Group #30638852 — subgroups

Subgroups — #30638852

Order 16256.

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

normal subgroups only

This group has 172 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×#16861144	C₂	1	1	yes	maximal, characteristic
2	#22549912	C₂	1	1	yes	maximal, characteristic
3	#22549832	C₂	1	1	yes	maximal, characteristic
4	#22549648	C₂	1	1	yes	maximal, characteristic
5	#22549646	C₂	1	1	yes	maximal, characteristic
6	#22549632	C₂	1	1	yes	maximal, characteristic
7	H(3,ℤ/4)×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523232	C₂²	1	1	yes	characteristic
9	C₂×#14523232	C₂²	1	1	yes	characteristic
10	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	#16861168	C₂²	1	1	yes	characteristic
12	#16861144	—	2	1	no	—
13	#16861144	—	2	1	no	—
14	#16861142	—	2	1	no	—
15	#16861140	—	2	1	no	—
16	#16861120	C₄	1	2	yes	complemented
17	C₄×C₁₂₇⋊C₈	C₄	1	2	yes	complemented
18	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
19	#117×C₁₂₇	C₂²	1	1	yes	characteristic
20	#117×C₁₂₇	C₂²	1	1	yes	characteristic
21	#14523232	—	2	1	no	—
22	#14523232	—	2	1	no	—
23	#14523232	—	2	1	no	—
24	#14523232	—	2	1	no	—
25	C₂×C₁₂₇⋊C₈	—	2	1	no	—
26	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
30	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
31	C₂×C₁₂₇⋊C₈	Q₈	1	1	yes	characteristic
32	C₂×C₁₂₇⋊C₈	Q₈	1	1	yes	characteristic
33	C₄⋊Dic₁₂₇	—	2	1	no	—
34	C₂²⋊Dic₁₂₇	—	2	1	no	—
35	C₄×Dic₁₂₇	—	2	1	no	—
36	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	M₁₆×C₁₂₇	—	2	1	no	—
38	M₁₆×C₁₂₇	—	2	1	no	—
39	C₂×C₈×C₁₂₇	—	2	1	no	—
40	C₄⋊C₄×C₁₂₇	—	2	1	no	—
41	C₄⋊C₄×C₁₂₇	—	2	1	no	—
42	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
43	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
44	C₄²×C₁₂₇	—	2	1	no	—
45	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
46	C₂×Dic₁₂₇	—	4	1	no	—
47	C₁₂₇⋊C₈	—	2	2	no	—
48	C₁₂₇⋊C₈	—	4	1	no	—
49	C₁₂₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₁₂₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₁₂₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
52	C₁₂₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₂³×C₁₂₇	#41	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	—	2	2	no	—
55	C₂×C₄×C₁₂₇	—	4	1	no	—
56	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
62	C₈×C₁₂₇	—	4	1	no	—
63	C₂²×C₁₂₇	—	4	1	no	—
64	C₂²×C₁₂₇	#117	1	1	yes	characteristic
65	C₂²×C₁₂₇	#124	1	1	yes	characteristic
66	C₂²×C₁₂₇	#125	1	1	yes	characteristic
67	C₄×C₁₂₇	—	4	2	no	—
68	C₄×C₁₂₇	—	8	1	no	—
69	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₄×C₁₂₇	#117	1	1	yes	characteristic
71	C₄×C₁₂₇	#124	1	1	yes	characteristic
72	C₄×C₁₂₇	#125	1	1	yes	characteristic
73	Dic₁₂₇	—	8	1	no	—
74	C₂×C₁₂₇	—	2	1	no	—
75	C₂×C₁₂₇	—	2	1	no	—
76	C₂×C₁₂₇	—	2	1	no	—
77	C₂×C₁₂₇	#382	1	1	yes	characteristic
78	#1833	—	127	1	no	maximal, Sylow
79	C₁₂₇	#1833	1	1	yes	characteristic, complemented, Sylow
80	C₂×#108	—	127	1	no	—
81	#427	—	127	1	no	—
82	#405	—	127	1	no	—
83	#344	—	127	1	no	—
84	#344	—	127	1	no	—
85	H(3,ℤ/4)	—	127	1	no	—
86	H(3,ℤ/4)	D₁₂₇	1	1	yes	characteristic
87	C₂×M₁₆	—	127	1	no	—
88	C₂×M₁₆	—	127	1	no	—
89	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
90	C₂²×C₈	—	127	1	no	—
91	#117	—	127	1	no	—
92	#117	C₂×D₁₂₇	1	1	yes	characteristic
93	#117	C₂×D₁₂₇	1	1	yes	characteristic
94	#108	—	254	1	no	—
95	#108	—	254	1	no	—
96	#107	—	254	1	no	—
97	#106	—	254	1	no	—
98	#97	—	127	2	no	—
99	C₄×C₈	—	127	2	no	—
100	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
101	M₁₆	—	2	1	no	—
102	M₁₆	—	2	1	no	—
103	M₁₆	—	254	1	no	—
104	M₁₆	—	254	1	no	—
105	M₁₆	—	254	1	no	—
106	M₁₆	—	254	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	127	1	no	—
109	C₂×C₈	—	127	1	no	—
110	C₂×C₈	—	127	1	no	—
111	C₂×C₈	—	127	1	no	—
112	C₂×C₈	—	127	1	no	—
113	C₂×C₈	—	127	1	no	—
114	C₂×C₈	—	127	2	no	—
115	C₂×C₈	—	254	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	254	1	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	2	1	no	—
121	C₂²⋊C₄	—	254	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	254	1	no	—
124	C₄²	C₄×D₁₂₇	1	2	yes	—
125	C₂³	#14523247	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	508	1	no	—
129	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
130	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
131	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
132	C₂×C₄	#14523247	1	1	yes	characteristic
133	C₂×C₄	#14523248	1	1	yes	characteristic
134	C₂×C₄	#14523248	1	1	yes	characteristic
135	C₈	—	4	1	no	—
136	C₈	—	127	1	no	—
137	C₈	—	127	1	no	—
138	C₈	—	127	1	no	—
139	C₈	—	127	1	no	—
140	C₈	—	508	1	no	—
141	C₈	—	254	2	no	—
142	C₂²	—	4	1	no	—
143	C₂²	#16861165	1	1	yes	characteristic
144	C₂²	#16861195	1	1	yes	characteristic
145	C₂²	#16861199	1	1	yes	characteristic
146	C₄	—	4	2	no	—
147	C₄	—	8	1	no	—
148	C₄	—	1016	1	no	—
149	C₄	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
150	C₄	#16861166	1	1	yes	characteristic
151	C₄	#16861195	1	1	yes	characteristic
152	C₄	#16861199	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#22549740	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861144

