Group #30638372 — subgroups

Subgroups — #30638372

Order 16256.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22550253	C₂	1	1	yes	maximal, characteristic
2	#22550250	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550224	C₂	1	1	yes	maximal, characteristic, complemented
4	#22550224	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550224	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549598	C₂	1	1	yes	maximal, characteristic, complemented
7	#487×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14523235	—	2	2	no	—
9	C₂×D₁₀₁₆	—	2	2	no	—
10	#16861206	C₂²	1	1	yes	characteristic
11	#16861206	C₂²	1	1	yes	characteristic
12	#16861129	—	2	2	no	—
13	#16861129	—	2	2	no	—
14	#16861129	—	2	2	no	—
15	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	#103×C₁₂₇	—	2	2	no	—
20	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×D₅₀₈	—	2	1	no	—
22	C₂×D₅₀₈	—	2	1	no	—
23	C₂×D₅₀₈	—	2	1	no	—
24	C₂×D₅₀₈	—	2	1	no	—
25	C₂×D₅₀₈	—	2	1	no	—
26	C₂×D₅₀₈	—	2	1	no	—
27	#14523235	—	4	4	no	—
28	D₁₀₁₆	—	4	4	no	—
29	C₂×C₁₂₇⋊C₈	D₄	1	2	yes	—
30	C₂×C₁₂₇⋊C₈	D₄	1	2	yes	—
31	C₂×Q₈×C₁₂₇	—	2	1	no	—
32	C₂×Q₈×C₁₂₇	—	2	1	no	—
33	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
34	C₄⋊C₄×C₁₂₇	—	2	1	no	—
35	C₄⋊C₄×C₁₂₇	—	2	1	no	—
36	C₄⋊C₄×C₁₂₇	—	2	1	no	—
37	C₄⋊C₄×C₁₂₇	—	2	1	no	—
38	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂²×D₁₂₇	—	4	1	no	—
40	C₂²×D₁₂₇	—	4	1	no	—
41	D₅₀₈	—	4	2	no	—
42	D₅₀₈	—	4	2	no	—
43	D₅₀₈	—	4	2	no	—
44	D₅₀₈	—	4	2	no	—
45	D₅₀₈	—	4	2	no	—
46	D₅₀₈	—	4	2	no	—
47	C₁₂₇⋊C₈	—	2	2	no	—
48	C₁₂₇⋊C₈	—	2	2	no	—
49	Q₈×C₁₂₇	—	4	2	no	—
50	Q₈×C₁₂₇	—	4	2	no	—
51	C₂×C₄×C₁₂₇	—	4	1	no	—
52	C₂×C₄×C₁₂₇	—	4	1	no	—
53	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
56	C₈×C₁₂₇	—	2	2	no	—
57	C₂²×C₁₂₇	#127	1	1	yes	characteristic
58	C₂×D₁₂₇	—	8	1	no	—
59	C₂×D₁₂₇	—	8	1	no	—
60	C₂×D₁₂₇	—	8	1	no	—
61	C₂×D₁₂₇	—	8	1	no	—
62	C₂×D₁₂₇	—	8	1	no	—
63	C₂×D₁₂₇	—	8	1	no	—
64	C₄×C₁₂₇	—	8	1	no	—
65	C₄×C₁₂₇	—	8	1	no	—
66	C₄×C₁₂₇	#135	1	2	yes	—
67	C₄×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
68	C₄×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×C₁₂₇	#495	1	1	yes	characteristic
70	C₂×C₁₂₇	#496	1	1	yes	characteristic
71	C₂×C₁₂₇	#496	1	1	yes	characteristic
72	D₁₂₇	—	16	1	no	—
73	D₁₂₇	—	16	1	no	—
74	#1717	—	127	1	no	maximal, Sylow
75	C₁₂₇	#1717	1	1	yes	characteristic, complemented, Sylow
76	#493	—	127	1	no	—
77	#492	—	127	1	no	—
78	#487	D₁₂₇	1	1	yes	characteristic, complemented
79	#486	—	127	1	no	—
80	#486	—	127	1	no	—
81	#486	—	127	1	no	—
82	#322	—	127	1	no	—
83	C₂×SD₁₆	—	254	2	no	—
84	C₂×D₈	—	254	2	no	—
85	C₄⋊Q₈	C₂×D₁₂₇	1	1	yes	characteristic
86	C₄⋊Q₈	C₂×D₁₂₇	1	1	yes	characteristic
87	#127	—	127	1	no	—
88	#127	—	127	1	no	—
89	#103	—	2	2	no	—
90	#102	—	254	2	no	—
91	#102	—	254	2	no	—
92	#102	—	254	2	no	—
93	C₄×C₈	—	127	1	no	—
94	C₄×C₈	—	127	1	no	—
95	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	2	1	no	—
98	C₂×D₄	—	254	1	no	—
99	C₂×D₄	—	254	1	no	—
100	C₂×D₄	—	254	1	no	—
101	C₂×D₄	—	254	1	no	—
102	C₂×D₄	—	254	1	no	—
103	C₂×D₄	—	254	1	no	—
104	SD₁₆	—	508	4	no	—
105	D₈	—	508	4	no	—
106	C₂×C₈	—	127	2	no	—
107	C₂×C₈	—	127	2	no	—
108	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
114	C₂³	—	508	1	no	—
115	C₂³	—	508	1	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	4	2	no	—
118	D₄	—	508	2	no	—
119	D₄	—	508	2	no	—
120	D₄	—	508	2	no	—
121	D₄	—	508	2	no	—
122	D₄	—	508	2	no	—
123	D₄	—	508	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
127	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	254	2	no	—
131	C₈	—	254	2	no	—
132	C₂²	—	1016	1	no	—
133	C₂²	—	1016	1	no	—
134	C₂²	—	1016	1	no	—
135	C₂²	—	1016	1	no	—
136	C₂²	—	1016	1	no	—
137	C₂²	—	1016	1	no	—
138	C₂²	#16861205	1	1	yes	characteristic
139	C₄	—	8	1	no	—
140	C₄	—	8	1	no	—
141	C₄	#16861232	1	2	yes	—
142	C₄	#16861234	1	2	yes	—
143	C₄	#16861236	1	2	yes	—
144	C₂	—	2032	1	no	—
145	C₂	—	2032	1	no	—
146	C₂	#22550261	1	1	yes	characteristic
147	C₂	#22550262	1	1	yes	characteristic
148	C₂	#22550266	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550253

