Group #44570869 — subgroups

Subgroups — C₁₂₇×#7196 #44570869

Order 32512.

210 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2163	C₂	1	2	yes	maximal
2	C₂×#322×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
4	C₂×C₄×C₈×C₁₂₇	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₄⋊C₁₆×C₁₂₇	—	2	2	no	—
7	C₂²⋊C₁₆×C₁₂₇	C₄	1	4	yes	—
8	#346×C₁₂₇	—	2	2	no	—
9	#322×C₁₂₇	—	2	2	no	—
10	C₂²×C₈×C₁₂₇	—	2	1	no	—
11	C₂²×C₈×C₁₂₇	—	2	1	no	—
12	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₁₂₇	—	2	4	no	—
15	C₄×C₈×C₁₂₇	—	2	1	no	—
16	C₄×C₈×C₁₂₇	—	2	1	no	—
17	C₄×C₈×C₁₂₇	C₈	1	2	yes	—
18	C₄×C₈×C₁₂₇	C₈	1	2	yes	—
19	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
20	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
21	C₄×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
23	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
24	C₂²×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
25	C₂×C₈×C₁₂₇	—	2	4	no	—
26	C₂×C₈×C₁₂₇	—	2	2	no	—
27	C₂×C₈×C₁₂₇	—	2	2	no	—
28	C₂×C₈×C₁₂₇	—	4	2	no	—
29	C₂×C₈×C₁₂₇	—	4	2	no	—
30	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
31	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
32	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₄²×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
34	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
36	C₁₆×C₁₂₇	—	4	4	no	—
37	C₂³×C₁₂₇	#97	1	1	yes	characteristic
38	C₂×C₄×C₁₂₇	—	2	2	no	—
39	C₂×C₄×C₁₂₇	—	2	2	no	—
40	C₂×C₄×C₁₂₇	—	2	2	no	—
41	C₂×C₄×C₁₂₇	#95	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₁₂₇	C₄⋊C₈	1	1	yes	characteristic
47	C₈×C₁₂₇	—	2	4	no	—
48	C₈×C₁₂₇	—	2	2	no	—
49	C₈×C₁₂₇	—	2	2	no	—
50	C₈×C₁₂₇	—	4	2	no	—
51	C₂²×C₁₂₇	—	2	1	no	—
52	C₂²×C₁₂₇	—	2	1	no	—
53	C₂²×C₁₂₇	#336	1	1	yes	characteristic
54	C₂²×C₁₂₇	#347	1	2	yes	—
55	C₄×C₁₂₇	—	2	1	no	—
56	C₄×C₁₂₇	—	2	1	no	—
57	C₄×C₁₂₇	—	2	1	no	—
58	C₄×C₁₂₇	—	2	1	no	—
59	C₄×C₁₂₇	#344	1	2	yes	—
60	C₄×C₁₂₇	#364	1	2	yes	—
61	#7196	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₁₂₇	—	2	2	no	—
63	C₂×C₁₂₇	#1281	1	1	yes	characteristic
64	C₂×C₁₂₇	#1375	1	1	yes	characteristic
65	C₂×C₁₂₇	#1376	1	1	yes	characteristic
66	#2163	C₂×C₁₂₇	1	2	yes	—
67	C₂×#322	C₂×C₁₂₇	1	1	yes	characteristic
68	C₁₂₇	#7196	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₄×C₈	C₄×C₁₂₇	1	1	yes	characteristic
70	C₂×C₄×C₈	C₄×C₁₂₇	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
72	C₄⋊C₁₆	—	2	2	no	—
73	C₂²⋊C₁₆	C₄×C₁₂₇	1	4	yes	—
74	#346	—	2	2	no	—
75	#322	—	2	2	no	—
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	—	2	1	no	—
78	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
79	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
80	C₂×C₁₆	—	2	4	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	—	2	1	no	—
83	C₄×C₈	C₈×C₁₂₇	1	2	yes	—
84	C₄×C₈	C₈×C₁₂₇	1	2	yes	—
85	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
86	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
87	C₄×C₈	Q₈×C₁₂₇	1	1	yes	characteristic
88	C₂²×C₄	C₄²×C₁₂₇	1	1	yes	characteristic
89	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
90	C₂²×C₄	M₁₆×C₁₂₇	1	1	yes	characteristic
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
97	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
98	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
99	C₄²	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
100	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
101	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
102	C₁₆	—	4	4	no	—
103	C₂³	#97×C₁₂₇	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	#95×C₁₂₇	1	1	yes	characteristic
108	C₂×C₄	C₄×C₈×C₁₂₇	1	1	yes	characteristic
109	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
110	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
111	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
112	C₂×C₄	C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
113	C₈	—	2	4	no	—
114	C₈	—	2	2	no	—
115	C₈	—	2	2	no	—
116	C₈	—	4	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#336×C₁₂₇	1	1	yes	characteristic
120	C₂²	#347×C₁₂₇	1	2	yes	—
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	#344×C₁₂₇	1	2	yes	—
126	C₄	#364×C₁₂₇	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	C₁₂₇×#1281	1	1	yes	characteristic
129	C₂	C₁₂₇×#1375	1	1	yes	characteristic
130	C₂	C₁₂₇×#1376	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₂₇×#2163

