Group #44578009 — subgroups

Subgroups — C₁₂₇×#14336 #44578009

Order 32512.

3998 conjugacy classes of subgroups; only the 702 normal subgroups are shown.

normal subgroups only

This group has 3998 conjugacy classes of subgroups — too many to draw. Only its 702 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	C₂⁴×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#381×C₁₂₇	C₂	1	6	yes	maximal
3	C₁₂₇×#1446	C₂	1	8	yes	maximal, complemented
4	C₂³×Q₈×C₁₂₇	—	2	24	no	—
5	C₂³×Q₈×C₁₂₇	C₄	1	8	yes	complemented
6	C₂³×Q₈×C₁₂₇	C₂²	1	4	yes	—
7	C₂⁴×C₄×C₁₂₇	C₂²	1	3	yes	—
8	C₂²×C₂²⋊C₄×C₁₂₇	C₂²	1	4	yes	—
9	#381×C₁₂₇	—	2	24	no	—
10	C₂×#95×C₁₂₇	C₂²	1	24	yes	—
11	C₂⁵×C₁₂₇	C₂³	1	1	yes	characteristic
12	C₂²×Q₈×C₁₂₇	—	2	128	no	—
13	C₂²×Q₈×C₁₂₇	—	2	96	no	—
14	C₂²×Q₈×C₁₂₇	—	2	48	no	—
15	C₂²×Q₈×C₁₂₇	C₂×C₄	1	16	yes	—
16	C₂³×C₄×C₁₂₇	—	2	24	no	—
17	C₂³×C₄×C₁₂₇	C₂×C₄	1	12	yes	—
18	C₂³×C₄×C₁₂₇	D₄	1	24	yes	—
19	C₂³×C₄×C₁₂₇	C₂³	1	8	yes	—
20	C₂³×C₄×C₁₂₇	C₂³	1	6	yes	—
21	C₂×C₂²⋊C₄×C₁₂₇	—	2	24	no	—
22	#95×C₁₂₇	—	2	48	no	—
23	C₂⁴×C₁₂₇	—	2	8	no	—
24	C₂⁴×C₁₂₇	C₂²×C₄	1	2	yes	—
25	C₂⁴×C₁₂₇	C₂×D₄	1	12	yes	—
26	C₂⁴×C₁₂₇	C₂⁴	1	1	yes	characteristic
27	C₂×Q₈×C₁₂₇	—	2	128	no	—
28	C₂×Q₈×C₁₂₇	—	2	256	no	—
29	C₂×Q₈×C₁₂₇	—	2	96	no	—
30	C₂²×C₄×C₁₂₇	—	2	48	no	—
31	C₂²×C₄×C₁₂₇	—	2	96	no	—
32	C₂²×C₄×C₁₂₇	—	2	24	no	—
33	C₂²×C₄×C₁₂₇	—	2	24	no	—
34	C₂²×C₄×C₁₂₇	—	2	32	no	—
35	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	48	yes	—
36	C₂²×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
37	C₂²×C₄×C₁₂₇	C₂×D₄	1	24	yes	—
38	C₂²⋊C₄×C₁₂₇	—	4	16	no	—
39	C₂³×C₁₂₇	—	2	8	no	—
40	C₂³×C₁₂₇	—	2	48	no	—
41	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	12	yes	—
42	C₂³×C₁₂₇	#120	1	16	yes	—
43	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
44	C₂³×C₁₂₇	C₂²×D₄	1	6	yes	—
45	C₂³×C₁₂₇	2₋¹⁺⁴	1	8	yes	—
46	Q₈×C₁₂₇	—	2	128	no	—
47	C₂×C₄×C₁₂₇	—	2	24	no	—
48	C₂×C₄×C₁₂₇	—	2	48	no	—
49	C₂×C₄×C₁₂₇	—	2	96	no	—
50	C₂×C₄×C₁₂₇	—	4	8	no	—
51	C₂×C₄×C₁₂₇	—	4	48	no	—
52	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	24	yes	—
53	C₂²×C₁₂₇	—	2	24	no	—
54	C₂²×C₁₂₇	—	2	32	no	—
55	C₂²×C₁₂₇	#379	1	8	yes	—
56	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	3	yes	—
57	C₂²×C₁₂₇	#519	1	4	yes	—
58	C₂²×C₁₂₇	C₂×#120	1	4	yes	—
59	C₂²×C₁₂₇	#536	1	24	yes	—
60	C₄×C₁₂₇	—	2	24	no	—
61	C₄×C₁₂₇	—	8	8	no	—
