Group #38794171 — subgroups

Subgroups — C₁₀₁×#12020 #38794171

Order 25856.

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×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₁₀₁×#1566	C₂	1	2	yes	maximal
3	C₁₀₁×#1368	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₀₁	C₄	1	2	yes	—
5	#423×C₁₀₁	—	2	2	no	—
6	C₂×C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
7	#346×C₁₀₁	C₂²	1	2	yes	—
8	C₄×C₁₆×C₁₀₁	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₁₀₁	C₄	1	1	yes	characteristic
10	#334×C₁₀₁	—	2	2	no	—
11	#334×C₁₀₁	C₂²	1	4	yes	—
12	C₂×M₁₆×C₁₀₁	—	4	2	no	—
13	C₂²×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₁	Q₈	1	1	yes	characteristic
15	C₂×C₄²×C₁₀₁	D₄	1	1	yes	characteristic
16	M₃₂×C₁₀₁	—	2	4	no	—
17	C₂×C₁₆×C₁₀₁	C₂×C₄	1	2	yes	—
18	C₂×C₁₆×C₁₀₁	C₂×C₄	1	2	yes	—
19	C₄⋊C₈×C₁₀₁	—	2	4	no	—
20	C₄⋊C₈×C₁₀₁	—	4	2	no	—
21	C₄×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₀₁	D₄	1	1	yes	characteristic
23	C₄×C₈×C₁₀₁	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₁	SD₁₆	1	2	yes	—
27	M₁₆×C₁₀₁	—	8	4	no	—
28	C₂×C₈×C₁₀₁	—	2	2	no	—
29	C₂×C₈×C₁₀₁	—	2	2	no	—
30	C₂×C₈×C₁₀₁	—	4	4	no	—
31	C₂×C₈×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
35	C₄²×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
36	C₄²×C₁₀₁	SD₁₆	1	2	yes	—
37	C₄²×C₁₀₁	C₂×D₄	1	1	yes	characteristic
38	C₁₆×C₁₀₁	—	2	2	no	—
39	C₁₆×C₁₀₁	M₁₆	1	4	yes	complemented
40	C₂³×C₁₀₁	#106	1	1	yes	characteristic
41	C₂×C₄×C₁₀₁	—	2	4	no	—
42	C₂×C₄×C₁₀₁	—	2	1	no	—
43	C₂×C₄×C₁₀₁	—	2	1	no	—
44	C₂×C₄×C₁₀₁	#106	1	1	yes	characteristic
45	C₂×C₄×C₁₀₁	#106	1	2	yes	—
46	C₂×C₄×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₁	C₂×SD₁₆	1	2	yes	—
48	C₈×C₁₀₁	—	2	1	no	—
49	C₈×C₁₀₁	—	2	1	no	—
50	C₈×C₁₀₁	—	8	4	no	—
51	C₈×C₁₀₁	#108	1	2	yes	—
52	C₈×C₁₀₁	C₂×M₁₆	1	2	yes	—
53	C₂²×C₁₀₁	—	2	1	no	—
54	C₂²×C₁₀₁	—	2	2	no	—
55	C₂²×C₁₀₁	C₂×#106	1	1	yes	characteristic
56	C₄×C₁₀₁	—	2	1	no	—
57	C₄×C₁₀₁	—	2	2	no	—
58	C₄×C₁₀₁	#365	1	4	yes	—
59	C₄×C₁₀₁	#423	1	1	yes	characteristic
60	C₄×C₁₀₁	C₂×#108	1	1	yes	characteristic
61	#12020	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₁₀₁	—	4	1	no	—
63	C₂×C₁₀₁	#1566	1	1	yes	characteristic
64	C₂×C₁₀₁	C₂×#365	1	2	yes	—
65	C₄×M₃₂	C₂×C₁₀₁	1	1	yes	characteristic
66	#1566	C₂×C₁₀₁	1	2	yes	—
67	#1368	C₂×C₁₀₁	1	4	yes	complemented
68	C₁₀₁	#12020	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×M₃₂	C₄×C₁₀₁	1	2	yes	—
70	#423	—	2	2	no	—
71	C₂×C₄×C₈	C₂²×C₁₀₁	1	1	yes	characteristic
72	#346	C₂²×C₁₀₁	1	2	yes	—
73	C₄×C₁₆	C₄×C₁₀₁	1	1	yes	characteristic
74	C₄×C₁₆	C₄×C₁₀₁	1	1	yes	characteristic
75	#334	—	2	2	no	—
76	#334	C₂²×C₁₀₁	1	4	yes	—
77	C₂×M₁₆	—	4	2	no	—
78	C₂²×C₈	C₂×C₄×C₁₀₁	1	1	yes	characteristic
79	C₂²×C₈	Q₈×C₁₀₁	1	1	yes	characteristic
80	C₂×C₄²	D₄×C₁₀₁	1	1	yes	characteristic
81	M₃₂	—	2	4	no	—
82	C₂×C₁₆	C₂×C₄×C₁₀₁	1	2	yes	—
83	C₂×C₁₆	C₂×C₄×C₁₀₁	1	2	yes	—
84	C₄⋊C₈	—	2	4	no	—
85	C₄⋊C₈	—	4	2	no	—
86	C₄×C₈	C₂×C₄×C₁₀₁	1	1	yes	characteristic
87	C₄×C₈	D₄×C₁₀₁	1	1	yes	characteristic
88	C₄×C₈	Q₈×C₁₀₁	1	1	yes	characteristic
89	C₄×C₈	C₂³×C₁₀₁	1	1	yes	characteristic
90	C₂²×C₄	C₄⋊C₄×C₁₀₁	1	1	yes	characteristic
91	C₂²×C₄	SD₁₆×C₁₀₁	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×C₁₀₁	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₄×C₁₀₁	1	1	yes	characteristic
98	C₂×C₈	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
99	C₂×C₈	C₂×Q₈×C₁₀₁	1	1	yes	characteristic
100	C₄²	C₄⋊C₄×C₁₀₁	1	1	yes	characteristic
101	C₄²	SD₁₆×C₁₀₁	1	2	yes	—
102	C₄²	C₂×D₄×C₁₀₁	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	M₁₆×C₁₀₁	1	4	yes	complemented
105	C₂³	#106×C₁₀₁	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	#106×C₁₀₁	1	1	yes	characteristic
110	C₂×C₄	#106×C₁₀₁	1	2	yes	—
111	C₂×C₄	C₂×C₄⋊C₄×C₁₀₁	1	1	yes	characteristic
112	C₂×C₄	C₂×SD₁₆×C₁₀₁	1	2	yes	—
113	C₈	—	2	1	no	—
114	C₈	—	2	1	no	—
115	C₈	—	8	4	no	—
116	C₈	#108×C₁₀₁	1	2	yes	—
117	C₈	C₂×M₁₆×C₁₀₁	1	2	yes	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂×#106×C₁₀₁	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	#365×C₁₀₁	1	4	yes	—
124	C₄	#423×C₁₀₁	1	1	yes	characteristic
125	C₄	C₂×#108×C₁₀₁	1	1	yes	characteristic
126	C₂	—	4	1	no	—
127	C₂	C₁₀₁×#1566	1	1	yes	characteristic
128	C₂	C₂×#365×C₁₀₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×M₃₂×C₁₀₁

