Group #30639851 — subgroups

Subgroups — #30639851

Order 16256.

342 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 73 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×#16861239	C₂	1	2	yes	maximal
2	C₂²×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16861127	C₂	1	2	yes	maximal, complemented
4	#22549641	C₂	1	1	yes	maximal, characteristic, complemented
5	#408×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₅₄	C₂²	1	2	yes	—
7	#16861239	—	2	4	no	—
8	#16861239	—	2	4	no	—
9	C₂×Dic₅₀₈	—	2	4	no	—
10	C₂×Dic₅₀₈	—	2	2	no	—
11	C₂×#14523238	—	2	2	no	—
12	C₂×#14523236	—	2	2	no	—
13	C₂×#14523232	C₂²	1	2	yes	—
14	#16861127	—	2	4	no	—
15	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
19	C₂×Dic₂₅₄	—	2	2	no	—
20	C₂×Dic₂₅₄	—	2	2	no	—
21	C₂×Dic₂₅₄	—	2	2	no	—
22	C₂×Dic₂₅₄	—	2	2	no	—
23	C₂×Dic₂₅₄	—	2	2	no	—
24	C₂×Dic₂₅₄	D₄	1	4	yes	—
25	C₂²×Dic₁₂₇	—	2	2	no	—
26	Dic₅₀₈	—	2	8	no	—
27	#14523238	—	2	8	no	—
28	#14523236	—	2	8	no	—
29	#14523232	—	2	2	no	—
30	#14523232	D₄	1	4	yes	—
31	C₂×C₁₂₇⋊C₈	—	2	2	no	—
32	#41×C₁₂₇	—	2	2	no	—
33	#41×C₁₂₇	—	2	2	no	—
34	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
35	C₂×D₄×C₁₂₇	—	2	1	no	—
36	C₂×D₄×C₁₂₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	—	2	1	no	—
38	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	M₁₆×C₁₂₇	—	2	2	no	—
40	C₂×C₈×C₁₂₇	—	2	2	no	—
41	C₂×C₈×C₁₂₇	—	2	1	no	—
42	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
43	Dic₂₅₄	—	2	4	no	—
44	Dic₂₅₄	—	2	4	no	—
45	Dic₂₅₄	—	4	2	no	—
46	Dic₂₅₄	—	4	2	no	—
47	C₂×Dic₁₂₇	—	2	4	no	—
48	C₂×Dic₁₂₇	—	4	2	no	—
49	C₂×Dic₁₂₇	—	4	2	no	—
50	C₁₂₇⋊C₈	—	2	4	no	—
51	C₂³×C₁₂₇	—	2	1	no	—
52	C₂³×C₁₂₇	#41	1	1	yes	characteristic
53	Q₈×C₁₂₇	—	2	2	no	—
54	D₄×C₁₂₇	—	2	2	no	—
55	D₄×C₁₂₇	—	4	2	no	—
56	C₂×C₄×C₁₂₇	—	2	1	no	—
57	C₂×C₄×C₁₂₇	—	4	1	no	—
58	C₂×C₄×C₁₂₇	—	4	1	no	—
59	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
63	C₈×C₁₂₇	—	2	2	no	—
64	C₈×C₁₂₇	—	4	1	no	—
65	C₂²×C₁₂₇	—	2	2	no	—
66	C₂²×C₁₂₇	—	4	1	no	—
67	C₂²×C₁₂₇	—	4	2	no	—
68	C₂²×C₁₂₇	#121	1	2	yes	—
69	C₂²×C₁₂₇	#121	1	1	yes	characteristic
70	C₄×C₁₂₇	—	4	1	no	—
71	C₄×C₁₂₇	#120	1	1	yes	characteristic
72	C₄×C₁₂₇	#120	1	2	yes	—
73	C₄×C₁₂₇	#127	1	1	yes	characteristic
74	Dic₁₂₇	—	4	4	no	—
75	C₂×C₁₂₇	—	2	2	no	—
76	C₂×C₁₂₇	—	8	1	no	—
77	C₂×C₁₂₇	#392	1	1	yes	characteristic
78	C₂×C₁₂₇	#473	1	2	yes	—
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	—	127	1	no	—
83	C₂×#101	—	127	2	no	—
84	#408	D₁₂₇	1	1	yes	characteristic
85	#341	—	127	1	no	—
86	C₂×#41	C₂×D₁₂₇	1	1	yes	characteristic
87	C₂²×Q₈	—	127	2	no	—
88	#137	—	254	4	no	—
89	#137	—	254	4	no	—
90	C₂×Q₁₆	—	254	4	no	—
91	C₂×Q₁₆	—	254	2	no	—
92	C₂×Q₁₆	—	254	2	no	—
93	C₂×SD₁₆	—	254	2	no	—
94	C₂×M₁₆	—	127	2	no	—
95	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
97	#101	—	254	4	no	—
98	C₂²⋊C₈	—	2	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	C₂×Q₈	—	127	4	no	—
102	C₂×Q₈	—	254	2	no	—
103	C₂×Q₈	—	254	2	no	—
104	C₂×Q₈	—	254	2	no	—
105	C₂×Q₈	—	254	2	no	—
106	C₂×Q₈	—	254	2	no	—
107	C₂×Q₈	D₅₀₈	1	1	yes	characteristic
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	D₅₀₈	1	1	yes	characteristic
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	254	2	no	—
112	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
113	Q₁₆	—	254	8	no	—
114	Q₁₆	—	254	8	no	—
115	SD₁₆	—	254	8	no	—
116	M₁₆	—	2	2	no	—
117	M₁₆	—	127	4	no	—
118	M₁₆	—	254	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	254	2	no	—
122	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
123	C₂³	—	2	1	no	—
124	C₂³	#14523248	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	Q₈	—	508	2	no	—
127	Q₈	—	508	2	no	—
128	Q₈	—	254	4	no	—
129	Q₈	—	254	4	no	—
130	D₄	—	2	2	no	—
131	D₄	—	4	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	508	2	no	—
136	C₂×C₄	—	508	2	no	—
137	C₂×C₄	—	254	4	no	—
138	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
139	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
141	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	4	1	no	—
144	C₈	—	254	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	#16861182	1	2	yes	—
149	C₂²	#16861183	1	1	yes	characteristic
150	C₄	—	4	1	no	—
151	C₄	—	508	4	no	—
152	C₄	#16861177	1	1	yes	characteristic
153	C₄	#16861179	1	2	yes	—
154	C₄	#16861205	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	8	1	no	—
157	C₂	#22549785	1	1	yes	characteristic
158	C₂	#22550151	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861239

