Group #27167401 — subgroups

Subgroups — C₁₀₁×#2047 #27167401

Order 12928.

256 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 84 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	C₂²×Q₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	#424×C₁₀₁	C₂	1	1	yes	maximal, characteristic
3	#419×C₁₀₁	C₂	1	2	yes	maximal
4	C₂×#101×C₁₀₁	C₂	1	2	yes	maximal
5	#341×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₀₁	C₂²	1	2	yes	—
7	C₂×Q₁₆×C₁₀₁	—	2	4	no	—
8	C₂×Q₁₆×C₁₀₁	—	2	2	no	—
9	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
12	#117×C₁₀₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₀₁	—	2	2	no	—
14	#103×C₁₀₁	—	2	4	no	—
15	#101×C₁₀₁	—	2	4	no	—
16	C₂×Q₈×C₁₀₁	—	2	2	no	—
17	C₂×Q₈×C₁₀₁	—	2	2	no	—
18	C₂×Q₈×C₁₀₁	—	2	2	no	—
19	C₂×Q₈×C₁₀₁	—	2	2	no	—
20	C₂×Q₈×C₁₀₁	—	2	2	no	—
21	C₂×Q₈×C₁₀₁	D₄	1	4	yes	—
22	C₂²×C₄×C₁₀₁	—	2	2	no	—
23	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
24	Q₁₆×C₁₀₁	—	2	8	no	—
25	M₁₆×C₁₀₁	—	2	2	no	—
26	M₁₆×C₁₀₁	—	2	2	no	—
27	M₁₆×C₁₀₁	—	2	2	no	—
28	C₂×C₈×C₁₀₁	—	2	2	no	—
29	C₂×C₈×C₁₀₁	—	2	2	no	—
30	C₂×C₈×C₁₀₁	—	2	1	no	—
31	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
32	C₄⋊C₄×C₁₀₁	D₄	1	2	yes	—
33	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
34	C₄²×C₁₀₁	—	2	1	no	—
35	C₂³×C₁₀₁	#41	1	1	yes	characteristic
36	Q₈×C₁₀₁	—	2	4	no	—
37	Q₈×C₁₀₁	—	2	4	no	—
38	Q₈×C₁₀₁	—	4	2	no	—
39	Q₈×C₁₀₁	—	4	2	no	—
40	C₂×C₄×C₁₀₁	—	2	2	no	—
41	C₂×C₄×C₁₀₁	—	2	4	no	—
42	C₂×C₄×C₁₀₁	—	4	2	no	—
43	C₂×C₄×C₁₀₁	—	4	2	no	—
44	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₁	#41	1	2	yes	—
48	C₈×C₁₀₁	—	2	2	no	—
49	C₈×C₁₀₁	—	4	1	no	—
50	C₈×C₁₀₁	—	4	2	no	—
51	C₂²×C₁₀₁	—	2	2	no	—
52	C₂²×C₁₀₁	#121	1	1	yes	characteristic
53	C₂²×C₁₀₁	#123	1	2	yes	—
54	C₄×C₁₀₁	—	4	4	no	—
55	C₄×C₁₀₁	—	4	2	no	—
56	C₄×C₁₀₁	#120	1	1	yes	characteristic
57	C₄×C₁₀₁	#121	1	2	yes	—
58	C₄×C₁₀₁	#124	1	1	yes	characteristic
59	C₂×C₁₀₁	—	2	2	no	—
60	C₂×C₁₀₁	#394	1	1	yes	characteristic
61	C₂×C₁₀₁	#473	1	2	yes	—
62	#2047	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₁₀₁	#2047	1	1	yes	characteristic, direct product factor, Sylow
64	C₂²×Q₁₆	C₂×C₁₀₁	1	1	yes	characteristic
65	#424	C₂×C₁₀₁	1	1	yes	characteristic
66	#419	C₂×C₁₀₁	1	2	yes	—
67	C₂×#101	C₂×C₁₀₁	1	2	yes	—
68	#341	C₂×C₁₀₁	1	1	yes	characteristic
69	C₂²×Q₈	C₂²×C₁₀₁	1	2	yes	—
70	C₂×Q₁₆	—	2	4	no	—
71	C₂×Q₁₆	—	2	2	no	—
72	C₂×M₁₆	C₂²×C₁₀₁	1	2	yes	—
73	C₂×M₁₆	C₂²×C₁₀₁	1	1	yes	characteristic
74	C₂²×C₈	C₂²×C₁₀₁	1	1	yes	characteristic
75	#117	C₂²×C₁₀₁	1	1	yes	characteristic
76	C₄⋊C₈	—	2	2	no	—
77	#103	—	2	4	no	—
78	#101	—	2	4	no	—
79	C₂×Q₈	—	2	2	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	D₄×C₁₀₁	1	4	yes	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
87	Q₁₆	—	2	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	D₄×C₁₀₁	1	2	yes	—
95	C₄⋊C₄	D₄×C₁₀₁	1	2	yes	—
96	C₂²⋊C₄	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	#41×C₁₀₁	1	1	yes	characteristic
99	Q₈	—	2	4	no	—
100	Q₈	—	2	4	no	—
101	Q₈	—	4	2	no	—
102	Q₈	—	4	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
108	C₂×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
109	C₂×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
110	C₂×C₄	#41×C₁₀₁	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	4	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#121×C₁₀₁	1	1	yes	characteristic
116	C₂²	#123×C₁₀₁	1	2	yes	—
117	C₄	—	4	2	no	—
118	C₄	—	4	4	no	—
119	C₄	#120×C₁₀₁	1	1	yes	characteristic
120	C₄	#121×C₁₀₁	1	2	yes	—
121	C₄	#124×C₁₀₁	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#394×C₁₀₁	1	1	yes	characteristic
124	C₂	#473×C₁₀₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₁₆×C₁₀₁

