Group #30647861 — subgroups

Subgroups — #30647861

Order 16256.

394 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 179 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×#14523249	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549955	C₂	1	3	yes	maximal, complemented
4	#22549953	C₂	1	3	yes	maximal, complemented
5	#22549927	C₂	1	3	yes	maximal, complemented
6	#22549898	C₂	1	3	yes	maximal, complemented
7	C₈×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14523249	C₄	1	1	yes	characteristic
9	C₂×#14523231	C₂²	1	6	yes	—
10	Q₈×Dic₁₂₇	C₄	1	1	yes	characteristic
11	C₄×D₅₀₈	C₂²	1	3	yes	—
12	#16861171	C₄	1	3	yes	—
13	#16861167	C₄	1	3	yes	—
14	C₄²×D₁₂₇	C₂²	1	3	yes	—
15	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	3	yes	—
16	Dic₁₂₇⋊C₈	C₂²	1	3	yes	—
17	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
18	(C₂×D₁₂₇)⋊C₈	C₂²	1	3	yes	—
19	(C₂×D₁₂₇)⋊C₈	C₂²	1	3	yes	—
20	C₄×C₁₂₇⋊C₈	C₂²	1	3	yes	—
21	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₂₇	C₂²	1	3	yes	—
23	C₄×C₈×C₁₂₇	C₂²	1	3	yes	—
24	#14523249	—	2	4	no	—
25	C₂×D₅₀₈	C₂×C₄	1	3	yes	—
26	C₂×C₄×D₁₂₇	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₁₂₇	C₂×C₄	1	3	yes	—
28	C₂×C₄×D₁₂₇	C₂³	1	3	yes	—
29	#14523231	—	4	6	no	—
30	C₂×C₁₂₇⋊C₈	C₂³	1	3	yes	—
31	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₁₂₇	C₂×C₄	1	3	yes	—
33	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	6	yes	—
34	C₄×Dic₁₂₇	C₂×C₄	1	3	yes	—
35	C₄×Dic₁₂₇	C₂×C₄	1	3	yes	—
36	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₂₇	C₂³	1	3	yes	—
39	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₁₂₇	C₂×C₄	1	3	yes	—
41	C₄²×C₁₂₇	C₂³	1	3	yes	—
42	C₂²×D₁₂₇	C₂²×C₄	1	3	yes	—
43	D₅₀₈	—	2	6	no	—
44	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₁₂₇	C₂²×C₄	1	3	yes	—
46	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₄×D₁₂₇	—	2	6	no	—
48	C₄×D₁₂₇	—	2	3	no	—
49	C₄×D₁₂₇	—	2	3	no	—
50	C₄×D₁₂₇	—	2	6	no	—
51	C₁₂₇⋊C₈	—	2	3	no	—
52	C₁₂₇⋊C₈	—	2	1	no	—
53	Q₈×C₁₂₇	M₁₆	1	4	yes	complemented
54	C₂×C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
55	C₂×C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
56	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
57	C₈×C₁₂₇	—	2	3	no	—
58	C₈×C₁₂₇	—	2	1	no	—
59	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
60	C₂×D₁₂₇	—	2	3	no	—
61	C₂×D₁₂₇	—	2	3	no	—
62	C₂×D₁₂₇	—	2	3	no	—
63	C₄×C₁₂₇	—	2	3	no	—
64	C₄×C₁₂₇	C₂×M₁₆	1	6	yes	—
65	C₄×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
66	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
67	Dic₁₂₇	—	2	3	no	—
68	Dic₁₂₇	—	2	1	no	—
69	Dic₁₂₇	#131	1	2	yes	—
70	C₂×C₁₂₇	#520	1	1	yes	characteristic
71	C₂×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
72	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
73	D₁₂₇	—	4	3	no	—
74	#2991	—	127	1	no	maximal, Sylow
75	C₁₂₇	#2991	1	1	yes	characteristic, complemented, Sylow
76	C₄×#41	—	127	1	no	—
77	C₈×Q₈	—	127	1	no	—
78	C₈×Q₈	D₁₂₇	1	1	yes	characteristic, complemented
79	#436	—	127	3	no	—
80	#436	—	127	3	no	—
81	#431	—	127	3	no	—
82	#423	—	127	3	no	—
83	C₂×#41	—	127	1	no	—
84	C₂×M₁₆	—	127	6	no	—
85	C₄×Q₈	—	127	1	no	—
86	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
87	C₄×D₄	—	127	3	no	—
88	C₄×D₄	—	127	3	no	—
89	#117	—	127	3	no	—
90	C₂×C₄²	—	127	3	no	—
91	C₄⋊C₈	—	127	3	no	—
92	C₄⋊C₈	—	127	1	no	—
93	C₄⋊C₈	—	127	3	no	—
94	C₄⋊C₈	C₂×D₁₂₇	1	3	yes	—
95	C₂²⋊C₈	—	127	3	no	—
96	C₂²⋊C₈	—	127	3	no	—
97	C₄×C₈	—	127	3	no	—
98	C₄×C₈	C₂×D₁₂₇	1	3	yes	—
99	#41	—	254	4	no	—
100	C₂×Q₈	C₄×D₁₂₇	1	1	yes	characteristic
101	C₂×D₄	—	127	3	no	—
102	C₂²×C₄	—	127	3	no	—
103	C₂²×C₄	—	127	3	no	—
104	C₂²×C₄	—	127	3	no	—
105	M₁₆	—	508	6	no	—
106	C₂×C₈	—	127	1	no	—
107	C₂×C₈	—	127	3	no	—
108	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₁₂₇	1	3	yes	—
110	C₄⋊C₄	—	127	3	no	—
111	C₄⋊C₄	C₄×D₁₂₇	1	3	yes	—
112	C₂²⋊C₄	—	127	6	no	—
113	C₄²	—	127	3	no	—
114	C₄²	—	127	3	no	—
115	C₄²	—	127	1	no	—
116	C₄²	C₂²×D₁₂₇	1	3	yes	—
117	C₂³	—	127	3	no	—
118	Q₈	#14523231	1	4	yes	complemented
119	D₄	—	254	6	no	—
120	C₂×C₄	—	127	1	no	—
121	C₂×C₄	—	127	1	no	—
122	C₂×C₄	—	127	3	no	—
123	C₂×C₄	—	254	6	no	—
124	C₂×C₄	—	254	6	no	—
125	C₂×C₄	—	254	3	no	—
126	C₂×C₄	—	254	3	no	—
127	C₂×C₄	C₂×C₄×D₁₂₇	1	3	yes	—
128	C₂×C₄	C₂×C₄×D₁₂₇	1	3	yes	—
129	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
130	C₈	—	2	3	no	—
131	C₈	—	2	1	no	—
132	C₈	—	254	3	no	—
133	C₈	—	254	1	no	—
134	C₂²	—	254	3	no	—
135	C₂²	—	254	3	no	—
136	C₂²	—	254	3	no	—
137	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	127	2	no	—
140	C₄	—	254	3	no	—
141	C₄	—	254	1	no	—
142	C₄	C₂×#14523231	1	6	yes	—
143	C₄	#16861255	1	1	yes	characteristic
144	C₄	#16861257	1	1	yes	characteristic
145	C₂	—	508	3	no	—
146	C₂	#22550358	1	1	yes	characteristic
147	C₂	C₂²×#14523231	1	1	yes	characteristic
148	C₂	#131×D₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14523249

