Group #30634909 — subgroups

Subgroups — C₁₂₇×#1957 #30634909

Order 16256.

436 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 160 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×#127×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#102×C₁₂₇	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#384×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄×C₁₂₇	—	2	2	no	—
6	C₂²×D₄×C₁₂₇	C₂²	1	2	yes	—
7	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
8	#127×C₁₂₇	—	2	2	no	—
9	#127×C₁₂₇	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
11	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
13	#102×C₁₂₇	—	2	8	no	—
14	C₄×C₈×C₁₂₇	—	2	2	no	—
15	C₂⁴×C₁₂₇	—	2	2	no	—
16	C₂×D₄×C₁₂₇	—	2	8	no	—
17	C₂×D₄×C₁₂₇	—	2	4	no	—
18	C₂×D₄×C₁₂₇	—	2	2	no	—
19	C₂×D₄×C₁₂₇	—	2	4	no	—
20	C₂×D₄×C₁₂₇	—	4	4	no	—
21	C₂×D₄×C₁₂₇	—	4	4	no	—
22	C₂×D₄×C₁₂₇	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₁₂₇	—	2	2	no	—
24	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₂₇	—	2	2	no	—
27	C₂×C₈×C₁₂₇	—	2	2	no	—
28	C₂×C₈×C₁₂₇	D₄	1	4	yes	complemented
29	C₄⋊C₄×C₁₂₇	—	2	4	no	—
30	C₄⋊C₄×C₁₂₇	—	4	4	no	—
31	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
32	C₄²×C₁₂₇	D₄	1	2	yes	—
33	C₂³×C₁₂₇	—	2	4	no	—
34	C₂³×C₁₂₇	—	4	2	no	—
35	C₂³×C₁₂₇	—	4	2	no	—
36	C₂³×C₁₂₇	—	4	4	no	—
37	C₂³×C₁₂₇	—	4	4	no	—
38	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	D₄×C₁₂₇	—	2	8	no	—
40	D₄×C₁₂₇	—	4	16	no	—
41	D₄×C₁₂₇	—	4	4	no	—
42	C₂×C₄×C₁₂₇	—	4	2	no	—
43	C₂×C₄×C₁₂₇	—	4	4	no	—
44	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
45	C₂×C₄×C₁₂₇	D₈	1	4	yes	—
46	C₂×C₄×C₁₂₇	SD₁₆	1	4	yes	—
47	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₂₇	#41	1	2	yes	—
51	C₈×C₁₂₇	—	2	4	no	—
52	C₂²×C₁₂₇	—	4	4	no	—
53	C₂²×C₁₂₇	—	4	8	no	—
54	C₂²×C₁₂₇	—	8	4	no	—
55	C₂²×C₁₂₇	—	8	2	no	—
56	C₂²×C₁₂₇	—	8	2	no	—
57	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₂₇	#124	1	1	yes	characteristic
59	C₂²×C₁₂₇	#127	1	1	yes	characteristic
60	C₂²×C₁₂₇	C₂×D₈	1	2	yes	—
61	C₂²×C₁₂₇	C₂×SD₁₆	1	2	yes	—
62	C₄×C₁₂₇	—	8	2	no	—
63	C₄×C₁₂₇	#102	1	8	yes	complemented
64	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
65	C₄×C₁₂₇	#121	1	2	yes	—
66	C₂×C₁₂₇	—	8	4	no	—
67	C₂×C₁₂₇	#390	1	1	yes	characteristic
68	C₂×C₁₂₇	C₂×#102	1	2	yes	—
69	C₂×C₁₂₇	#486	1	2	yes	—
70	C₂×C₁₂₇	#492	1	1	yes	characteristic
71	C₂×C₁₂₇	#493	1	1	yes	characteristic
