Group #30647323 — subgroups

Subgroups — #30647323

Order 16256.

476 conjugacy classes of subgroups; only the 253 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 253 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×#16861216	C₂	1	1	yes	maximal, characteristic
2	#22550359	C₂	1	1	yes	maximal, characteristic
3	#22549919	C₂	1	3	yes	maximal, complemented
4	#22549916	C₂	1	3	yes	maximal, complemented
5	#22549912	C₂	1	3	yes	maximal, complemented
6	C₂×#16861140	C₂	1	3	yes	maximal, complemented
7	#520×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	#16861216	C₄	1	8	yes	complemented
9	C₂×#14523232	C₂²	1	3	yes	—
10	C₂²×C₁₂₇⋊C₈	C₂²	1	3	yes	—
11	Q₈×Dic₁₂₇	—	2	1	no	—
12	D₄×Dic₁₂₇	—	2	3	no	—
13	#16861168	C₂²	1	3	yes	—
14	C₂×C₄⋊Dic₁₂₇	C₂²	1	3	yes	—
15	#16861142	C₂²	1	3	yes	—
16	#16861142	C₂²	1	3	yes	—
17	#16861140	C₂²	1	3	yes	—
18	#16861140	C₂²	1	6	yes	—
19	#16861140	C₂²	1	1	yes	characteristic
20	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₁₂₇	—	2	1	no	—
22	C₄×D₄×C₁₂₇	—	2	3	no	—
23	#117×C₁₂₇	C₂²	1	3	yes	—
24	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	3	yes	—
25	C₂²×Dic₁₂₇	—	2	3	no	—
26	#14523232	C₂×C₄	1	12	yes	complemented
27	C₂×C₁₂₇⋊C₈	C₂×C₄	1	12	yes	complemented
28	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
29	C₂×C₁₂₇⋊C₈	C₂³	1	3	yes	—
30	C₄⋊Dic₁₂₇	—	2	3	no	—
31	C₄⋊Dic₁₂₇	C₂³	1	3	yes	—
32	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
33	C₂²⋊Dic₁₂₇	—	2	3	no	—
34	C₄×Dic₁₂₇	—	2	3	no	—
35	#41×C₁₂₇	C₂×C₄	1	4	yes	—
36	#41×C₁₂₇	Q₈	1	4	yes	—
37	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
38	C₂×D₄×C₁₂₇	D₄	1	3	yes	—
39	C₂²×C₄×C₁₂₇	—	2	3	no	—
40	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
41	C₄⋊C₄×C₁₂₇	—	2	3	no	—
42	C₄⋊C₄×C₁₂₇	C₂³	1	3	yes	—
43	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₂²⋊C₄×C₁₂₇	—	2	3	no	—
45	C₄²×C₁₂₇	—	2	3	no	—
46	C₂×Dic₁₂₇	—	2	1	no	—
47	C₂×Dic₁₂₇	—	2	3	no	—
48	C₂×Dic₁₂₇	—	8	3	no	—
49	C₁₂₇⋊C₈	C₂²×C₄	1	2	yes	—
50	C₁₂₇⋊C₈	C₂²×C₄	1	6	yes	—
51	C₂³×C₁₂₇	C₂×D₄	1	3	yes	—
52	Q₈×C₁₂₇	C₄⋊C₄	1	4	yes	—
53	D₄×C₁₂₇	C₄⋊C₄	1	12	yes	—
54	C₂×C₄×C₁₂₇	—	2	1	no	—
55	C₂×C₄×C₁₂₇	—	2	3	no	—
56	C₂×C₄×C₁₂₇	—	8	3	no	—
57	C₂×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
58	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
59	C₂×C₄×C₁₂₇	C₂×Q₈	1	6	yes	—
60	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
61	C₂²×C₁₂₇	—	2	3	no	—
62	C₂²×C₁₂₇	—	2	3	no	—
63	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	6	yes	—
64	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₂₇	—	4	1	no	—
