2	4	no	—
48	C₄×D₁₂₇	C₂×Q₈	1	4	yes	—
49	C₁₂₇⋊C₈	C₂×Q₈	1	4	yes	—
50	Q₈×C₁₂₇	—	4	4	no	—
51	C₂×C₄×C₁₂₇	—	2	4	no	—
52	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
54	C₈×C₁₂₇	C₂×Q₈	1	4	yes	—
55	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
56	C₂×D₁₂₇	—	2	2	no	—
57	C₂×D₁₂₇	C₄⋊Q₈	1	2	yes	—
58	C₄×C₁₂₇	—	4	4	no	—
59	C₄×C₁₂₇	#135	1	4	yes	—
60	C₄×C₁₂₇	C₂²×Q₈	1	2	yes	—
61	Dic₁₂₇	—	2	2	no	—
62	Dic₁₂₇	—	4	4	no	—
63	Dic₁₂₇	C₄⋊Q₈	1	2	yes	—
64	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₁₂₇	C₂×#135	1	2	yes	—
66	D₁₂₇	—	2	2	no	—
67	#3160	—	127	1	no	maximal, Sylow
68	C₁₂₇	#3160	1	1	yes	characteristic, complemented, Sylow
69	#533	—	127	2	no	—
70	#500	—	127	2	no	—
71	#499	—	127	4	no	—
72	#498	—	127	1	no	—
73	#498	D₁₂₇	1	1	yes	characteristic, complemented
74	#427	—	127	4	no	—
75	C₂×C₄×C₈	—	127	1	no	—
76	C₂²×C₈	—	127	2	no	—
77	C₄⋊Q₈	—	127	2	no	—
78	C₄⋊Q₈	C₂×D₁₂₇	1	2	yes	—
79	#125	—	127	4	no	—
80	C₂²⋊Q₈	—	254	2	no	—
81	C₂²⋊Q₈	—	254	2	no	—
82	C₄×Q₈	—	254	2	no	—
83	C₄×Q₈	—	254	2	no	—
84	#117	—	127	4	no	—
85	C₂×C₄²	—	127	1	no	—
86	#107	—	127	8	no	—
87	#106	—	127	4	no	—
88	#106	C₂×D₁₂₇	1	4	yes	—
89	C₄×C₈	—	127	2	no	—
90	C₄×C₈	—	127	1	no	—
91	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	254	2	no	—
94	C₂²×C₄	—	127	1	no	—
95	C₂²×C₄	—	127	2	no	—
96	C₂×C₈	—	127	2	no	—
97	C₂×C₈	—	127	8	no	—
98	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	127	4	no	—
101	C₄⋊C₄	—	254	4	no	—
102	C₄⋊C₄	—	254	4	no	—
103	C₄⋊C₄	—	254	2	no	—
104	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
105	C₂²⋊C₄	—	254	4	no	—
106	C₄²	—	127	2	no	—
107	C₄²	—	127	1	no	—
108	C₄²	—	254	4	no	—
109	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
110	C₂³	—	127	1	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	508	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	127	1	no	—
115	C₂×C₄	—	127	4	no	—
116	C₂×C₄	—	127	2	no	—
117	C₂×C₄	—	254	4	no	—
118	C₂×C₄	—	254	4	no	—
119	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
120	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
121	C₈	—	127	4	no	—
122	C₈	Q₈×D₁₂₇	1	4	yes	—
123	C₂²	—	127	2	no	—
124	C₂²	—	254	2	no	—
125	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
126	C₄	—	4	4	no	—
127	C₄	—	127	2	no	—
128	C₄	—	508	4	no	—
129	C₄	—	254	2	no	—
130	C₄	#16861230	1	4	yes	—
131	C₄	C₂×Q₈×D₁₂₇	1	2	yes	—
132	C₂	—	254	2	no	—
133	C₂	C₄⋊Q₈×D₁₂₇	1	1	yes	characteristic
134	C₂	C₂×#16861230	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550431