C₂

yes

maximal, characteristic

#424×C₁₀₁

C₂

yes

maximal, characteristic

#419×C₁₀₁

C₂

yes

maximal

C₂×#101×C₁₀₁

C₂

yes

maximal

#341×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₀₁

C₂²

yes

—

C₂×Q₁₆×C₁₀₁

—

C₂×Q₁₆×C₁₀₁

—

C₂×M₁₆×C₁₀₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₁

C₂²

yes

—

C₂²×C₈×C₁₀₁

C₂²

yes

characteristic

#117×C₁₀₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₁

—

#103×C₁₀₁

—

#101×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

Q₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

D₄

yes

—

C₄⋊C₄×C₁₀₁

D₄

yes

—

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

#41

yes

characteristic

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₈×C₁₀₁

—

C₈×C₁₀₁

—

C₈×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#121

yes

characteristic

C₂²×C₁₀₁

#123

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#120

yes

characteristic

C₄×C₁₀₁

#121

yes

—

C₄×C₁₀₁

#124

yes

characteristic

C₂×C₁₀₁

—

C₂×C₁₀₁

#394

yes

characteristic

C₂×C₁₀₁

#473

yes

—

#2047

C₁₀₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₁

#2047

yes

characteristic, direct product factor, Sylow

C₂²×Q₁₆

C₂×C₁₀₁

yes

characteristic

#424

C₂×C₁₀₁

yes

characteristic

#419

C₂×C₁₀₁

yes

—

C₂×#101

C₂×C₁₀₁

yes

—

#341

C₂×C₁₀₁

yes

characteristic

C₂²×Q₈

C₂²×C₁₀₁

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂²×C₁₀₁

yes

—

C₂×M₁₆

C₂²×C₁₀₁

yes

characteristic

C₂²×C₈

C₂²×C₁₀₁

yes

characteristic

#117

C₂²×C₁₀₁

yes

characteristic

C₄⋊C₈

—

#103

—

#101

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄×C₁₀₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂³×C₁₀₁

yes

characteristic

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₁₀₁

yes

—

C₄⋊C₄

D₄×C₁₀₁

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#41×C₁₀₁

yes

characteristic

Q₈

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×D₄×C₁₀₁

yes

—

108

C₂×C₄

C₂×D₄×C₁₀₁

yes

characteristic

109

C₂×C₄

C₂×D₄×C₁₀₁

yes

characteristic

110

C₂×C₄

#41×C₁₀₁

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₂²

—

115

C₂²

#121×C₁₀₁

yes

characteristic

116

C₂²

#123×C₁₀₁

yes

—

117

C₄

—

118

C₄

—

119

C₄

#120×C₁₀₁

yes

characteristic

120

C₄

#121×C₁₀₁

yes

—

121

C₄

#124×C₁₀₁

yes

characteristic

122

C₂

—

123

C₂

#394×C₁₀₁

yes

characteristic

124

C₂

#473×C₁₀₁

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₁	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₁
prime radical	C₂³×C₁₀₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₂²×C₄×C₁₀₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₁

#121

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₀₁

C₂³×C₁₀₁

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂²×C₁₀₁ ◁ C₂²×C₄×C₁₀₁ ◁ G

N	H ≅ G/N	type
62. #2047	C₁₀₁	direct (N × H)
63. C₁₀₁	#2047	direct (N × H)

H ≅ G/N

type

62. #2047

C₁₀₁

direct (N × H)

63. C₁₀₁

#2047

direct (N × H)

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

subgroup	index
C₂²×Q₁₆×C₁₀₁	2
#424×C₁₀₁	2
#419×C₁₀₁	2
C₂×#101×C₁₀₁	2
#341×C₁₀₁	2
#2047	101

subgroup

index

C₂²×Q₁₆×C₁₀₁

#424×C₁₀₁

#419×C₁₀₁

C₂×#101×C₁₀₁

#341×C₁₀₁

#2047

101

Subgroups — C₁₀₁×#2047 #27167401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C101×#2047 #27167401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#2047 #27167401