Group #28332756 — subgroups

Subgroups — C₁₀₇×#2073 #28332756

Order 13696.

258 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 92 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×#102×C₁₀₇	C₂	1	2	yes	maximal
2	#390×C₁₀₇	C₂	1	1	yes	maximal, characteristic
3	#382×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#340×C₁₀₇	C₂	1	2	yes	maximal, complemented
5	#336×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₁₀₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₀₇	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
10	C₂×C₂²⋊C₄×C₁₀₇	—	2	2	no	—
11	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
12	#102×C₁₀₇	—	2	4	no	—
13	#95×C₁₀₇	—	2	2	no	—
14	C₂⁴×C₁₀₇	—	2	1	no	—
15	C₂×D₄×C₁₀₇	—	2	2	no	—
16	C₂×D₄×C₁₀₇	—	2	2	no	—
17	C₂×D₄×C₁₀₇	—	2	1	no	—
18	C₂×D₄×C₁₀₇	D₄	1	2	yes	—
19	C₂²×C₄×C₁₀₇	—	2	2	no	—
20	C₂²×C₄×C₁₀₇	—	2	1	no	—
21	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
22	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₁₀₇	—	2	2	no	—
24	C₂×C₈×C₁₀₇	—	2	2	no	—
25	C₂×C₈×C₁₀₇	—	2	2	no	—
26	C₄⋊C₄×C₁₀₇	—	2	2	no	—
27	C₄⋊C₄×C₁₀₇	—	2	2	no	—
28	C₄⋊C₄×C₁₀₇	—	2	2	no	—
29	C₂²⋊C₄×C₁₀₇	—	4	4	no	—
30	C₄²×C₁₀₇	—	2	2	no	—
31	C₂³×C₁₀₇	—	2	2	no	—
32	C₂³×C₁₀₇	—	4	1	no	—
33	C₂³×C₁₀₇	—	4	1	no	—
34	C₂³×C₁₀₇	—	4	2	no	—
35	C₂³×C₁₀₇	—	4	2	no	—
36	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
37	D₄×C₁₀₇	—	2	4	no	—
38	D₄×C₁₀₇	—	4	2	no	—
39	C₂×C₄×C₁₀₇	—	2	2	no	—
40	C₂×C₄×C₁₀₇	—	2	2	no	—
41	C₂×C₄×C₁₀₇	—	2	2	no	—
42	C₂×C₄×C₁₀₇	—	4	2	no	—
43	C₂×C₄×C₁₀₇	—	4	1	no	—
44	C₂×C₄×C₁₀₇	—	4	2	no	—
45	C₂×C₄×C₁₀₇	—	4	2	no	—
46	C₂×C₄×C₁₀₇	—	4	2	no	—
47	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₇	#41	1	2	yes	—
49	C₂×C₄×C₁₀₇	#41	1	2	yes	—
50	C₂×C₄×C₁₀₇	#41	1	1	yes	characteristic
51	C₈×C₁₀₇	—	4	2	no	—
52	C₂²×C₁₀₇	—	4	2	no	—
53	C₂²×C₁₀₇	—	4	4	no	—
54	C₂²×C₁₀₇	—	8	2	no	—
55	C₂²×C₁₀₇	—	8	1	no	—
56	C₂²×C₁₀₇	—	8	1	no	—
57	C₂²×C₁₀₇	#121	1	1	yes	characteristic
58	C₂²×C₁₀₇	#123	1	1	yes	characteristic
59	C₂²×C₁₀₇	#124	1	1	yes	characteristic
60	C₂²×C₁₀₇	#135	1	2	yes	—
61	C₂²×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₁₀₇	—	4	2	no	—
63	C₄×C₁₀₇	—	8	2	no	—
64	C₄×C₁₀₇	—	8	1	no	—
