Group #44575169 — subgroups

Subgroups — C₁₂₇×#11496 #44575169

Order 32512.

296 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 138 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	C₁₂₇×#2960	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2162	C₂	1	2	yes	maximal
3	C₁₂₇×#1323	C₂	1	4	yes	maximal
4	#533×C₁₂₇	C₄	1	1	yes	characteristic
5	#446×C₁₂₇	—	2	2	no	—
6	C₈×Q₈×C₁₂₇	—	2	2	no	—
7	#424×C₁₂₇	C₂²	1	2	yes	—
8	#423×C₁₂₇	C₄	1	1	yes	characteristic
9	#405×C₁₂₇	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₁₆×C₁₂₇	—	2	2	no	—
12	C₂²⋊C₁₆×C₁₂₇	C₂²	1	4	yes	—
13	C₄×C₁₆×C₁₂₇	—	2	2	no	—
14	C₂×M₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₂₇	C₈	1	2	yes	—
18	#125×C₁₂₇	C₈	1	2	yes	—
19	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
20	C₄×Q₈×C₁₂₇	—	2	2	no	—
21	#117×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₂₇	—	2	4	no	—
24	C₄⋊C₈×C₁₂₇	—	2	4	no	—
25	C₄⋊C₈×C₁₂₇	—	2	2	no	—
26	#97×C₁₂₇	—	2	2	no	—
27	C₄×C₈×C₁₂₇	—	2	2	no	—
28	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
29	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
30	C₂×Q₈×C₁₂₇	—	2	2	no	—
31	C₂²×C₄×C₁₂₇	M₁₆	1	2	yes	—
32	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
33	M₁₆×C₁₂₇	—	4	2	no	—
34	C₂×C₈×C₁₂₇	—	2	2	no	—
35	C₂×C₈×C₁₂₇	—	2	2	no	—
36	C₂×C₈×C₁₂₇	—	4	1	no	—
37	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
40	C₄⋊C₄×C₁₂₇	—	2	4	no	—
41	C₄⋊C₄×C₁₂₇	—	2	2	no	—
42	C₄⋊C₄×C₁₂₇	C₂×C₈	1	4	yes	—
43	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
44	C₄²×C₁₂₇	—	2	2	no	—
45	C₄²×C₁₂₇	C₂²⋊C₄	1	2	yes	—
46	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
48	C₁₆×C₁₂₇	—	4	4	no	—
49	C₂³×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
50	Q₈×C₁₂₇	—	4	4	no	—
51	C₂×C₄×C₁₂₇	—	2	4	no	—
52	C₂×C₄×C₁₂₇	—	2	1	no	—
53	C₂×C₄×C₁₂₇	—	2	1	no	—
54	C₂×C₄×C₁₂₇	—	2	4	no	—
55	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₁₂₇	—	2	2	no	—
60	C₈×C₁₂₇	—	2	1	no	—
61	C₈×C₁₂₇	—	4	2	no	—
62	C₈×C₁₂₇	#101	1	2	yes	—
63	C₂²×C₁₂₇	—	2	2	no	—
64	C₂²×C₁₂₇	—	2	1	no	—
65	C₂²×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
66	C₄×C₁₂₇	—	2	2	no	—
67	C₄×C₁₂₇	—	2	1	no	—
68	C₄×C₁₂₇	—	4	4	no	—
69	C₄×C₁₂₇	#350	1	4	yes	—
70	C₄×C₁₂₇	#410	1	1	yes	characteristic
71	C₄×C₁₂₇	C₂×#101	1	1	yes	characteristic
72	#11496	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂×C₁₂₇	—	4	1	no	—
