Group #30639830 — subgroups

Subgroups — #30639830

Order 16256.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂²×#14523238	C₂	1	2	yes	maximal
2	C₂×#16861133	C₂	1	2	yes	maximal
3	#22549782	C₂	1	1	yes	maximal, characteristic
4	#22549631	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₅₄	C₂²	1	1	yes	characteristic
7	C₂×#14523238	—	2	4	no	—
8	C₂×#14523238	—	2	4	no	—
9	C₂×#14523238	—	2	4	no	—
10	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
12	#16861133	—	2	4	no	—
13	C₁₂₇⋊#95	—	2	2	no	—
14	C₂²×Q₈×C₁₂₇	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
16	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
17	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₅₄	—	2	2	no	—
19	C₂×Dic₂₅₄	—	2	2	no	—
20	C₂×Dic₂₅₄	—	2	1	no	—
21	C₂×Dic₂₅₄	D₄	1	2	yes	—
22	C₂²×Dic₁₂₇	—	2	1	no	—
23	C₂²×Dic₁₂₇	—	2	1	no	—
24	#14523238	—	2	16	no	—
25	C₂×C₁₂₇⋊C₈	—	2	2	no	—
26	C₂×C₁₂₇⋊C₈	—	2	2	no	—
27	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	—
28	C₄⋊Dic₁₂₇	—	2	2	no	—
29	C₂×Q₈×C₁₂₇	—	2	2	no	—
30	C₂×Q₈×C₁₂₇	—	2	2	no	—
31	C₂×Q₈×C₁₂₇	—	2	2	no	—
32	C₂×Q₈×C₁₂₇	—	2	2	no	—
33	C₂×Q₈×C₁₂₇	—	2	2	no	—
34	C₂×Q₈×C₁₂₇	D₄	1	4	yes	—
35	C₂²×C₄×C₁₂₇	—	2	2	no	—
36	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
37	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₂₇	—	2	8	no	—
39	C₄²×C₁₂₇	—	2	2	no	—
40	Dic₂₅₄	—	2	4	no	—
41	Dic₂₅₄	—	4	2	no	—
42	C₂×Dic₁₂₇	—	2	2	no	—
43	C₂×Dic₁₂₇	—	4	1	no	—
44	C₂×Dic₁₂₇	—	4	1	no	—
45	C₂×Dic₁₂₇	—	4	1	no	—
46	C₂×Dic₁₂₇	—	4	2	no	—
47	C₁₂₇⋊C₈	—	2	4	no	—
48	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
49	Q₈×C₁₂₇	—	2	4	no	—
50	Q₈×C₁₂₇	—	2	4	no	—
51	Q₈×C₁₂₇	—	4	2	no	—
52	Q₈×C₁₂₇	—	4	2	no	—
53	C₂×C₄×C₁₂₇	—	2	2	no	—
54	C₂×C₄×C₁₂₇	—	2	2	no	—
55	C₂×C₄×C₁₂₇	—	2	4	no	—
56	C₂×C₄×C₁₂₇	—	4	2	no	—
57	C₂×C₄×C₁₂₇	—	4	2	no	—
58	C₂×C₄×C₁₂₇	Q₁₆	1	4	yes	complemented
59	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
63	C₂²×C₁₂₇	#120	1	1	yes	characteristic
64	C₂²×C₁₂₇	#121	1	1	yes	characteristic
65	C₂²×C₁₂₇	#127	1	1	yes	characteristic
66	C₂²×C₁₂₇	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₁₂₇	#137	1	2	yes	—
68	C₄×C₁₂₇	—	2	4	no	—
69	C₄×C₁₂₇	—	4	4	no	—
70	C₄×C₁₂₇	#120	1	2	yes	—
71	C₄×C₁₂₇	#121	1	2	yes	—
72	Dic₁₂₇	—	4	2	no	—
73	Dic₁₂₇	—	8	1	no	—
74	C₂×C₁₂₇	#392	1	1	yes	characteristic
75	C₂×C₁₂₇	#451	1	2	yes	—
76	C₂×C₁₂₇	#462	1	2	yes	—
77	C₂×C₁₂₇	#494	1	1	yes	characteristic
78	C₂×C₁₂₇	#497	1	1	yes	characteristic
79	#2016	—	127	1	no	maximal, Sylow
80	C₁₂₇	#2016	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₁₆	—	127	2	no	—
82	C₂×C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic
83	C₂×#103	—	127	2	no	—
84	#391	—	127	1	no	—
85	#336	—	127	1	no	—
86	C₂²×Q₈	—	127	1	no	—
87	C₂²×Q₈	C₂×D₁₂₇	1	2	yes	—
88	C₂×Q₁₆	—	254	4	no	—
89	C₂×Q₁₆	—	254	4	no	—
90	C₂×Q₁₆	—	254	4	no	—
91	C₂²×C₈	—	127	2	no	—
92	C₄⋊Q₈	—	2	4	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	—	127	1	no	—
95	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
96	#103	—	254	4	no	—
97	#95	—	254	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	127	2	no	—
104	C₂×Q₈	—	254	2	no	—
105	C₂×Q₈	—	254	2	no	—
106	C₂×Q₈	—	254	1	no	—
107	C₂×Q₈	C₂²⋊D₁₂₇	1	4	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	254	1	no	—
110	C₂²×C₄	—	254	1	no	—
111	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
113	Q₁₆	—	254	16	no	—
114	C₂×C₈	—	127	4	no	—
115	C₂×C₈	—	254	2	no	—
116	C₂×C₈	—	254	2	no	—
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	—	254	2	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	2	4	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	508	2	no	—
126	Q₈	—	254	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	508	1	no	—
133	C₂×C₄	—	508	2	no	—
134	C₂×C₄	—	508	1	no	—
135	C₂×C₄	—	508	1	no	—
136	C₂×C₄	—	254	2	no	—
137	C₂×C₄	#14523238	1	4	yes	complemented
138	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
139	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
141	C₂×C₄	#14523247	1	1	yes	characteristic
142	C₈	—	254	4	no	—
143	C₂²	#16861178	1	1	yes	characteristic
144	C₂²	#16861180	1	1	yes	characteristic
145	C₂²	#16861205	1	1	yes	characteristic
146	C₂²	C₂×#14523238	1	2	yes	—
147	C₂²	#16861240	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	4	4	no	—
150	C₄	—	1016	1	no	—
151	C₄	—	508	2	no	—
152	C₄	#16861177	1	2	yes	—
153	C₄	#16861181	1	2	yes	—
154	C₂	#22549784	1	1	yes	characteristic
155	C₂	#22550032	1	2	yes	—
156	C₂	#22550083	1	2	yes	—
157	C₂	#22550255	1	1	yes	characteristic
158	C₂	#22550268	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14523238

