Group #30647372 — subgroups

Subgroups — #30647372

Order 16256.

400 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₄×#14523248	C₂	1	1	yes	maximal, characteristic
2	#22549935	C₂	1	4	yes	maximal, complemented
3	#22549927	C₂	1	4	yes	maximal, complemented
4	#22549902	C₂	1	2	yes	maximal, complemented
5	#22549842	C₂	1	2	yes	maximal
6	C₄×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14523248	C₂²	1	1	yes	characteristic
9	C₂×#14523232	C₂²	1	2	yes	—
10	C₄×Dic₂₅₄	C₄	1	2	yes	—
11	C₄×D₅₀₈	C₄	1	2	yes	—
12	C₄×C₂²⋊D₁₂₇	C₄	1	4	yes	—
13	#16861168	C₂²	1	1	yes	characteristic
14	#16861166	C₂²	1	2	yes	—
15	C₄²×D₁₂₇	C₂²	1	2	yes	—
16	Dic₁₂₇⋊C₈	C₂²	1	8	yes	—
17	(C₂×D₁₂₇)⋊C₈	C₂²	1	8	yes	—
18	#16861120	C₂²	1	2	yes	—
19	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
20	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
23	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
24	#14523248	—	2	4	no	—
25	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
26	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₁₂₇	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
30	#14523232	—	2	4	no	—
31	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
32	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
33	Dic₁₂₇⋊C₄	C₂×C₄	1	4	yes	—
34	C₂²⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
35	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	4	yes	—
36	C₄×Dic₁₂₇	C₂×C₄	1	4	yes	—
37	C₄×Dic₁₂₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₁₂₇	—	2	4	no	—
41	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
42	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
43	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
46	Dic₂₅₄	—	2	2	no	—
47	D₅₀₈	—	2	2	no	—
48	C₂²⋊D₁₂₇	—	2	4	no	—
49	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
50	C₂×Dic₁₂₇	C₂²×C₄	1	4	yes	—
51	C₄×D₁₂₇	—	2	8	no	—
52	C₄×D₁₂₇	—	2	4	no	—
53	C₁₂₇⋊C₈	—	2	4	no	—
54	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₂₇	—	2	4	no	—
56	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₁₂₇	#41	1	4	yes	—
60	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
61	C₈×C₁₂₇	—	2	4	no	—
62	C₂²×C₁₂₇	—	2	2	no	—
63	C₂²×C₁₂₇	#117	1	2	yes	—
64	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₁₂₇	—	2	2	no	—
66	C₂×D₁₂₇	—	2	4	no	—
67	C₄×C₁₂₇	—	2	2	no	—
68	C₄×C₁₂₇	#117	1	2	yes	—
69	C₄×C₁₂₇	#131	1	4	yes	—
70	C₄×C₁₂₇	C₂×#41	1	2	yes	—
71	Dic₁₂₇	—	2	2	no	—
72	Dic₁₂₇	—	2	4	no	—
73	C₂×C₁₂₇	—	2	2	no	—
74	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
75	C₂×C₁₂₇	C₂×#131	1	2	yes	—
76	D₁₂₇	—	4	2	no	—
77	#2922	—	127	1	no	maximal, Sylow
78	C₁₂₇	#2922	1	1	yes	characteristic, complemented, Sylow
79	C₄×#41	—	127	1	no	—
80	#433	—	127	4	no	—
81	#431	—	127	4	no	—
82	#424	—	127	2	no	—
83	#408	—	127	2	no	—
84	C₄×M₁₆	—	127	1	no	—
85	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic, complemented
86	C₂×#41	—	127	1	no	—
87	C₂×M₁₆	—	127	2	no	—
88	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
89	C₄×Q₈	—	127	2	no	—
90	C₄×D₄	—	127	4	no	—
91	C₄×D₄	—	127	2	no	—
92	#117	—	127	1	no	—
93	#117	—	127	2	no	—
94	C₂×C₄²	—	127	2	no	—
95	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
96	C₄⋊C₈	—	127	8	no	—
97	C₂²⋊C₈	—	127	8	no	—
98	#97	—	127	2	no	—
99	C₄×C₈	—	127	2	no	—
100	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
101	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
102	#41	—	254	4	no	—
103	C₂×Q₈	—	127	1	no	—
104	C₂×D₄	—	127	2	no	—
105	C₂×D₄	—	127	1	no	—
106	C₂²×C₄	—	127	4	no	—
107	C₂²×C₄	—	127	2	no	—
108	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
110	M₁₆	—	254	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	127	4	no	—
113	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
114	C₄⋊C₄	—	127	4	no	—
115	C₄⋊C₄	—	127	2	no	—
116	C₂²⋊C₄	—	127	2	no	—
117	C₂²⋊C₄	—	127	4	no	—
118	C₄²	—	127	4	no	—
119	C₄²	—	127	2	no	—
120	C₄²	C₄×D₁₂₇	1	2	yes	—
121	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
122	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
123	C₂³	—	127	2	no	—
124	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
125	Q₈	—	254	2	no	—
126	D₄	—	254	4	no	—
127	D₄	—	254	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	127	4	no	—
130	C₂×C₄	—	127	2	no	—
131	C₂×C₄	—	254	8	no	—
132	C₂×C₄	—	254	4	no	—
133	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
135	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
136	C₂×C₄	#14523248	1	4	yes	—
137	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	254	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	254	2	no	—
142	C₂²	—	254	4	no	—
143	C₂²	#16861166	1	2	yes	—
144	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	254	2	no	—
147	C₄	—	254	4	no	—
148	C₄	#16861166	1	2	yes	—
149	C₄	#16861215	1	4	yes	—
150	C₄	C₂×#14523248	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	508	2	no	—
153	C₂	C₂×#16861166	1	1	yes	characteristic
154	C₂	C₂×#16861215	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14523248

