Group #44580754 — subgroups

Subgroups — C₁₂₇×#17081 #44580754

Order 32512.

1002 conjugacy classes of subgroups; only the 334 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 334 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×#1442	C₂	1	12	yes	maximal
2	C₁₂₇×C₄⋊#95	C₂	1	3	yes	maximal
3	C₂²×C₄⋊C₄×C₁₂₇	C₂²	1	6	yes	—
4	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
5	C₂×#95×C₁₂₇	C₂²	1	12	yes	—
6	C₂×#95×C₁₂₇	C₂²	1	12	yes	—
7	C₂×#95×C₁₂₇	C₂²	1	4	yes	—
8	C₂³×C₄×C₁₂₇	C₂³	1	12	yes	—
9	C₂³×C₄×C₁₂₇	C₂³	1	3	yes	—
10	C₂×C₄⋊C₄×C₁₂₇	—	2	12	no	—
11	C₂×C₄⋊C₄×C₁₂₇	—	4	6	no	—
12	C₂×C₄⋊C₄×C₁₂₇	—	4	6	no	—
13	C₂×C₄²×C₁₂₇	—	4	2	no	—
14	C₂×C₄²×C₁₂₇	Q₈	1	4	yes	—
15	#95×C₁₂₇	—	2	24	no	—
16	#95×C₁₂₇	—	4	12	no	—
17	#95×C₁₂₇	—	4	4	no	—
18	C₂⁴×C₁₂₇	C₂⁴	1	1	yes	characteristic
19	C₂²×C₄×C₁₂₇	—	2	6	no	—
20	C₂²×C₄×C₁₂₇	—	2	3	no	—
21	C₂²×C₄×C₁₂₇	—	2	3	no	—
22	C₂²×C₄×C₁₂₇	—	2	12	no	—
23	C₂²×C₄×C₁₂₇	—	2	12	no	—
24	C₂²×C₄×C₁₂₇	—	2	12	no	—
25	C₂²×C₄×C₁₂₇	—	2	12	no	—
26	C₂²×C₄×C₁₂₇	—	2	12	no	—
27	C₂²×C₄×C₁₂₇	—	2	12	no	—
28	C₂²×C₄×C₁₂₇	—	2	12	no	—
29	C₂²×C₄×C₁₂₇	C₂×Q₈	1	6	yes	—
30	C₂²×C₄×C₁₂₇	#41	1	12	yes	—
31	C₄⋊C₄×C₁₂₇	—	4	24	no	—
32	C₄²×C₁₂₇	—	4	4	no	—
33	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
34	C₂³×C₁₂₇	C₂×#41	1	6	yes	—
35	C₂³×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
36	C₂³×C₁₂₇	H(5,𝔽₂)	1	3	yes	—
37	C₂³×C₁₂₇	2₋¹⁺⁴	1	3	yes	—
38	C₂×C₄×C₁₂₇	—	2	12	no	—
39	C₂×C₄×C₁₂₇	—	2	12	no	—
40	C₂×C₄×C₁₂₇	—	2	6	no	—
41	C₂×C₄×C₁₂₇	—	2	6	no	—
42	C₂×C₄×C₁₂₇	—	4	12	no	—
43	C₂×C₄×C₁₂₇	—	4	12	no	—
44	C₂×C₄×C₁₂₇	—	4	12	no	—
45	C₂×C₄×C₁₂₇	—	4	12	no	—
46	C₂×C₄×C₁₂₇	—	4	12	no	—
47	C₂×C₄×C₁₂₇	—	4	12	no	—
48	C₂×C₄×C₁₂₇	—	4	12	no	—
49	C₂×C₄×C₁₂₇	#125	1	12	yes	—
50	C₂²×C₁₂₇	C₂×#125	1	3	yes	—
51	C₂²×C₁₂₇	#538	1	6	yes	—
52	C₂²×C₁₂₇	#539	1	6	yes	—
53	C₂²×C₁₂₇	#543	1	1	yes	characteristic
54	C₂²×C₁₂₇	#544	1	3	yes	—
55	C₂²×C₁₂₇	#555	1	3	yes	—
56	C₂²×C₁₂₇	#556	1	6	yes	—
57	C₂²×C₁₂₇	#559	1	3	yes	—
58	C₂²×C₁₂₇	#562	1	3	yes	—
59	C₂²×C₁₂₇	#564	1	1	yes	characteristic
60	C₄×C₁₂₇	—	2	12	no	—
61	C₄×C₁₂₇	—	8	12	no	—