C₂

yes

maximal

C₂×#16861133

C₂

yes

maximal

#22549782

C₂

yes

maximal, characteristic

#22549631

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

—

—

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

characteristic

#16861133

—

C₁₂₇⋊#95

—

C₂²×Q₈×C₁₂₇

C₂²

yes

—

C₄⋊Q₈×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

D₄

yes

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

#14523238

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₄⋊Dic₁₂₇

—

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

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

Q₁₆

yes

complemented

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

C₂×Q₁₆

yes

—

C₂²×C₁₂₇

#137

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#120

yes

—

C₄×C₁₂₇

#121

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#392

yes

characteristic

C₂×C₁₂₇

#451

yes

—

C₂×C₁₂₇

#462

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#2016

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

127

—

C₂×C₄⋊Q₈

D₁₂₇

yes

characteristic

C₂×#103

—

127

—

#391

—

127

—

#336

—

127

—

C₂²×Q₈

—

127

—

C₂²×Q₈

C₂×D₁₂₇

yes

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

—

254

—

C₂²×C₈

—

127

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#103

—

254

—

#95

—

254

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

127

—

104

C₂×Q₈

—

254

—

105

C₂×Q₈

—

254

—

106

C₂×Q₈

—

254

—

107

C₂×Q₈

C₂²⋊D₁₂₇

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

254

—

110

C₂²×C₄

—

254

—

111

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

112

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

113

Q₁₆

—

254

—

114

C₂×C₈

—

127

—

115

C₂×C₈

—

254

—

116

C₂×C₈

—

254

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

254

—

119

C₄²

—

120

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

508

—

126

Q₈

—

254

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

508

—

133

C₂×C₄

—

508

—

134

C₂×C₄

—

508

—

135

C₂×C₄

—

508

—

136

C₂×C₄

—

254

—

137

C₂×C₄

#14523238

yes

complemented

138

C₂×C₄

D₄×D₁₂₇

yes

characteristic

139

C₂×C₄

D₄×D₁₂₇

yes

—

140

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

141

C₂×C₄

#14523247

yes

characteristic

142

C₈

—

254

—

143

C₂²

#16861178

yes

characteristic

144

C₂²

#16861180

yes

characteristic

145

C₂²

#16861205

yes

characteristic

146

C₂²

C₂×#14523238

yes

—

147

C₂²

#16861240

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

1016

—

151

C₄

—

508

—

152

C₄

#16861177

yes

—

153

C₄

#16861181

yes

—

154

C₂

#22549784

yes

characteristic

155

C₂

#22550032

yes

—

156

C₂

#22550083

yes

—

157

C₂

#22550255

yes

characteristic

158

C₂

#22550268

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×C₄⋊Q₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	#2016
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×C₄⋊Q₈×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₂₇

hypocenter

C₁₂₇

#2016

LCS

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
58. C₂×C₄×C₁₂₇	Q₁₆	semidirect
80. C₁₂₇	#2016	semidirect
137. C₂×C₄	#14523238	semidirect

H ≅ G/N

type

58. C₂×C₄×C₁₂₇

Q₁₆

semidirect

80. C₁₂₇

#2016

semidirect

137. C₂×C₄

#14523238

semidirect

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

subgroup	index
C₂²×#14523238	2
C₂×#16861133	2
#22549782	2
#22549631	2
C₂×C₄⋊Q₈×C₁₂₇	2
#2016	127

subgroup

index

C₂×C₄⋊Q₈×C₁₂₇

127

Subgroups — #30639830

Conjugacy classes of subgroups

Complements

Maximal subgroups