72	#1957	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₁₂₇	#1957	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×#127	C₂×C₁₂₇	1	1	yes	characteristic
75	C₂×#102	C₂×C₁₂₇	1	4	yes	complemented
76	C₂×C₄×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
77	#384	C₂×C₁₂₇	1	1	yes	characteristic, complemented
78	C₂²×D₄	—	2	2	no	—
79	C₂²×D₄	C₂²×C₁₂₇	1	2	yes	—
80	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
81	#127	—	2	2	no	—
82	#127	C₄×C₁₂₇	1	4	yes	complemented
83	C₂×C₄⋊C₄	—	2	2	no	—
84	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
85	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
86	#102	—	2	8	no	—
87	C₄×C₈	—	2	2	no	—
88	C₂⁴	—	2	2	no	—
89	C₂×D₄	—	2	8	no	—
90	C₂×D₄	—	2	4	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	4	no	—
93	C₂×D₄	—	4	4	no	—
94	C₂×D₄	—	4	4	no	—
95	C₂×D₄	C₂×C₄×C₁₂₇	1	4	yes	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	D₄×C₁₂₇	1	2	yes	—
98	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	D₄×C₁₂₇	1	4	yes	complemented
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	4	4	no	—
104	C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
105	C₄²	D₄×C₁₂₇	1	2	yes	—
106	C₂³	—	2	4	no	—
107	C₂³	—	4	2	no	—
108	C₂³	—	4	2	no	—
109	C₂³	—	4	4	no	—
110	C₂³	—	4	4	no	—
111	C₂³	C₂×D₄×C₁₂₇	1	1	yes	characteristic
112	D₄	—	2	8	no	—
113	D₄	—	4	16	no	—
114	D₄	—	4	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
118	C₂×C₄	D₈×C₁₂₇	1	4	yes	—
119	C₂×C₄	SD₁₆×C₁₂₇	1	4	yes	—
120	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
121	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
122	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
123	C₂×C₄	#41×C₁₂₇	1	2	yes	—
124	C₈	—	2	4	no	—
125	C₂²	—	4	4	no	—
126	C₂²	—	4	8	no	—
127	C₂²	—	8	4	no	—
128	C₂²	—	8	2	no	—
129	C₂²	—	8	2	no	—
130	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
131	C₂²	#124×C₁₂₇	1	1	yes	characteristic
132	C₂²	#127×C₁₂₇	1	1	yes	characteristic
133	C₂²	C₂×D₈×C₁₂₇	1	2	yes	—
134	C₂²	C₂×SD₁₆×C₁₂₇	1	2	yes	—
135	C₄	—	8	2	no	—
136	C₄	#102×C₁₂₇	1	8	yes	complemented
137	C₄	C₄×D₄×C₁₂₇	1	2	yes	—
138	C₄	#121×C₁₂₇	1	2	yes	—
139	C₂	—	8	4	no	—
140	C₂	#390×C₁₂₇	1	1	yes	characteristic
141	C₂	C₂×#102×C₁₂₇	1	2	yes	—
142	C₂	#486×C₁₂₇	1	2	yes	—
143	C₂	#492×C₁₂₇	1	1	yes	characteristic
144	C₂	#493×C₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#127×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#102×C₁₂₇

