Group #44577191 — subgroups

Subgroups — C₁₂₇×#13518 #44577191

Order 32512.

250 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 110 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×#367×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2151	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1952	C₂	1	2	yes	maximal
4	C₁₂₇×#1387	C₂	1	2	yes	maximal
5	C₁₂₇×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
9	#428×C₁₂₇	—	2	2	no	—
10	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
11	#405×C₁₂₇	C₂²	1	1	yes	characteristic
12	#367×C₁₂₇	—	2	2	no	—
13	C₄⋊C₁₆×C₁₂₇	—	2	2	no	—
14	#343×C₁₂₇	—	2	2	no	—
15	C₂×M₁₆×C₁₂₇	—	2	2	no	—
16	C₂×M₁₆×C₁₂₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
18	#117×C₁₂₇	D₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
20	M₃₂×C₁₂₇	—	2	2	no	—
21	C₂×C₁₆×C₁₂₇	—	2	2	no	—
22	C₂×C₁₆×C₁₂₇	—	2	1	no	—
23	C₂×C₁₆×C₁₂₇	D₄	1	2	yes	—
24	#108×C₁₂₇	—	2	2	no	—
25	#108×C₁₂₇	Q₈	1	4	yes	—
26	#107×C₁₂₇	—	2	2	no	—
27	#107×C₁₂₇	—	2	2	no	—
28	#107×C₁₂₇	—	4	2	no	—
29	#106×C₁₂₇	—	4	2	no	—
30	#97×C₁₂₇	—	2	1	no	—
31	C₄×C₈×C₁₂₇	—	2	1	no	—
32	C₂²×C₄×C₁₂₇	—	4	2	no	—
33	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
34	M₁₆×C₁₂₇	—	2	4	no	—
35	M₁₆×C₁₂₇	—	2	2	no	—
36	M₁₆×C₁₂₇	—	4	2	no	—
37	C₂×C₈×C₁₂₇	—	2	1	no	—
38	C₂×C₈×C₁₂₇	—	4	2	no	—
39	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
41	C₂×C₈×C₁₂₇	#41	1	2	yes	—
42	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
43	C₄⋊C₄×C₁₂₇	—	4	2	no	—
44	C₄⋊C₄×C₁₂₇	—	4	2	no	—
45	C₄⋊C₄×C₁₂₇	D₈	1	2	yes	—
46	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
47	C₄²×C₁₂₇	—	2	1	no	—
48	C₁₆×C₁₂₇	—	2	2	no	—
49	C₁₆×C₁₂₇	—	4	1	no	—
50	C₂³×C₁₂₇	#135	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	—	2	2	no	—
52	C₂×C₄×C₁₂₇	—	8	2	no	—
53	C₂×C₄×C₁₂₇	—	16	2	no	—
54	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
55	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
56	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	#137	1	2	yes	—
58	C₈×C₁₂₇	—	4	1	no	—
59	C₈×C₁₂₇	—	4	4	no	—
60	C₈×C₁₂₇	#123	1	1	yes	characteristic
61	C₈×C₁₂₇	#125	1	2	yes	—
62	C₈×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
63	C₂²×C₁₂₇	—	2	2	no	—
64	C₂²×C₁₂₇	#466	1	1	yes	characteristic
65	C₂²×C₁₂₇	#479	1	2	yes	—
66	C₄×C₁₂₇	—	4	2	no	—
67	C₄×C₁₂₇	—	16	2	no	—
68	C₄×C₁₂₇	#398	1	1	yes	characteristic
69	C₄×C₁₂₇	#470	1	1	yes	characteristic
70	C₄×C₁₂₇	#474	1	2	yes	—
71	#13518	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₁₂₇	—	2	2	no	—
73	C₂×C₁₂₇	#2086	1	1	yes	characteristic
74	C₂×C₁₂₇	#2223	1	2	yes	—
75	C₂×#367	C₂×C₁₂₇	1	1	yes	characteristic
76	#2151	C₂×C₁₂₇	1	1	yes	characteristic
77	#1952	C₂×C₁₂₇	1	2	yes	—
78	#1387	C₂×C₁₂₇	1	2	yes	—
79	#1381	C₂×C₁₂₇	1	1	yes	characteristic
80	C₁₂₇	#13518	1	1	yes	characteristic, direct product factor, Sylow
81	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
82	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
83	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
84	#428	—	2	2	no	—
85	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
86	#405	C₂²×C₁₂₇	1	1	yes	characteristic
87	#367	—	2	2	no	—
88	C₄⋊C₁₆	—	2	2	no	—
89	#343	—	2	2	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	D₄×C₁₂₇	1	1	yes	characteristic
92	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
93	#117	D₄×C₁₂₇	1	1	yes	characteristic
94	C₂×C₄⋊C₄	—	2	2	no	—
95	M₃₂	—	2	2	no	—
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	D₄×C₁₂₇	1	2	yes	—
99	#108	—	2	2	no	—
100	#108	Q₈×C₁₂₇	1	4	yes	—
101	#107	—	2	2	no	—
102	#107	—	2	2	no	—
103	#107	—	4	2	no	—
104	#106	—	4	2	no	—
105	#97	—	2	1	no	—
106	C₄×C₈	—	2	1	no	—
107	C₂²×C₄	—	4	2	no	—
108	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
109	M₁₆	—	2	4	no	—
110	M₁₆	—	2	2	no	—
111	M₁₆	—	4	2	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
115	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
116	C₂×C₈	#41×C₁₂₇	1	2	yes	—
117	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
118	C₄⋊C₄	—	4	2	no	—
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	D₈×C₁₂₇	1	2	yes	—
121	C₂²⋊C₄	—	2	1	no	—
122	C₄²	—	2	1	no	—
123	C₁₆	—	2	2	no	—
124	C₁₆	—	4	1	no	—
125	C₂³	#135×C₁₂₇	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	8	2	no	—
128	C₂×C₄	—	16	2	no	—
129	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	2	yes	—
131	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
132	C₂×C₄	#137×C₁₂₇	1	2	yes	—
133	C₈	—	4	1	no	—
134	C₈	—	4	4	no	—
135	C₈	#123×C₁₂₇	1	1	yes	characteristic
136	C₈	#125×C₁₂₇	1	2	yes	—
137	C₈	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	#466×C₁₂₇	1	1	yes	characteristic
140	C₂²	#479×C₁₂₇	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	16	2	no	—
143	C₄	#398×C₁₂₇	1	1	yes	characteristic
144	C₄	#470×C₁₂₇	1	1	yes	characteristic
145	C₄	#474×C₁₂₇	1	2	yes	—
146	C₂	—	2	2	no	—
147	C₂	C₁₂₇×#2086	1	1	yes	characteristic
148	C₂	C₁₂₇×#2223	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#367×C₁₂₇

