Group #44576557 — subgroups

Subgroups — C₁₂₇×#12884 #44576557

Order 32512.

302 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 166 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×#2150	C₂	1	2	yes	maximal
2	C₁₂₇×#1726	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1376	C₂	1	4	yes	maximal
4	C₂×M₃₂×C₁₂₇	C₂²	1	4	yes	—
5	C₂×#97×C₁₂₇	C₄	1	2	yes	—
6	C₂×#97×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	#376×C₁₂₇	C₄	1	1	yes	characteristic
10	C₄⋊C₁₆×C₁₂₇	—	2	4	no	—
11	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	M₃₂×C₁₂₇	—	2	8	no	—
17	C₂×C₁₆×C₁₂₇	—	2	4	no	—
18	#97×C₁₂₇	—	2	4	no	—
19	#97×C₁₂₇	C₈	1	4	yes	—
20	C₄×C₈×C₁₂₇	—	2	4	no	—
21	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
22	C₄×C₈×C₁₂₇	Q₈	1	2	yes	—
23	#95×C₁₂₇	—	2	2	no	—
24	C₂²×C₄×C₁₂₇	—	2	2	no	—
25	C₂²×C₄×C₁₂₇	M₁₆	1	2	yes	—
26	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₂₇	—	2	2	no	—
28	C₂×C₈×C₁₂₇	—	2	2	no	—
29	C₂×C₈×C₁₂₇	—	2	2	no	—
30	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₁₂₇	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₁₂₇	#41	1	4	yes	—
35	C₄²×C₁₂₇	—	2	2	no	—
36	C₄²×C₁₂₇	C₄⋊C₄	1	2	yes	—
37	C₄²×C₁₂₇	C₂×C₈	1	2	yes	—
38	C₄²×C₁₂₇	M₁₆	1	4	yes	—
39	C₁₆×C₁₂₇	—	4	4	no	—
40	C₂³×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	—	2	4	no	—
42	C₂×C₄×C₁₂₇	—	4	2	no	—
43	C₂×C₄×C₁₂₇	—	4	4	no	—
44	C₂×C₄×C₁₂₇	C₄⋊C₈	1	4	yes	—
45	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	#117	1	2	yes	—
47	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
49	C₈×C₁₂₇	—	2	2	no	—
50	C₈×C₁₂₇	—	2	4	no	—
51	C₈×C₁₂₇	#124	1	2	yes	—
52	C₈×C₁₂₇	#125	1	2	yes	—
53	C₂²×C₁₂₇	—	2	2	no	—
54	C₂²×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
55	C₂²×C₁₂₇	#432	1	2	yes	—
56	C₄×C₁₂₇	—	2	4	no	—
57	C₄×C₁₂₇	—	4	4	no	—
58	C₄×C₁₂₇	#382	1	1	yes	characteristic
59	C₄×C₁₂₇	#423	1	1	yes	characteristic
60	C₄×C₁₂₇	#431	1	2	yes	—
61	#12884	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₁₂₇	—	2	2	no	—
63	C₂×C₁₂₇	#1831	1	1	yes	characteristic
64	C₂×C₁₂₇	#2153	1	2	yes	—
65	#2150	C₂×C₁₂₇	1	2	yes	—
66	#1726	C₂×C₁₂₇	1	1	yes	characteristic
67	#1376	C₂×C₁₂₇	1	4	yes	—
68	C₁₂₇	#12884	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×M₃₂	C₂²×C₁₂₇	1	4	yes	—
70	C₂×#97	C₄×C₁₂₇	1	2	yes	—
71	C₂×#97	C₄×C₁₂₇	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
73	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
74	#376	C₄×C₁₂₇	1	1	yes	characteristic
75	C₄⋊C₁₆	—	2	4	no	—
76	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
77	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
78	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
79	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
80	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
81	M₃₂	—	2	8	no	—
82	C₂×C₁₆	—	2	4	no	—
83	#97	—	2	4	no	—
84	#97	C₈×C₁₂₇	1	4	yes	—
85	C₄×C₈	—	2	4	no	—
86	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
87	C₄×C₈	Q₈×C₁₂₇	1	2	yes	—
88	#95	—	2	2	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	M₁₆×C₁₂₇	1	2	yes	—
91	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
96	C₂×C₈	C₂×C₈×C₁₂₇	1	4	yes	—
97	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
98	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
99	C₂×C₈	#41×C₁₂₇	1	4	yes	—
100	C₄²	—	2	2	no	—
101	C₄²	C₄⋊C₄×C₁₂₇	1	2	yes	—
102	C₄²	C₂×C₈×C₁₂₇	1	2	yes	—
103	C₄²	M₁₆×C₁₂₇	1	4	yes	—
104	C₁₆	—	4	4	no	—
105	C₂³	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	C₄⋊C₈×C₁₂₇	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
111	C₂×C₄	#117×C₁₂₇	1	2	yes	—
112	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
113	C₂×C₄	C₂×M₁₆×C₁₂₇	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	2	4	no	—
116	C₈	#124×C₁₂₇	1	2	yes	—
117	C₈	#125×C₁₂₇	1	2	yes	—
118	C₂²	—	2	2	no	—
119	C₂²	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
120	C₂²	#432×C₁₂₇	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	#382×C₁₂₇	1	1	yes	characteristic
124	C₄	#423×C₁₂₇	1	1	yes	characteristic
125	C₄	#431×C₁₂₇	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	C₁₂₇×#1831	1	1	yes	characteristic
128	C₂	C₁₂₇×#2153	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₂₇×#2150

