Group #44574341 — subgroups

Subgroups — C₁₂₇×#10668 #44574341

Order 32512.

316 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 114 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×#1995	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1843	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1571	C₂	1	2	yes	maximal
4	C₁₂₇×#1481	C₂	1	2	yes	maximal
5	C₁₂₇×#1274	C₂	1	1	yes	maximal, characteristic
6	#533×C₁₂₇	C₂²	1	1	yes	characteristic
7	#431×C₁₂₇	C₂²	1	2	yes	—
8	#423×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₁₂₇	—	2	2	no	—
10	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
11	#376×C₁₂₇	C₂²	1	1	yes	characteristic
12	#339×C₁₂₇	—	2	2	no	—
13	#335×C₁₂₇	—	2	4	no	—
14	Q₈⋊C₈×C₁₂₇	—	2	4	no	—
15	C₂×M₁₆×C₁₂₇	—	2	1	no	—
16	C₂²×C₈×C₁₂₇	—	2	2	no	—
17	C₄⋊Q₈×C₁₂₇	Q₈	1	2	yes	—
18	#125×C₁₂₇	—	2	1	no	—
19	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
20	C₄×Q₈×C₁₂₇	—	2	2	no	—
21	#117×C₁₂₇	—	2	1	no	—
22	C₂×C₄²×C₁₂₇	D₄	1	2	yes	—
23	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₈×C₁₂₇	—	2	2	no	—
25	C₄⋊C₈×C₁₂₇	—	2	2	no	—
26	C₄⋊C₈×C₁₂₇	—	2	2	no	—
27	C₄⋊C₈×C₁₂₇	—	2	2	no	—
28	C₄⋊C₈×C₁₂₇	—	4	4	no	—
29	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
30	C₄×C₈×C₁₂₇	D₄	1	4	yes	—
31	#95×C₁₂₇	—	2	2	no	—
32	C₂×Q₈×C₁₂₇	—	2	2	no	—
33	C₂²×C₄×C₁₂₇	—	2	2	no	—
34	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
36	M₁₆×C₁₂₇	—	4	2	no	—
37	C₂×C₈×C₁₂₇	—	2	4	no	—
38	C₂×C₈×C₁₂₇	—	4	2	no	—
39	C₂×C₈×C₁₂₇	—	4	1	no	—
40	C₂×C₈×C₁₂₇	—	4	4	no	—
41	C₄⋊C₄×C₁₂₇	—	2	2	no	—
42	C₄⋊C₄×C₁₂₇	—	2	2	no	—
43	C₄⋊C₄×C₁₂₇	—	4	2	no	—
44	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
45	C₄²×C₁₂₇	—	2	2	no	—
46	C₄²×C₁₂₇	—	4	1	no	—
47	C₄²×C₁₂₇	Q₁₆	1	4	yes	—
48	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
49	C₄²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₁₂₇	#41	1	1	yes	characteristic
51	C₂³×C₁₂₇	#123	1	1	yes	characteristic
52	Q₈×C₁₂₇	—	4	4	no	—
53	C₂×C₄×C₁₂₇	—	2	4	no	—
54	C₂×C₄×C₁₂₇	—	4	2	no	—
55	C₂×C₄×C₁₂₇	—	4	4	no	—
56	C₂×C₄×C₁₂₇	—	4	2	no	—
57	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	#123	1	2	yes	—
60	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	2	yes	—
62	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
63	C₈×C₁₂₇	—	4	4	no	—
64	C₈×C₁₂₇	—	8	2	no	—
65	C₈×C₁₂₇	—	8	1	no	—
66	C₂²×C₁₂₇	—	2	2	no	—
67	C₂²×C₁₂₇	#399	1	1	yes	characteristic
68	C₂²×C₁₂₇	#484	1	2	yes	—
69	C₄×C₁₂₇	—	2	2	no	—
70	C₄×C₁₂₇	—	2	2	no	—
71	C₄×C₁₂₇	—	4	4	no	—
72	C₄×C₁₂₇	—	8	2	no	—