C₂

yes

maximal, characteristic

C₁₂₇×#2151

C₂

yes

maximal, characteristic

C₁₂₇×#1952

C₂

yes

maximal

C₁₂₇×#1387

C₂

yes

maximal

C₁₂₇×#1381

C₂

yes

maximal, characteristic

C₂×M₃₂×C₁₂₇

C₂²

yes

characteristic

C₂²×C₁₆×C₁₂₇

C₂²

yes

characteristic

C₂×#108×C₁₂₇

C₂²

yes

—

#428×C₁₂₇

—

C₂×#107×C₁₂₇

C₂²

yes

—

#405×C₁₂₇

C₂²

yes

characteristic

#367×C₁₂₇

—

C₄⋊C₁₆×C₁₂₇

—

#343×C₁₂₇

—

C₂×M₁₆×C₁₂₇

—

C₂×M₁₆×C₁₂₇

D₄

yes

characteristic

C₂²×C₈×C₁₂₇

C₂³

yes

characteristic

#117×C₁₂₇

D₄

yes

characteristic

C₂×C₄⋊C₄×C₁₂₇

—

M₃₂×C₁₂₇

—

C₂×C₁₆×C₁₂₇

—

C₂×C₁₆×C₁₂₇

—

C₂×C₁₆×C₁₂₇

D₄

yes

—

#108×C₁₂₇

—

#108×C₁₂₇

Q₈

yes

—

#107×C₁₂₇

—

#107×C₁₂₇

—

#107×C₁₂₇

—

#106×C₁₂₇

—

#97×C₁₂₇

—

C₄×C₈×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂×D₄

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×Q₈

yes

—

C₂×C₈×C₁₂₇

#41

yes

—

C₂×C₈×C₁₂₇

#41

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

D₈

yes

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₁₆×C₁₂₇

—

C₁₆×C₁₂₇

—

C₂³×C₁₂₇

#135

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

#121

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₂₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₂₇

