Group #27186632 — subgroups

Subgroups — D₁₀₁⋊#333 #27186632

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	D₁₀₁⋊(C₄⋊C₈)	C₂	1	2	yes	maximal
2	#20266037	C₂	1	4	yes	maximal, complemented
3	C₄⋊Q₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂×D₁₀₁⋊C₈	—	2	2	no	—
5	#15692322	C₂²	1	4	yes	—
6	C₄⋊#11717581	—	2	2	no	—
7	C₂×Q₈×D₁₀₁	C₄	1	2	yes	—
8	#15692255	C₄	1	1	yes	characteristic
9	#15692254	—	2	2	no	—
10	#15692253	C₂²	1	2	yes	—
11	C₄⋊C₄×D₁₀₁	—	2	2	no	—
12	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₁₀₁	C₄	1	1	yes	characteristic
14	D₁₀₁⋊C₈	—	4	4	no	—
15	C₂×#11717581	—	2	4	no	—
16	C₂×Dic₂₀₂	—	2	2	no	—
17	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	—
18	Q₈×D₁₀₁	—	4	4	no	—
19	C₂×C₄×D₁₀₁	—	2	2	no	—
20	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
22	C₄⋊Dic₁₀₁	—	2	2	no	—
23	#13108053	—	2	4	no	—
24	C₄×Dic₁₀₁	D₄	1	2	yes	—
25	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
26	C₂×Q₈×C₁₀₁	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₁₀₁	—	2	2	no	—
28	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
30	Q₈×C₁₀₁	—	4	2	no	—
31	C₂×C₄×C₁₀₁	—	2	2	no	—
32	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
33	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₁₀₁	—	2	2	no	—
35	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
37	C₄×D₁₀₁	—	2	2	no	—
38	C₄×D₁₀₁	—	4	4	no	—
39	C₄×D₁₀₁	SD₁₆	1	4	yes	—
40	C₄×D₁₀₁	Q₁₆	1	4	yes	—
41	Dic₂₀₂	—	4	4	no	—
42	Dic₂₀₂	—	4	2	no	—
43	#11717581	—	4	4	no	—
44	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×D₁₀₁	#100	1	2	yes	—
46	C₂×D₁₀₁	#103	1	4	yes	—
47	C₄×C₁₀₁	—	2	1	no	—
48	C₄×C₁₀₁	—	4	2	no	—
49	C₄×C₁₀₁	#103	1	4	yes	—
50	Dic₁₀₁	—	2	1	no	—
51	Dic₁₀₁	—	4	2	no	—
52	Dic₁₀₁	C₂×SD₁₆	1	2	yes	—
53	Dic₁₀₁	C₂×Q₁₆	1	2	yes	—
54	C₂×C₁₀₁	C₂×#100	1	1	yes	characteristic
55	C₂×C₁₀₁	C₂×#103	1	2	yes	—
56	D₁₀₁	#333	1	4	yes	complemented
57	C₂×#333	—	101	1	no	maximal, Sylow
58	C₁₀₁	C₂×#333	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₄⋊Q₈	—	101	1	no	—
60	C₂×C₄⋊C₈	—	101	2	no	—
61	#333	—	101	4	no	—
62	C₂²×Q₈	—	101	2	no	—
63	C₂²×C₈	—	202	2	no	—
64	C₄⋊Q₈	—	101	2	no	—
65	C₄⋊Q₈	—	101	1	no	—
66	C₄⋊Q₈	—	202	2	no	—
67	C₄⋊Q₈	#87938	1	1	yes	characteristic
68	C₂×C₄⋊C₄	—	202	2	no	—
69	C₂×C₄²	—	101	1	no	—
70	C₄⋊C₈	—	101	4	no	—
71	C₄⋊C₈	—	202	2	no	—
72	C₂×Q₈	—	101	2	no	—
73	C₂×Q₈	—	404	4	no	—
74	C₂×Q₈	—	202	2	no	—
75	C₂×Q₈	C₂×#87938	1	2	yes	—
76	C₂²×C₄	—	101	2	no	—
77	C₂²×C₄	—	101	1	no	—
78	C₂²×C₄	—	202	2	no	—
79	C₂×C₈	—	404	4	no	—
80	C₂×C₈	—	202	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	202	4	no	—
83	C₄⋊C₄	—	202	2	no	—
84	C₄²	—	101	2	no	—
85	C₄²	—	101	1	no	—
86	C₄²	C₂×#87938	1	1	yes	characteristic
87	C₂³	—	101	1	no	—
88	Q₈	—	4	2	no	—
89	Q₈	—	404	2	no	—
90	Q₈	—	404	4	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	101	2	no	—
93	C₂×C₄	—	101	4	no	—
94	C₂×C₄	—	101	4	no	—
95	C₂×C₄	—	101	1	no	—
96	C₂×C₄	—	404	4	no	—
97	C₂×C₄	—	202	2	no	—
98	C₂×C₄	—	202	2	no	—
99	C₂×C₄	C₂²⋊#87938	1	2	yes	—
100	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
101	C₈	—	404	4	no	—
102	C₂²	—	101	2	no	—
103	C₂²	—	101	4	no	—
104	C₂²	C₂×C₂²⋊#87938	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	4	2	no	—
107	C₄	—	101	2	no	—
108	C₄	—	101	2	no	—
109	C₄	—	404	2	no	—
110	C₄	—	202	1	no	—
111	C₄	#15692318	1	4	yes	—
112	C₂	—	101	4	no	—
113	C₂	D₁₀₁⋊#100	1	1	yes	characteristic
114	C₂	C₂×#15692318	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₀₁⋊(C₄⋊C₈)