C₂

yes

maximal, characteristic

#22549935

C₂

yes

maximal, complemented

#22549927

C₂

yes

maximal, complemented

#22549902

C₂

yes

maximal, complemented

#22549842

C₂

yes

maximal

C₄×#14523232

C₂

yes

maximal, characteristic, complemented

C₂×C₄×C₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523248

C₂²

yes

characteristic

C₂×#14523232

C₂²

yes

—

C₄×Dic₂₅₄

C₄

yes

—

C₄×D₅₀₈

C₄

yes

—

C₄×C₂²⋊D₁₂₇

C₄

yes

—

#16861168

C₂²

yes

characteristic

#16861166

C₂²

yes

—

C₄²×D₁₂₇

C₂²

yes

—

Dic₁₂₇⋊C₈

C₂²

yes

—

(C₂×D₁₂₇)⋊C₈

C₂²

yes

—

#16861120

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

—

#14523248

—

C₂×Dic₂₅₄

C₂×C₄

yes

characteristic

C₂×D₅₀₈

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₁₂₇

C₂×C₄

yes

—

C₂×C₄×D₁₂₇

C₂×C₄

yes

—

C₂×C₄×D₁₂₇

C₂³

yes

—

#14523232

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

Dic₁₂₇⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₁₂₇

C₂×C₄

yes

—

(C₂×D₁₂₇)⋊C₄

C₂×C₄

yes

—

C₄×Dic₁₂₇

C₂×C₄

yes

—

C₄×Dic₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

characteristic

C₂²×D₁₂₇

C₂²×C₄

yes

—

Dic₂₅₄

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

C₂²×C₄

yes

—

C₂×Dic₁₂₇

C₂²×C₄

yes

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#117

yes

—

C₂²×C₁₂₇

C₂³×C₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#117

yes

—

C₄×C₁₂₇

#131

yes

—

C₄×C₁₂₇

C₂×#41

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×#117

yes

characteristic

C₂×C₁₂₇

C₂×#131

yes

—

D₁₂₇

—

#2922

—

127

maximal, Sylow

C₁₂₇

#2922

yes

characteristic, complemented, Sylow

C₄×#41

—

127

—

#433

—

127

—

#431

—

127

—

#424

—

127

—

#408

—

127

—

C₄×M₁₆

—

127

—

C₂×C₄×C₈

D₁₂₇

yes

characteristic, complemented

C₂×#41

—

127

—

C₂×M₁₆

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

—

C₄×Q₈

—

127

—

C₄×D₄

—

127

—

C₄×D₄

—

127

—

#117

—

127

—

#117

—

127

—

C₂×C₄²

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

127

—

C₂²⋊C₈

—

127

—

#97

—

127

—

C₄×C₈

—

127

—

100

C₄×C₈

C₂×D₁₂₇

yes

—

101

C₄×C₈

C₂×D₁₂₇

yes

—

102

#41

—

254

—

103

C₂×Q₈

—

127

—

104

C₂×D₄

—

127

—

105

C₂×D₄

—

127

—

106

C₂²×C₄

—

127

—

107

C₂²×C₄

—

127

—

108

C₂²×C₄

C₄×D₁₂₇

yes

—

109

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

110

M₁₆

—