C₂

yes

maximal, characteristic

C₁₀₁×#1566

C₂

yes

maximal

C₁₀₁×#1368

C₂

yes

maximal, complemented

C₂×M₃₂×C₁₀₁

C₄

yes

—

#423×C₁₀₁

—

C₂×C₄×C₈×C₁₀₁

C₂²

yes

characteristic

#346×C₁₀₁

C₂²

yes

—

C₄×C₁₆×C₁₀₁

C₄

yes

characteristic

C₄×C₁₆×C₁₀₁

C₄

yes

characteristic

#334×C₁₀₁

—

#334×C₁₀₁

C₂²

yes

—

C₂×M₁₆×C₁₀₁

—

C₂²×C₈×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₀₁

Q₈

yes

characteristic

C₂×C₄²×C₁₀₁

D₄

yes

characteristic

M₃₂×C₁₀₁

—

C₂×C₁₆×C₁₀₁

C₂×C₄

yes

—

C₂×C₁₆×C₁₀₁

C₂×C₄

yes

—

C₄⋊C₈×C₁₀₁

—

C₄⋊C₈×C₁₀₁

—

C₄×C₈×C₁₀₁

C₂×C₄

yes

characteristic

C₄×C₈×C₁₀₁

D₄

yes

characteristic

C₄×C₈×C₁₀₁

Q₈

yes

characteristic

C₄×C₈×C₁₀₁

C₂³

yes

characteristic

C₂²×C₄×C₁₀₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₀₁

SD₁₆

yes

—

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₀₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₀₁