C₂

yes

maximal

#20266037

C₂

yes

maximal, complemented

C₄⋊Q₈×D₁₀₁

C₂

yes

maximal, characteristic

C₂×D₁₀₁⋊C₈

—

#15692322

C₂²

yes

—

C₄⋊#11717581

—

C₂×Q₈×D₁₀₁

C₄

yes

—

#15692255

C₄

yes

characteristic

#15692254

—

#15692253

C₂²

yes

—

C₄⋊C₄×D₁₀₁

—

C₄²×D₁₀₁

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₁

C₄

yes

characteristic

D₁₀₁⋊C₈

—

C₂×#11717581

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

C₂×C₄

yes

—

Q₈×D₁₀₁

—

C₂×C₄×D₁₀₁

—

C₂×C₄×D₁₀₁

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₁

D₄

yes

—

C₄⋊Dic₁₀₁

—

#13108053

—

C₄×Dic₁₀₁

D₄

yes

—

C₄×Dic₁₀₁

C₂³

yes

characteristic

C₂×Q₈×C₁₀₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×D₁₀₁

C₂²⋊C₄

yes

characteristic

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₁

C₂×D₄

yes

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₄×D₁₀₁

SD₁₆

yes

—

C₄×D₁₀₁

Q₁₆

yes

—

Dic₂₀₂

—

Dic₂₀₂

—

#11717581

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₁

#100

yes

—

C₂×D₁₀₁

#103

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#103

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

C₂×SD₁₆

yes

—

Dic₁₀₁

C₂×Q₁₆

yes

—

C₂×C₁₀₁

C₂×#100

yes

characteristic

C₂×C₁₀₁

C₂×#103

yes

—

D₁₀₁

#333

yes

complemented

C₂×#333

—

101

maximal, Sylow

C₁₀₁

C₂×#333

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

101

—

C₂×C₄⋊C₈

—

101

—

#333

—

101

—

C₂²×Q₈

—

101

—

C₂²×C₈

—

202

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

—

202

—

C₄⋊Q₈

#87938

yes

characteristic

C₂×C₄⋊C₄

—

202

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

101

—

C₄⋊C₈

—

202

—

C₂×Q₈

—

101

—

C₂×Q₈

—

404

—

C₂×Q₈

—

202

—

C₂×Q₈

C₂×#87938

yes

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

202

—

C₂×C₈

—

404

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

C₂×#87938

yes

characteristic

C₂³

—

101

—

Q₈

—

Q₈

—

404

—

Q₈

—

404

—

C₂×C₄

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

404

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

C₂²⋊#87938

yes

—

100

C₂×C₄

C₂²×#87938

yes

characteristic

101

C₈

—

404

—

102

C₂²

—

101

—

103

C₂²

—

101

—

104

C₂²

C₂×C₂²⋊#87938

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

101

—

108

C₄

—

101

—

109

C₄

—

404

—

110

C₄

—

202

—

111

C₄

#15692318

yes

—

112

C₂

—

101

—

113

C₂

D₁₀₁⋊#100

yes

characteristic

114

C₂

C₂×#15692318

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#87938
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87938
Fitting	C₄⋊Q₈×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₀₁	C₂²⋊C₄
cosocle	C₄×Dic₁₀₁	C₂³
hypercenter	C₄⋊Q₈	#87938
hypocenter	C₁₀₁	C₂×#333
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#87938

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#87938

Fitting

C₄⋊Q₈×C₁₀₁

C₄

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₀₁

C₂²⋊C₄

cosocle

C₄×Dic₁₀₁

C₂³

C₄⋊Q₈

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #20266037	C₂	semidirect
56. D₁₀₁	#333	semidirect
58. C₁₀₁	C₂×#333	semidirect

H ≅ G/N

type

2. #20266037

C₂

semidirect

56. D₁₀₁

#333

semidirect

58. C₁₀₁

C₂×#333

semidirect

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

subgroup	index
D₁₀₁⋊(C₄⋊C₈)	2
#20266037	2
C₄⋊Q₈×D₁₀₁	2
C₂×#333	101

subgroup

index

D₁₀₁⋊(C₄⋊C₈)

#20266037

C₄⋊Q₈×D₁₀₁

C₂×#333

101

Subgroups — D₁₀₁⋊#333 #27186632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D101⋊#333 #27186632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₀₁⋊#333 #27186632