73	C₄×C₁₂₇	#477	1	2	yes	—
74	C₄×C₁₂₇	#494	1	1	yes	characteristic
75	C₄×C₁₂₇	#496	1	1	yes	characteristic
76	#10668	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×C₁₂₇	—	2	2	no	—
78	C₂×C₁₂₇	#2095	1	1	yes	characteristic
79	C₂×C₁₂₇	#2098	1	2	yes	—
80	#1995	C₂×C₁₂₇	1	1	yes	characteristic
81	#1843	C₂×C₁₂₇	1	1	yes	characteristic
82	#1571	C₂×C₁₂₇	1	2	yes	—
83	#1481	C₂×C₁₂₇	1	2	yes	—
84	#1274	C₂×C₁₂₇	1	1	yes	characteristic
85	C₁₂₇	#10668	1	1	yes	characteristic, direct product factor, Sylow
86	#533	C₂²×C₁₂₇	1	1	yes	characteristic
87	#431	C₂²×C₁₂₇	1	2	yes	—
88	#423	C₂²×C₁₂₇	1	1	yes	characteristic
89	C₂×C₄⋊C₈	—	2	2	no	—
90	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
91	#376	C₂²×C₁₂₇	1	1	yes	characteristic
92	#339	—	2	2	no	—
93	#335	—	2	4	no	—
94	Q₈⋊C₈	—	2	4	no	—
95	C₂×M₁₆	—	2	1	no	—
96	C₂²×C₈	—	2	2	no	—
97	C₄⋊Q₈	Q₈×C₁₂₇	1	2	yes	—
98	#125	—	2	1	no	—
99	C₂²⋊Q₈	—	2	2	no	—
100	C₄×Q₈	—	2	2	no	—
101	#117	—	2	1	no	—
102	C₂×C₄²	D₄×C₁₂₇	1	2	yes	—
103	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	4	4	no	—
109	C₂²⋊C₈	—	2	2	no	—
110	C₄×C₈	D₄×C₁₂₇	1	4	yes	—
111	#95	—	2	2	no	—
112	C₂×Q₈	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
115	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
116	M₁₆	—	4	2	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	4	4	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₂²⋊C₄	—	4	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	4	1	no	—
127	C₄²	Q₁₆×C₁₂₇	1	4	yes	—
128	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
129	C₄²	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
130	C₄²	#41×C₁₂₇	1	1	yes	characteristic
131	C₂³	#123×C₁₂₇	1	1	yes	characteristic
132	Q₈	—	4	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	2	yes	—
138	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
139	C₂×C₄	#123×C₁₂₇	1	2	yes	—
140	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
141	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	2	yes	—
142	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
143	C₈	—	4	4	no	—
144	C₈	—	8	2	no	—
145	C₈	—	8	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#399×C₁₂₇	1	1	yes	characteristic
148	C₂²	#484×C₁₂₇	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	8	2	no	—
153	C₄	#477×C₁₂₇	1	2	yes	—
154	C₄	#494×C₁₂₇	1	1	yes	characteristic
155	C₄	#496×C₁₂₇	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	C₁₂₇×#2095	1	1	yes	characteristic
158	C₂	C₁₂₇×#2098	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₂₇×#1995

