Group #30649243 — subgroups

Subgroups — #30649243

Order 16256.

356 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 141 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	2	yes	maximal
2	#22550280	C₂	1	2	yes	maximal, complemented
3	#22550278	C₂	1	1	yes	maximal, characteristic, complemented
4	#22550275	C₂	1	2	yes	maximal, complemented
5	#22550273	C₂	1	2	yes	maximal, complemented
6	#22549913	C₂	1	2	yes	maximal
7	#22549911	C₂	1	2	yes	maximal
8	C₄×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic
9	#499×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
12	#16861208	C₂²	1	4	yes	—
13	#16861201	C₂²	1	2	yes	—
14	#16861187	—	2	4	no	—
15	#16861174	—	2	4	no	—
16	#16861167	C₂²	1	4	yes	—
17	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
18	#16861143	C₂²	1	2	yes	—
19	#16861142	C₂²	1	4	yes	—
20	#16861141	C₂²	1	2	yes	—
21	#16861140	C₂²	1	4	yes	—
22	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
23	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
24	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
25	#125×C₁₂₇	C₂²	1	2	yes	—
26	#107×C₁₂₇	C₂²	1	2	yes	—
27	#106×C₁₂₇	C₂²	1	2	yes	—
28	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
29	C₂×Dic₂₅₄	—	2	4	no	—
30	C₂×C₄×D₁₂₇	D₄	1	2	yes	—
31	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
32	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
33	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
34	C₈×D₁₂₇	Q₈	1	4	yes	complemented
35	C₈×D₁₂₇	Q₈	1	4	yes	complemented
36	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
37	Dic₁₂₇⋊C₄	—	2	4	no	—
38	Dic₁₂₇⋊C₄	—	2	4	no	—
39	(C₂×D₁₂₇)⋊C₄	—	2	4	no	—
40	C₄×Dic₁₂₇	—	2	4	no	—
41	C₄×Dic₁₂₇	D₄	1	2	yes	—
42	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₁₂₇	—	2	4	no	—
46	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
47	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
48	C₂²×D₁₂₇	C₂×D₄	1	1	yes	characteristic
49	Dic₂₅₄	—	4	8	no	—
50	C₂×Dic₁₂₇	—	2	4	no	—
51	C₂×Dic₁₂₇	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₁₂₇	C₂×D₄	1	2	yes	—
53	C₄×D₁₂₇	—	2	4	no	—
54	C₄×D₁₂₇	C₂×Q₈	1	2	yes	—
55	C₄×D₁₂₇	C₂×Q₈	1	2	yes	—
56	C₁₂₇⋊C₈	C₂×Q₈	1	2	yes	—
57	C₁₂₇⋊C₈	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₁₂₇	—	2	4	no	—
59	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
61	C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
62	C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
63	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
64	C₂×D₁₂₇	—	2	2	no	—
