Group #41037984 — subgroups

Subgroups — C₁₀₇×#12020 #41037984

Order 27392.

232 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 134 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	C₄×M₃₂×C₁₀₇	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1566	C₂	1	2	yes	maximal
3	C₁₀₇×#1368	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₀₇	C₄	1	2	yes	—
5	#423×C₁₀₇	—	2	2	no	—
6	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
7	#346×C₁₀₇	C₂²	1	2	yes	—
8	C₄×C₁₆×C₁₀₇	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₁₀₇	C₄	1	1	yes	characteristic
10	#334×C₁₀₇	—	2	2	no	—
11	#334×C₁₀₇	C₂²	1	4	yes	—
12	C₂×M₁₆×C₁₀₇	—	4	2	no	—
13	C₂²×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₇	Q₈	1	1	yes	characteristic
15	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
16	M₃₂×C₁₀₇	—	2	4	no	—
17	C₂×C₁₆×C₁₀₇	C₂×C₄	1	2	yes	—
18	C₂×C₁₆×C₁₀₇	C₂×C₄	1	2	yes	—
19	C₄⋊C₈×C₁₀₇	—	2	4	no	—
20	C₄⋊C₈×C₁₀₇	—	4	2	no	—
21	C₄×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₀₇	D₄	1	1	yes	characteristic
23	C₄×C₈×C₁₀₇	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₁₀₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₇	SD₁₆	1	2	yes	—
27	M₁₆×C₁₀₇	—	8	4	no	—
28	C₂×C₈×C₁₀₇	—	2	2	no	—
29	C₂×C₈×C₁₀₇	—	2	2	no	—
30	C₂×C₈×C₁₀₇	—	4	4	no	—
31	C₂×C₈×C₁₀₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₀₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
35	C₄²×C₁₀₇	C₄⋊C₄	1	1	yes	characteristic
36	C₄²×C₁₀₇	SD₁₆	1	2	yes	—
37	C₄²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
38	C₁₆×C₁₀₇	—	2	2	no	—
39	C₁₆×C₁₀₇	M₁₆	1	4	yes	complemented
40	C₂³×C₁₀₇	#106	1	1	yes	characteristic
41	C₂×C₄×C₁₀₇	—	2	4	no	—
42	C₂×C₄×C₁₀₇	—	2	1	no	—
43	C₂×C₄×C₁₀₇	—	2	1	no	—
44	C₂×C₄×C₁₀₇	#106	1	1	yes	characteristic
45	C₂×C₄×C₁₀₇	#106	1	2	yes	—
46	C₂×C₄×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₇	C₂×SD₁₆	1	2	yes	—
48	C₈×C₁₀₇	—	2	1	no	—
49	C₈×C₁₀₇	—	2	1	no	—
50	C₈×C₁₀₇	—	8	4	no	—
51	C₈×C₁₀₇	#108	1	2	yes	—
52	C₈×C₁₀₇	C₂×M₁₆	1	2	yes	—
53	C₂²×C₁₀₇	—	2	1	no	—
54	C₂²×C₁₀₇	—	2	2	no	—
55	C₂²×C₁₀₇	C₂×#106	1	1	yes	characteristic
56	C₄×C₁₀₇	—	2	1	no	—
57	C₄×C₁₀₇	—	2	2	no	—
58	C₄×C₁₀₇	#365	1	4	yes	—
59	C₄×C₁₀₇	#423	1	1	yes	characteristic
60	C₄×C₁₀₇	C₂×#108	1	1	yes	characteristic
61	#12020	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₁₀₇	—	4	1	no	—
63	C₂×C₁₀₇	#1566	1	1	yes	characteristic
64	C₂×C₁₀₇	C₂×#365	1	2	yes	—
65	C₄×M₃₂	C₂×C₁₀₇	1	1	yes	characteristic
66	#1566	C₂×C₁₀₇	1	2	yes	—
67	#1368	C₂×C₁₀₇	1	4	yes	complemented
68	C₁₀₇	#12020	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×M₃₂	C₄×C₁₀₇	1	2	yes	—
70	#423	—	2	2	no	—
71	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
72	#346	C₂²×C₁₀₇	1	2	yes	—
73	C₄×C₁₆	C₄×C₁₀₇	1	1	yes	characteristic
74	C₄×C₁₆	C₄×C₁₀₇	1	1	yes	characteristic
75	#334	—	2	2	no	—
76	#334	C₂²×C₁₀₇	1	4	yes	—
77	C₂×M₁₆	—	4	2	no	—
78	C₂²×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
79	C₂²×C₈	Q₈×C₁₀₇	1	1	yes	characteristic
80	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
81	M₃₂	—	2	4	no	—
82	C₂×C₁₆	C₂×C₄×C₁₀₇	1	2	yes	—
83	C₂×C₁₆	C₂×C₄×C₁₀₇	1	2	yes	—
84	C₄⋊C₈	—	2	4	no	—
85	C₄⋊C₈	—	4	2	no	—
86	C₄×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
87	C₄×C₈	D₄×C₁₀₇	1	1	yes	characteristic
88	C₄×C₈	Q₈×C₁₀₇	1	1	yes	characteristic
89	C₄×C₈	C₂³×C₁₀₇	1	1	yes	characteristic
90	C₂²×C₄	C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
91	C₂²×C₄	SD₁₆×C₁₀₇	1	2	yes	—
92	M₁₆	—	8	4	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	4	4	no	—
96	C₂×C₈	C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
98	C₂×C₈	C₂²×C₄×C₁₀₇	1	1	yes	characteristic
99	C₂×C₈	C₂×Q₈×C₁₀₇	1	1	yes	characteristic
100	C₄²	C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
101	C₄²	SD₁₆×C₁₀₇	1	2	yes	—
102	C₄²	C₂×D₄×C₁₀₇	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	M₁₆×C₁₀₇	1	4	yes	complemented
105	C₂³	#106×C₁₀₇	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	#106×C₁₀₇	1	1	yes	characteristic
110	C₂×C₄	#106×C₁₀₇	1	2	yes	—
111	C₂×C₄	C₂×C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
112	C₂×C₄	C₂×SD₁₆×C₁₀₇	1	2	yes	—
113	C₈	—	2	1	no	—
114	C₈	—	2	1	no	—
115	C₈	—	8	4	no	—
116	C₈	#108×C₁₀₇	1	2	yes	—
117	C₈	C₂×M₁₆×C₁₀₇	1	2	yes	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂×#106×C₁₀₇	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	#365×C₁₀₇	1	4	yes	—
124	C₄	#423×C₁₀₇	1	1	yes	characteristic
125	C₄	C₂×#108×C₁₀₇	1	1	yes	characteristic
126	C₂	—	4	1	no	—
127	C₂	C₁₀₇×#1566	1	1	yes	characteristic
128	C₂	C₂×#365×C₁₀₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×M₃₂×C₁₀₇

