Group #30639849 — subgroups

Subgroups — #30639849

Order 16256.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16861240	C₂	1	2	yes	maximal
2	C₂×#16861128	C₂	1	2	yes	maximal, complemented
3	#22549842	C₂	1	1	yes	maximal, characteristic
4	#22549641	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523248	C₂²	1	1	yes	characteristic
7	#16861240	—	2	4	no	—
8	#16861240	—	2	4	no	—
9	C₂×#14523238	—	2	2	no	—
10	C₂×#14523235	—	2	2	no	—
11	C₂×#14523232	C₂²	1	1	yes	characteristic
12	C₂×#14523232	C₂²	1	2	yes	—
13	#16861128	—	2	4	no	—
14	(C₂×D₁₂₇)⋊C₈	—	2	2	no	—
15	C₂²×Q₈×C₁₂₇	C₂²	1	2	yes	—
16	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
17	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
18	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	#14523248	—	2	2	no	—
20	#14523248	—	2	2	no	—
21	C₂×Dic₂₅₄	D₄	1	1	yes	characteristic
22	C₂×D₅₀₈	D₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
24	C₂×C₄×D₁₂₇	—	2	1	no	—
25	#14523238	—	2	8	no	—
26	#14523235	—	2	8	no	—
27	#14523232	—	2	2	no	—
28	#14523232	—	2	2	no	—
29	#14523232	D₄	1	4	yes	—
30	C₂×C₁₂₇⋊C₈	—	2	2	no	—
31	C₂×C₁₂₇⋊C₈	—	2	1	no	—
32	C₂×Q₈×C₁₂₇	—	2	2	no	—
33	C₂×Q₈×C₁₂₇	—	2	2	no	—
34	C₂×Q₈×C₁₂₇	—	2	2	no	—
35	C₂×Q₈×C₁₂₇	—	2	2	no	—
36	C₂×Q₈×C₁₂₇	—	2	2	no	—
37	C₂×Q₈×C₁₂₇	D₄	1	4	yes	—
38	C₂²×C₄×C₁₂₇	—	2	2	no	—
39	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
40	Q₁₆×C₁₂₇	—	2	8	no	—
41	C₂×C₈×C₁₂₇	—	2	2	no	—
42	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
43	C₂²×D₁₂₇	—	2	1	no	—
44	Dic₂₅₄	—	2	2	no	—
45	D₅₀₈	—	2	2	no	—
46	C₂²⋊D₁₂₇	—	4	2	no	—
47	C₂×Dic₁₂₇	—	2	1	no	—
48	C₄×D₁₂₇	—	4	1	no	—
49	C₄×D₁₂₇	—	4	1	no	—
50	C₁₂₇⋊C₈	—	2	4	no	—
51	C₁₂₇⋊C₈	—	4	1	no	—
52	C₂³×C₁₂₇	#41	1	1	yes	characteristic
53	Q₈×C₁₂₇	—	2	4	no	—
54	Q₈×C₁₂₇	—	2	4	no	—
55	Q₈×C₁₂₇	—	4	2	no	—
56	Q₈×C₁₂₇	—	4	2	no	—
57	C₂×C₄×C₁₂₇	—	2	4	no	—
58	C₂×C₄×C₁₂₇	—	4	2	no	—
59	C₂×C₄×C₁₂₇	—	4	2	no	—
60	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
64	C₈×C₁₂₇	—	2	2	no	—
65	C₂²×C₁₂₇	—	2	2	no	—
66	C₂²×C₁₂₇	#121	1	1	yes	characteristic
67	C₂²×C₁₂₇	#121	1	2	yes	—
68	C₂×D₁₂₇	—	4	1	no	—
69	C₂×D₁₂₇	—	4	2	no	—
70	C₄×C₁₂₇	—	4	4	no	—
71	C₄×C₁₂₇	#120	1	2	yes	—
72	C₄×C₁₂₇	#120	1	1	yes	characteristic
73	C₄×C₁₂₇	#127	1	1	yes	characteristic