C₂

yes

maximal

#22550279

C₂

yes

maximal, complemented

#22550276

C₂

yes

maximal, complemented

#22550274

C₂

yes

maximal, characteristic, complemented

#22549911

C₂

yes

maximal

C₄×C₈×D₁₂₇

C₂

yes

maximal, characteristic

#498×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₂₇

C₂²

yes

—

#16861209

C₂²

yes

—

#16861200

C₂²

yes

—

#16861189

—

#16861185

—

Q₈×Dic₁₂₇

—

#16861174

—

#16861167

C₂²

yes

—

C₄²×D₁₂₇

C₂²

yes

characteristic

#16861142

C₂²

yes

—

#16861141

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₁₂₇

C₂²

yes

—

C₄⋊Q₈×C₁₂₇

C₂²

yes

—

#106×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×C₄×D₁₂₇

D₄

yes

—

C₂×C₄×D₁₂₇

C₂³

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₈×D₁₂₇

Q₈

yes

complemented

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

—

(C₂×D₁₂₇)⋊C₄

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

D₄

yes

—

C₄×Dic₁₂₇

C₂³

yes

characteristic

C₂×Q₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₂²×D₁₂₇

C₂×D₄

yes

characteristic

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×D₄

yes

—

C₂×Dic₁₂₇

C₂×D₄

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

C₂×Q₈

yes

—

C₁₂₇⋊C₈

C₂×Q₈

yes

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

C₂×Q₈

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

C₄⋊Q₈

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#135

yes

—

C₄×C₁₂₇

C₂²×Q₈

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₄⋊Q₈

yes

—

C₂×C₁₂₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₂₇

C₂×#135

yes

—

D₁₂₇

—

#3160

—

127

maximal, Sylow

C₁₂₇

#3160

yes

characteristic, complemented, Sylow

#533

—

127

—

#500

—

127

—

#499

—

127

—

#498

—

127

—

#498

D₁₂₇

yes

characteristic, complemented

#427

—

127

—

C₂×C₄×C₈

—

127

—

C₂²×C₈

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

C₂×D₁₂₇

yes

—

#125

—

127

—

C₂²⋊Q₈

—

254

—

C₂²⋊Q₈

—

254

—

C₄×Q₈

—

254

—

C₄×Q₈

—

254

—

#117

—

127

—

C₂×C₄²

—

127

—

#107

—

127

—

#106

—

127

—

#106

C₂×D₁₂₇

yes

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

254

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

C₂²×D₁₂₇

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

127

—

101

C₄⋊C₄

—

254

—

102

C₄⋊C₄

—

254

—

103

C₄⋊C₄

—

254

—

104

C₄⋊C₄

C₂²×D₁₂₇

yes

—

105

C₂²⋊C₄

—

254

—

106

C₄²

—

127

—

107

C₄²

—

127

—

108

C₄²

—

254

—

109

C₄²

C₂²×D₁₂₇

yes

characteristic

110

C₂³

—

127

—

111

Q₈

—

112

Q₈

—

508

—

113

C₂×C₄

—

114

C₂×C₄

—

127

—

115

C₂×C₄

—

127

—

116

C₂×C₄

—

127

—

117

C₂×C₄

—

254

—

118

C₂×C₄

—

254

—

119

C₂×C₄

D₄×D₁₂₇

yes

—

120

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

121

C₈

—

127

—

122

C₈

Q₈×D₁₂₇

yes

—

123

C₂²

—

127

—

124

C₂²

—

254

—

125

C₂²

C₂×D₄×D₁₂₇

yes

characteristic

126

C₄

—

127

C₄

—

127

—

128

C₄

—

508

—

129

C₄

—

254

—

130

C₄

#16861230

yes

—

131

C₄

C₂×Q₈×D₁₂₇

yes

—

132

C₂

—

254

—

133

C₂

C₄⋊Q₈×D₁₂₇

yes

characteristic

134

C₂

C₂×#16861230

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#498×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂²×D₁₂₇	C₂×D₄
hypercenter	#498	D₁₂₇
hypocenter	C₁₂₇	#3160
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₁₂₇

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

#498×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×D₄

prime radical

C₂²×D₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #22550279	C₂	semidirect
3. #22550276	C₂	semidirect
4. #22550274	C₂	semidirect
7. #498×C₁₂₇	C₂	semidirect
28. C₈×D₁₂₇	Q₈	semidirect
68. C₁₂₇	#3160	semidirect
73. #498	D₁₂₇	semidirect

H ≅ G/N

type

2. #22550279

C₂

semidirect

3. #22550276

C₂

semidirect

4. #22550274

C₂

semidirect

7. #498×C₁₂₇

C₂

semidirect

28. C₈×D₁₂₇

Q₈

semidirect

68. C₁₂₇

#3160

semidirect

73. #498

D₁₂₇

semidirect

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

subgroup	index
#22550431	2
#22550279	2
#22550276	2
#22550274	2
#22549911	2
C₄×C₈×D₁₂₇	2
#498×C₁₂₇	2
#3160	127

subgroup

index

C₄×C₈×D₁₂₇

#498×C₁₂₇

#3160

127

Subgroups — #30649239

Conjugacy classes of subgroups

Complements

Maximal subgroups