Group #41059692 — subgroups

Subgroups — C₁₀₇×#33728 #41059692

Order 27392.

482 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 270 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	C₁₀₇×#3160	C₂	1	2	yes	maximal
2	C₁₀₇×#2255	C₂	1	8	yes	maximal, complemented
3	C₁₀₇×#2158	C₂	1	4	yes	maximal
4	C₂×#346×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	#533×C₁₀₇	—	2	2	no	—
6	C₂²×C₁₆×C₁₀₇	C₂²	1	2	yes	—
7	#500×C₁₀₇	C₂²	1	4	yes	—
8	#499×C₁₀₇	—	2	2	no	—
9	#499×C₁₀₇	C₂²	1	4	yes	—
10	#498×C₁₀₇	—	2	2	no	—
11	#427×C₁₀₇	—	2	2	no	—
12	#427×C₁₀₇	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
14	#367×C₁₀₇	C₂²	1	8	yes	—
15	#366×C₁₀₇	C₂²	1	8	yes	—
16	#346×C₁₀₇	C₂²	1	4	yes	—
17	C₂²×C₈×C₁₀₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₀₇	—	2	4	no	—
20	#125×C₁₀₇	—	2	4	no	—
21	C₂²⋊Q₈×C₁₀₇	—	4	4	no	—
22	C₄×Q₈×C₁₀₇	—	4	4	no	—
23	#117×C₁₀₇	—	2	4	no	—
24	C₂×C₄²×C₁₀₇	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₁₀₇	Q₈	1	8	yes	complemented
26	C₂×C₁₆×C₁₀₇	C₂³	1	4	yes	—
27	#107×C₁₀₇	—	2	4	no	—
28	#107×C₁₀₇	C₂³	1	8	yes	—
29	#106×C₁₀₇	—	2	4	no	—
30	#106×C₁₀₇	—	2	4	no	—
31	C₄×C₈×C₁₀₇	D₄	1	2	yes	—
32	C₄×C₈×C₁₀₇	C₂³	1	2	yes	—
33	C₂×Q₈×C₁₀₇	—	4	4	no	—
34	C₂²×C₄×C₁₀₇	D₈	1	2	yes	—
35	C₂²×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₁₀₇	Q₁₆	1	4	yes	—
37	C₂×C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₀₇	C₂×D₄	1	2	yes	—
39	C₂×C₈×C₁₀₇	C₂×Q₈	1	4	yes	—
40	C₂×C₈×C₁₀₇	C₂⁴	1	1	yes	characteristic
41	C₄⋊C₄×C₁₀₇	—	2	8	no	—
42	C₄⋊C₄×C₁₀₇	—	4	8	no	—
43	C₄⋊C₄×C₁₀₇	—	4	4	no	—
44	C₂²⋊C₄×C₁₀₇	—	4	4	no	—
45	C₄²×C₁₀₇	—	4	4	no	—
46	C₄²×C₁₀₇	D₈	1	2	yes	—
47	C₄²×C₁₀₇	C₂×D₄	1	2	yes	—
48	C₁₆×C₁₀₇	C₂×Q₈	1	8	yes	—
49	C₂³×C₁₀₇	C₂×D₈	1	1	yes	characteristic
50	Q₈×C₁₀₇	—	8	8	no	—
51	C₂×C₄×C₁₀₇	—	4	2	no	—
52	C₂×C₄×C₁₀₇	—	4	8	no	—
53	C₂×C₄×C₁₀₇	C₄⋊Q₈	1	2	yes	—
54	C₂×C₄×C₁₀₇	C₂×D₈	1	4	yes	—
55	C₂×C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₁₀₇	C₂×Q₁₆	1	2	yes	—
57	C₂×C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
58	C₈×C₁₀₇	C₄⋊Q₈	1	2	yes	—
59	C₈×C₁₀₇	C₂×Q₁₆	1	4	yes	—
60	C₈×C₁₀₇	C₂²×Q₈	1	2	yes	—
61	C₂²×C₁₀₇	—	2	2	no	—
62	C₂²×C₁₀₇	#500	1	2	yes	—
63	C₂²×C₁₀₇	C₂²×D₈	1	1	yes	characteristic
64	C₄×C₁₀₇	—	2	4	no	—
65	C₄×C₁₀₇	—	8	8	no	—
66	C₄×C₁₀₇	#500	1	2	yes	—
67	C₄×C₁₀₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₄×C₁₀₇	C₂²×Q₁₆	1	1	yes	characteristic
69	#33728	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×C₁₀₇	—	2	2	no	—
