Group #41038555 — subgroups

Subgroups — C₁₀₇×#12591 #41038555

Order 27392.

390 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 254 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×#346×C₁₀₇	C₂	1	2	yes	maximal
2	C₄²×C₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
3	C₁₀₇×#1374	C₂	1	4	yes	maximal
4	C₂²×C₁₆×C₁₀₇	C₂²	1	4	yes	—
5	C₂×C₄×C₈×C₁₀₇	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₁₀₇	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₀₇	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
9	C₄³×C₁₀₇	C₄	1	1	yes	characteristic
10	#346×C₁₀₇	C₄	1	8	yes	complemented
11	C₂²×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
12	C₂²×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₁₀₇	C₈	1	4	yes	—
15	C₂×C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆×C₁₀₇	—	2	8	no	—
18	C₂×C₁₆×C₁₀₇	C₂×C₄	1	8	yes	—
19	C₄×C₈×C₁₀₇	—	2	4	no	—
20	C₄×C₈×C₁₀₇	—	2	4	no	—
21	C₄×C₈×C₁₀₇	C₈	1	4	yes	—
22	C₄×C₈×C₁₀₇	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₁₀₇	C₂×C₈	1	2	yes	—
24	C₂²×C₄×C₁₀₇	C₂×C₈	1	4	yes	—
25	C₂²×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₀₇	—	2	2	no	—
27	C₂×C₈×C₁₀₇	—	2	2	no	—
28	C₂×C₈×C₁₀₇	C₄²	1	2	yes	—
29	C₂×C₈×C₁₀₇	C₂×C₈	1	4	yes	—
30	C₂×C₈×C₁₀₇	M₁₆	1	4	yes	—
31	C₂×C₈×C₁₀₇	C₂²×C₄	1	2	yes	—
32	C₂×C₈×C₁₀₇	#41	1	4	yes	—
33	C₄²×C₁₀₇	—	2	8	no	—
34	C₄²×C₁₀₇	—	2	2	no	—
35	C₄²×C₁₀₇	—	2	2	no	—
36	C₄²×C₁₀₇	C₄²	1	2	yes	—
37	C₄²×C₁₀₇	C₂×C₈	1	2	yes	—
38	C₁₆×C₁₀₇	—	2	8	no	—
39	C₂³×C₁₀₇	C₂²×C₈	1	1	yes	characteristic
40	C₂×C₄×C₁₀₇	—	2	4	no	—
41	C₂×C₄×C₁₀₇	—	2	4	no	—
42	C₂×C₄×C₁₀₇	—	2	4	no	—
43	C₂×C₄×C₁₀₇	C₄×C₈	1	4	yes	—
44	C₂×C₄×C₁₀₇	M₃₂	1	8	yes	—
45	C₂×C₄×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄×C₁₀₇	#117	1	2	yes	—
47	C₂×C₄×C₁₀₇	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₁₀₇	C₂×M₁₆	1	2	yes	—
49	C₈×C₁₀₇	—	2	4	no	—
50	C₈×C₁₀₇	#97	1	4	yes	—
51	C₈×C₁₀₇	#117	1	4	yes	—
52	C₂²×C₁₀₇	C₂×C₄×C₈	1	1	yes	characteristic
53	C₂²×C₁₀₇	#431	1	2	yes	—
54	C₂²×C₁₀₇	C₂×M₃₂	1	4	yes	—
55	C₄×C₁₀₇	—	2	4	no	—
56	C₄×C₁₀₇	—	2	8	no	—
57	C₄×C₁₀₇	#376	1	1	yes	characteristic
58	C₄×C₁₀₇	C₂×#97	1	1	yes	characteristic
59	C₄×C₁₀₇	#431	1	2	yes	—