C₂

yes

maximal, characteristic

C₁₀₇×#1566

C₂

yes

maximal

C₁₀₇×#1368

C₂

yes

maximal, complemented

C₂×M₃₂×C₁₀₇

C₄

yes

—

#423×C₁₀₇

—

C₂×C₄×C₈×C₁₀₇

C₂²

yes

characteristic

#346×C₁₀₇

C₂²

yes

—

C₄×C₁₆×C₁₀₇

C₄

yes

characteristic

C₄×C₁₆×C₁₀₇

C₄

yes

characteristic

#334×C₁₀₇

—

#334×C₁₀₇

C₂²

yes

—

C₂×M₁₆×C₁₀₇

—

C₂²×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₀₇

Q₈

yes

characteristic

C₂×C₄²×C₁₀₇

D₄

yes

characteristic

M₃₂×C₁₀₇

—

C₂×C₁₆×C₁₀₇

C₂×C₄

yes

—

C₂×C₁₆×C₁₀₇

C₂×C₄

yes

—

C₄⋊C₈×C₁₀₇

—

C₄⋊C₈×C₁₀₇

—

C₄×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₀₇

D₄

yes

characteristic

C₄×C₈×C₁₀₇

Q₈

yes

characteristic

C₄×C₈×C₁₀₇

C₂³

yes

characteristic

C₂²×C₄×C₁₀₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₀₇

SD₁₆

yes

—

M₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₀₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₀₇

