Group #38790274 — subgroups

Subgroups — C₂×C₁₀₁×#1308 #38790274

Order 25856.

526 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 238 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×#432×C₁₀₁	C₂	1	3	yes	maximal
2	C₁₀₁×#1308	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
4	#432×C₁₀₁	—	2	3	no	—
5	#432×C₁₀₁	—	2	3	no	—
6	#432×C₁₀₁	C₄	1	6	yes	—
7	#432×C₁₀₁	C₂²	1	6	yes	—
8	C₂×C₄⋊C₈×C₁₀₁	—	2	3	no	—
9	C₂×C₂²⋊C₈×C₁₀₁	C₄	1	6	yes	—
10	C₂×#97×C₁₀₁	—	2	3	no	—
11	C₂³×C₄×C₁₀₁	C₂×C₄	1	3	yes	—
12	C₂²×C₈×C₁₀₁	—	2	6	no	—
13	C₂×C₄²×C₁₀₁	C₂×C₄	1	3	yes	—
14	C₂×C₄²×C₁₀₁	D₄	1	3	yes	—
15	C₂×C₄²×C₁₀₁	D₄	1	3	yes	—
16	C₂×C₄²×C₁₀₁	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₁₀₁	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
19	C₄⋊C₈×C₁₀₁	—	2	6	no	—
20	C₄⋊C₈×C₁₀₁	—	2	6	no	—
21	C₂²⋊C₈×C₁₀₁	—	2	6	no	—
22	C₂²⋊C₈×C₁₀₁	C₂×C₄	1	12	yes	—
23	#97×C₁₀₁	—	2	6	no	—
24	#97×C₁₀₁	—	2	6	no	—
25	C₂⁴×C₁₀₁	C₄²	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₁	—	4	6	no	—
27	C₂²×C₄×C₁₀₁	C₄²	1	3	yes	—
28	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	3	yes	—
29	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	3	yes	—
30	C₂²×C₄×C₁₀₁	C₄⋊C₄	1	3	yes	—
31	C₂²×C₄×C₁₀₁	C₄⋊C₄	1	3	yes	—
32	C₂²×C₄×C₁₀₁	C₂²×C₄	1	3	yes	—
33	C₂×C₈×C₁₀₁	—	2	12	no	—
34	C₂×C₈×C₁₀₁	—	4	12	no	—
35	C₄²×C₁₀₁	C₂²⋊C₄	1	3	yes	—
36	C₄²×C₁₀₁	C₂²⋊C₄	1	3	yes	—
37	C₄²×C₁₀₁	C₄⋊C₄	1	3	yes	—
38	C₄²×C₁₀₁	C₄⋊C₄	1	3	yes	—
39	C₄²×C₁₀₁	C₂×D₄	1	3	yes	—
40	C₄²×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₁₀₁	—	2	6	no	—
42	C₂³×C₁₀₁	#95	1	1	yes	characteristic
43	C₂³×C₁₀₁	#95	1	1	yes	characteristic
44	C₂³×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
45	C₂×C₄×C₁₀₁	—	2	6	no	—
46	C₂×C₄×C₁₀₁	—	2	6	no	—
47	C₂×C₄×C₁₀₁	—	4	3	no	—
48	C₂×C₄×C₁₀₁	—	4	3	no	—
49	C₂×C₄×C₁₀₁	—	4	3	no	—
50	C₂×C₄×C₁₀₁	—	4	3	no	—
51	C₂×C₄×C₁₀₁	#95	1	3	yes	—
52	C₂×C₄×C₁₀₁	#95	1	3	yes	—
53	C₂×C₄×C₁₀₁	C₂×C₂²⋊C₄	1	3	yes	—
54	C₂×C₄×C₁₀₁	C₂×C₄⋊C₄	1	3	yes	—
55	C₈×C₁₀₁	—	4	12	no	—
56	C₂²×C₁₀₁	—	2	3	no	—
57	C₂²×C₁₀₁	—	2	3	no	—
58	C₂²×C₁₀₁	—	4	4	no	—
59	C₂²×C₁₀₁	#344	1	6	yes	—
60	C₂²×C₁₀₁	C₂×#95	1	1	yes	characteristic
61	C₄×C₁₀₁	—	2	3	no	—
62	C₄×C₁₀₁	—	2	3	no	—
63	C₄×C₁₀₁	—	2	3	no	—
64	C₄×C₁₀₁	—	2	3	no	—
65	C₂×#1308	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×C₁₀₁	—	4	1	no	—