62	#17081	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×C₁₂₇	#2571	1	3	yes	—
64	C₂×C₁₂₇	#2646	1	3	yes	—
65	C₂×C₁₂₇	#2808	1	6	yes	—
66	C₂×C₁₂₇	#2810	1	2	yes	—
67	C₂×C₁₂₇	#2845	1	1	yes	characteristic
68	#1442	C₂×C₁₂₇	1	12	yes	—
69	C₄⋊#95	C₂×C₁₂₇	1	3	yes	—
70	C₁₂₇	#17081	1	1	yes	characteristic, direct product factor, Sylow
71	C₂²×C₄⋊C₄	C₂²×C₁₂₇	1	6	yes	—
72	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
73	C₂×#95	C₂²×C₁₂₇	1	12	yes	—
74	C₂×#95	C₂²×C₁₂₇	1	12	yes	—
75	C₂×#95	C₂²×C₁₂₇	1	4	yes	—
76	C₂³×C₄	C₂³×C₁₂₇	1	12	yes	—
77	C₂³×C₄	C₂³×C₁₂₇	1	3	yes	—
78	C₂×C₄⋊C₄	—	2	12	no	—
79	C₂×C₄⋊C₄	—	4	6	no	—
80	C₂×C₄⋊C₄	—	4	6	no	—
81	C₂×C₄²	—	4	2	no	—
82	C₂×C₄²	Q₈×C₁₂₇	1	4	yes	—
83	#95	—	2	24	no	—
84	#95	—	4	12	no	—
85	#95	—	4	4	no	—
86	C₂⁴	C₂⁴×C₁₂₇	1	1	yes	characteristic
87	C₂²×C₄	—	2	6	no	—
88	C₂²×C₄	—	2	3	no	—
89	C₂²×C₄	—	2	3	no	—
90	C₂²×C₄	—	2	12	no	—
91	C₂²×C₄	—	2	12	no	—
92	C₂²×C₄	—	2	12	no	—
93	C₂²×C₄	—	2	12	no	—
94	C₂²×C₄	—	2	12	no	—
95	C₂²×C₄	—	2	12	no	—
96	C₂²×C₄	—	2	12	no	—
97	C₂²×C₄	C₂×Q₈×C₁₂₇	1	6	yes	—
98	C₂²×C₄	#41×C₁₂₇	1	12	yes	—
99	C₄⋊C₄	—	4	24	no	—
100	C₄²	—	4	4	no	—
101	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
102	C₂³	C₂×#41×C₁₂₇	1	6	yes	—
103	C₂³	H(5,𝔽₂)×C₁₂₇	1	2	yes	—
104	C₂³	H(5,𝔽₂)×C₁₂₇	1	3	yes	—
105	C₂³	2₋¹⁺⁴×C₁₂₇	1	3	yes	—
106	C₂×C₄	—	2	12	no	—
107	C₂×C₄	—	2	12	no	—
108	C₂×C₄	—	2	6	no	—
109	C₂×C₄	—	2	6	no	—
110	C₂×C₄	—	4	12	no	—
111	C₂×C₄	—	4	12	no	—
112	C₂×C₄	—	4	12	no	—
113	C₂×C₄	—	4	12	no	—
114	C₂×C₄	—	4	12	no	—
115	C₂×C₄	—	4	12	no	—
116	C₂×C₄	—	4	12	no	—
117	C₂×C₄	#125×C₁₂₇	1	12	yes	—
118	C₂²	C₂×#125×C₁₂₇	1	3	yes	—
119	C₂²	#538×C₁₂₇	1	6	yes	—
120	C₂²	#539×C₁₂₇	1	6	yes	—
121	C₂²	#543×C₁₂₇	1	1	yes	characteristic
122	C₂²	#544×C₁₂₇	1	3	yes	—
123	C₂²	#555×C₁₂₇	1	3	yes	—
124	C₂²	#556×C₁₂₇	1	6	yes	—
125	C₂²	#559×C₁₂₇	1	3	yes	—
126	C₂²	#562×C₁₂₇	1	3	yes	—
127	C₂²	#564×C₁₂₇	1	1	yes	characteristic
128	C₄	—	2	12	no	—
129	C₄	—	8	12	no	—
130	C₂	C₁₂₇×#2571	1	3	yes	—
131	C₂	C₁₂₇×#2646	1	3	yes	—
132	C₂	C₁₂₇×#2808	1	6	yes	—
133	C₂	C₁₂₇×#2810	1	2	yes	—
134	C₂	C₁₂₇×#2845	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₂₇×#1442