74	C₂×C₁₂₇	#1461	1	1	yes	characteristic
75	C₂×C₁₂₇	C₂×#350	1	2	yes	—
76	#2960	C₂×C₁₂₇	1	1	yes	characteristic
77	#2162	C₂×C₁₂₇	1	2	yes	—
78	#1323	C₂×C₁₂₇	1	4	yes	—
79	C₁₂₇	#11496	1	1	yes	characteristic, direct product factor, Sylow
80	#533	C₄×C₁₂₇	1	1	yes	characteristic
81	#446	—	2	2	no	—
82	C₈×Q₈	—	2	2	no	—
83	#424	C₂²×C₁₂₇	1	2	yes	—
84	#423	C₄×C₁₂₇	1	1	yes	characteristic
85	#405	C₄×C₁₂₇	1	2	yes	—
86	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
87	C₄⋊C₁₆	—	2	2	no	—
88	C₂²⋊C₁₆	C₂²×C₁₂₇	1	4	yes	—
89	C₄×C₁₆	—	2	2	no	—
90	C₂×M₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
91	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
92	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
93	C₄⋊Q₈	C₈×C₁₂₇	1	2	yes	—
94	#125	C₈×C₁₂₇	1	2	yes	—
95	C₂²⋊Q₈	—	2	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	#117	C₂×C₄×C₁₂₇	1	2	yes	—
98	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
99	C₂×C₁₆	—	2	4	no	—
100	C₄⋊C₈	—	2	4	no	—
101	C₄⋊C₈	—	2	2	no	—
102	#97	—	2	2	no	—
103	C₄×C₈	—	2	2	no	—
104	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
105	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
106	C₂×Q₈	—	2	2	no	—
107	C₂²×C₄	M₁₆×C₁₂₇	1	2	yes	—
108	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
109	M₁₆	—	4	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
114	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
115	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	C₂×C₈×C₁₂₇	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	C₂²⋊C₄×C₁₂₇	1	2	yes	—
122	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
123	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
124	C₁₆	—	4	4	no	—
125	C₂³	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
126	Q₈	—	4	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
132	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
134	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	2	1	no	—
137	C₈	—	4	2	no	—
138	C₈	#101×C₁₂₇	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	4	no	—
145	C₄	#350×C₁₂₇	1	4	yes	—
146	C₄	#410×C₁₂₇	1	1	yes	characteristic
147	C₄	C₂×#101×C₁₂₇	1	1	yes	characteristic
148	C₂	—	4	1	no	—
149	C₂	C₁₂₇×#1461	1	1	yes	characteristic
150	C₂	C₂×#350×C₁₂₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₂₇×#2960