67	C₂×C₁₀₁	—	4	1	no	—
68	C₂×C₁₀₁	#1308	1	4	yes	direct product factor
69	C₂×C₁₀₁	C₂×#344	1	3	yes	—
70	C₂×#432	C₂×C₁₀₁	1	3	yes	—
71	#1308	C₂×C₁₀₁	1	4	yes	direct product factor
72	C₁₀₁	C₂×#1308	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²×C₄²	C₂²×C₁₀₁	1	1	yes	characteristic
74	#432	—	2	3	no	—
75	#432	—	2	3	no	—
76	#432	C₄×C₁₀₁	1	6	yes	—
77	#432	C₂²×C₁₀₁	1	6	yes	—
78	C₂×C₄⋊C₈	—	2	3	no	—
79	C₂×C₂²⋊C₈	C₄×C₁₀₁	1	6	yes	—
80	C₂×#97	—	2	3	no	—
81	C₂³×C₄	C₂×C₄×C₁₀₁	1	3	yes	—
82	C₂²×C₈	—	2	6	no	—
83	C₂×C₄²	C₂×C₄×C₁₀₁	1	3	yes	—
84	C₂×C₄²	D₄×C₁₀₁	1	3	yes	—
85	C₂×C₄²	D₄×C₁₀₁	1	3	yes	—
86	C₂×C₄²	Q₈×C₁₀₁	1	1	yes	characteristic
87	C₂×C₄²	Q₈×C₁₀₁	1	1	yes	characteristic
88	C₂×C₄²	C₂³×C₁₀₁	1	1	yes	characteristic
89	C₄⋊C₈	—	2	6	no	—
90	C₄⋊C₈	—	2	6	no	—
91	C₂²⋊C₈	—	2	6	no	—
92	C₂²⋊C₈	C₂×C₄×C₁₀₁	1	12	yes	—
93	#97	—	2	6	no	—
94	#97	—	2	6	no	—
95	C₂⁴	C₄²×C₁₀₁	1	1	yes	characteristic
96	C₂²×C₄	—	4	6	no	—
97	C₂²×C₄	C₄²×C₁₀₁	1	3	yes	—
98	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	3	yes	—
99	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	3	yes	—
100	C₂²×C₄	C₄⋊C₄×C₁₀₁	1	3	yes	—
101	C₂²×C₄	C₄⋊C₄×C₁₀₁	1	3	yes	—
102	C₂²×C₄	C₂²×C₄×C₁₀₁	1	3	yes	—
103	C₂×C₈	—	2	12	no	—
104	C₂×C₈	—	4	12	no	—
105	C₄²	C₂²⋊C₄×C₁₀₁	1	3	yes	—
106	C₄²	C₂²⋊C₄×C₁₀₁	1	3	yes	—
107	C₄²	C₄⋊C₄×C₁₀₁	1	3	yes	—
108	C₄²	C₄⋊C₄×C₁₀₁	1	3	yes	—
109	C₄²	C₂×D₄×C₁₀₁	1	3	yes	—
110	C₄²	C₂×Q₈×C₁₀₁	1	1	yes	characteristic
111	C₂³	—	2	6	no	—
112	C₂³	#95×C₁₀₁	1	1	yes	characteristic
113	C₂³	#95×C₁₀₁	1	1	yes	characteristic
114	C₂³	C₂×C₄²×C₁₀₁	1	1	yes	characteristic
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	2	6	no	—
117	C₂×C₄	—	4	3	no	—
118	C₂×C₄	—	4	3	no	—
119	C₂×C₄	—	4	3	no	—
120	C₂×C₄	—	4	3	no	—
121	C₂×C₄	#95×C₁₀₁	1	3	yes	—
122	C₂×C₄	#95×C₁₀₁	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₁	1	3	yes	—
124	C₂×C₄	C₂×C₄⋊C₄×C₁₀₁	1	3	yes	—
125	C₈	—	4	12	no	—
126	C₂²	—	2	3	no	—
127	C₂²	—	2	3	no	—
128	C₂²	—	4	4	no	—
129	C₂²	#344×C₁₀₁	1	6	yes	—
130	C₂²	C₂×#95×C₁₀₁	1	1	yes	characteristic
131	C₄	—	2	3	no	—
132	C₄	—	2	3	no	—
133	C₄	—	2	3	no	—
134	C₄	—	2	3	no	—
135	C₂	—	4	1	no	—
136	C₂	—	4	1	no	—
137	C₂	C₁₀₁×#1308	1	4	yes	direct product factor
138	C₂	C₂×#344×C₁₀₁	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#432×C₁₀₁

