Group #27172126 — subgroups

Subgroups — #27172126

Order 12928.

228 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20265201	C₂	1	1	yes	maximal, characteristic
2	C₄×#13108058	C₂	1	1	yes	maximal, characteristic
3	#20264946	C₂	1	4	yes	maximal
4	C₂×#125×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#13108058	C₂²	1	2	yes	—
6	C₂×#13108058	C₂²	1	2	yes	—
7	C₄⋊#11717579	—	2	2	no	—
8	#15692166	—	2	1	no	—
9	C₄×#11717579	—	2	1	no	—
10	#125×C₁₀₁	—	2	1	no	—
11	#125×C₁₀₁	—	2	1	no	—
12	#125×C₁₀₁	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₀₁	C₄	1	4	yes	—
14	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
16	#13108058	—	2	2	no	—
17	#13108058	—	2	4	no	—
18	#13108058	Q₈	1	4	yes	—
19	C₂×#11717579	—	2	2	no	—
20	C₂×#11717579	—	2	2	no	—
21	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₀₁	—	2	4	no	—
25	C₄⋊C₄×C₁₀₁	—	2	4	no	—
26	C₄⋊C₄×C₁₀₁	—	2	2	no	—
27	C₄⋊C₄×C₁₀₁	—	2	2	no	—
28	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
29	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
30	C₄²×C₁₀₁	D₄	1	2	yes	—
31	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₁	—	2	4	no	—
33	C₂×C₄×C₁₀₁	—	4	2	no	—
34	C₂×C₄×C₁₀₁	—	4	4	no	—
35	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₀₁	#41	1	2	yes	—
41	#11717579	—	2	4	no	—
42	#11717579	—	4	2	no	—
43	C₂²×C₁₀₁	—	2	2	no	—
44	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	—
46	C₄×C₁₀₁	—	2	2	no	—
47	C₄×C₁₀₁	—	2	2	no	—
48	C₄×C₁₀₁	—	4	4	no	—
49	C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₁₀₁	#124	1	1	yes	characteristic
51	C₄×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₁₀₁	—	2	2	no	—
53	C₂×C₁₀₁	#391	1	1	yes	characteristic
54	C₂×C₁₀₁	#413	1	2	yes	—
55	#1998	—	101	1	no	maximal, Sylow
56	C₁₀₁	#1998	1	1	yes	characteristic, complemented, Sylow
57	C₂×#125	D₁₀₁	1	1	yes	characteristic
58	#423	—	101	1	no	—
59	C₄×M₁₆	—	101	1	no	—
60	#343	—	101	4	no	—
61	C₂×M₁₆	—	101	2	no	—
62	C₂×M₁₆	—	101	2	no	—
63	#125	—	2	1	no	—
64	#125	—	2	1	no	—
65	#125	—	2	2	no	—
66	C₂×C₄⋊C₄	Dic₁₀₁	1	4	yes	—
67	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
68	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
69	C₄⋊C₈	—	202	2	no	—
70	#97	—	202	1	no	—
71	C₄×C₈	—	202	1	no	—
72	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	—
73	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
75	M₁₆	—	101	4	no	—
76	M₁₆	—	202	4	no	—
77	M₁₆	—	202	2	no	—
78	C₂×C₈	—	202	2	no	—
79	C₂×C₈	—	202	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
85	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
86	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
87	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	C₂²⋊Dic₁₀₁	1	2	yes	—
92	C₂×C₄	C₂²⋊Dic₁₀₁	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
94	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
95	C₂×C₄	Q₈×D₁₀₁	1	2	yes	—
96	C₂×C₄	#13108075	1	2	yes	—
97	C₈	—	404	2	no	—
98	C₈	—	202	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
101	C₂²	Q₈×Dic₁₀₁	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	#15692231	1	2	yes	—
106	C₄	#15692244	1	1	yes	characteristic
107	C₄	#15692253	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#20265084	1	1	yes	characteristic
110	C₂	#20265161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265201

C₂

yes

maximal, characteristic

C₄×#13108058

C₂

yes

maximal, characteristic

#20264946

C₂

yes

maximal

C₂×#125×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

—

C₂×#13108058

C₂²

yes

—

C₄⋊#11717579

—

#15692166

—

C₄×#11717579

—

#125×C₁₀₁

—

#125×C₁₀₁

—

#125×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₄

yes

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

—

—

Q₈

yes

—

C₂×#11717579

—

C₂×#11717579

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

D₄

yes

characteristic

C₄²×C₁₀₁

D₄

yes

characteristic

C₄²×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

#11717579

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₄×Q₈

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂²⋊Q₈

yes

—

C₄×C₁₀₁

#124

yes

characteristic

C₄×C₁₀₁

C₄⋊Q₈

yes

characteristic

C₂×C₁₀₁

—

C₂×C₁₀₁

#391

yes

characteristic

C₂×C₁₀₁

#413

yes

—

#1998

—

101

maximal, Sylow

C₁₀₁

#1998

yes

characteristic, complemented, Sylow

C₂×#125

D₁₀₁

yes

characteristic

#423

—

101

—

C₄×M₁₆

—

101

—

#343

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

101

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

Dic₁₀₁

yes

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

202

—

#97

—

202

—

C₄×C₈

—

202

—

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

101

—

M₁₆

—

202

—

M₁₆

—

202

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₁₀₁

yes

characteristic

C₄²

C₂²⋊D₁₀₁

yes

characteristic

C₄²

C₂²⋊D₁₀₁

yes

—

C₂³

C₂²×Dic₁₀₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

C₂×C₄

Q₈×D₁₀₁

yes

—

C₂×C₄

#13108075

yes

—

C₈

—

404

—

C₈

—

202

—

C₂²

—

100

C₂²

C₂×C₂²⋊Dic₁₀₁

yes

characteristic

101

C₂²

Q₈×Dic₁₀₁

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#15692231

yes

—

106

C₄

#15692244

yes

characteristic

107

C₄

#15692253

yes

characteristic

108

C₂

—

109

C₂

#20265084

yes

characteristic

110

C₂

#20265161

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₀₁
commutator, prime radical	C₂³×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂×#125×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×#125	D₁₀₁
hypocenter	C₁₀₁	#1998
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₀₁

commutator, prime radical

C₂³×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂×#125×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
56. C₁₀₁	#1998	semidirect

H ≅ G/N

type

56. C₁₀₁

#1998

semidirect

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

subgroup	index
#20265201	2
C₄×#13108058	2
#20264946	2
C₂×#125×C₁₀₁	2
#1998	101

subgroup

index

C₂×#125×C₁₀₁

101

Subgroups — #27172126

Conjugacy classes of subgroups

Complements

Maximal subgroups