65	C₄×C₁₀₇	#123	1	2	yes	—
66	C₄×C₁₀₇	#126	1	2	yes	—
67	C₂×C₁₀₇	—	8	2	no	—
68	C₂×C₁₀₇	#397	1	1	yes	characteristic
69	C₂×C₁₀₇	#463	1	2	yes	—
70	C₂×C₁₀₇	#482	1	2	yes	—
71	C₂×C₁₀₇	#488	1	1	yes	characteristic
72	C₂×C₁₀₇	#490	1	1	yes	characteristic
73	#2073	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₁₀₇	#2073	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×#102	C₂×C₁₀₇	1	2	yes	—
76	#390	C₂×C₁₀₇	1	1	yes	characteristic
77	#382	C₂×C₁₀₇	1	1	yes	characteristic, complemented
78	#340	C₂×C₁₀₇	1	2	yes	complemented
79	#336	C₂×C₁₀₇	1	1	yes	characteristic, complemented
80	C₂²×D₄	C₂²×C₁₀₇	1	1	yes	characteristic
81	C₂²×C₈	C₂²×C₁₀₇	1	2	yes	—
82	C₂×C₄⋊C₄	C₂²×C₁₀₇	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₂²×C₁₀₇	1	2	yes	—
84	C₂×C₂²⋊C₄	—	2	2	no	—
85	C₂×C₄²	C₂²×C₁₀₇	1	1	yes	characteristic
86	#102	—	2	4	no	—
87	#95	—	2	2	no	—
88	C₂⁴	—	2	1	no	—
89	C₂×D₄	—	2	2	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	D₄×C₁₀₇	1	2	yes	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	D₄×C₁₀₇	1	2	yes	—
96	C₂²×C₄	C₂³×C₁₀₇	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₂²⋊C₄	—	4	4	no	—
104	C₄²	—	2	2	no	—
105	C₂³	—	2	2	no	—
106	C₂³	—	4	1	no	—
107	C₂³	—	4	1	no	—
108	C₂³	—	4	2	no	—
109	C₂³	—	4	2	no	—
110	C₂³	C₂×D₄×C₁₀₇	1	1	yes	characteristic
111	D₄	—	2	4	no	—
112	D₄	—	4	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
122	C₂×C₄	#41×C₁₀₇	1	2	yes	—
123	C₂×C₄	#41×C₁₀₇	1	2	yes	—
124	C₂×C₄	#41×C₁₀₇	1	1	yes	characteristic
125	C₈	—	4	2	no	—
126	C₂²	—	4	2	no	—
127	C₂²	—	4	4	no	—
128	C₂²	—	8	2	no	—
129	C₂²	—	8	1	no	—
130	C₂²	—	8	1	no	—
131	C₂²	#121×C₁₀₇	1	1	yes	characteristic
132	C₂²	#123×C₁₀₇	1	1	yes	characteristic
133	C₂²	#124×C₁₀₇	1	1	yes	characteristic
134	C₂²	#135×C₁₀₇	1	2	yes	—
135	C₂²	AGL(1,ℤ/8)×C₁₀₇	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	—	8	2	no	—
138	C₄	—	8	1	no	—
139	C₄	#123×C₁₀₇	1	2	yes	—
140	C₄	#126×C₁₀₇	1	2	yes	—
141	C₂	—	8	2	no	—
142	C₂	#397×C₁₀₇	1	1	yes	characteristic
143	C₂	#463×C₁₀₇	1	2	yes	—
144	C₂	#482×C₁₀₇	1	2	yes	—
145	C₂	#488×C₁₀₇	1	1	yes	characteristic
146	C₂	#490×C₁₀₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#102×C₁₀₇