C₂

yes

maximal, characteristic

#22550250

C₂

yes

maximal, characteristic, complemented

#22550224

C₂

yes

maximal, characteristic, complemented

#22550224

C₂

yes

maximal, characteristic, complemented

#22550224

C₂

yes

maximal, characteristic, complemented

#22549598

C₂

yes

maximal, characteristic, complemented

#487×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523235

—

C₂×D₁₀₁₆

—

#16861206

C₂²

yes

characteristic

#16861206

C₂²

yes

—

—

—

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊Q₈×C₁₂₇

C₂²

yes

characteristic

#103×C₁₂₇

—

C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

#14523235

—

D₁₀₁₆

—

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

#127

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#135

yes

—

C₄×C₁₂₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₂₇

AGL(1,ℤ/8)

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

D₁₂₇

—

D₁₂₇

—

#1717

—

127

maximal, Sylow

C₁₂₇

#1717

yes

characteristic, complemented, Sylow

#493

—

127

—

#492

—

127

—

#487

D₁₂₇

yes

characteristic, complemented

#486

—

127

—

#486

—

127

—

#486

—

127

—

#322

—

127

—

C₂×SD₁₆

—

254

—

C₂×D₈

—

254

—

C₄⋊Q₈

C₂×D₁₂₇

yes

characteristic

C₄⋊Q₈

C₂×D₁₂₇

yes

characteristic

#127

—

127

—

#127

—

127

—

#103

—

#102

—

254

—

#102

—

254

—

#102

—

254

—

C₄×C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

100

C₂×D₄

—

254

—

101

C₂×D₄

—

254

—

102

C₂×D₄

—

254

—

103

C₂×D₄

—

254

—

104

SD₁₆

—

508

—

105

D₈

—

508

—

106

C₂×C₈

—

127

—

107

C₂×C₈

—

127

—

108

C₂×C₈

C₂²⋊D₁₂₇

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

C₂²×D₁₂₇

yes

characteristic

114

C₂³

—

508

—

115

C₂³

—

508

—

116

Q₈

—

117

Q₈

—

118

D₄

—

508

—

119

D₄

—

508

—

120

D₄

—

508

—

121

D₄

—

508

—

122

D₄

—

508

—

123

D₄

—

508

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

D₄×D₁₂₇

yes

characteristic

127

C₂×C₄

D₄×D₁₂₇

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

129

C₈

—

130

C₈

—

254

—

131

C₈

—

254

—

132

C₂²

—

1016

—

133

C₂²

—

1016

—

134

C₂²

—

1016

—

135

C₂²

—

1016

—

136

C₂²

—

1016

—

137

C₂²

—

1016

—

138

C₂²

#16861205

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

#16861232

yes

—

142

C₄

#16861234

yes

—

143

C₄

#16861236

yes

—

144

C₂

—

2032

—

145

C₂

—

2032

—

146

C₂

#22550261

yes

characteristic

147

C₂

#22550262

yes

characteristic

148

C₂

#22550266

yes

characteristic

	subgroup	G/H
center	C₂²	#16861205
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#487×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#127
prime radical	#22550253	C₂
hypercenter	#487	D₁₂₇
hypocenter	C₁₂₇	#1717
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#16861205

commutator, cosocle

C₄²×C₁₂₇

C₂³

Frattini

C₄²

C₂²×D₁₂₇

Fitting

#487×C₁₂₇

C₂

C₂²×C₁₂₇

C₂

D₁₂₇

C₁₂₇

G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
2. #22550250	C₂	semidirect
3. #22550224	C₂	semidirect
4. #22550224	C₂	semidirect
5. #22550224	C₂	semidirect
6. #22549598	C₂	semidirect
7. #487×C₁₂₇	C₂	semidirect
75. C₁₂₇	#1717	semidirect
78. #487	D₁₂₇	semidirect

H ≅ G/N

type

2. #22550250

C₂

semidirect

3. #22550224

C₂

semidirect

4. #22550224

C₂

semidirect

5. #22550224

C₂

semidirect

6. #22549598

C₂

semidirect

7. #487×C₁₂₇

C₂

semidirect

75. C₁₂₇

#1717

semidirect

78. #487

D₁₂₇

semidirect

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

subgroup	index
#22550253	2
#22550250	2
#22550224	2
#22549598	2
#487×C₁₂₇	2
#1717	127

subgroup

index

#487×C₁₂₇

127

Subgroups — #30638372

Conjugacy classes of subgroups

Complements

Maximal subgroups