C₂

yes

maximal, characteristic

Q₈×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic, complemented

#22549955

C₂

yes

maximal, complemented

#22549953

C₂

yes

maximal, complemented

#22549927

C₂

yes

maximal, complemented

#22549898

C₂

yes

maximal, complemented

C₈×Q₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523249

C₄

yes

characteristic

C₂×#14523231

C₂²

yes

—

Q₈×Dic₁₂₇

C₄

yes

characteristic

C₄×D₅₀₈

C₂²

yes

—

#16861171

C₄

yes

—

#16861167

C₄

yes

—

C₄²×D₁₂₇

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

Dic₁₂₇⋊C₈

C₂²

yes

—

Dic₁₂₇⋊C₈

C₂²

yes

characteristic

(C₂×D₁₂₇)⋊C₈

C₂²

yes

—

(C₂×D₁₂₇)⋊C₈

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₄×Q₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

C₂²

yes

—

C₄×C₈×C₁₂₇

C₂²

yes

—

#14523249

—

C₂×D₅₀₈

C₂×C₄

yes

—

C₂×C₄×D₁₂₇

C₂×C₄

yes

—

C₂×C₄×D₁₂₇

C₂×C₄

yes

—

C₂×C₄×D₁₂₇

C₂³

yes

—

#14523231

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₂×C₁₂₇⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

(C₂×D₁₂₇)⋊C₄

C₂×C₄

yes

—

C₄×Dic₁₂₇

C₂×C₄

yes

—

C₄×Dic₁₂₇

C₂×C₄

yes

—

C₄×Dic₁₂₇

C₂³

yes

characteristic

C₂×Q₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂³

yes

—

C₂²×D₁₂₇

C₂²×C₄

yes

—

D₅₀₈

—

C₂×Dic₁₂₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₂₇

C₂²×C₄

yes

—

C₂×Dic₁₂₇

C₂²×C₄

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

Q₈×C₁₂₇

M₁₆

yes

complemented

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

C₂³×C₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×M₁₆

yes

—

C₄×C₁₂₇

H(5,𝔽₂)