C₂

yes

maximal

C₂²×Dic₅₀₈

C₂

yes

maximal, characteristic, complemented

C₂×#16861127

C₂

yes

maximal, complemented

#22549641

C₂

yes

maximal, characteristic, complemented

#408×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

—

#16861239

—

#16861239

—

C₂×Dic₅₀₈

—

C₂×Dic₅₀₈

—

C₂×#14523238

—

C₂×#14523236

—

C₂×#14523232

C₂²

yes

—

#16861127

—

C₂×#41×C₁₂₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂²⋊C₈×C₁₂₇

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

D₄

yes

—

C₂²×Dic₁₂₇

—

Dic₅₀₈

—

—

—

—

D₄

yes

—

C₂×C₁₂₇⋊C₈

—

#41×C₁₂₇

—

#41×C₁₂₇

—

C₂×Q₈×C₁₂₇

D₄

yes

characteristic

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

#41

yes

characteristic

Q₈×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#121

yes

—

C₂²×C₁₂₇

#121

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

#120

yes

characteristic

C₄×C₁₂₇

#120

yes

—

C₄×C₁₂₇

#127

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#392

yes

characteristic

C₂×C₁₂₇

#473

yes

—

#2020

—

127

maximal, Sylow

C₁₂₇

#2020

yes

characteristic, complemented, Sylow

C₂×#137

—

127

—

C₂²×Q₁₆

—

127

—

C₂×#101

—

127

—

#408

D₁₂₇

yes

characteristic

#341

—

127

—

C₂×#41

C₂×D₁₂₇

yes

characteristic

C₂²×Q₈

—

127

—

#137

—

254

—

#137

—

254

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

—

254

—

C₂×SD₁₆

—

254

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#101

—

254

—

C₂²⋊C₈

—

#41

—

100

#41

—

101

C₂×Q₈

—

127

—

102

C₂×Q₈

—

254

—

103

C₂×Q₈

—

254

—

104

C₂×Q₈

—

254

—

105

C₂×Q₈

—

254

—

106

C₂×Q₈

—

254

—

107

C₂×Q₈

D₅₀₈

yes

characteristic

108

C₂×D₄

—

109

C₂×D₄

D₅₀₈

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

—

254

—

112

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

113

Q₁₆

—

254

—

114

Q₁₆

—

254

—

115

SD₁₆

—

254

—

116

M₁₆

—

117

M₁₆

—

127

—

118

M₁₆

—

254

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

254

—

122

C₂×C₈

C₂²⋊D₁₂₇

yes

—

123

C₂³

—

124

C₂³

#14523248

yes

characteristic

125

Q₈

—

126

Q₈

—

508

—

127

Q₈

—

508

—

128

Q₈

—

254

—

129

Q₈

—

254

—

130

D₄

—

131

D₄

—

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

—

140

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

141

C₂×C₄

C₂×D₅₀₈

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

254

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

#16861182

yes

—

149

C₂²

#16861183

yes

characteristic

150

C₄

—

151

C₄

—

508

—

152

C₄

#16861177

yes

characteristic

153

C₄

#16861179

yes

—

154

C₄

#16861205

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#22549785

yes

characteristic

158

C₂

#22550151

yes

—

	subgroup	G/H
center	C₂²	#16861183
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#408×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#121
prime radical	C₂×#41×C₁₂₇	C₂²
hypercenter	#408	D₁₂₇
hypocenter	C₁₂₇	#2020
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₂²

#16861183

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#408×C₁₂₇

C₂

socle

C₂²×C₁₂₇

#121

prime radical

C₂×#41×C₁₂₇

C₂²

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. C₂²×Dic₅₀₈	C₂	semidirect
3. C₂×#16861127	C₂	semidirect
4. #22549641	C₂	semidirect
80. C₁₂₇	#2020	semidirect

H ≅ G/N

type

2. C₂²×Dic₅₀₈

C₂

semidirect

3. C₂×#16861127

C₂

semidirect

4. #22549641

C₂

semidirect

80. C₁₂₇

#2020

semidirect

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

subgroup	index
C₂×#16861239	2
C₂²×Dic₅₀₈	2
C₂×#16861127	2
#22549641	2
#408×C₁₂₇	2
#2020	127

subgroup

index

C₂²×Dic₅₀₈

#408×C₁₂₇

127

Subgroups — #30639851

Conjugacy classes of subgroups

Complements

Maximal subgroups