C₂

yes

maximal

C₁₀₁×#1308

C₂

yes

maximal, direct product factor

C₂²×C₄²×C₁₀₁

C₂²

yes

characteristic

#432×C₁₀₁

—

#432×C₁₀₁

—

#432×C₁₀₁

C₄

yes

—

#432×C₁₀₁

C₂²

yes

—

C₂×C₄⋊C₈×C₁₀₁

—

C₂×C₂²⋊C₈×C₁₀₁

C₄

yes

—

C₂×#97×C₁₀₁

—

C₂³×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₈×C₁₀₁

—

C₂×C₄²×C₁₀₁

C₂×C₄

yes

—

C₂×C₄²×C₁₀₁

D₄

yes

—

C₂×C₄²×C₁₀₁

D₄

yes

—

C₂×C₄²×C₁₀₁

Q₈

yes

characteristic

C₂×C₄²×C₁₀₁

Q₈

yes

characteristic

C₂×C₄²×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₈×C₁₀₁

—

C₄⋊C₈×C₁₀₁

—

C₂²⋊C₈×C₁₀₁

—

C₂²⋊C₈×C₁₀₁

C₂×C₄

yes

—

#97×C₁₀₁

—

#97×C₁₀₁

—

C₂⁴×C₁₀₁

C₄²

yes

characteristic

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₄²

yes

—

C₂²×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂²×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂²×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₄²×C₁₀₁