C₂

yes

maximal, characteristic

#22549912

C₂

yes

maximal, characteristic

#22549832

C₂

yes

maximal, characteristic

#22549648

C₂

yes

maximal, characteristic

#22549646

C₂

yes

maximal, characteristic

#22549632

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

characteristic

C₂×#14523232

C₂²

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

#16861168

C₂²

yes

—

—

—

—

C₄

yes

complemented

C₄×C₁₂₇⋊C₈

C₄

yes

complemented

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

C₂²

yes

—

—

—

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

D₄

yes

characteristic

C₂×C₁₂₇⋊C₈

D₄

yes

characteristic

C₂×C₁₂₇⋊C₈

Q₈

yes

characteristic

C₂×C₁₂₇⋊C₈

Q₈

yes

characteristic

C₄⋊Dic₁₂₇

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₂²⋊C₄×C₁₂₇

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

C₄⋊C₄

yes

characteristic

C₁₂₇⋊C₈

C₄⋊C₄

yes

characteristic

C₁₂₇⋊C₈

C₄⋊C₄

yes

characteristic

C₁₂₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂³×C₁₂₇

#41

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

C₂×C₄×C₁₂₇

yes

C₂×C₄×C₁₂₇

yes

C₂×C₄×C₁₂₇

yes

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#382

yes

characteristic

#1833

—

127

maximal, Sylow

C₁₂₇

#1833

yes

characteristic, complemented, Sylow

C₂×#108

—

127

—

#427

—

127

—

#405

—

127

—

#344

—

127

—

#344

—

127

—

H(3,ℤ/4)