62	#14336	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×C₁₂₇	—	2	8	no	—
64	C₂×C₁₂₇	C₂×#379	1	1	yes	characteristic
65	C₂×C₁₂₇	#2310	1	6	yes	—
66	C₂×C₁₂₇	#2614	1	8	yes	—
67	C₂⁴×Q₈	C₂×C₁₂₇	1	1	yes	characteristic
68	C₂×#381	C₂×C₁₂₇	1	6	yes	—
69	#1446	C₂×C₁₂₇	1	8	yes	complemented
70	C₁₂₇	#14336	1	1	yes	characteristic, direct product factor, Sylow
71	C₂³×Q₈	—	2	24	no	—
72	C₂³×Q₈	C₄×C₁₂₇	1	8	yes	complemented
73	C₂³×Q₈	C₂²×C₁₂₇	1	4	yes	—
74	C₂⁴×C₄	C₂²×C₁₂₇	1	3	yes	—
75	C₂²×C₂²⋊C₄	C₂²×C₁₂₇	1	4	yes	—
76	#381	—	2	24	no	—
77	C₂×#95	C₂²×C₁₂₇	1	24	yes	—
78	C₂⁵	C₂³×C₁₂₇	1	1	yes	characteristic
79	C₂²×Q₈	—	2	128	no	—
80	C₂²×Q₈	—	2	96	no	—
81	C₂²×Q₈	—	2	48	no	—
82	C₂²×Q₈	C₂×C₄×C₁₂₇	1	16	yes	—
83	C₂³×C₄	—	2	24	no	—
84	C₂³×C₄	C₂×C₄×C₁₂₇	1	12	yes	—
85	C₂³×C₄	D₄×C₁₂₇	1	24	yes	—
86	C₂³×C₄	C₂³×C₁₂₇	1	8	yes	—
87	C₂³×C₄	C₂³×C₁₂₇	1	6	yes	—
88	C₂×C₂²⋊C₄	—	2	24	no	—
89	#95	—	2	48	no	—
90	C₂⁴	—	2	8	no	—
91	C₂⁴	C₂²×C₄×C₁₂₇	1	2	yes	—
92	C₂⁴	C₂×D₄×C₁₂₇	1	12	yes	—
93	C₂⁴	C₂⁴×C₁₂₇	1	1	yes	characteristic
94	C₂×Q₈	—	2	128	no	—
95	C₂×Q₈	—	2	256	no	—
96	C₂×Q₈	—	2	96	no	—
97	C₂²×C₄	—	2	48	no	—
98	C₂²×C₄	—	2	96	no	—
99	C₂²×C₄	—	2	24	no	—
100	C₂²×C₄	—	2	24	no	—
101	C₂²×C₄	—	2	32	no	—
102	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	48	yes	—
103	C₂²×C₄	C₂²×C₄×C₁₂₇	1	12	yes	—
104	C₂²×C₄	C₂×D₄×C₁₂₇	1	24	yes	—
105	C₂²⋊C₄	—	4	16	no	—
106	C₂³	—	2	8	no	—
107	C₂³	—	2	48	no	—
108	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	12	yes	—
109	C₂³	#120×C₁₂₇	1	16	yes	—
110	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
111	C₂³	C₂²×D₄×C₁₂₇	1	6	yes	—
112	C₂³	2₋¹⁺⁴×C₁₂₇	1	8	yes	—
113	Q₈	—	2	128	no	—
114	C₂×C₄	—	2	24	no	—
115	C₂×C₄	—	2	48	no	—
116	C₂×C₄	—	2	96	no	—
117	C₂×C₄	—	4	8	no	—
118	C₂×C₄	—	4	48	no	—
119	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	24	yes	—
120	C₂²	—	2	32	no	—
121	C₂²	—	2	24	no	—
122	C₂²	#379×C₁₂₇	1	8	yes	—
123	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	3	yes	—
124	C₂²	#519×C₁₂₇	1	4	yes	—
125	C₂²	C₂×#120×C₁₂₇	1	4	yes	—
126	C₂²	#536×C₁₂₇	1	24	yes	—
127	C₄	—	2	24	no	—
128	C₄	—	8	8	no	—
129	C₂	—	2	8	no	—
130	C₂	C₂×#379×C₁₂₇	1	1	yes	characteristic
131	C₂	C₁₂₇×#2310	1	6	yes	—
132	C₂	C₁₂₇×#2614	1	8	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁴×Q₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#381×C₁₂₇