C₂

yes

maximal

C₁₂₇×#1726

C₂

yes

maximal, characteristic

C₁₂₇×#1376

C₂

yes

maximal

C₂×M₃₂×C₁₂₇

C₂²

yes

—

C₂×#97×C₁₂₇

C₄

yes

—

C₂×#97×C₁₂₇

C₄

yes

characteristic

C₂×C₄×C₈×C₁₂₇

C₂²

yes

—

C₂×C₄×C₈×C₁₂₇

C₂²

yes

characteristic

#376×C₁₂₇

C₄

yes

characteristic

C₄⋊C₁₆×C₁₂₇

—

C₂²×C₈×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₂₇

C₂×C₄

yes

—

C₂²×C₈×C₁₂₇

C₂³

yes

characteristic

C₂×C₄²×C₁₂₇

C₂×C₄

yes

—

C₂×C₄²×C₁₂₇

C₂×C₄

yes

characteristic

M₃₂×C₁₂₇

—

C₂×C₁₆×C₁₂₇

—

#97×C₁₂₇

—

#97×C₁₂₇

C₈

yes

—

C₄×C₈×C₁₂₇

—

C₄×C₈×C₁₂₇

D₄

yes

—

C₄×C₈×C₁₂₇

Q₈

yes

—

#95×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

M₁₆

yes

—

C₂²×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₂₇

C₂×C₈

yes

—

C₂×C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₂₇

#41

yes

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

C₄⋊C₄

yes

—

C₄²×C₁₂₇

C₂×C₈

yes

—

C₄²×C₁₂₇

M₁₆

yes

—

C₁₆×C₁₂₇

—

C₂³×C₁₂₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄⋊C₈

yes

—

C₂×C₄×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

#117

yes

—

C₂×C₄×C₁₂₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₂×M₁₆

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

#124

yes

—

C₈×C₁₂₇

#125

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₁₂₇

#432

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

—

#12884

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

—

C₂×C₁₂₇

#1831

yes

characteristic

C₂×C₁₂₇

#2153

yes

—

#2150

C₂×C₁₂₇

yes

—

#1726

C₂×C₁₂₇

yes

characteristic

#1376

C₂×C₁₂₇

yes

—

C₁₂₇

#12884

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₂²×C₁₂₇

yes

—

C₂×#97

C₄×C₁₂₇

yes

—

C₂×#97

C₄×C₁₂₇

yes

characteristic

C₂×C₄×C₈

C₂²×C₁₂₇

yes

—

C₂×C₄×C₈

C₂²×C₁₂₇

yes

characteristic

#376

C₄×C₁₂₇

yes

characteristic

C₄⋊C₁₆

—

C₂²×C₈

C₂×C₄×C₁₂₇

yes

characteristic

C₂²×C₈

C₂×C₄×C₁₂₇

yes

—

C₂²×C₈

C₂³×C₁₂₇

yes

characteristic

C₂×C₄²

C₂×C₄×C₁₂₇

yes

—

C₂×C₄²

C₂×C₄×C₁₂₇

yes