—

127

—

H(3,ℤ/4)

D₁₂₇

yes

characteristic

C₂×M₁₆

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

—

127

—

#117

—

127

—

#117

C₂×D₁₂₇

yes

characteristic

#117

C₂×D₁₂₇

yes

characteristic

#108

—

254

—

#108

—

254

—

#107

—

254

—

#106

—

254

—

#97

—

127

—

C₄×C₈

—

127

—

100

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

254

—

104

M₁₆

—

254

—

105

M₁₆

—

254

—

106

M₁₆

—

254

—

107

C₂×C₈

—

108

C₂×C₈

—

127

—

109

C₂×C₈

—

127

—

110

C₂×C₈

—

127

—

111

C₂×C₈

—

127

—

112

C₂×C₈

—

127

—

113

C₂×C₈

—

127

—

114

C₂×C₈

—

127

—

115

C₂×C₈

—

254

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

254

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

254

—

122

C₄²

—

123

C₄²

—

254

—

124

C₄²

C₄×D₁₂₇

yes

—

125

C₂³

#14523247

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

508

—

129

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

130

C₂×C₄

D₄×D₁₂₇

yes

characteristic

131

C₂×C₄

Q₈×D₁₂₇

yes

characteristic

132

C₂×C₄

#14523247

yes

characteristic

133

C₂×C₄

#14523248

yes

characteristic

134

C₂×C₄

#14523248

yes

characteristic

135

C₈

—

136

C₈

—

127

—

137

C₈

—

127

—

138

C₈

—

127

—

139

C₈

—

127

—

140

C₈

—

508

—

141

C₈

—

254

—

142

C₂²

—

143

C₂²

#16861165

yes

characteristic

144

C₂²

#16861195

yes

characteristic

145

C₂²

#16861199

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

1016

—

149

C₄

C₄⋊C₄×D₁₂₇

yes

characteristic

150

C₄

#16861166

yes

characteristic

151

C₄

#16861195

yes

characteristic

152

C₄

#16861199

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

#22549740

yes

characteristic

	subgroup	G/H
center	C₄	#16861166
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	H(3,ℤ/4)×C₁₂₇	C₂
socle	C₂×C₁₂₇	#382
prime radical	C₂³×C₁₂₇	#41
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	H(3,ℤ/4)	D₁₂₇
hypocenter	C₁₂₇	#1833
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ #117 ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#16861166

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

H(3,ℤ/4)×C₁₂₇

C₂

C₂×C₁₂₇

C₂³×C₁₂₇

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₄ ◁ #117 ◁ H(3,ℤ/4)

N	H ≅ G/N	type
16. #16861120	C₄	semidirect
17. C₄×C₁₂₇⋊C₈	C₄	semidirect
79. C₁₂₇	#1833	semidirect

H ≅ G/N

type

16. #16861120

C₄

semidirect

17. C₄×C₁₂₇⋊C₈

C₄

semidirect

79. C₁₂₇

#1833

semidirect

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

subgroup	index
C₂×#16861144	2
#22549912	2
#22549832	2
#22549648	2
#22549646	2
#22549632	2
H(3,ℤ/4)×C₁₂₇	2
#1833	127

subgroup

index

H(3,ℤ/4)×C₁₂₇

127

Subgroups — #30638852

Conjugacy classes of subgroups

Complements

Maximal subgroups