C₂

yes

maximal

C₁₂₇×#1446

C₂

yes

maximal, complemented

C₂³×Q₈×C₁₂₇

—

C₂³×Q₈×C₁₂₇

C₄

yes

complemented

C₂³×Q₈×C₁₂₇

C₂²

yes

—

C₂⁴×C₄×C₁₂₇

C₂²

yes

—

C₂²×C₂²⋊C₄×C₁₂₇

C₂²

yes

—

#381×C₁₂₇

—

C₂×#95×C₁₂₇

C₂²

yes

—

C₂⁵×C₁₂₇

C₂³

yes

characteristic

C₂²×Q₈×C₁₂₇

—

128

—

C₂²×Q₈×C₁₂₇

—

C₂²×Q₈×C₁₂₇

—

C₂²×Q₈×C₁₂₇

C₂×C₄

yes

—

C₂³×C₄×C₁₂₇

—

C₂³×C₄×C₁₂₇

C₂×C₄

yes

—

C₂³×C₄×C₁₂₇

D₄

yes

—

C₂³×C₄×C₁₂₇

C₂³

yes

—

C₂³×C₄×C₁₂₇

C₂³

yes

—

C₂×C₂²⋊C₄×C₁₂₇

—

#95×C₁₂₇

—

C₂⁴×C₁₂₇

—

C₂⁴×C₁₂₇

C₂²×C₄

yes

—

C₂⁴×C₁₂₇

C₂×D₄

yes

—

C₂⁴×C₁₂₇

C₂⁴

yes

characteristic

C₂×Q₈×C₁₂₇

—

128

—

C₂×Q₈×C₁₂₇

—

256

—

C₂×Q₈×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×D₄

yes

—

C₂²⋊C₄×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₂×C₂²⋊C₄

yes

—

C₂³×C₁₂₇

#120

yes

—

C₂³×C₁₂₇

C₂³×C₄

yes

characteristic

C₂³×C₁₂₇

C₂²×D₄

yes

—

C₂³×C₁₂₇

2₋¹⁺⁴

yes

—

Q₈×C₁₂₇

—

128

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#379

yes

—

C₂²×C₁₂₇

C₂²×C₂²⋊C₄

yes

—

C₂²×C₁₂₇

#519

yes

—

C₂²×C₁₂₇

C₂×#120

yes

—

C₂²×C₁₂₇

#536

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

#14336

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×#379

yes

characteristic

C₂×C₁₂₇

#2310

yes

—

C₂×C₁₂₇

#2614

yes

—

C₂⁴×Q₈

C₂×C₁₂₇

yes

characteristic

C₂×#381

C₂×C₁₂₇

yes

—

#1446

C₂×C₁₂₇

yes

complemented

C₁₂₇

#14336

yes

characteristic, direct product factor, Sylow

C₂³×Q₈

—

C₂³×Q₈

C₄×C₁₂₇

yes

complemented

C₂³×Q₈

C₂²×C₁₂₇

yes

—

C₂⁴×C₄

C₂²×C₁₂₇

yes

—

C₂²×C₂²⋊C₄

C₂²×C₁₂₇

yes

—

#381

—

C₂×#95

C₂²×C₁₂₇

yes

—

C₂⁵

C₂³×C₁₂₇

yes

characteristic

C₂²×Q₈

—

128

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×C₄×C₁₂₇

yes

—

C₂³×C₄

—

C₂³×C₄

C₂×C₄×C₁₂₇

yes

—

C₂³×C₄

D₄×C₁₂₇

yes

—

C₂³×C₄

C₂³×C₁₂₇

yes

—

C₂³×C₄

C₂³×C₁₂₇

yes

—

C₂×C₂²⋊C₄

—

#95

—

C₂⁴

—

C₂⁴

C₂²×C₄×C₁₂₇

yes

—

C₂⁴

C₂×D₄×C₁₂₇

yes

—

C₂⁴

C₂⁴×C₁₂₇

yes

characteristic