characteristic

M₃₂

—

C₂×C₁₆

—

#97

—

#97

C₈×C₁₂₇

yes

—

C₄×C₈

—

C₄×C₈

D₄×C₁₂₇

yes

—

C₄×C₈

Q₈×C₁₂₇

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

M₁₆×C₁₂₇

yes

—

C₂²×C₄

C₂²×C₄×C₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₁₂₇

yes

—

C₂×C₈

C₂×C₈×C₁₂₇

yes

—

C₂×C₈

C₂×D₄×C₁₂₇

yes

characteristic

C₂×C₈

C₂×Q₈×C₁₂₇

yes

characteristic

C₂×C₈

#41×C₁₂₇

yes

—

100

C₄²

—

101

C₄²

C₄⋊C₄×C₁₂₇

yes

—

102

C₄²

C₂×C₈×C₁₂₇

yes

—

103

C₄²

M₁₆×C₁₂₇

yes

—

104

C₁₆

—

105

C₂³

C₂×M₁₆×C₁₂₇

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊C₈×C₁₂₇

yes

—

110

C₂×C₄

C₂×C₄⋊C₄×C₁₂₇

yes

characteristic

111

C₂×C₄

#117×C₁₂₇

yes

—

112

C₂×C₄

C₂²×C₈×C₁₂₇

yes

characteristic

113

C₂×C₄

C₂×M₁₆×C₁₂₇

yes

—

114

C₈

—

115

C₈

—

116

C₈

#124×C₁₂₇

yes

—

117

C₈

#125×C₁₂₇

yes

—

118

C₂²

—

119

C₂²

C₂×C₄⋊C₈×C₁₂₇

yes

characteristic

120

C₂²

#432×C₁₂₇

yes

—

121

C₄

—

122

C₄

—

123

C₄

#382×C₁₂₇

yes

characteristic

124

C₄

#423×C₁₂₇

yes

characteristic

125

C₄

#431×C₁₂₇

yes

—

126

C₂

—

127

C₂

C₁₂₇×#1831

yes

characteristic

128

C₂

C₁₂₇×#2153

yes

—

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₂₇	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

subgroup

G/H

center

C₂×C₈×C₁₂₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₈×C₁₂₇

Frattini, cosocle

C₂²×C₈

C₂³×C₁₂₇

socle

C₂²×C₁₂₇

C₂×C₄⋊C₈

prime radical

C₂³×C₁₂₇

C₂×M₁₆

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

N	H ≅ G/N	type
61. #12884	C₁₂₇	direct (N × H)
68. C₁₂₇	#12884	direct (N × H)

H ≅ G/N

type

61. #12884

C₁₂₇

direct (N × H)

68. C₁₂₇

#12884

direct (N × H)

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

subgroup	index
C₁₂₇×#2150	2
C₁₂₇×#1726	2
C₁₂₇×#1376	2
#12884	127

subgroup

index

C₁₂₇×#2150

C₁₂₇×#1726

C₁₂₇×#1376

#12884

127

Subgroups — C₁₂₇×#12884 #44576557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#12884 #44576557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12884 #44576557