74	Dic₁₂₇	—	4	1	no	—
75	C₂×C₁₂₇	—	2	2	no	—
76	C₂×C₁₂₇	#392	1	1	yes	characteristic
77	C₂×C₁₂₇	#473	1	2	yes	—
78	D₁₂₇	—	8	1	no	—
79	#2020	—	127	1	no	maximal, Sylow
80	C₁₂₇	#2020	1	1	yes	characteristic, complemented, Sylow
81	C₂×#137	—	127	2	no	—
82	C₂²×Q₁₆	D₁₂₇	1	1	yes	characteristic, complemented
83	C₂×#101	—	127	2	no	—
84	#408	—	127	1	no	—
85	#341	—	127	1	no	—
86	C₂×#41	—	127	1	no	—
87	C₂²×Q₈	C₂×D₁₂₇	1	2	yes	—
88	#137	—	254	4	no	—
89	#137	—	254	4	no	—
90	C₂×Q₁₆	—	2	4	no	—
91	C₂×Q₁₆	—	2	2	no	—
92	C₂×Q₁₆	—	254	2	no	—
93	C₂×SD₁₆	—	254	2	no	—
94	C₂×M₁₆	—	127	2	no	—
95	C₂×M₁₆	—	127	1	no	—
96	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
97	#101	—	254	4	no	—
98	C₂²⋊C₈	—	254	2	no	—
99	#41	—	254	2	no	—
100	#41	—	254	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	127	1	no	—
107	C₂×Q₈	C₂²⋊D₁₂₇	1	4	yes	—
108	C₂×D₄	—	127	1	no	—
109	C₂×D₄	—	254	1	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	—	254	1	no	—
112	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
113	Q₁₆	—	2	8	no	—
114	Q₁₆	—	254	8	no	—
115	SD₁₆	—	254	8	no	—
116	M₁₆	—	127	4	no	—
117	M₁₆	—	254	2	no	—
118	M₁₆	—	254	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	254	1	no	—
121	C₂×C₈	—	254	2	no	—
122	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
123	C₂³	—	254	1	no	—
124	C₂³	#14523247	1	1	yes	characteristic
125	Q₈	—	2	4	no	—
126	Q₈	—	2	4	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	254	2	no	—
130	D₄	—	508	2	no	—
131	D₄	—	254	2	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	508	1	no	—
136	C₂×C₄	—	508	1	no	—
137	C₂×C₄	—	254	1	no	—
138	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
139	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
142	C₈	—	2	2	no	—
143	C₈	—	508	1	no	—
144	C₈	—	254	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	508	1	no	—
147	C₂²	—	508	2	no	—
148	C₂²	#16861180	1	1	yes	characteristic
149	C₂²	#16861181	1	2	yes	—
150	C₄	—	4	4	no	—
151	C₄	—	508	1	no	—
152	C₄	#16861177	1	2	yes	—
153	C₄	#16861178	1	1	yes	characteristic
154	C₄	#16861205	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	1016	1	no	—
157	C₂	#22549784	1	1	yes	characteristic
158	C₂	#22550148	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861240