60	#12591	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₁₀₇	#1744	1	1	yes	characteristic
62	C₂×C₁₀₇	C₄×M₃₂	1	2	yes	—
63	C₂×C₁₀₇	#2163	1	4	yes	—
64	C₂×#346	C₂×C₁₀₇	1	2	yes	—
65	C₄²×C₈	C₂×C₁₀₇	1	1	yes	characteristic
66	#1374	C₂×C₁₀₇	1	4	yes	—
67	C₁₀₇	#12591	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²×C₁₆	C₂²×C₁₀₇	1	4	yes	—
69	C₂×C₄×C₈	C₄×C₁₀₇	1	2	yes	—
70	C₂×C₄×C₈	C₄×C₁₀₇	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂²×C₁₀₇	1	2	yes	—
72	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
73	C₄³	C₄×C₁₀₇	1	1	yes	characteristic
74	#346	C₄×C₁₀₇	1	8	yes	complemented
75	C₂²×C₈	C₂×C₄×C₁₀₇	1	2	yes	—
76	C₂²×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
77	C₂²×C₈	C₂³×C₁₀₇	1	1	yes	characteristic
78	C₂×C₄²	C₈×C₁₀₇	1	4	yes	—
79	C₂×C₄²	C₂×C₄×C₁₀₇	1	2	yes	—
80	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
81	C₂×C₁₆	—	2	8	no	—
82	C₂×C₁₆	C₂×C₄×C₁₀₇	1	8	yes	—
83	C₄×C₈	—	2	4	no	—
84	C₄×C₈	—	2	4	no	—
85	C₄×C₈	C₈×C₁₀₇	1	4	yes	—
86	C₄×C₈	C₂×C₄×C₁₀₇	1	4	yes	—
87	C₂²×C₄	C₂×C₈×C₁₀₇	1	2	yes	—
88	C₂²×C₄	C₂×C₈×C₁₀₇	1	4	yes	—
89	C₂²×C₄	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	C₄²×C₁₀₇	1	2	yes	—
93	C₂×C₈	C₂×C₈×C₁₀₇	1	4	yes	—
94	C₂×C₈	M₁₆×C₁₀₇	1	4	yes	—
95	C₂×C₈	C₂²×C₄×C₁₀₇	1	2	yes	—
96	C₂×C₈	#41×C₁₀₇	1	4	yes	—
97	C₄²	—	2	8	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	2	2	no	—
100	C₄²	C₄²×C₁₀₇	1	2	yes	—
101	C₄²	C₂×C₈×C₁₀₇	1	2	yes	—
102	C₁₆	—	2	8	no	—
103	C₂³	C₂²×C₈×C₁₀₇	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	C₄×C₈×C₁₀₇	1	4	yes	—
108	C₂×C₄	M₃₂×C₁₀₇	1	8	yes	—
109	C₂×C₄	C₂×C₄²×C₁₀₇	1	1	yes	characteristic
110	C₂×C₄	#117×C₁₀₇	1	2	yes	—
111	C₂×C₄	C₂²×C₈×C₁₀₇	1	1	yes	characteristic
112	C₂×C₄	C₂×M₁₆×C₁₀₇	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₈	#97×C₁₀₇	1	4	yes	—
115	C₈	#117×C₁₀₇	1	4	yes	—
116	C₂²	C₂×C₄×C₈×C₁₀₇	1	1	yes	characteristic
117	C₂²	#431×C₁₀₇	1	2	yes	—
118	C₂²	C₂×M₃₂×C₁₀₇	1	4	yes	—
119	C₄	—	2	8	no	—
120	C₄	—	2	4	no	—
121	C₄	#376×C₁₀₇	1	1	yes	characteristic
122	C₄	C₂×#97×C₁₀₇	1	1	yes	characteristic
123	C₄	#431×C₁₀₇	1	2	yes	—
124	C₂	C₁₀₇×#1744	1	1	yes	characteristic
125	C₂	C₄×M₃₂×C₁₀₇	1	2	yes	—
126	C₂	C₁₀₇×#2163	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#346×C₁₀₇