C₂×Q₈

—

128

—

C₂×Q₈

—

256

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²⋊C₄×C₁₂₇

yes

—

103

C₂²×C₄

C₂²×C₄×C₁₂₇

yes

—

104

C₂²×C₄

C₂×D₄×C₁₂₇

yes

—

105

C₂²⋊C₄

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₂×C₂²⋊C₄×C₁₂₇

yes

—

109

C₂³

#120×C₁₂₇

yes

—

110

C₂³

C₂³×C₄×C₁₂₇

yes

characteristic

111

C₂³

C₂²×D₄×C₁₂₇

yes

—

112

C₂³

2₋¹⁺⁴×C₁₂₇

yes

—

113

Q₈

—

128

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂×C₂²⋊C₄×C₁₂₇

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

#379×C₁₂₇

yes

—

123

C₂²

C₂²×C₂²⋊C₄×C₁₂₇

yes

—

124

C₂²

#519×C₁₂₇

yes

—

125

C₂²

C₂×#120×C₁₂₇

yes

—

126

C₂²

#536×C₁₂₇

yes

—

127

C₄

—

128

C₄

—

129

C₂

—

130

C₂

C₂×#379×C₁₂₇

yes

characteristic

131

C₂

C₁₂₇×#2310

yes

—

132

C₂

C₁₂₇×#2614

yes

—

	subgroup	G/H
center, socle	C₂⁴×C₁₂₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂⁴	C₂⁴×C₁₂₇
prime radical	C₂⁵×C₁₂₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₂₇ ◁ G

subgroup

G/H

center, socle

C₂⁴×C₁₂₇

C₂⁴

commutator

C₂³

C₂³×C₄×C₁₂₇

Frattini, cosocle

C₂⁴

C₂⁴×C₁₂₇

prime radical

C₂⁵×C₁₂₇

C₂³

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂⁴×C₁₂₇ ◁ G

N	H ≅ G/N	type
3. C₁₂₇×#1446	C₂	semidirect
5. C₂³×Q₈×C₁₂₇	C₄	semidirect
62. #14336	C₁₂₇	direct (N × H)
69. #1446	C₂×C₁₂₇	semidirect
70. C₁₂₇	#14336	direct (N × H)
72. C₂³×Q₈	C₄×C₁₂₇	semidirect

H ≅ G/N

type

3. C₁₂₇×#1446

C₂

semidirect

5. C₂³×Q₈×C₁₂₇

C₄

semidirect

62. #14336

C₁₂₇

direct (N × H)

69. #1446

C₂×C₁₂₇

semidirect

70. C₁₂₇

#14336

direct (N × H)

72. C₂³×Q₈

C₄×C₁₂₇

semidirect

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

subgroup	index
C₂⁴×Q₈×C₁₂₇	2
C₂×#381×C₁₂₇	2
C₁₂₇×#1446	2
#14336	127

subgroup

index

C₂⁴×Q₈×C₁₂₇

C₂×#381×C₁₂₇

C₁₂₇×#1446

#14336

127

Subgroups — C₁₂₇×#14336 #44578009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#14336 #44578009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#14336 #44578009