66	C₄×C₁₂₇	—	4	3	no	—
67	C₄×C₁₂₇	C₂×C₄⋊C₄	1	6	yes	—
68	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
70	Dic₁₂₇	—	4	1	no	—
71	Dic₁₂₇	—	4	3	no	—
72	C₂×C₁₂₇	—	2	6	no	—
73	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₁₂₇	#577	1	1	yes	characteristic
75	C₂×C₁₂₇	#577	1	1	yes	characteristic
76	#2912	—	127	1	no	maximal, Sylow
77	C₁₂₇	#2912	1	1	yes	characteristic, complemented, Sylow
78	C₂×#131	—	127	1	no	—
79	#520	—	127	1	no	—
80	#520	D₁₂₇	1	1	yes	characteristic
81	#428	—	127	3	no	—
82	#428	—	127	3	no	—
83	#427	—	127	3	no	—
84	C₂×#106	—	127	3	no	—
85	C₂×#41	C₂×D₁₂₇	1	1	yes	characteristic
86	#131	—	127	8	no	—
87	C₂×M₁₆	—	127	3	no	—
88	C₂²×C₈	—	127	3	no	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	—	254	1	no	—
91	C₄×D₄	—	2	3	no	—
92	C₄×D₄	—	254	3	no	—
93	#117	—	127	3	no	—
94	#117	C₂×D₁₂₇	1	3	yes	—
95	C₂×C₄⋊C₄	—	127	3	no	—
96	C₂×C₄⋊C₄	C₂×D₁₂₇	1	3	yes	—
97	#107	—	127	3	no	—
98	#107	—	127	3	no	—
99	#106	—	127	3	no	—
100	#106	—	127	6	no	—
101	#106	—	127	1	no	—
102	#41	C₄×D₁₂₇	1	4	yes	—
103	#41	Dic₂₅₄	1	4	yes	—
104	C₂×Q₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
105	C₂×D₄	C₂²⋊D₁₂₇	1	3	yes	—
106	C₂²×C₄	—	2	3	no	—
107	C₂²×C₄	—	254	3	no	—
108	C₂²×C₄	C₂²×D₁₂₇	1	3	yes	—
109	M₁₆	—	127	12	no	—
110	C₂×C₈	—	127	12	no	—
111	C₂×C₈	—	127	1	no	—
112	C₂×C₈	—	127	3	no	—
113	C₄⋊C₄	—	2	3	no	—
114	C₄⋊C₄	—	127	3	no	—
115	C₄⋊C₄	—	127	1	no	—
116	C₄⋊C₄	—	254	3	no	—
117	C₄⋊C₄	C₂²×D₁₂₇	1	3	yes	—
118	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
119	C₂²⋊C₄	—	2	3	no	—
120	C₂²⋊C₄	—	254	3	no	—
121	C₄²	—	2	3	no	—
122	C₄²	—	254	3	no	—
123	C₂³	C₂×C₂²⋊D₁₂₇	1	3	yes	—
124	Q₈	Dic₁₂₇⋊C₄	1	4	yes	—
125	D₄	Dic₁₂₇⋊C₄	1	12	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	8	3	no	—
129	C₂×C₄	—	1016	3	no	—
130	C₂×C₄	—	254	1	no	—
131	C₂×C₄	—	254	3	no	—
132	C₂×C₄	C₂×C₄×D₁₂₇	1	6	yes	—
133	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	3	yes	—
134	C₂×C₄	C₂×Dic₂₅₄	1	6	yes	—
135	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
136	C₈	—	127	2	no	—
137	C₈	—	127	6	no	—
138	C₂²	—	2	3	no	—
139	C₂²	—	2	3	no	—
140	C₂²	C₂×Dic₁₂₇⋊C₄	1	6	yes	—
141	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
142	C₄	—	4	1	no	—
143	C₄	—	4	3	no	—
144	C₄	—	508	3	no	—
145	C₄	—	508	1	no	—
146	C₄	C₂×Dic₁₂₇⋊C₄	1	6	yes	—
147	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
148	C₄	C₂²×Dic₂₅₄	1	1	yes	characteristic
149	C₂	—	2	6	no	—
150	C₂	C₂²×Dic₁₂₇⋊C₄	1	1	yes	characteristic
151	C₂	#22550684	1	1	yes	characteristic
152	C₂	#22550687	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861216

