Group #41033197 — subgroups

Subgroups — C₁₀₇×#7233 #41033197

Order 27392.

308 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2112	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1807	C₂	1	2	yes	maximal
3	C₂³×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
4	C₂×M₃₂×C₁₀₇	C₄	1	2	yes	complemented
5	C₂×#106×C₁₀₇	—	2	2	no	—
6	#380×C₁₀₇	—	2	2	no	—
7	C₂²⋊C₁₆×C₁₀₇	—	2	2	no	—
8	#340×C₁₀₇	C₄	1	4	yes	complemented
9	C₂³×C₄×C₁₀₇	D₄	1	1	yes	characteristic
10	C₂²×C₈×C₁₀₇	—	4	2	no	—
11	C₂²×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₁₀₇	D₄	1	2	yes	—
14	C₂²×C₈×C₁₀₇	Q₈	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₀₇	—	2	4	no	—
16	C₂×C₂²⋊C₄×C₁₀₇	—	4	2	no	—
17	M₃₂×C₁₀₇	—	4	2	no	—
18	C₂×C₁₆×C₁₀₇	—	2	2	no	—
19	#106×C₁₀₇	—	4	4	no	—
20	#95×C₁₀₇	—	4	2	no	—
21	C₂⁴×C₁₀₇	D₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₇	—	4	1	no	—
23	C₂²×C₄×C₁₀₇	—	4	1	no	—
24	C₂²×C₄×C₁₀₇	—	4	4	no	—
25	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₄×C₁₀₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₇	SD₁₆	1	2	yes	—
28	C₂²×C₄×C₁₀₇	Q₁₆	1	1	yes	characteristic
29	C₂×C₈×C₁₀₇	—	2	4	no	—
30	C₂×C₈×C₁₀₇	—	2	2	no	—
31	C₂×C₈×C₁₀₇	—	4	2	no	—
32	C₂×C₈×C₁₀₇	—	4	1	no	—
33	C₂×C₈×C₁₀₇	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₀₇	C₄⋊C₄	1	2	yes	—
36	C₄⋊C₄×C₁₀₇	—	4	4	no	—
37	C₄⋊C₄×C₁₀₇	—	4	4	no	—
38	C₂²⋊C₄×C₁₀₇	—	8	4	no	—
39	C₁₆×C₁₀₇	—	4	2	no	—
40	C₂³×C₁₀₇	—	2	4	no	—
41	C₂³×C₁₀₇	—	4	1	no	—
42	C₂³×C₁₀₇	#102	1	2	yes	—
43	C₂³×C₁₀₇	#107	1	1	yes	characteristic
44	C₂×C₄×C₁₀₇	—	2	2	no	—
45	C₂×C₄×C₁₀₇	—	2	1	no	—
46	C₂×C₄×C₁₀₇	—	2	2	no	—
47	C₂×C₄×C₁₀₇	—	2	1	no	—
48	C₂×C₄×C₁₀₇	—	4	2	no	—
49	C₂×C₄×C₁₀₇	—	4	1	no	—
50	C₂×C₄×C₁₀₇	—	8	4	no	—
51	C₂×C₄×C₁₀₇	—	8	4	no	—
52	C₂×C₄×C₁₀₇	—	8	4	no	—
53	C₂×C₄×C₁₀₇	#95	1	1	yes	characteristic
54	C₂×C₄×C₁₀₇	#103	1	2	yes	—
55	C₂×C₄×C₁₀₇	#106	1	1	yes	characteristic
56	C₈×C₁₀₇	—	2	1	no	—
57	C₈×C₁₀₇	—	2	2	no	—
58	C₈×C₁₀₇	#99	1	2	yes	—
59	C₂²×C₁₀₇	—	2	2	no	—
60	C₂²×C₁₀₇	—	2	2	no	—
61	C₂²×C₁₀₇	—	2	1	no	—
62	C₂²×C₁₀₇	—	2	2	no	—
63	C₂²×C₁₀₇	—	4	4	no	—
64	C₂²×C₁₀₇	#340	1	1	yes	characteristic
65	C₂²×C₁₀₇	#360	1	4	yes	—
66	C₄×C₁₀₇	—	2	2	no	—
67	C₄×C₁₀₇	—	2	1	no	—
68	C₄×C₁₀₇	—	16	4	no	—
69	C₄×C₁₀₇	H(3,ℤ/4)	1	1	yes	characteristic
70	C₄×C₁₀₇	#342	1	1	yes	characteristic