C₂²⋊C₄

yes

—

C₄²×C₁₀₁

C₂²⋊C₄

yes

—

C₄²×C₁₀₁

C₄⋊C₄

yes

—

C₄²×C₁₀₁

C₄⋊C₄

yes

—

C₄²×C₁₀₁

C₂×D₄

yes

—

C₄²×C₁₀₁

C₂×Q₈

yes

characteristic

C₂³×C₁₀₁

—

C₂³×C₁₀₁

yes

C₂³×C₁₀₁

yes

C₂³×C₁₀₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

#95

yes

—

C₂×C₄×C₁₀₁

#95

yes

—

C₂×C₄×C₁₀₁

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂×C₄⋊C₄

yes

—

C₈×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#344

yes

—

C₂²×C₁₀₁

C₂×#95

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₂×#1308

C₁₀₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#1308

yes

direct product factor

C₂×C₁₀₁

C₂×#344

yes

—

C₂×#432

C₂×C₁₀₁

yes

—

#1308

C₂×C₁₀₁

yes

direct product factor

C₁₀₁

C₂×#1308

yes

characteristic, direct product factor, Sylow

C₂²×C₄²

C₂²×C₁₀₁

yes

—

—

C₄×C₁₀₁

yes

—

#432

C₂²×C₁₀₁

yes

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

C₄×C₁₀₁

yes

—

C₂×#97

—

C₂³×C₄

C₂×C₄×C₁₀₁

yes

—

C₂²×C₈

—

C₂×C₄²

C₂×C₄×C₁₀₁

yes

—

C₂×C₄²

D₄×C₁₀₁

yes

—

C₂×C₄²

D₄×C₁₀₁

yes

—

C₂×C₄²

Q₈×C₁₀₁

yes

characteristic

C₂×C₄²

Q₈×C₁₀₁

yes

characteristic

C₂×C₄²

C₂³×C₁₀₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×C₄×C₁₀₁

yes

—

#97

—

#97

—

C₂⁴

C₄²×C₁₀₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄²×C₁₀₁

yes

—

C₂²×C₄

C₂²⋊C₄×C₁₀₁

yes

—

C₂²×C₄

C₂²⋊C₄×C₁₀₁

yes

—

100

C₂²×C₄

C₄⋊C₄×C₁₀₁

yes

—

101

C₂²×C₄

C₄⋊C₄×C₁₀₁

yes

—

102

C₂²×C₄

C₂²×C₄×C₁₀₁

yes

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₄²

C₂²⋊C₄×C₁₀₁

yes

—

106

C₄²

C₂²⋊C₄×C₁₀₁

yes

—

107

C₄²

C₄⋊C₄×C₁₀₁

yes

—

108

C₄²

C₄⋊C₄×C₁₀₁

yes

—

109

C₄²

C₂×D₄×C₁₀₁

yes

—

110

C₄²

C₂×Q₈×C₁₀₁

yes

characteristic

111

C₂³

—

112

C₂³

#95×C₁₀₁

yes

characteristic

113

C₂³

#95×C₁₀₁

yes

characteristic

114

C₂³

C₂×C₄²×C₁₀₁

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

#95×C₁₀₁

yes

—

122

C₂×C₄

#95×C₁₀₁

yes

—

123

C₂×C₄

C₂×C₂²⋊C₄×C₁₀₁

yes

—

124

C₂×C₄

C₂×C₄⋊C₄×C₁₀₁

yes

—

125

C₈

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#344×C₁₀₁

yes

—

130

C₂²

C₂×#95×C₁₀₁

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₂

—

136

C₂

—

137

C₂

C₁₀₁×#1308

yes

direct product factor

138

C₂

C₂×#344×C₁₀₁

yes

—

	subgroup	G/H
center, socle	C₂³×C₁₀₁	#95
commutator	C₂³	C₂×C₄²×C₁₀₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₁
prime radical	C₂⁴×C₁₀₁	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₁ ◁ C₂²×C₄²×C₁₀₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₀₁

#95

commutator

C₂³

C₂×C₄²×C₁₀₁

Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₁

prime radical

C₂⁴×C₁₀₁

C₄²

LCS

G ▷ C₂³ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₁ ◁ C₂²×C₄²×C₁₀₁ ◁ G

N	H ≅ G/N	type
2. C₁₀₁×#1308	C₂	direct (N × H)
65. C₂×#1308	C₁₀₁	direct (N × H)
68. C₂×C₁₀₁	#1308	direct (N × H)
71. #1308	C₂×C₁₀₁	direct (N × H)
72. C₁₀₁	C₂×#1308	direct (N × H)
137. C₂	C₁₀₁×#1308	direct (N × H)

H ≅ G/N

type

2. C₁₀₁×#1308

C₂

direct (N × H)

65. C₂×#1308

C₁₀₁

direct (N × H)

68. C₂×C₁₀₁

#1308

direct (N × H)

71. #1308

C₂×C₁₀₁

direct (N × H)

72. C₁₀₁

C₂×#1308

direct (N × H)

137. C₂

C₁₀₁×#1308

direct (N × H)

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

subgroup	index
C₂×#432×C₁₀₁	2
C₁₀₁×#1308	2
C₂×#1308	101

subgroup

index

C₂×#432×C₁₀₁

C₁₀₁×#1308

C₂×#1308

101

Subgroups — C₂×C₁₀₁×#1308 #38790274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C101×#1308 #38790274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₁×#1308 #38790274