65	C₂×D₁₂₇	C₄⋊Q₈	1	2	yes	—
66	C₄×C₁₂₇	—	2	2	no	—
67	C₄×C₁₂₇	—	4	4	no	—
68	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
70	Dic₁₂₇	—	4	4	no	—
71	Dic₁₂₇	C₄⋊Q₈	1	2	yes	—
72	Dic₁₂₇	#135	1	4	yes	—
73	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₁₂₇	C₂×#135	1	2	yes	—
75	D₁₂₇	—	2	2	no	—
76	#3160	—	127	1	no	maximal, Sylow
77	C₁₂₇	#3160	1	1	yes	characteristic, complemented, Sylow
78	#533	—	127	2	no	—
79	#500	—	127	2	no	—
80	#499	—	127	2	no	—
81	#499	—	127	1	no	—
82	#499	D₁₂₇	1	1	yes	characteristic, complemented
83	#498	—	127	2	no	—
84	#427	—	127	2	no	—
85	#427	—	127	2	no	—
86	C₂×C₄×C₈	—	127	1	no	—
87	C₂²×C₈	—	127	1	no	—
88	C₂²×C₈	—	127	1	no	—
89	C₄⋊Q₈	—	127	4	no	—
90	#125	—	127	2	no	—
91	#125	C₂×D₁₂₇	1	2	yes	—
92	C₂²⋊Q₈	—	254	4	no	—
93	C₄×Q₈	—	254	4	no	—
94	#117	—	127	4	no	—
95	C₂×C₄²	—	127	1	no	—
96	#107	—	127	2	no	—
97	#107	—	127	4	no	—
98	#107	C₂×D₁₂₇	1	2	yes	—
99	#106	—	127	2	no	—
100	#106	—	127	4	no	—
101	#106	C₂×D₁₂₇	1	2	yes	—
102	C₄×C₈	—	127	1	no	—
103	C₄×C₈	—	127	1	no	—
104	C₄×C₈	—	127	1	no	—
105	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
106	C₂×Q₈	—	254	4	no	—
107	C₂²×C₄	—	127	1	no	—
108	C₂²×C₄	—	127	2	no	—
109	C₂×C₈	—	127	1	no	—
110	C₂×C₈	—	127	1	no	—
111	C₂×C₈	—	127	4	no	—
112	C₂×C₈	—	127	4	no	—
113	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
114	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	127	4	no	—
117	C₄⋊C₄	—	254	4	no	—
118	C₄⋊C₄	—	254	4	no	—
119	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
120	C₂²⋊C₄	—	254	4	no	—
121	C₄²	—	127	2	no	—
122	C₄²	—	127	1	no	—
123	C₄²	—	254	4	no	—
124	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
125	C₂³	—	127	1	no	—
126	Q₈	—	508	8	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	127	1	no	—
129	C₂×C₄	—	127	2	no	—
130	C₂×C₄	—	127	2	no	—
131	C₂×C₄	—	127	2	no	—
132	C₂×C₄	—	254	4	no	—
133	C₂×C₄	—	254	4	no	—
134	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
135	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
136	C₈	—	127	2	no	—
137	C₈	—	127	2	no	—
138	C₈	Q₈×D₁₂₇	1	2	yes	—
139	C₈	Q₈×D₁₂₇	1	2	yes	—
140	C₂²	—	127	2	no	—
141	C₂²	—	254	2	no	—
142	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	4	no	—
145	C₄	—	127	4	no	—
146	C₄	—	127	2	no	—
147	C₄	—	508	4	no	—
148	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
149	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
150	C₂	—	254	2	no	—
151	C₂	C₄⋊Q₈×D₁₂₇	1	1	yes	characteristic
152	C₂	#135×D₁₂₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550432