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#384×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₂₇

—

C₂²×D₄×C₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

#127×C₁₂₇

—

#127×C₁₂₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#102×C₁₂₇

—

C₄×C₈×C₁₂₇

—

C₂⁴×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

complemented

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

D₈

yes

—

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

C₂×D₈

yes

—

C₂²×C₁₂₇

C₂×SD₁₆

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#102

yes

complemented

C₄×C₁₂₇

C₄×D₄

yes

—

C₄×C₁₂₇

#121

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#390

yes

characteristic

C₂×C₁₂₇

C₂×#102

yes

—

C₂×C₁₂₇

#486

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₂₇

#1957

yes

characteristic, direct product factor, Sylow

C₂×#127

C₂×C₁₂₇

yes

characteristic

C₂×#102

C₂×C₁₂₇

yes

complemented

C₂×C₄×C₈

C₂×C₁₂₇

yes

characteristic, complemented

#384

C₂×C₁₂₇

yes

characteristic, complemented

C₂²×D₄

—

C₂²×D₄

C₂²×C₁₂₇

yes

—

C₂²×C₈

C₂²×C₁₂₇

yes

—

#127

—

#127

C₄×C₁₂₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₂₇

yes

—

C₂×C₄²

C₂²×C₁₂₇

yes

characteristic

#102

—

C₄×C₈

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₄×C₁₂₇

yes

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₂₇

yes

—

C₂²×C₄

C₂³×C₁₂₇

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

D₄×C₁₂₇

yes

complemented

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₂×C₄×C₁₂₇

yes

—

105

C₄²

D₄×C₁₂₇

yes

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

C₂×D₄×C₁₂₇

yes

characteristic

112

D₄

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²⋊C₄×C₁₂₇

yes

—

118

C₂×C₄

D₈×C₁₂₇

yes

—

119

C₂×C₄

SD₁₆×C₁₂₇

yes

—

120

C₂×C₄

C₂²×C₄×C₁₂₇

yes

characteristic

121

C₂×C₄

C₂×D₄×C₁₂₇

yes

—

122

C₂×C₄

C₂×D₄×C₁₂₇

yes

characteristic

123

C₂×C₄

#41×C₁₂₇

yes

—

124

C₈

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×C₂²⋊C₄×C₁₂₇

yes

characteristic

131

C₂²

#124×C₁₂₇

yes

characteristic

132

C₂²

#127×C₁₂₇

yes

characteristic

133

C₂²

C₂×D₈×C₁₂₇

yes

—

134

C₂²

C₂×SD₁₆×C₁₂₇

yes

—

135

C₄

—

136

C₄

#102×C₁₂₇

yes

complemented

137

C₄

C₄×D₄×C₁₂₇

yes

—

138

C₄

#121×C₁₂₇

yes

—

139

C₂

—

140

C₂

#390×C₁₂₇

yes

characteristic

141

C₂

C₂×#102×C₁₂₇

yes

—

142

C₂

#486×C₁₂₇

yes

—

143

C₂

#492×C₁₂₇

yes

characteristic

144

C₂

#493×C₁₂₇

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₁₂₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₂₇
prime radical	C₂×#127×C₁₂₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₂₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₁₂₇

Frattini, cosocle

C₂²×C₄

C₂³×C₁₂₇

prime radical

C₂×#127×C₁₂₇

C₂

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

N	H ≅ G/N	type
2. C₂×#102×C₁₂₇	C₂	semidirect
3. C₂×C₄×C₈×C₁₂₇	C₂	semidirect
4. #384×C₁₂₇	C₂	semidirect
9. #127×C₁₂₇	C₄	semidirect
28. C₂×C₈×C₁₂₇	D₄	semidirect
63. C₄×C₁₂₇	#102	semidirect
72. #1957	C₁₂₇	direct (N × H)
73. C₁₂₇	#1957	direct (N × H)
75. C₂×#102	C₂×C₁₂₇	semidirect
76. C₂×C₄×C₈	C₂×C₁₂₇	semidirect
77. #384	C₂×C₁₂₇	semidirect
82. #127	C₄×C₁₂₇	semidirect
101. C₂×C₈	D₄×C₁₂₇	semidirect
136. C₄	#102×C₁₂₇	semidirect

H ≅ G/N

type

2. C₂×#102×C₁₂₇

C₂

semidirect

3. C₂×C₄×C₈×C₁₂₇

C₂

semidirect

4. #384×C₁₂₇

C₂

semidirect

9. #127×C₁₂₇

C₄

semidirect

28. C₂×C₈×C₁₂₇

D₄

semidirect

63. C₄×C₁₂₇

#102

semidirect

72. #1957

C₁₂₇

direct (N × H)

73. C₁₂₇

#1957

direct (N × H)

75. C₂×#102

C₂×C₁₂₇

semidirect

76. C₂×C₄×C₈

C₂×C₁₂₇

semidirect

77. #384

C₂×C₁₂₇

semidirect

82. #127

C₄×C₁₂₇

semidirect

101. C₂×C₈

D₄×C₁₂₇

semidirect

136. C₄

#102×C₁₂₇

semidirect

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

subgroup	index
C₂×#127×C₁₂₇	2
C₂×#102×C₁₂₇	2
C₂×C₄×C₈×C₁₂₇	2
#384×C₁₂₇	2
#1957	127

subgroup

index

C₂×#127×C₁₂₇

C₂×#102×C₁₂₇

C₂×C₄×C₈×C₁₂₇

#384×C₁₂₇

#1957

127

Subgroups — C₁₂₇×#1957 #30634909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#1957 #30634909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#1957 #30634909