C₂

yes

maximal

#390×C₁₀₇

C₂

yes

maximal, characteristic

#382×C₁₀₇

C₂

yes

maximal, characteristic, complemented

#340×C₁₀₇

C₂

yes

maximal, complemented

#336×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₀₇

C₂²

yes

characteristic

C₂²×C₈×C₁₀₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₀₇

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

#102×C₁₀₇

—

#95×C₁₀₇

—

C₂⁴×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

C₂×D₄

yes

characteristic

D₄×C₁₀₇

—

D₄×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

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

—

C₂×C₄×C₁₀₇

#41

yes

—

C₂×C₄×C₁₀₇

#41

yes

characteristic

C₈×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

—

C₂²×C₁₀₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#123

yes

—

C₄×C₁₀₇

#126

yes

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#397

yes

characteristic

C₂×C₁₀₇

#463

yes

—

C₂×C₁₀₇

#482

yes

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₇

#2073

yes

characteristic, direct product factor, Sylow

C₂×#102

C₂×C₁₀₇

yes

—

#390

C₂×C₁₀₇

yes

characteristic

#382

C₂×C₁₀₇

yes

characteristic, complemented

#340

C₂×C₁₀₇

yes

complemented

#336

C₂×C₁₀₇

yes

characteristic, complemented

C₂²×D₄

C₂²×C₁₀₇

yes

characteristic

C₂²×C₈

C₂²×C₁₀₇

yes

—

C₂×C₄⋊C₄

C₂²×C₁₀₇

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₀₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂²×C₁₀₇

yes

characteristic

#102

—

#95

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄×C₁₀₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₀₇

yes

—

C₂²×C₄

C₂³×C₁₀₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂×D₄×C₁₀₇

yes

characteristic

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂×D₄×C₁₀₇

yes

characteristic

122

C₂×C₄

#41×C₁₀₇

yes

—

123

C₂×C₄

#41×C₁₀₇

yes

—

124

C₂×C₄

#41×C₁₀₇

yes

characteristic

125

C₈

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

#121×C₁₀₇

yes

characteristic

132

C₂²

#123×C₁₀₇

yes

characteristic

133

C₂²

#124×C₁₀₇

yes

characteristic

134

C₂²

#135×C₁₀₇

yes

—

135

C₂²

AGL(1,ℤ/8)×C₁₀₇

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#123×C₁₀₇

yes

—

140

C₄

#126×C₁₀₇

yes

—

141

C₂

—

142

C₂

#397×C₁₀₇

yes

characteristic

143

C₂

#463×C₁₀₇

yes

—

144

C₂

#482×C₁₀₇

yes

—

145

C₂

#488×C₁₀₇

yes

characteristic

146

C₂

#490×C₁₀₇

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₇
prime radical	C₂²×D₄×C₁₀₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₀₇

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₀₇

prime radical

C₂²×D₄×C₁₀₇

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ G

N	H ≅ G/N	type
3. #382×C₁₀₇	C₂	semidirect
4. #340×C₁₀₇	C₂	semidirect
5. #336×C₁₀₇	C₂	semidirect
73. #2073	C₁₀₇	direct (N × H)
74. C₁₀₇	#2073	direct (N × H)
77. #382	C₂×C₁₀₇	semidirect
78. #340	C₂×C₁₀₇	semidirect
79. #336	C₂×C₁₀₇	semidirect

H ≅ G/N

type

3. #382×C₁₀₇

C₂

semidirect

4. #340×C₁₀₇

C₂

semidirect

5. #336×C₁₀₇

C₂

semidirect

73. #2073

C₁₀₇

direct (N × H)

74. C₁₀₇

#2073

direct (N × H)

77. #382

C₂×C₁₀₇

semidirect

78. #340

C₂×C₁₀₇

semidirect

79. #336

C₂×C₁₀₇

semidirect

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

subgroup	index
C₂×#102×C₁₀₇	2
#390×C₁₀₇	2
#382×C₁₀₇	2
#340×C₁₀₇	2
#336×C₁₀₇	2
#2073	107

subgroup

index

C₂×#102×C₁₀₇

#390×C₁₀₇

#382×C₁₀₇

#340×C₁₀₇

#336×C₁₀₇

#2073

107

Subgroups — C₁₀₇×#2073 #28332756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#2073 #28332756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#2073 #28332756