71	#7233	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₁₀₇	—	2	4	no	—
73	C₂×C₁₀₇	—	4	1	no	—
74	C₂×C₁₀₇	#1289	1	1	yes	characteristic
75	C₂×C₁₀₇	#1386	1	2	yes	—
76	#2112	C₂×C₁₀₇	1	1	yes	characteristic
77	#1807	C₂×C₁₀₇	1	2	yes	—
78	C₁₀₇	#7233	1	1	yes	characteristic, direct product factor, Sylow
79	C₂³×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
80	C₂×M₃₂	C₄×C₁₀₇	1	2	yes	complemented
81	C₂×#106	—	2	2	no	—
82	#380	—	2	2	no	—
83	C₂²⋊C₁₆	—	2	2	no	—
84	#340	C₄×C₁₀₇	1	4	yes	complemented
85	C₂³×C₄	D₄×C₁₀₇	1	1	yes	characteristic
86	C₂²×C₈	—	4	2	no	—
87	C₂²×C₈	C₂×C₄×C₁₀₇	1	1	yes	characteristic
88	C₂²×C₈	C₂×C₄×C₁₀₇	1	2	yes	—
89	C₂²×C₈	D₄×C₁₀₇	1	2	yes	—
90	C₂²×C₈	Q₈×C₁₀₇	1	1	yes	characteristic
91	C₂×C₄⋊C₄	—	2	4	no	—
92	C₂×C₂²⋊C₄	—	4	2	no	—
93	M₃₂	—	4	2	no	—
94	C₂×C₁₆	—	2	2	no	—
95	#106	—	4	4	no	—
96	#95	—	4	2	no	—
97	C₂⁴	D₈×C₁₀₇	1	1	yes	characteristic
98	C₂²×C₄	—	4	1	no	—
99	C₂²×C₄	—	4	1	no	—
100	C₂²×C₄	—	4	4	no	—
101	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	2	yes	—
102	C₂²×C₄	C₄⋊C₄×C₁₀₇	1	1	yes	characteristic
103	C₂²×C₄	SD₁₆×C₁₀₇	1	2	yes	—
104	C₂²×C₄	Q₁₆×C₁₀₇	1	1	yes	characteristic
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	4	1	no	—
109	C₂×C₈	C₄²×C₁₀₇	1	1	yes	characteristic
110	C₂×C₈	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
111	C₂×C₈	C₄⋊C₄×C₁₀₇	1	2	yes	—
112	C₄⋊C₄	—	4	4	no	—
113	C₄⋊C₄	—	4	4	no	—
114	C₂²⋊C₄	—	8	4	no	—
115	C₁₆	—	4	2	no	—
116	C₂³	—	2	4	no	—
117	C₂³	—	4	1	no	—
118	C₂³	#102×C₁₀₇	1	2	yes	—
119	C₂³	#107×C₁₀₇	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	8	4	no	—
127	C₂×C₄	—	8	4	no	—
128	C₂×C₄	—	8	4	no	—
129	C₂×C₄	#95×C₁₀₇	1	1	yes	characteristic
130	C₂×C₄	#103×C₁₀₇	1	2	yes	—
131	C₂×C₄	#106×C₁₀₇	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	2	no	—
134	C₈	#99×C₁₀₇	1	2	yes	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	2	2	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	4	4	no	—
140	C₂²	#340×C₁₀₇	1	1	yes	characteristic
141	C₂²	#360×C₁₀₇	1	4	yes	—
142	C₄	—	2	2	no	—
143	C₄	—	2	1	no	—
144	C₄	—	16	4	no	—
145	C₄	H(3,ℤ/4)×C₁₀₇	1	1	yes	characteristic
146	C₄	#342×C₁₀₇	1	1	yes	characteristic
147	C₂	—	2	4	no	—
148	C₂	—	4	1	no	—
149	C₂	C₁₀₇×#1289	1	1	yes	characteristic
150	C₂	C₁₀₇×#1386	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₇×#2112