C₂

yes

maximal, characteristic

#22550359

C₂

yes

maximal, characteristic

#22549919

C₂

yes

maximal, complemented

#22549916

C₂

yes

maximal, complemented

#22549912

C₂

yes

maximal, complemented

C₂×#16861140

C₂

yes

maximal, complemented

#520×C₁₂₇

C₂

yes

maximal, characteristic

#16861216

C₄

yes

complemented

C₂×#14523232

C₂²

yes

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

Q₈×Dic₁₂₇

—

D₄×Dic₁₂₇

—

#16861168

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861142

C₂²

yes

—

#16861142

C₂²

yes

—

#16861140

C₂²

yes

—

#16861140

C₂²

yes

—

#16861140

C₂²

yes

characteristic

C₂×#41×C₁₂₇

C₂²

yes

characteristic

C₄×Q₈×C₁₂₇

—

C₄×D₄×C₁₂₇

—

#117×C₁₂₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

—

#14523232

C₂×C₄

yes

complemented

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

complemented

C₂×C₁₂₇⋊C₈

C₂³

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂³

yes

—

C₄⋊Dic₁₂₇

C₂³

yes

characteristic

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

#41×C₁₂₇

C₂×C₄

yes

—

#41×C₁₂₇

Q₈

yes

—

C₂×Q₈×C₁₂₇

D₄

yes

characteristic

C₂×D₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

C₂³

yes

characteristic

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

C₂²×C₄

yes

—

C₁₂₇⋊C₈

C₂²×C₄

yes

—

C₂³×C₁₂₇

C₂×D₄

yes

—

Q₈×C₁₂₇

C₄⋊C₄

yes

—

D₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

—

C₄×C₁₂₇

C₂³×C₄

yes

characteristic

C₄×C₁₂₇

C₂²×Q₈

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×C₄⋊C₄

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#2912

yes

characteristic, complemented, Sylow

C₂×#131

—

127

—

#520

—

127

—

#520

D₁₂₇

yes

characteristic

#428

—

127

—

#428

—

127

—

#427

—

127

—

C₂×#106

—

127

—

C₂×#41

C₂×D₁₂₇

yes

characteristic

#131

—

127

—

C₂×M₁₆

—

127

—

C₂²×C₈

—

127

—

C₄×Q₈

—

C₄×Q₈

—

254

—

C₄×D₄

—

C₄×D₄

—

254

—

#117

—

127

—

#117

C₂×D₁₂₇

yes

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

#107

—

127

—

#107

—

127

—

#106

—

127

—

100

#106

—

127

—

101

#106

—

127

—

102

#41

C₄×D₁₂₇

yes

—

103

#41

Dic₂₅₄

yes

—

104

C₂×Q₈

C₂²⋊D₁₂₇

yes

characteristic

105

C₂×D₄

C₂²⋊D₁₂₇

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

254

—

108

C₂²×C₄

C₂²×D₁₂₇

yes

—

109

M₁₆

—

127

—

110

C₂×C₈

—

127

—

111

C₂×C₈

—

127

—

112

C₂×C₈

—

127

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

127

—

115

C₄⋊C₄

—

127

—

116

C₄⋊C₄

—

254

—

117

C₄⋊C₄

C₂²×D₁₂₇

yes

—

118

C₄⋊C₄

C₂²×D₁₂₇

yes

characteristic

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

254

—

121

C₄²

—

122

C₄²

—

254

—

123

C₂³

C₂×C₂²⋊D₁₂₇

yes

—

124

Q₈

Dic₁₂₇⋊C₄

yes

—

125

D₄

Dic₁₂₇⋊C₄

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

1016

—

130

C₂×C₄

—

254

—

131

C₂×C₄

—

254

—

132

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

133

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

134

C₂×C₄

C₂×Dic₂₅₄

yes

—

135

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

136

C₈

—

127

—

137

C₈

—

127

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×Dic₁₂₇⋊C₄

yes

—

141

C₂²

C₂²×C₂²⋊D₁₂₇

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

508

—

145

C₄

—

508

—

146

C₄

C₂×Dic₁₂₇⋊C₄

yes

—

147

C₄

C₂²×C₄×D₁₂₇

yes

characteristic

148

C₄

C₂²×Dic₂₅₄

yes

characteristic

149

C₂

—

150

C₂

C₂²×Dic₁₂₇⋊C₄

yes

characteristic

151

C₂

#22550684

yes

characteristic

152

C₂

#22550687

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#520×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂×#41×C₁₂₇	C₂²
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#520	D₁₂₇
hypocenter	C₁₂₇	#2912
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #520

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₂₇

commutator

C₄×C₁₂₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

#520×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×D₄

prime radical

C₂×#41×C₁₂₇

C₂²

cosocle

C₂×C₄×C₁₂₇

C₂⁴

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ #520

N	H ≅ G/N	type
3. #22549919	C₂	semidirect
4. #22549916	C₂	semidirect
5. #22549912	C₂	semidirect
6. C₂×#16861140	C₂	semidirect
8. #16861216	C₄	semidirect
26. #14523232	C₂×C₄	semidirect
27. C₂×C₁₂₇⋊C₈	C₂×C₄	semidirect
77. C₁₂₇	#2912	semidirect

H ≅ G/N

type

3. #22549919

C₂

semidirect

4. #22549916

C₂

semidirect

5. #22549912

C₂

semidirect

6. C₂×#16861140

C₂

semidirect

8. #16861216

C₄

semidirect

26. #14523232

C₂×C₄

semidirect

27. C₂×C₁₂₇⋊C₈

C₂×C₄

semidirect

77. C₁₂₇

#2912

semidirect

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

subgroup	index
C₂×#16861216	2
#22550359	2
#22549919	2
#22549916	2
#22549912	2
C₂×#16861140	2
#520×C₁₂₇	2
#2912	127

subgroup

index

#520×C₁₂₇

127

Subgroups — #30647323

Conjugacy classes of subgroups

Complements

Maximal subgroups