71	C₂×C₁₀₇	C₂×#500	1	1	yes	characteristic
72	C₂×C₁₀₇	#3414	1	2	yes	—
73	#3160	C₂×C₁₀₇	1	2	yes	—
74	#2255	C₂×C₁₀₇	1	8	yes	complemented
75	#2158	C₂×C₁₀₇	1	4	yes	—
76	C₂×#346	C₂×C₁₀₇	1	1	yes	characteristic
77	C₁₀₇	#33728	1	1	yes	characteristic, direct product factor, Sylow
78	#533	—	2	2	no	—
79	C₂²×C₁₆	C₂²×C₁₀₇	1	2	yes	—
80	#500	C₂²×C₁₀₇	1	4	yes	—
81	#499	—	2	2	no	—
82	#499	C₂²×C₁₀₇	1	4	yes	—
83	#498	—	2	2	no	—
84	#427	—	2	2	no	—
85	#427	C₂²×C₁₀₇	1	4	yes	—
86	C₂×C₄×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
87	#367	C₂²×C₁₀₇	1	8	yes	—
88	#366	C₂²×C₁₀₇	1	8	yes	—
89	#346	C₂²×C₁₀₇	1	4	yes	—
90	C₂²×C₈	D₄×C₁₀₇	1	1	yes	characteristic
91	C₂²×C₈	C₂³×C₁₀₇	1	1	yes	characteristic
92	C₄⋊Q₈	—	2	4	no	—
93	#125	—	2	4	no	—
94	C₂²⋊Q₈	—	4	4	no	—
95	C₄×Q₈	—	4	4	no	—
96	#117	—	2	4	no	—
97	C₂×C₄²	D₄×C₁₀₇	1	1	yes	characteristic
98	C₂×C₁₆	Q₈×C₁₀₇	1	8	yes	complemented
99	C₂×C₁₆	C₂³×C₁₀₇	1	4	yes	—
100	#107	—	2	4	no	—
101	#107	C₂³×C₁₀₇	1	8	yes	—
102	#106	—	2	4	no	—
103	#106	—	2	4	no	—
104	C₄×C₈	D₄×C₁₀₇	1	2	yes	—
105	C₄×C₈	C₂³×C₁₀₇	1	2	yes	—
106	C₂×Q₈	—	4	4	no	—
107	C₂²×C₄	D₈×C₁₀₇	1	2	yes	—
108	C₂²×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
109	C₂×C₈	Q₁₆×C₁₀₇	1	4	yes	—
110	C₂×C₈	C₂×D₄×C₁₀₇	1	1	yes	characteristic
111	C₂×C₈	C₂×D₄×C₁₀₇	1	2	yes	—
112	C₂×C₈	C₂×Q₈×C₁₀₇	1	4	yes	—
113	C₂×C₈	C₂⁴×C₁₀₇	1	1	yes	characteristic
114	C₄⋊C₄	—	2	8	no	—
115	C₄⋊C₄	—	4	8	no	—
116	C₄⋊C₄	—	4	4	no	—
117	C₂²⋊C₄	—	4	4	no	—
118	C₄²	—	4	4	no	—
119	C₄²	D₈×C₁₀₇	1	2	yes	—
120	C₄²	C₂×D₄×C₁₀₇	1	2	yes	—
121	C₁₆	C₂×Q₈×C₁₀₇	1	8	yes	—
122	C₂³	C₂×D₈×C₁₀₇	1	1	yes	characteristic
123	Q₈	—	8	8	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	8	no	—
126	C₂×C₄	C₄⋊Q₈×C₁₀₇	1	2	yes	—
127	C₂×C₄	C₂×D₈×C₁₀₇	1	4	yes	—
128	C₂×C₄	C₂×D₈×C₁₀₇	1	1	yes	characteristic
129	C₂×C₄	C₂×Q₁₆×C₁₀₇	1	2	yes	—
130	C₂×C₄	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
131	C₈	C₄⋊Q₈×C₁₀₇	1	2	yes	—
132	C₈	C₂×Q₁₆×C₁₀₇	1	4	yes	—
133	C₈	C₂²×Q₈×C₁₀₇	1	2	yes	—
134	C₂²	—	2	2	no	—
135	C₂²	#500×C₁₀₇	1	2	yes	—
136	C₂²	C₂²×D₈×C₁₀₇	1	1	yes	characteristic
137	C₄	—	2	4	no	—
138	C₄	—	8	8	no	—
139	C₄	#500×C₁₀₇	1	2	yes	—
140	C₄	C₂×C₄⋊Q₈×C₁₀₇	1	1	yes	characteristic
141	C₄	C₂²×Q₁₆×C₁₀₇	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#500×C₁₀₇	1	1	yes	characteristic
144	C₂	C₁₀₇×#3414	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₇×#3160