#137

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

#123

yes

characteristic

C₈×C₁₂₇

#125

yes

—

C₈×C₁₂₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#466

yes

characteristic

C₂²×C₁₂₇

#479

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

—

#13518

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

—

C₂×C₁₂₇

#2086

yes

characteristic

C₂×C₁₂₇

#2223

yes

—

C₂×#367

C₂×C₁₂₇

yes

characteristic

#2151

C₂×C₁₂₇

yes

characteristic

#1952

C₂×C₁₂₇

yes

—

#1387

C₂×C₁₂₇

yes

—

#1381

C₂×C₁₂₇

yes

characteristic

C₁₂₇

#13518

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₂²×C₁₂₇

yes

characteristic

C₂²×C₁₆

C₂²×C₁₂₇

yes

characteristic

C₂×#108

C₂²×C₁₂₇

yes

—

#428

—

C₂×#107

C₂²×C₁₂₇

yes

—

#405

C₂²×C₁₂₇

yes

characteristic

#367

—

C₄⋊C₁₆

—

#343

—

C₂×M₁₆

—

C₂×M₁₆

D₄×C₁₂₇

yes

characteristic

C₂²×C₈

C₂³×C₁₂₇

yes

characteristic

#117

D₄×C₁₂₇

yes

characteristic

C₂×C₄⋊C₄

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₁₂₇

yes

—

#108

—

100

#108

Q₈×C₁₂₇

yes

—

101

#107

—

102

#107

—

103

#107

—

104

#106

—

105

#97

—

106

C₄×C₈

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂×D₄×C₁₂₇

yes

characteristic

109

M₁₆

—

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₂×D₄×C₁₂₇

yes

characteristic

115

C₂×C₈

C₂×Q₈×C₁₂₇

yes

—

116

C₂×C₈

#41×C₁₂₇

yes

—

117

C₂×C₈

#41×C₁₂₇

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

D₈×C₁₂₇

yes

—

121

C₂²⋊C₄

—

122

C₄²

—

123

C₁₆

—

124

C₁₆

—

125

C₂³

#135×C₁₂₇

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#121×C₁₂₇

yes

characteristic

130

C₂×C₄

C₂²⋊Q₈×C₁₂₇

yes

—

131

C₂×C₄

C₂×D₈×C₁₂₇

yes

characteristic

132

C₂×C₄

#137×C₁₂₇

yes

—

133

C₈

—

134

C₈

—

135

C₈

#123×C₁₂₇

yes

characteristic

136

C₈

#125×C₁₂₇

yes

—

137

C₈

C₄⋊Q₈×C₁₂₇

yes

characteristic

138

C₂²

—

139

C₂²

#466×C₁₂₇

yes

characteristic

140

C₂²

#479×C₁₂₇

yes

—

141

C₄

—

142

C₄

—

143

C₄

#398×C₁₂₇

yes

characteristic

144

C₄

#470×C₁₂₇

yes

characteristic

145

C₄

#474×C₁₂₇

yes

—

146

C₂

—

147

C₂

C₁₂₇×#2086

yes

characteristic

148

C₂

C₁₂₇×#2223

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #405×C₁₂₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₂₇

#466

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₁₂₇

prime radical

C₂³×C₁₂₇

#135

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #405×C₁₂₇ ◁ G

N	H ≅ G/N	type
71. #13518	C₁₂₇	direct (N × H)
80. C₁₂₇	#13518	direct (N × H)

H ≅ G/N

type

71. #13518

C₁₂₇

direct (N × H)

80. C₁₂₇

#13518

direct (N × H)

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

subgroup	index
C₂×#367×C₁₂₇	2
C₁₂₇×#2151	2
C₁₂₇×#1952	2
C₁₂₇×#1387	2
C₁₂₇×#1381	2
#13518	127

subgroup

index

C₂×#367×C₁₂₇

C₁₂₇×#2151

C₁₂₇×#1952

C₁₂₇×#1387

C₁₂₇×#1381

#13518

127

Subgroups — C₁₂₇×#13518 #44577191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#13518 #44577191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13518 #44577191