C₂

yes

maximal

C₂×#322×C₁₂₇

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₂₇

C₄

yes

characteristic

C₂×C₄×C₈×C₁₂₇

C₄

yes

characteristic

C₂×C₄×C₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₁₂₇

—

C₂²⋊C₁₆×C₁₂₇

C₄

yes

—

#346×C₁₂₇

—

#322×C₁₂₇

—

C₂²×C₈×C₁₂₇

—

C₂²×C₈×C₁₂₇

—

C₂²×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₄²×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₂₇

—

C₄×C₈×C₁₂₇

—

C₄×C₈×C₁₂₇

—

C₄×C₈×C₁₂₇

C₈

yes

—

C₄×C₈×C₁₂₇

C₈

yes

—

C₄×C₈×C₁₂₇

D₄

yes

—

C₄×C₈×C₁₂₇

D₄

yes

characteristic

C₄×C₈×C₁₂₇

Q₈

yes

characteristic

C₂²×C₄×C₁₂₇

C₄²

yes

characteristic

C₂²×C₄×C₁₂₇

M₁₆

yes

characteristic

C₂²×C₄×C₁₂₇

M₁₆

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₂₇

C₄⋊C₄

yes

—

C₄²×C₁₂₇

C₂²⋊C₄

yes

characteristic

C₄²×C₁₂₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₂₇

C₂×C₈

yes

characteristic

C₄²×C₁₂₇

C₂×C₈

yes

characteristic

C₁₆×C₁₂₇

—

C₂³×C₁₂₇

#97

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

#95

yes

characteristic

C₂×C₄×C₁₂₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₂₇

C₄⋊C₈

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#336

yes

characteristic

C₂²×C₁₂₇

#347

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#344

yes

—

C₄×C₁₂₇

#364

yes

—

#7196

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

C₂×#322

C₂×C₁₂₇

yes

characteristic

C₁₂₇

#7196

yes

characteristic, direct product factor, Sylow

C₂×C₄×C₈

C₄×C₁₂₇

yes

characteristic

C₂×C₄×C₈

C₄×C₁₂₇

yes

characteristic

C₂×C₄×C₈

C₂²×C₁₂₇

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

C₄×C₁₂₇

yes

—

#346

—

#322

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₁₂₇

yes

—

C₂×C₄²

C₂×C₄×C₁₂₇

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₈×C₁₂₇

yes

—

C₄×C₈

C₈×C₁₂₇

yes

—

C₄×C₈

D₄×C₁₂₇

yes

—

C₄×C₈

D₄×C₁₂₇

yes

characteristic

C₄×C₈

Q₈×C₁₂₇

yes

characteristic

C₂²×C₄

C₄²×C₁₂₇

yes

characteristic

C₂²×C₄

M₁₆×C₁₂₇

yes

characteristic

C₂²×C₄

M₁₆×C₁₂₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄×C₁₂₇

yes

—

C₂×C₈

C₄⋊C₄×C₁₂₇

yes

—

C₄²

C₂²⋊C₄×C₁₂₇

yes

characteristic

C₄²

C₄⋊C₄×C₁₂₇

yes

characteristic

100

C₄²

C₂×C₈×C₁₂₇

yes

characteristic

101

C₄²

C₂×C₈×C₁₂₇

yes

characteristic

102

C₁₆

—

103

C₂³

#97×C₁₂₇

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#95×C₁₂₇

yes

characteristic

108

C₂×C₄

C₄×C₈×C₁₂₇

yes

characteristic

109

C₂×C₄

C₂²⋊C₈×C₁₂₇

yes

characteristic

110

C₂×C₄

C₂²⋊C₈×C₁₂₇

yes

characteristic

111

C₂×C₄

C₄⋊C₈×C₁₂₇

yes

characteristic

112

C₂×C₄

C₄⋊C₈×C₁₂₇

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

#336×C₁₂₇

yes

characteristic

120

C₂²

#347×C₁₂₇

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#344×C₁₂₇

yes

—

126

C₄

#364×C₁₂₇

yes

—

127

C₂

—

128

C₂

C₁₂₇×#1281

yes

characteristic

129

C₂

C₁₂₇×#1375

yes

characteristic

130

C₂

C₁₂₇×#1376

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#95
commutator	C₂×C₄	C₄×C₈×C₁₂₇
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₂₇
socle	C₂²×C₁₂₇	#336
prime radical	C₂³×C₁₂₇	#97
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₂₇

#95

commutator

C₂×C₄

C₄×C₈×C₁₂₇

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₁₂₇

C₂²×C₁₂₇

C₂³×C₁₂₇

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

N	H ≅ G/N	type
61. #7196	C₁₂₇	direct (N × H)
68. C₁₂₇	#7196	direct (N × H)

H ≅ G/N

type

61. #7196

C₁₂₇

direct (N × H)

68. C₁₂₇

#7196

direct (N × H)

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

subgroup	index
C₁₂₇×#2163	2
C₂×#322×C₁₂₇	2
#7196	127

subgroup

index

C₁₂₇×#2163

C₂×#322×C₁₂₇

#7196

127

Subgroups — C₁₂₇×#7196 #44570869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#7196 #44570869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#7196 #44570869