C₂

yes

maximal

C₁₀₇×#2255

C₂

yes

maximal, complemented

C₁₀₇×#2158

C₂

yes

maximal

C₂×#346×C₁₀₇

C₂

yes

maximal, characteristic

#533×C₁₀₇

—

C₂²×C₁₆×C₁₀₇

C₂²

yes

—

#500×C₁₀₇

C₂²

yes

—

#499×C₁₀₇

—

#499×C₁₀₇

C₂²

yes

—

#498×C₁₀₇

—

#427×C₁₀₇

—

#427×C₁₀₇

C₂²

yes

—

C₂×C₄×C₈×C₁₀₇

C₂²

yes

characteristic

#367×C₁₀₇

C₂²

yes

—

#366×C₁₀₇

C₂²

yes

—

#346×C₁₀₇

C₂²

yes

—

C₂²×C₈×C₁₀₇

D₄

yes

characteristic

C₂²×C₈×C₁₀₇

C₂³

yes

characteristic

C₄⋊Q₈×C₁₀₇

—

#125×C₁₀₇

—

C₂²⋊Q₈×C₁₀₇

—

C₄×Q₈×C₁₀₇

—

#117×C₁₀₇

—

C₂×C₄²×C₁₀₇

D₄

yes

characteristic

C₂×C₁₆×C₁₀₇

Q₈

yes

complemented

C₂×C₁₆×C₁₀₇

C₂³

yes

—

#107×C₁₀₇

—

#107×C₁₀₇

C₂³

yes

—

#106×C₁₀₇

—

#106×C₁₀₇

—

C₄×C₈×C₁₀₇

D₄

yes

—

C₄×C₈×C₁₀₇

C₂³

yes

—

C₂×Q₈×C₁₀₇

—

C₂²×C₄×C₁₀₇

D₈

yes

—

C₂²×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₀₇

Q₁₆

yes

—

C₂×C₈×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₀₇

C₂×D₄

yes

—

C₂×C₈×C₁₀₇

C₂×Q₈

yes

—

C₂×C₈×C₁₀₇

C₂⁴

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

D₈

yes

—

C₄²×C₁₀₇

C₂×D₄

yes

—

C₁₆×C₁₀₇

C₂×Q₈

yes

—

C₂³×C₁₀₇

C₂×D₈

yes

characteristic

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₄⋊Q₈

yes

—

C₂×C₄×C₁₀₇

C₂×D₈

yes

—

C₂×C₄×C₁₀₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₁₆

yes

—

C₂×C₄×C₁₀₇

C₂²×D₄

yes

characteristic

C₈×C₁₀₇

C₄⋊Q₈

yes

—

C₈×C₁₀₇

C₂×Q₁₆

yes

—

C₈×C₁₀₇

C₂²×Q₈

yes

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#500

yes

—

C₂²×C₁₀₇

C₂²×D₈

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#500

yes

—

C₄×C₁₀₇

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₁₀₇

C₂²×Q₁₆

yes

characteristic

#33728

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂×#500

yes

characteristic

C₂×C₁₀₇

#3414

yes

—

#3160

C₂×C₁₀₇

yes

—

#2255

C₂×C₁₀₇

yes

complemented

#2158

C₂×C₁₀₇

yes

—

C₂×#346

C₂×C₁₀₇

yes

characteristic

C₁₀₇

#33728

yes

characteristic, direct product factor, Sylow

#533

—

C₂²×C₁₆

C₂²×C₁₀₇

yes

—

#500

C₂²×C₁₀₇

yes

—

#499

—

#499

C₂²×C₁₀₇

yes

—

#498

—

#427

—

#427

C₂²×C₁₀₇

yes

—

C₂×C₄×C₈

C₂²×C₁₀₇

yes

characteristic

#367

C₂²×C₁₀₇

yes

—

#366

C₂²×C₁₀₇

yes

—

#346

C₂²×C₁₀₇

yes

—

C₂²×C₈

D₄×C₁₀₇

yes

characteristic

C₂²×C₈

C₂³×C₁₀₇

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

D₄×C₁₀₇

yes

characteristic

C₂×C₁₆

Q₈×C₁₀₇

yes

complemented

C₂×C₁₆