C₂

yes

maximal

C₄²×C₈×C₁₀₇

C₂

yes

maximal, characteristic

C₁₀₇×#1374

C₂

yes

maximal

C₂²×C₁₆×C₁₀₇

C₂²

yes

—

C₂×C₄×C₈×C₁₀₇

C₄

yes

—

C₂×C₄×C₈×C₁₀₇

C₄

yes

characteristic

C₂×C₄×C₈×C₁₀₇

C₂²

yes

—

C₂×C₄×C₈×C₁₀₇

C₂²

yes

characteristic

C₄³×C₁₀₇

C₄

yes

characteristic

#346×C₁₀₇

C₄

yes

complemented

C₂²×C₈×C₁₀₇

C₂×C₄

yes

—

C₂²×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₀₇

C₂³

yes

characteristic

C₂×C₄²×C₁₀₇

C₈

yes

—

C₂×C₄²×C₁₀₇

C₂×C₄

yes

—

C₂×C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₀₇

—

C₂×C₁₆×C₁₀₇

C₂×C₄

yes

—

C₄×C₈×C₁₀₇

—

C₄×C₈×C₁₀₇

—

C₄×C₈×C₁₀₇

C₈

yes

—

C₄×C₈×C₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂×C₈

yes

—

C₂²×C₄×C₁₀₇

C₂×C₈

yes

—

C₂²×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

C₄²

yes

—

C₂×C₈×C₁₀₇

C₂×C₈

yes

—

C₂×C₈×C₁₀₇

M₁₆

yes

—

C₂×C₈×C₁₀₇

C₂²×C₄

yes

—

C₂×C₈×C₁₀₇

#41

yes

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

C₄²

yes

—

C₄²×C₁₀₇

C₂×C₈

yes

—

C₁₆×C₁₀₇

—

C₂³×C₁₀₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₄×C₈

yes

—

C₂×C₄×C₁₀₇

M₃₂

yes

—

C₂×C₄×C₁₀₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₀₇

#117

yes

—

C₂×C₄×C₁₀₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₀₇

C₂×M₁₆

yes

—

C₈×C₁₀₇

—

C₈×C₁₀₇

#97

yes

—

C₈×C₁₀₇

#117

yes

—

C₂²×C₁₀₇

C₂×C₄×C₈

yes

characteristic

C₂²×C₁₀₇

#431

yes

—

C₂²×C₁₀₇

C₂×M₃₂

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#376

yes

characteristic

C₄×C₁₀₇

C₂×#97

yes

characteristic

C₄×C₁₀₇

#431

yes

—

#12591

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

#1744

yes

characteristic

C₂×C₁₀₇

C₄×M₃₂

yes

—

C₂×C₁₀₇

#2163

yes

—

C₂×#346

C₂×C₁₀₇

yes

—

C₄²×C₈

C₂×C₁₀₇

yes

characteristic

#1374

C₂×C₁₀₇

yes

—

C₁₀₇

#12591

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

C₂²×C₁₀₇

yes

—

C₂×C₄×C₈

C₄×C₁₀₇

yes

—

C₂×C₄×C₈

C₄×C₁₀₇

yes

characteristic

C₂×C₄×C₈

C₂²×C₁₀₇

yes

—

C₂×C₄×C₈

C₂²×C₁₀₇

yes

characteristic

C₄³

C₄×C₁₀₇

yes

characteristic

#346

C₄×C₁₀₇

yes

complemented

C₂²×C₈

C₂×C₄×C₁₀₇

yes

—

C₂²×C₈

C₂×C₄×C₁₀₇

yes

characteristic

C₂²×C₈

C₂³×C₁₀₇

yes

characteristic

C₂×C₄²

C₈×C₁₀₇

yes

—

C₂×C₄²

C₂×C₄×C₁₀₇

yes

—

C₂×C₄²

C₂×C₄×C₁₀₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₁₀₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₈×C₁₀₇

yes

—

C₄×C₈

C₂×C₄×C₁₀₇

yes

—

C₂²×C₄

C₂×C₈×C₁₀₇

yes

—

C₂²×C₄

C₂×C₈×C₁₀₇

yes

—

C₂²×C₄

C₂²×C₄×C₁₀₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₁₀₇

yes

—

C₂×C₈

C₂×C₈×C₁₀₇

yes

—

C₂×C₈

M₁₆×C₁₀₇

yes

—

C₂×C₈

C₂²×C₄×C₁₀₇

yes

—

C₂×C₈

#41×C₁₀₇

yes

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

C₄²×C₁₀₇

yes

—

101

C₄²

C₂×C₈×C₁₀₇

yes

—

102

C₁₆

—

103

C₂³

C₂²×C₈×C₁₀₇

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄×C₈×C₁₀₇

yes

—

108

C₂×C₄

M₃₂×C₁₀₇

yes

—

109

C₂×C₄

C₂×C₄²×C₁₀₇

yes

characteristic

110

C₂×C₄

#117×C₁₀₇

yes

—

111

C₂×C₄

C₂²×C₈×C₁₀₇

yes

characteristic

112

C₂×C₄

C₂×M₁₆×C₁₀₇

yes

—

113

C₈

—

114

C₈

#97×C₁₀₇

yes

—

115

C₈

#117×C₁₀₇

yes

—

116

C₂²

C₂×C₄×C₈×C₁₀₇

yes

characteristic

117

C₂²

#431×C₁₀₇

yes

—

118

C₂²

C₂×M₃₂×C₁₀₇

yes

—

119

C₄

—

120

C₄

—

121

C₄

#376×C₁₀₇

yes

characteristic

122

C₄

C₂×#97×C₁₀₇

yes

characteristic

123

C₄

#431×C₁₀₇

yes

—

124

C₂

C₁₀₇×#1744

yes

characteristic

125

C₂

C₄×M₃₂×C₁₀₇

yes

—

126

C₂

C₁₀₇×#2163

yes

—

	subgroup	G/H
center	C₂²×C₈×C₁₀₇	C₂³
commutator	C₂²	C₂×C₄×C₈×C₁₀₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₇
socle, prime radical	C₂³×C₁₀₇	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₀₇ ◁ G

subgroup

G/H

center

C₂²×C₈×C₁₀₇

C₂³

commutator

C₂²

C₂×C₄×C₈×C₁₀₇

Frattini, cosocle

C₂²×C₈

C₂³×C₁₀₇

socle, prime radical

C₂³×C₁₀₇

C₂²×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₈×C₁₀₇ ◁ G

N	H ≅ G/N	type
10. #346×C₁₀₇	C₄	semidirect
60. #12591	C₁₀₇	direct (N × H)
67. C₁₀₇	#12591	direct (N × H)
74. #346	C₄×C₁₀₇	semidirect

H ≅ G/N

type

10. #346×C₁₀₇

C₄

semidirect

60. #12591

C₁₀₇

direct (N × H)

67. C₁₀₇

#12591

direct (N × H)

74. #346

C₄×C₁₀₇

semidirect

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

subgroup	index
C₂×#346×C₁₀₇	2
C₄²×C₈×C₁₀₇	2
C₁₀₇×#1374	2
#12591	107

subgroup

index

C₂×#346×C₁₀₇

C₄²×C₈×C₁₀₇

C₁₀₇×#1374

#12591

107

Subgroups — C₁₀₇×#12591 #41038555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#12591 #41038555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#12591 #41038555