C₂×Q₈

yes

characteristic

C₄²×C₁₀₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₀₇

SD₁₆

yes

—

C₄²×C₁₀₇

C₂×D₄

yes

characteristic

C₁₆×C₁₀₇

—

C₁₆×C₁₀₇

M₁₆

yes

complemented

C₂³×C₁₀₇

#106

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

#106

yes

characteristic

C₂×C₄×C₁₀₇

#106

yes

—

C₂×C₄×C₁₀₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×SD₁₆

yes

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

#108

yes

—

C₈×C₁₀₇

C₂×M₁₆

yes

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×#106

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#365

yes

—

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

—

C₂×C₁₀₇

#1566

yes

characteristic

C₂×C₁₀₇

C₂×#365

yes

—

C₄×M₃₂

C₂×C₁₀₇

yes

characteristic

#1566

C₂×C₁₀₇

yes

—

#1368

C₂×C₁₀₇

yes

complemented

C₁₀₇

#12020

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₄×C₁₀₇

yes

—

#423

—

C₂×C₄×C₈

C₂²×C₁₀₇

yes

characteristic

#346

C₂²×C₁₀₇

yes

—

C₄×C₁₆

C₄×C₁₀₇

yes

characteristic

C₄×C₁₆

C₄×C₁₀₇

yes

characteristic

#334

—

#334

C₂²×C₁₀₇

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×C₄×C₁₀₇

yes

characteristic

C₂²×C₈

Q₈×C₁₀₇

yes

characteristic

C₂×C₄²

D₄×C₁₀₇

yes

characteristic

M₃₂

—

C₂×C₁₆

C₂×C₄×C₁₀₇

yes

—

C₂×C₁₆

C₂×C₄×C₁₀₇

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₁₀₇

yes

characteristic

C₄×C₈

D₄×C₁₀₇

yes

characteristic

C₄×C₈

Q₈×C₁₀₇

yes

characteristic

C₄×C₈

C₂³×C₁₀₇

yes

characteristic

C₂²×C₄

C₄⋊C₄×C₁₀₇

yes

characteristic

C₂²×C₄

SD₁₆×C₁₀₇

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₁₀₇

yes

characteristic

C₂×C₈

C₄⋊C₄×C₁₀₇

yes

characteristic

C₂×C₈

C₂²×C₄×C₁₀₇

yes

characteristic

C₂×C₈

C₂×Q₈×C₁₀₇

yes

characteristic

100

C₄²

C₄⋊C₄×C₁₀₇

yes

characteristic

101

C₄²

SD₁₆×C₁₀₇

yes

—

102

C₄²

C₂×D₄×C₁₀₇

yes

characteristic

103

C₁₆

—

104

C₁₆

M₁₆×C₁₀₇

yes

complemented

105

C₂³

#106×C₁₀₇

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#106×C₁₀₇

yes

characteristic

110

C₂×C₄

#106×C₁₀₇

yes

—

111

C₂×C₄

C₂×C₄⋊C₄×C₁₀₇

yes

characteristic

112

C₂×C₄

C₂×SD₁₆×C₁₀₇

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

#108×C₁₀₇

yes

—

117

C₈

C₂×M₁₆×C₁₀₇

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#106×C₁₀₇

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

#365×C₁₀₇

yes

—

124

C₄

#423×C₁₀₇

yes

characteristic

125

C₄

C₂×#108×C₁₀₇

yes

characteristic

126

C₂

—

127

C₂

C₁₀₇×#1566

yes

characteristic

128

C₂

C₂×#365×C₁₀₇

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₇	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₁₀₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₀₇
prime radical	C₂³×C₁₀₇	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ C₂×C₄×C₈×C₁₀₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₇

C₂×#106

commutator

C₂×C₈

C₂²×C₄×C₁₀₇

Frattini, cosocle

C₄×C₈

C₂³×C₁₀₇

prime radical

C₂³×C₁₀₇

#106

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ C₂×C₄×C₈×C₁₀₇ ◁ G

N	H ≅ G/N	type
3. C₁₀₇×#1368	C₂	semidirect
39. C₁₆×C₁₀₇	M₁₆	semidirect
61. #12020	C₁₀₇	direct (N × H)
67. #1368	C₂×C₁₀₇	semidirect
68. C₁₀₇	#12020	direct (N × H)
104. C₁₆	M₁₆×C₁₀₇	semidirect

H ≅ G/N

type

3. C₁₀₇×#1368

C₂

semidirect

39. C₁₆×C₁₀₇

M₁₆

semidirect

61. #12020

C₁₀₇

direct (N × H)

67. #1368

C₂×C₁₀₇

semidirect

68. C₁₀₇

#12020

direct (N × H)

104. C₁₆

M₁₆×C₁₀₇

semidirect

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

subgroup	index
C₄×M₃₂×C₁₀₇	2
C₁₀₇×#1566	2
C₁₀₇×#1368	2
#12020	107

subgroup

index

C₄×M₃₂×C₁₀₇

C₁₀₇×#1566

C₁₀₇×#1368

#12020

107

Subgroups — C₁₀₇×#12020 #41037984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#12020 #41037984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#12020 #41037984