C₂

yes

maximal, characteristic

C₁₂₇×#1843

C₂

yes

maximal, characteristic

C₁₂₇×#1571

C₂

yes

maximal

C₁₂₇×#1481

C₂

yes

maximal

C₁₂₇×#1274

C₂

yes

maximal, characteristic

#533×C₁₂₇

C₂²

yes

characteristic

#431×C₁₂₇

C₂²

yes

—

#423×C₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₁₂₇

—

C₂×C₄⋊C₈×C₁₂₇

C₂²

yes

—

#376×C₁₂₇

C₂²

yes

characteristic

#339×C₁₂₇

—

#335×C₁₂₇

—

Q₈⋊C₈×C₁₂₇

—

C₂×M₁₆×C₁₂₇

—

C₂²×C₈×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

Q₈

yes

—

#125×C₁₂₇

—

C₂²⋊Q₈×C₁₂₇

—

C₄×Q₈×C₁₂₇

—

#117×C₁₂₇

—

C₂×C₄²×C₁₂₇

D₄

yes

—

C₂×C₄²×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₈×C₁₂₇

—

C₄⋊C₈×C₁₂₇

—

C₄⋊C₈×C₁₂₇

—

C₄⋊C₈×C₁₂₇

—

C₄⋊C₈×C₁₂₇

—

C₂²⋊C₈×C₁₂₇

—

C₄×C₈×C₁₂₇

D₄

yes

—

#95×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₂₇

#41

yes

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

Q₁₆

yes

—

C₄²×C₁₂₇

C₂×D₄

yes

—

C₄²×C₁₂₇

C₂×Q₈

yes

C₄²×C₁₂₇

yes

C₂³×C₁₂₇

yes

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#123

yes

—

C₂×C₄×C₁₂₇

#127

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₁₆

yes

—

C₂×C₄×C₁₂₇

AGL(1,ℤ/8)

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#399

yes

characteristic

C₂²×C₁₂₇

#484

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#477

yes

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

—

C₂×C₁₂₇

#2095

yes

characteristic

C₂×C₁₂₇

#2098

yes

—

#1995

C₂×C₁₂₇

yes

characteristic

#1843

C₂×C₁₂₇

yes

characteristic

#1571

C₂×C₁₂₇

yes

—

#1481

C₂×C₁₂₇

yes

—

#1274

C₂×C₁₂₇

yes

characteristic

C₁₂₇

#10668

yes

characteristic, direct product factor, Sylow

#533

C₂²×C₁₂₇

yes

characteristic

#431

C₂²×C₁₂₇

yes

—

#423

C₂²×C₁₂₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²×C₁₂₇

yes

—

#376

C₂²×C₁₂₇

yes

characteristic

#339

—

#335

—

Q₈⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

Q₈×C₁₂₇

yes

—

#125

—

C₂²⋊Q₈

—

100

C₄×Q₈

—

101

#117

—

102

C₂×C₄²

D₄×C₁₂₇

yes

—

103

C₂×C₄²

C₂³×C₁₂₇

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₂²⋊C₈

—

110

C₄×C₈

D₄×C₁₂₇

yes

—

111

#95

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×D₄×C₁₂₇

yes

characteristic

115

C₂²×C₄

#41×C₁₂₇

yes

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

Q₁₆×C₁₂₇

yes

—

128

C₄²

C₂×D₄×C₁₂₇

yes

—

129

C₄²

C₂×Q₈×C₁₂₇

yes

characteristic

130

C₄²

#41×C₁₂₇

yes

characteristic

131

C₂³

#123×C₁₂₇

yes

characteristic

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂²⋊Q₈×C₁₂₇

yes

—

138

C₂×C₄

C₂²⋊Q₈×C₁₂₇

yes

characteristic

139

C₂×C₄

#123×C₁₂₇

yes

—

140

C₂×C₄

#127×C₁₂₇

yes

characteristic

141

C₂×C₄

C₂×Q₁₆×C₁₂₇

yes

—

142

C₂×C₄

AGL(1,ℤ/8)×C₁₂₇

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₂²

—

147

C₂²

#399×C₁₂₇

yes

characteristic

148

C₂²

#484×C₁₂₇

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#477×C₁₂₇

yes

—

154

C₄

#494×C₁₂₇

yes

characteristic

155

C₄

#496×C₁₂₇

yes

characteristic

156

C₂

—

157

C₂

C₁₂₇×#2095

yes

characteristic

158

C₂

C₁₂₇×#2098

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	#399
prime radical	C₂³×C₁₂₇	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₂₇

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₂₇

C₂²×C₁₂₇

C₂³×C₁₂₇

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

N	H ≅ G/N	type
76. #10668	C₁₂₇	direct (N × H)
85. C₁₂₇	#10668	direct (N × H)

H ≅ G/N

type

76. #10668

C₁₂₇

direct (N × H)

85. C₁₂₇

#10668

direct (N × H)

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

subgroup	index
C₁₂₇×#1995	2
C₁₂₇×#1843	2
C₁₂₇×#1571	2
C₁₂₇×#1481	2
C₁₂₇×#1274	2
#10668	127

subgroup

index

C₁₂₇×#1995

C₁₂₇×#1843

C₁₂₇×#1571

C₁₂₇×#1481

C₁₂₇×#1274

#10668

127

Subgroups — C₁₂₇×#10668 #44574341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#10668 #44574341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#10668 #44574341