C₂

yes

maximal, characteristic

C₁₂₇×#2162

C₂

yes

maximal

C₁₂₇×#1323

C₂

yes

maximal

#533×C₁₂₇

C₄

yes

characteristic

#446×C₁₂₇

—

C₈×Q₈×C₁₂₇

—

#424×C₁₂₇

C₂²

yes

—

#423×C₁₂₇

C₄

yes

characteristic

#405×C₁₂₇

C₄

yes

—

C₂×C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₁₂₇

—

C₂²⋊C₁₆×C₁₂₇

C₂²

yes

—

C₄×C₁₆×C₁₂₇

—

C₂×M₁₆×C₁₂₇

C₂×C₄

yes

—

C₂²×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₂₇

C₂³

yes

characteristic

C₄⋊Q₈×C₁₂₇

C₈

yes

—

#125×C₁₂₇

C₈

yes

—

C₂²⋊Q₈×C₁₂₇

—

C₄×Q₈×C₁₂₇

—

#117×C₁₂₇

C₂×C₄

yes

—

C₂×C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₂₇

—

C₄⋊C₈×C₁₂₇

—

C₄⋊C₈×C₁₂₇

—

#97×C₁₂₇

—

C₄×C₈×C₁₂₇

—

C₄×C₈×C₁₂₇

D₄

yes

—

C₄×C₈×C₁₂₇

D₄

yes

—

C₂×Q₈×C₁₂₇

—

C₂²×C₄×C₁₂₇

M₁₆

yes

—

C₂²×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×D₄

yes

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂×C₈

yes

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

C₂²⋊C₄

yes

—

C₄²×C₁₂₇

C₂×C₈

yes

characteristic

C₄²×C₁₂₇

C₂×C₈

yes

characteristic

C₁₆×C₁₂₇

—

C₂³×C₁₂₇

C₂×M₁₆

yes

characteristic

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₈

yes

—

C₂×C₄×C₁₂₇

C₂²⋊C₈

yes

—

C₂×C₄×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₈

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

#101

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#350

yes

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

—

C₂×C₁₂₇

#1461

yes

characteristic

C₂×C₁₂₇

C₂×#350

yes

—

#2960

C₂×C₁₂₇

yes

characteristic

#2162

C₂×C₁₂₇

yes

—

#1323

C₂×C₁₂₇

yes

—

C₁₂₇

#11496

yes

characteristic, direct product factor, Sylow

#533

C₄×C₁₂₇

yes

characteristic

#446

—

C₈×Q₈

—

#424

C₂²×C₁₂₇

yes

—

#423

C₄×C₁₂₇

yes

characteristic

#405

C₄×C₁₂₇

yes

—

C₂×C₄×C₈

C₂²×C₁₂₇

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

C₂²×C₁₂₇

yes

—

C₄×C₁₆

—

C₂×M₁₆

C₂×C₄×C₁₂₇

yes

—

C₂²×C₈

C₂×C₄×C₁₂₇

yes

characteristic

C₂²×C₈

C₂³×C₁₂₇

yes

characteristic

C₄⋊Q₈

C₈×C₁₂₇

yes

—

#125

C₈×C₁₂₇

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×C₄×C₁₂₇

yes

—

C₂×C₄²

C₂×C₄×C₁₂₇

yes

characteristic

C₂×C₁₆

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

#97

—

103

C₄×C₈

—

104

C₄×C₈

D₄×C₁₂₇

yes

—

105

C₄×C₈

D₄×C₁₂₇

yes

—

106

C₂×Q₈

—

107

C₂²×C₄

M₁₆×C₁₂₇

yes

—

108

C₂²×C₄

C₂²×C₄×C₁₂₇

yes

characteristic

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²⋊C₄×C₁₂₇

yes

characteristic

114

C₂×C₈

C₂²⋊C₄×C₁₂₇

yes

characteristic

115

C₂×C₈

C₂×D₄×C₁₂₇

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂×C₈×C₁₂₇

yes

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

C₂²⋊C₄×C₁₂₇

yes

—

122

C₄²

C₂×C₈×C₁₂₇

yes

characteristic

123

C₄²

C₂×C₈×C₁₂₇

yes

characteristic

124

C₁₆

—

125

C₂³

C₂×M₁₆×C₁₂₇

yes

characteristic

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²⋊C₈×C₁₂₇

yes

—

132

C₂×C₄

C₂²⋊C₈×C₁₂₇

yes

—

133

C₂×C₄

C₂×C₂²⋊C₄×C₁₂₇

yes

characteristic

134

C₂×C₄

C₂²×C₈×C₁₂₇

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

#101×C₁₂₇

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₂²⋊C₈×C₁₂₇

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#350×C₁₂₇

yes

—

146

C₄

#410×C₁₂₇

yes

characteristic

147

C₄

C₂×#101×C₁₂₇

yes

characteristic

148

C₂

—

149

C₂

C₁₂₇×#1461

yes

characteristic

150

C₂

C₂×#350×C₁₂₇

yes

—

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₁₂₇	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

subgroup

G/H

center

C₂×C₈×C₁₂₇

C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₈×C₁₂₇

Frattini, cosocle

C₂²×C₈

C₂³×C₁₂₇

socle

C₂²×C₁₂₇

C₂×C₂²⋊C₈

prime radical

C₂³×C₁₂₇

C₂×M₁₆

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

N	H ≅ G/N	type
72. #11496	C₁₂₇	direct (N × H)
79. C₁₂₇	#11496	direct (N × H)

H ≅ G/N

type

72. #11496

C₁₂₇

direct (N × H)

79. C₁₂₇

#11496

direct (N × H)

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

subgroup	index
C₁₂₇×#2960	2
C₁₂₇×#2162	2
C₁₂₇×#1323	2
#11496	127

subgroup

index

C₁₂₇×#2960

C₁₂₇×#2162

C₁₂₇×#1323

#11496

127

Subgroups — C₁₂₇×#11496 #44575169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#11496 #44575169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11496 #44575169