C₂

yes

maximal, characteristic

C₁₀₇×#1807

C₂

yes

maximal

C₂³×C₈×C₁₀₇

C₂²

yes

characteristic

C₂×M₃₂×C₁₀₇

C₄

yes

complemented

C₂×#106×C₁₀₇

—

#380×C₁₀₇

—

C₂²⋊C₁₆×C₁₀₇

—

#340×C₁₀₇

C₄

yes

complemented

C₂³×C₄×C₁₀₇

D₄

yes

characteristic

C₂²×C₈×C₁₀₇

—

C₂²×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₀₇

C₂×C₄

yes

—

C₂²×C₈×C₁₀₇

D₄

yes

—

C₂²×C₈×C₁₀₇

Q₈

yes

characteristic

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₂²⋊C₄×C₁₀₇

—

M₃₂×C₁₀₇

—

C₂×C₁₆×C₁₀₇

—

#106×C₁₀₇

—

#95×C₁₀₇

—

C₂⁴×C₁₀₇

D₈

yes

characteristic

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₀₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₀₇

SD₁₆

yes

—

C₂²×C₄×C₁₀₇

Q₁₆

yes

characteristic

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

C₄²

yes

characteristic

C₂×C₈×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₀₇

C₄⋊C₄

yes

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₂²⋊C₄×C₁₀₇

—

C₁₆×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

#102

yes

—

C₂³×C₁₀₇

#107

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

#95

yes

characteristic

C₂×C₄×C₁₀₇

#103

yes

—

C₂×C₄×C₁₀₇

#106

yes

characteristic

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

#99

yes

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#340

yes

characteristic

C₂²×C₁₀₇

#360

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#1289

yes

characteristic

C₂×C₁₀₇

#1386

yes

—

#2112

C₂×C₁₀₇

yes

characteristic

#1807

C₂×C₁₀₇

yes

—

C₁₀₇

#7233

yes

characteristic, direct product factor, Sylow

C₂³×C₈

C₂²×C₁₀₇

yes

characteristic

C₂×M₃₂

C₄×C₁₀₇

yes

complemented

C₂×#106

—

#380

—

C₂²⋊C₁₆

—

#340

C₄×C₁₀₇

yes

complemented

C₂³×C₄

D₄×C₁₀₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₁₀₇

yes

characteristic

C₂²×C₈

C₂×C₄×C₁₀₇

yes

—

C₂²×C₈

D₄×C₁₀₇

yes

—

C₂²×C₈

Q₈×C₁₀₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

M₃₂

—

C₂×C₁₆

—

#106

—

#95

—

C₂⁴

D₈×C₁₀₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²⋊C₄×C₁₀₇

yes

—

102

C₂²×C₄

C₄⋊C₄×C₁₀₇

yes

characteristic

103

C₂²×C₄

SD₁₆×C₁₀₇

yes

—

104

C₂²×C₄

Q₁₆×C₁₀₇

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₄²×C₁₀₇

yes

characteristic

110

C₂×C₈

C₂²⋊C₄×C₁₀₇

yes

characteristic

111

C₂×C₈

C₄⋊C₄×C₁₀₇

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₁₆

—

116

C₂³

—

117

C₂³

—

118

C₂³

#102×C₁₀₇

yes

—

119

C₂³

#107×C₁₀₇

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#95×C₁₀₇

yes

characteristic

130

C₂×C₄

#103×C₁₀₇

yes

—

131

C₂×C₄

#106×C₁₀₇

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

#99×C₁₀₇

yes

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#340×C₁₀₇

yes

characteristic

141

C₂²

#360×C₁₀₇

yes

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

H(3,ℤ/4)×C₁₀₇

yes

characteristic

146

C₄

#342×C₁₀₇

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

C₁₀₇×#1289

yes

characteristic

150

C₂

C₁₀₇×#1386

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₇	#340
commutator	C₂×C₈	C₄²×C₁₀₇
Frattini, cosocle	C₂³×C₈	C₂²×C₁₀₇
prime radical	C₂⁴×C₁₀₇	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₂³×C₄×C₁₀₇ ◁ C₂³×C₈×C₁₀₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₇

#340

commutator

C₂×C₈

C₄²×C₁₀₇

Frattini, cosocle

C₂³×C₈

C₂²×C₁₀₇

prime radical

C₂⁴×C₁₀₇

D₈

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₇ ◁ C₂³×C₄×C₁₀₇ ◁ C₂³×C₈×C₁₀₇ ◁ G

N	H ≅ G/N	type
4. C₂×M₃₂×C₁₀₇	C₄	semidirect
8. #340×C₁₀₇	C₄	semidirect
71. #7233	C₁₀₇	direct (N × H)
78. C₁₀₇	#7233	direct (N × H)
80. C₂×M₃₂	C₄×C₁₀₇	semidirect
84. #340	C₄×C₁₀₇	semidirect

H ≅ G/N

type

4. C₂×M₃₂×C₁₀₇

C₄

semidirect

8. #340×C₁₀₇

C₄

semidirect

71. #7233

C₁₀₇

direct (N × H)

78. C₁₀₇

#7233

direct (N × H)

80. C₂×M₃₂

C₄×C₁₀₇

semidirect

84. #340

C₄×C₁₀₇

semidirect

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

subgroup	index
C₁₀₇×#2112	2
C₁₀₇×#1807	2
#7233	107

subgroup

index

C₁₀₇×#2112

C₁₀₇×#1807

#7233

107

Subgroups — C₁₀₇×#7233 #41033197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#7233 #41033197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#7233 #41033197