254

—

111

C₂×C₈

—

112

C₂×C₈

—

127

—

113

C₂×C₈

C₂²×D₁₂₇

yes

—

114

C₄⋊C₄

—

127

—

115

C₄⋊C₄

—

127

—

116

C₂²⋊C₄

—

127

—

117

C₂²⋊C₄

—

127

—

118

C₄²

—

127

—

119

C₄²

—

127

—

120

C₄²

C₄×D₁₂₇

yes

—

121

C₄²

C₂²×D₁₂₇

yes

characteristic

122

C₄²

C₂²×D₁₂₇

yes

characteristic

123

C₂³

—

127

—

124

C₂³

C₂×C₄×D₁₂₇

yes

characteristic

125

Q₈

—

254

—

126

D₄

—

254

—

127

D₄

—

254

—

128

C₂×C₄

—

129

C₂×C₄

—

127

—

130

C₂×C₄

—

127

—

131

C₂×C₄

—

254

—

132

C₂×C₄

—

254

—

133

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

134

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

135

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

136

C₂×C₄

#14523248

yes

—

137

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

138

C₈

—

139

C₈

—

254

—

140

C₂²

—

141

C₂²

—

254

—

142

C₂²

—

254

—

143

C₂²

#16861166

yes

—

144

C₂²

C₂²×C₄×D₁₂₇

yes

characteristic

145

C₄

—

146

C₄

—

254

—

147

C₄

—

254

—

148

C₄

#16861166

yes

—

149

C₄

#16861215

yes

—

150

C₄

C₂×#14523248

yes

—

151

C₂

—

152

C₂

—

508

—

153

C₂

C₂×#16861166

yes

characteristic

154

C₂

C₂×#16861215

yes

—

	subgroup	G/H
center	C₄²	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
prime radical	C₂×#14523248	C₂²
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#2922
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄²

C₂²×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

C₂×C₄×C₈×C₁₂₇

C₂

prime radical

C₂×#14523248

C₂²

cosocle

C₂×C₄×C₁₂₇

C₂⁴

hypercenter

C₂×C₄×C₈

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #22549935	C₂	semidirect
3. #22549927	C₂	semidirect
4. #22549902	C₂	semidirect
6. C₄×#14523232	C₂	semidirect
7. C₂×C₄×C₈×C₁₂₇	C₂	semidirect
78. C₁₂₇	#2922	semidirect
85. C₂×C₄×C₈	D₁₂₇	semidirect

H ≅ G/N

type

2. #22549935

C₂

semidirect

3. #22549927

C₂

semidirect

4. #22549902

C₂

semidirect

6. C₄×#14523232

C₂

semidirect

7. C₂×C₄×C₈×C₁₂₇

C₂

semidirect

78. C₁₂₇

#2922

semidirect

85. C₂×C₄×C₈

D₁₂₇

semidirect

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

subgroup	index
C₄×#14523248	2
#22549935	2
#22549927	2
#22549902	2
#22549842	2
C₄×#14523232	2
C₂×C₄×C₈×C₁₂₇	2
#2922	127

subgroup

index

C₂×C₄×C₈×C₁₂₇

127

Subgroups — #30647372

Conjugacy classes of subgroups

Complements

Maximal subgroups