C₂

yes

maximal

#22550280

C₂

yes

maximal, complemented

#22550278

C₂

yes

maximal, characteristic, complemented

#22550275

C₂

yes

maximal, complemented

#22550273

C₂

yes

maximal, complemented

#22549913

C₂

yes

maximal

#22549911

C₂

yes

maximal

C₄×C₈×D₁₂₇

C₂

yes

maximal, characteristic

#499×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₂₇

C₂²

yes

characteristic

C₂×C₈×D₁₂₇

C₂²

yes

characteristic

#16861208

C₂²

yes

—

#16861201

C₂²

yes

—

#16861187

—

#16861174

—

#16861167

C₂²

yes

—

C₄²×D₁₂₇

C₂²

yes

characteristic

#16861143

C₂²

yes

—

#16861142

C₂²

yes

—

#16861141

C₂²

yes

—

#16861140

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₁₂₇

C₂²

yes

characteristic

C₈×Dic₁₂₇

C₂²

yes

characteristic

#125×C₁₂₇

C₂²

yes

—

#107×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

characteristic

C₂×C₁₂₇⋊C₈

C₂³

yes

characteristic

C₈×D₁₂₇

Q₈

yes

complemented

C₈×D₁₂₇

Q₈

yes

complemented

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

(C₂×D₁₂₇)⋊C₄

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

D₄

yes

—

C₄×Dic₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

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

characteristic

C₂×Dic₁₂₇

C₂×D₄

yes

—

C₄×D₁₂₇

—

C₄×D₁₂₇

C₂×Q₈

yes

—

C₄×D₁₂₇

C₂×Q₈

yes

—

C₁₂₇⋊C₈

C₂×Q₈

yes

—

C₁₂₇⋊C₈

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

C₂×Q₈

yes

—

C₈×C₁₂₇

C₂×Q₈

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

C₄⋊Q₈

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²×Q₈

yes

characteristic

C₄×C₁₂₇

C₂²×Q₈

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

C₄⋊Q₈

yes

—

Dic₁₂₇

#135

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

—

#499

—

127

—

#499

D₁₂₇

yes

characteristic, complemented

#498

—

127

—

#427

—

127

—

#427

—

127

—

C₂×C₄×C₈

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₄⋊Q₈

—

127

—

#125

—

127

—

#125

C₂×D₁₂₇

yes

—

C₂²⋊Q₈

—

254

—

C₄×Q₈

—

254

—

#117

—

127

—

C₂×C₄²

—

127

—

#107

—

127

—

#107

—

127

—

#107

C₂×D₁₂₇

yes

—

#106

—

127

—

100

#106

—

127

—

101

#106

C₂×D₁₂₇

yes

—

102

C₄×C₈

—

127

—

103

C₄×C₈

—

127

—

104

C₄×C₈

—

127

—

105

C₄×C₈

C₂×D₁₂₇

yes

characteristic

106

C₂×Q₈

—

254

—

107

C₂²×C₄

—

127

—

108

C₂²×C₄

—

127

—

109

C₂×C₈

—

127

—

110

C₂×C₈

—

127

—

111

C₂×C₈

—

127

—

112

C₂×C₈

—

127

—

113

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

114

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

127

—

117

C₄⋊C₄

—

254

—

118

C₄⋊C₄

—

254

—

119

C₄⋊C₄

C₂²×D₁₂₇

yes

—

120

C₂²⋊C₄

—

254

—

121

C₄²

—

127

—

122

C₄²

—

127

—

123

C₄²

—

254

—

124

C₄²

C₂²×D₁₂₇

yes

characteristic

125

C₂³

—

127

—

126

Q₈

—

508

—

127

C₂×C₄

—

128

C₂×C₄

—

127

—

129

C₂×C₄

—

127

—

130

C₂×C₄

—

127

—

131

C₂×C₄

—

127

—

132

C₂×C₄

—

254

—

133

C₂×C₄

—

254

—

134

C₂×C₄

D₄×D₁₂₇

yes

—

135

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

136

C₈

—

127

—

137

C₈

—

127

—

138

C₈

Q₈×D₁₂₇

yes

—

139

C₈

Q₈×D₁₂₇

yes

—

140

C₂²

—

127

—

141

C₂²

—

254

—

142

C₂²

C₂×D₄×D₁₂₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

127

—

146

C₄

—

127

—

147

C₄

—

508

—

148

C₄

C₂×Q₈×D₁₂₇

yes

characteristic

149

C₄

C₂×Q₈×D₁₂₇

yes

characteristic

150

C₂

—

254

—

151

C₂

C₄⋊Q₈×D₁₂₇

yes

characteristic

152

C₂

#135×D₁₂₇

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#499×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂²×D₁₂₇	C₂×D₄
hypercenter	#499	D₁₂₇
hypocenter	C₁₂₇	#3160
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

subgroup

G/H

center

C₂²

C₂×D₄×D₁₂₇

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

#499×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 ◁ #499

N	H ≅ G/N	type
2. #22550280	C₂	semidirect
3. #22550278	C₂	semidirect
4. #22550275	C₂	semidirect
5. #22550273	C₂	semidirect
9. #499×C₁₂₇	C₂	semidirect
34. C₈×D₁₂₇	Q₈	semidirect
35. C₈×D₁₂₇	Q₈	semidirect
77. C₁₂₇	#3160	semidirect
82. #499	D₁₂₇	semidirect

H ≅ G/N

type

2. #22550280

C₂

semidirect

3. #22550278

C₂

semidirect

4. #22550275

C₂

semidirect

5. #22550273

C₂

semidirect

9. #499×C₁₂₇

C₂

semidirect

34. C₈×D₁₂₇

Q₈

semidirect

35. C₈×D₁₂₇

Q₈

semidirect

77. C₁₂₇

#3160

semidirect

82. #499

D₁₂₇

semidirect

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

subgroup	index
#22550432	2
#22550280	2
#22550278	2
#22550275	2
#22550273	2
#22549913	2
#22549911	2
C₄×C₈×D₁₂₇	2
#499×C₁₂₇	2
#3160	127

subgroup

index

C₄×C₈×D₁₂₇

#499×C₁₂₇

#3160

127

Subgroups — #30649243

Conjugacy classes of subgroups

Complements

Maximal subgroups