C₂³×C₁₀₇

yes

—

100

#107

—

101

#107

C₂³×C₁₀₇

yes

—

102

#106

—

103

#106

—

104

C₄×C₈

D₄×C₁₀₇

yes

—

105

C₄×C₈

C₂³×C₁₀₇

yes

—

106

C₂×Q₈

—

107

C₂²×C₄

D₈×C₁₀₇

yes

—

108

C₂²×C₄

C₂×D₄×C₁₀₇

yes

characteristic

109

C₂×C₈

Q₁₆×C₁₀₇

yes

—

110

C₂×C₈

C₂×D₄×C₁₀₇

yes

characteristic

111

C₂×C₈

C₂×D₄×C₁₀₇

yes

—

112

C₂×C₈

C₂×Q₈×C₁₀₇

yes

—

113

C₂×C₈

C₂⁴×C₁₀₇

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

D₈×C₁₀₇

yes

—

120

C₄²

C₂×D₄×C₁₀₇

yes

—

121

C₁₆

C₂×Q₈×C₁₀₇

yes

—

122

C₂³

C₂×D₈×C₁₀₇

yes

characteristic

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₄⋊Q₈×C₁₀₇

yes

—

127

C₂×C₄

C₂×D₈×C₁₀₇

yes

—

128

C₂×C₄

C₂×D₈×C₁₀₇

yes

characteristic

129

C₂×C₄

C₂×Q₁₆×C₁₀₇

yes

—

130

C₂×C₄

C₂²×D₄×C₁₀₇

yes

characteristic

131

C₈

C₄⋊Q₈×C₁₀₇

yes

—

132

C₈

C₂×Q₁₆×C₁₀₇

yes

—

133

C₈

C₂²×Q₈×C₁₀₇

yes

—

134

C₂²

—

135

C₂²

#500×C₁₀₇

yes

—

136

C₂²

C₂²×D₈×C₁₀₇

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

#500×C₁₀₇

yes

—

140

C₄

C₂×C₄⋊Q₈×C₁₀₇

yes

characteristic

141

C₄

C₂²×Q₁₆×C₁₀₇

yes

characteristic

142

C₂

—

143

C₂

C₂×#500×C₁₀₇

yes

characteristic

144

C₂

C₁₀₇×#3414

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₀₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₀₇
socle	C₂²×C₁₀₇	C₂²×D₈
prime radical	C₂³×C₁₀₇	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ C₂×C₄×C₈×C₁₀₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₀₇

C₂×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴×C₁₀₇

socle

C₂²×C₁₀₇

C₂²×D₈

prime radical

C₂³×C₁₀₇

C₂×D₈

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
2. C₁₀₇×#2255	C₂	semidirect
25. C₂×C₁₆×C₁₀₇	Q₈	semidirect
69. #33728	C₁₀₇	direct (N × H)
74. #2255	C₂×C₁₀₇	semidirect
77. C₁₀₇	#33728	direct (N × H)
98. C₂×C₁₆	Q₈×C₁₀₇	semidirect

H ≅ G/N

type

2. C₁₀₇×#2255

C₂

semidirect

25. C₂×C₁₆×C₁₀₇

Q₈

semidirect

69. #33728

C₁₀₇

direct (N × H)

74. #2255

C₂×C₁₀₇

semidirect

77. C₁₀₇

#33728

direct (N × H)

98. C₂×C₁₆

Q₈×C₁₀₇

semidirect

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

subgroup	index
C₁₀₇×#3160	2
C₁₀₇×#2255	2
C₁₀₇×#2158	2
C₂×#346×C₁₀₇	2
#33728	107

subgroup

index

C₁₀₇×#3160

C₁₀₇×#2255

C₁₀₇×#2158

C₂×#346×C₁₀₇

#33728

107

Subgroups — C₁₀₇×#33728 #41059692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#33728 #41059692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#33728 #41059692