C₂×Q₈

yes

characteristic

C₄²×C₁₀₁

C₄⋊C₄

yes

characteristic

C₄²×C₁₀₁

SD₁₆

yes

—

C₄²×C₁₀₁

C₂×D₄

yes

characteristic

C₁₆×C₁₀₁

—

C₁₆×C₁₀₁

M₁₆

yes

complemented

C₂³×C₁₀₁

#106

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

#106

yes

characteristic

C₂×C₄×C₁₀₁

#106

yes

—

C₂×C₄×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×SD₁₆

yes

—

C₈×C₁₀₁

—

C₈×C₁₀₁

—

C₈×C₁₀₁

—

C₈×C₁₀₁

#108

yes

—

C₈×C₁₀₁

C₂×M₁₆

yes

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×#106

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#365

yes

—

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

C₁₀₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₁

—

C₂×C₁₀₁

#1566

yes

characteristic

C₂×C₁₀₁

C₂×#365

yes

—

C₄×M₃₂

C₂×C₁₀₁

yes

characteristic

#1566

C₂×C₁₀₁

yes

—

#1368

C₂×C₁₀₁

yes

complemented

C₁₀₁

#12020

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₄×C₁₀₁

yes

—

#423

—

C₂×C₄×C₈

C₂²×C₁₀₁

yes

characteristic

#346

C₂²×C₁₀₁

yes

—

C₄×C₁₆

C₄×C₁₀₁

yes

characteristic

C₄×C₁₆

C₄×C₁₀₁

yes

characteristic

#334

—

#334

C₂²×C₁₀₁

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×C₄×C₁₀₁

yes

characteristic

C₂²×C₈

Q₈×C₁₀₁

yes

characteristic

C₂×C₄²

D₄×C₁₀₁

yes

characteristic

M₃₂

—

C₂×C₁₆

C₂×C₄×C₁₀₁

yes

—

C₂×C₁₆

C₂×C₄×C₁₀₁

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₁₀₁

yes

characteristic

C₄×C₈

D₄×C₁₀₁

yes

characteristic

C₄×C₈

Q₈×C₁₀₁

yes

characteristic

C₄×C₈

C₂³×C₁₀₁

yes

characteristic

C₂²×C₄

C₄⋊C₄×C₁₀₁

yes

characteristic

C₂²×C₄

SD₁₆×C₁₀₁

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₁₀₁

yes

characteristic

C₂×C₈

C₄⋊C₄×C₁₀₁

yes

characteristic

C₂×C₈

C₂²×C₄×C₁₀₁

yes

characteristic

C₂×C₈

C₂×Q₈×C₁₀₁

yes

characteristic

100

C₄²

C₄⋊C₄×C₁₀₁

yes

characteristic

101

C₄²

SD₁₆×C₁₀₁

yes

—

102

C₄²

C₂×D₄×C₁₀₁

yes

characteristic

103

C₁₆

—

104

C₁₆

M₁₆×C₁₀₁

yes

complemented

105

C₂³

#106×C₁₀₁

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#106×C₁₀₁

yes

characteristic

110

C₂×C₄

#106×C₁₀₁

yes

—

111

C₂×C₄

C₂×C₄⋊C₄×C₁₀₁

yes

characteristic

112

C₂×C₄

C₂×SD₁₆×C₁₀₁

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

#108×C₁₀₁

yes

—

117

C₈

C₂×M₁₆×C₁₀₁

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#106×C₁₀₁

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

#365×C₁₀₁

yes

—

124

C₄

#423×C₁₀₁

yes

characteristic

125

C₄

C₂×#108×C₁₀₁

yes

characteristic

126

C₂

—

127

C₂

C₁₀₁×#1566

yes

characteristic

128

C₂

C₂×#365×C₁₀₁

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₁	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₄×C₈	C₂³×C₁₀₁
prime radical	C₂³×C₁₀₁	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₂×C₄²×C₁₀₁ ◁ C₂×C₄×C₈×C₁₀₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₁

C₂×#106

commutator

C₂×C₈

C₂²×C₄×C₁₀₁

Frattini, cosocle

C₄×C₈

C₂³×C₁₀₁

prime radical

C₂³×C₁₀₁

#106

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₁ ◁ C₂×C₄²×C₁₀₁ ◁ C₂×C₄×C₈×C₁₀₁ ◁ G

N	H ≅ G/N	type
3. C₁₀₁×#1368	C₂	semidirect
39. C₁₆×C₁₀₁	M₁₆	semidirect
61. #12020	C₁₀₁	direct (N × H)
67. #1368	C₂×C₁₀₁	semidirect
68. C₁₀₁	#12020	direct (N × H)
104. C₁₆	M₁₆×C₁₀₁	semidirect

H ≅ G/N

type

3. C₁₀₁×#1368

C₂

semidirect

39. C₁₆×C₁₀₁

M₁₆

semidirect

61. #12020

C₁₀₁

direct (N × H)

67. #1368

C₂×C₁₀₁

semidirect

68. C₁₀₁

#12020

direct (N × H)

104. C₁₆

M₁₆×C₁₀₁

semidirect

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

subgroup	index
C₄×M₃₂×C₁₀₁	2
C₁₀₁×#1566	2
C₁₀₁×#1368	2
#12020	101

subgroup

index

C₄×M₃₂×C₁₀₁

C₁₀₁×#1566

C₁₀₁×#1368

#12020

101

Subgroups — C₁₀₁×#12020 #38794171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C101×#12020 #38794171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#12020 #38794171