yes

characteristic

C₄×C₁₂₇

2₋¹⁺⁴

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

#131

yes

—

C₂×C₁₂₇

#520

yes

characteristic

C₂×C₁₂₇

C₂²×M₁₆

yes

characteristic

C₂×C₁₂₇

C₂×#131

yes

characteristic

D₁₂₇

—

#2991

—

127

maximal, Sylow

C₁₂₇

#2991

yes

characteristic, complemented, Sylow

C₄×#41

—

127

—

C₈×Q₈

—

127

—

C₈×Q₈

D₁₂₇

yes

characteristic, complemented

#436

—

127

—

#436

—

127

—

#431

—

127

—

#423

—

127

—

C₂×#41

—

127

—

C₂×M₁₆

—

127

—

C₄×Q₈

—

127

—

C₄×Q₈

C₂×D₁₂₇

yes

characteristic

C₄×D₄

—

127

—

C₄×D₄

—

127

—

#117

—

127

—

C₂×C₄²

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

—

127

—

C₄⋊C₈

C₂×D₁₂₇

yes

—

C₂²⋊C₈

—

127

—

C₂²⋊C₈

—

127

—

C₄×C₈

—

127

—

C₄×C₈

C₂×D₁₂₇

yes

—

#41

—

254

—

100

C₂×Q₈

C₄×D₁₂₇

yes

characteristic

101

C₂×D₄

—

127

—

102

C₂²×C₄

—

127

—

103

C₂²×C₄

—

127

—

104

C₂²×C₄

—

127

—

105

M₁₆

—

508

—

106

C₂×C₈

—

127

—

107

C₂×C₈

—

127

—

108

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

109

C₂×C₈

C₂²×D₁₂₇

yes

—

110

C₄⋊C₄

—

127

—

111

C₄⋊C₄

C₄×D₁₂₇

yes

—

112

C₂²⋊C₄

—

127

—

113

C₄²

—

127

—

114

C₄²

—

127

—

115

C₄²

—

127

—

116

C₄²

C₂²×D₁₂₇

yes

—

117

C₂³

—

127

—

118

Q₈

#14523231

yes

complemented

119

D₄

—

254

—

120

C₂×C₄

—

127

—

121

C₂×C₄

—

127

—

122

C₂×C₄

—

127

—

123

C₂×C₄

—

254

—

124

C₂×C₄

—

254

—

125

C₂×C₄

—

254

—

126

C₂×C₄

—

254

—

127

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

128

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

129

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

254

—

133

C₈

—

254

—

134

C₂²

—

254

—

135

C₂²

—

254

—

136

C₂²

—

254

—

137

C₂²

C₂²×C₄×D₁₂₇

yes

characteristic

138

C₄

—

139

C₄

—

127

—

140

C₄

—

254

—

141

C₄

—

254

—

142

C₄

C₂×#14523231

yes

—

143

C₄

#16861255

yes

characteristic

144

C₄

#16861257

yes

characteristic

145

C₂

—

508

—

146

C₂

#22550358

yes

characteristic

147

C₂

C₂²×#14523231

yes

characteristic

148

C₂

#131×D₁₂₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	C₈×Q₈×C₁₂₇	C₂
prime radical	C₂×#14523249	C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₈×Q₈	D₁₂₇
hypocenter	C₁₂₇	#2991
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂³×C₄

Fitting

C₈×Q₈×C₁₂₇

C₂

prime radical

C₂×#14523249

C₄

cosocle

C₂×C₄×C₁₂₇

C₂⁴

hypercenter

C₈×Q₈

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₄ ◁ C₈×Q₈

N	H ≅ G/N	type
2. Q₈×C₁₂₇⋊C₈	C₂	semidirect
3. #22549955	C₂	semidirect
4. #22549953	C₂	semidirect
5. #22549927	C₂	semidirect
6. #22549898	C₂	semidirect
7. C₈×Q₈×C₁₂₇	C₂	semidirect
53. Q₈×C₁₂₇	M₁₆	semidirect
75. C₁₂₇	#2991	semidirect
78. C₈×Q₈	D₁₂₇	semidirect
118. Q₈	#14523231	semidirect

H ≅ G/N

type

2. Q₈×C₁₂₇⋊C₈

C₂

semidirect

3. #22549955

C₂

semidirect

4. #22549953

C₂

semidirect

5. #22549927

C₂

semidirect

6. #22549898

C₂

semidirect

7. C₈×Q₈×C₁₂₇

C₂

semidirect

53. Q₈×C₁₂₇

M₁₆

semidirect

75. C₁₂₇

#2991

semidirect

78. C₈×Q₈

D₁₂₇

semidirect

118. Q₈

#14523231

semidirect

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

subgroup	index
C₄×#14523249	2
Q₈×C₁₂₇⋊C₈	2
#22549955	2
#22549953	2
#22549927	2
#22549898	2
C₈×Q₈×C₁₂₇	2
#2991	127

subgroup

index

Q₈×C₁₂₇⋊C₈

C₈×Q₈×C₁₂₇

127

Subgroups — #30647861

Conjugacy classes of subgroups

Complements

Maximal subgroups