C₂

yes

maximal

C₁₂₇×C₄⋊#95

C₂

yes

maximal

C₂²×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂²×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×#95×C₁₂₇

C₂²

yes

—

C₂×#95×C₁₂₇

C₂²

yes

—

C₂×#95×C₁₂₇

C₂²

yes

—

C₂³×C₄×C₁₂₇

C₂³

yes

—

C₂³×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄²×C₁₂₇

—

C₂×C₄²×C₁₂₇

Q₈

yes

—

#95×C₁₂₇

—

#95×C₁₂₇

—

#95×C₁₂₇

—

C₂⁴×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂²×C₄×C₁₂₇

#41

yes

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₂³×C₁₂₇

C₂²×Q₈

yes

characteristic

C₂³×C₁₂₇

C₂×#41

yes

—

C₂³×C₁₂₇

H(5,𝔽₂)

yes

—

C₂³×C₁₂₇

H(5,𝔽₂)

yes

—

C₂³×C₁₂₇

2₋¹⁺⁴

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

#125

yes

—

C₂²×C₁₂₇

C₂×#125

yes

—

C₂²×C₁₂₇

#538

yes

—

C₂²×C₁₂₇

#539

yes

—

C₂²×C₁₂₇

#543

yes

characteristic

C₂²×C₁₂₇

#544

yes

—

C₂²×C₁₂₇

#555

yes

—

C₂²×C₁₂₇

#556

yes

—

C₂²×C₁₂₇

#559

yes

—

C₂²×C₁₂₇

#562

yes

—

C₂²×C₁₂₇

#564

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

#17081

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

#2571

yes

—

C₂×C₁₂₇

#2646

yes

—

C₂×C₁₂₇

#2808

yes

—

C₂×C₁₂₇

#2810

yes

—

C₂×C₁₂₇

#2845

yes

characteristic

#1442

C₂×C₁₂₇

yes

—

C₄⋊#95

C₂×C₁₂₇

yes

—

C₁₂₇

#17081

yes

characteristic, direct product factor, Sylow

C₂²×C₄⋊C₄

C₂²×C₁₂₇

yes

—

C₂²×C₄²

C₂²×C₁₂₇

yes

characteristic

C₂×#95

C₂²×C₁₂₇

yes

—

C₂×#95

C₂²×C₁₂₇

yes

—

C₂×#95

C₂²×C₁₂₇

yes

—

C₂³×C₄

C₂³×C₁₂₇

yes

—

C₂³×C₄

C₂³×C₁₂₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

Q₈×C₁₂₇

yes

—

#95

—

#95

—

#95

—

C₂⁴

C₂⁴×C₁₂₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Q₈×C₁₂₇

yes

—

C₂²×C₄

#41×C₁₂₇

yes

—

C₄⋊C₄

—

100

C₄²

—

101

C₂³

C₂²×Q₈×C₁₂₇

yes

characteristic

102

C₂³

C₂×#41×C₁₂₇

yes

—

103

C₂³

H(5,𝔽₂)×C₁₂₇

yes

—

104

C₂³

H(5,𝔽₂)×C₁₂₇

yes

—

105

C₂³

2₋¹⁺⁴×C₁₂₇

yes

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#125×C₁₂₇

yes

—

118

C₂²

C₂×#125×C₁₂₇

yes

—

119

C₂²

#538×C₁₂₇

yes

—

120

C₂²

#539×C₁₂₇

yes

—

121

C₂²

#543×C₁₂₇

yes

characteristic

122

C₂²

#544×C₁₂₇

yes

—

123

C₂²

#555×C₁₂₇

yes

—

124

C₂²

#556×C₁₂₇

yes

—

125

C₂²

#559×C₁₂₇

yes

—

126

C₂²

#562×C₁₂₇

yes

—

127

C₂²

#564×C₁₂₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₂

C₁₂₇×#2571

yes

—

131

C₂

C₁₂₇×#2646

yes

—

132

C₂

C₁₂₇×#2808

yes

—

133

C₂

C₁₂₇×#2810

yes

—

134

C₂

C₁₂₇×#2845

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₂₇	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₁₂₇
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₂₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂⁴×C₁₂₇

C₂⁴

commutator, Frattini, cosocle

C₂⁴

C₂⁴×C₁₂₇

LCS

G ▷ C₂⁴ ▷ C₁

UCS

C₁ ◁ C₂⁴×C₁₂₇ ◁ G

N	H ≅ G/N	type
62. #17081	C₁₂₇	direct (N × H)
70. C₁₂₇	#17081	direct (N × H)

H ≅ G/N

type

62. #17081

C₁₂₇

direct (N × H)

70. C₁₂₇

#17081

direct (N × H)

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

subgroup	index
C₁₂₇×#1442	2
C₁₂₇×C₄⋊#95	2
#17081	127

subgroup

index

C₁₂₇×#1442

C₁₂₇×C₄⋊#95

#17081

127

Subgroups — C₁₂₇×#17081 #44580754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#17081 #44580754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#17081 #44580754