C₂

yes

maximal

C₂×#16861128

C₂

yes

maximal, complemented

#22549842

C₂

yes

maximal, characteristic

#22549641

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523248

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#14523232

C₂²

yes

—

#16861128

—

(C₂×D₁₂₇)⋊C₈

—

C₂²×Q₈×C₁₂₇

C₂²

yes

—

C₂×Q₁₆×C₁₂₇

—

C₂×Q₁₆×C₁₂₇

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

#14523248

—

#14523248

—

C₂×Dic₂₅₄

D₄

yes

characteristic

C₂×D₅₀₈

D₄

yes

characteristic

C₂×C₂²⋊D₁₂₇

—

C₂×C₄×D₁₂₇

—

—

—

—

—

D₄

yes

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

Q₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂²×D₁₂₇

—

Dic₂₅₄

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

#41

yes

characteristic

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#121

yes

characteristic

C₂²×C₁₂₇

#121

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#120

yes

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#392

yes

characteristic

C₂×C₁₂₇

#473

yes

—

D₁₂₇

—

#2020

—

127

maximal, Sylow

C₁₂₇

#2020

yes

characteristic, complemented, Sylow

C₂×#137

—

127

—

C₂²×Q₁₆

D₁₂₇

yes

characteristic, complemented

C₂×#101

—

127

—

#408

—

127

—

#341

—

127

—

C₂×#41

—

127

—

C₂²×Q₈

C₂×D₁₂₇

yes

—

#137

—

254

—

#137

—

254

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

254

—

C₂×SD₁₆

—

254

—

C₂×M₁₆

—

127

—

C₂×M₁₆

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#101

—

254

—

C₂²⋊C₈

—

254

—

#41

—

254

—

100

#41

—

254

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

127

—

107

C₂×Q₈

C₂²⋊D₁₂₇

yes

—

108

C₂×D₄

—

127

—

109

C₂×D₄

—

254

—

110

C₂²×C₄

—

111

C₂²×C₄

—

254

—

112

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

113

Q₁₆

—

114

Q₁₆

—

254

—

115

SD₁₆

—

254

—

116

M₁₆

—

127

—

117

M₁₆

—

254

—

118

M₁₆

—

254

—

119

C₂×C₈

—

120

C₂×C₈

—

254

—

121

C₂×C₈

—

254

—

122

C₂×C₈

C₂²⋊D₁₂₇

yes

—

123

C₂³

—

254

—

124

C₂³

#14523247

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

Q₈

—

254

—

130

D₄

—

508

—

131

D₄

—

254

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

508

—

136

C₂×C₄

—

508

—

137

C₂×C₄

—

254

—

138

C₂×C₄

D₄×D₁₂₇

yes

—

139

C₂×C₄

D₄×D₁₂₇

yes

characteristic

140

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

142

C₈

—

143

C₈

—

508

—

144

C₈

—

254

—

145

C₂²

—

146

C₂²

—

508

—

147

C₂²

—

508

—

148

C₂²

#16861180

yes

characteristic

149

C₂²

#16861181

yes

—

150

C₄

—

151

C₄

—

508

—

152

C₄

#16861177

yes

—

153

C₄

#16861178

yes

characteristic

154

C₄

#16861205

yes

characteristic

155

C₂

—

156

C₂

—

1016

—

157

C₂

#22549784

yes

characteristic

158

C₂

#22550148

yes

—

	subgroup	G/H
center	C₂²	#16861180
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂²×Q₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#121
prime radical	C₂×#14523248	C₂²
hypercenter	C₂²×Q₁₆	D₁₂₇
hypocenter	C₁₂₇	#2020
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#16861180

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂²×Q₁₆×C₁₂₇

C₂

C₂²×C₁₂₇

C₂²

C₂²×Q₁₆

D₁₂₇

hypocenter

C₁₂₇

#2020

LCS

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. C₂×#16861128	C₂	semidirect
4. #22549641	C₂	semidirect
5. C₂²×Q₁₆×C₁₂₇	C₂	semidirect
80. C₁₂₇	#2020	semidirect
82. C₂²×Q₁₆	D₁₂₇	semidirect

H ≅ G/N

type

2. C₂×#16861128

C₂

semidirect

4. #22549641

C₂

semidirect

5. C₂²×Q₁₆×C₁₂₇

C₂

semidirect

80. C₁₂₇

#2020

semidirect

82. C₂²×Q₁₆

D₁₂₇

semidirect

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

subgroup	index
C₂×#16861240	2
C₂×#16861128	2
#22549842	2
#22549641	2
C₂²×Q₁₆×C₁₂₇	2
#2020	127

subgroup

index

C₂²×Q₁₆×C₁₂₇

127

Subgroups — #30639849

Conjugacy classes of subgroups

Complements

Maximal subgroups