Group #27169015 — subgroups

Subgroups — #27169015

Order 12928.

188 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 105 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×C₄⋊#11717579	C₂	1	2	yes	maximal
2	#384×C₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11717579	—	2	2	no	—
4	C₄⋊#11717579	—	2	2	no	—
5	C₄⋊#11717579	C₄	1	4	yes	complemented
6	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
7	C₂×C₄⋊C₄×C₁₀₁	C₄	1	2	yes	—
8	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
9	C₂×#11717579	—	2	4	no	—
10	C₂×#11717579	—	4	2	no	—
11	C₂×#11717579	—	4	2	no	—
12	C₂²×C₄×C₁₀₁	—	2	2	no	—
13	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
15	C₄⋊C₄×C₁₀₁	—	4	2	no	—
16	C₄⋊C₄×C₁₀₁	—	4	2	no	—
17	C₄⋊C₄×C₁₀₁	—	4	2	no	—
18	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
19	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
20	C₄²×C₁₀₁	Q₈	1	1	yes	characteristic
21	C₂³×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₀₁	—	2	1	no	—
23	C₂×C₄×C₁₀₁	—	2	1	no	—
24	C₂×C₄×C₁₀₁	—	4	2	no	—
25	C₂×C₄×C₁₀₁	—	4	4	no	—
26	C₂×C₄×C₁₀₁	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₁	D₈	1	2	yes	—
31	C₂×C₄×C₁₀₁	SD₁₆	1	2	yes	—
32	C₂×C₄×C₁₀₁	SD₁₆	1	2	yes	—
33	C₂×C₄×C₁₀₁	Q₁₆	1	2	yes	—
34	#11717579	—	4	4	no	—
35	C₂²×C₁₀₁	#95	1	1	yes	characteristic
36	C₂²×C₁₀₁	#100	1	1	yes	characteristic
37	C₂²×C₁₀₁	#101	1	1	yes	characteristic
38	C₂²×C₁₀₁	#102	1	2	yes	—
39	C₂²×C₁₀₁	#103	1	2	yes	—
40	C₄×C₁₀₁	—	2	1	no	—
41	C₄×C₁₀₁	—	2	1	no	—
42	C₄×C₁₀₁	—	8	2	no	—
43	C₄×C₁₀₁	#102	1	2	yes	—
44	C₄×C₁₀₁	#103	1	2	yes	—
45	C₄×C₁₀₁	#106	1	2	yes	—
46	C₄×C₁₀₁	#107	1	2	yes	—
47	C₂×C₁₀₁	#331	1	1	yes	characteristic
48	C₂×C₁₀₁	#332	1	2	yes	—
49	C₂×C₁₀₁	#333	1	1	yes	characteristic
50	C₂×C₁₀₁	#340	1	2	yes	—
51	C₂×C₁₀₁	#343	1	1	yes	characteristic
52	#1296	—	101	1	no	maximal, Sylow
53	C₁₀₁	#1296	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₄⋊C₈	—	101	2	no	—
55	#384	D₁₀₁	1	1	yes	characteristic
56	C₂²×C₈	—	202	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	Dic₁₀₁	1	2	yes	—
59	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
60	C₄⋊C₈	—	101	4	no	—
61	C₄⋊C₈	—	202	2	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×Dic₁₀₁	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
65	C₂×C₈	—	404	2	no	—
66	C₂×C₈	—	404	2	no	—
67	C₂×C₈	—	202	4	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	Dic₂₀₂	1	1	yes	characteristic
72	C₄²	C₄×D₁₀₁	1	2	yes	—
73	C₄²	D₄₀₄	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₁₀₁	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	C₄×Dic₁₀₁	1	1	yes	characteristic
80	C₂×C₄	#13108051	1	2	yes	—
81	C₂×C₄	#13108053	1	2	yes	—
82	C₂×C₄	C₄⋊Dic₁₀₁	1	1	yes	characteristic
83	C₂×C₄	#13108059	1	2	yes	—
84	C₂×C₄	#13108061	1	2	yes	—
85	C₂×C₄	#13108062	1	2	yes	—
86	C₂×C₄	#13108064	1	2	yes	—
87	C₈	—	404	4	no	—
88	C₂²	#15692162	1	1	yes	characteristic
89	C₂²	#15692172	1	1	yes	characteristic
90	C₂²	#15692174	1	1	yes	characteristic
91	C₂²	#15692176	1	2	yes	—
92	C₂²	#15692179	1	2	yes	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	8	2	no	—
96	C₄	#15692175	1	2	yes	—
97	C₄	#15692178	1	2	yes	—
98	C₄	#15692186	1	2	yes	—
99	C₄	#15692188	1	2	yes	—
100	C₂	#20264921	1	1	yes	characteristic
101	C₂	#20264923	1	2	yes	—
102	C₂	#20264926	1	1	yes	characteristic
103	C₂	#20264939	1	2	yes	—
104	C₂	#20264946	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11717579

C₂

yes

maximal

#384×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

—

C₄⋊#11717579

—

C₄⋊#11717579

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₄

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

—

—

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

D₄

yes

characteristic

C₄²×C₁₀₁

Q₈

yes

characteristic

C₂³×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄²

yes

characteristic

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

D₈

yes

—

C₂×C₄×C₁₀₁

SD₁₆

yes

—

C₂×C₄×C₁₀₁

SD₁₆

yes

—

C₂×C₄×C₁₀₁

Q₁₆

yes

—

#11717579

—

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

—

C₂²×C₁₀₁

#103

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#102

yes

—

C₄×C₁₀₁

#103

yes

—

C₄×C₁₀₁

#106

yes

—

C₄×C₁₀₁

#107

yes

—

C₂×C₁₀₁

#331

yes

characteristic

C₂×C₁₀₁

#332

yes

—

C₂×C₁₀₁

#333

yes

characteristic

C₂×C₁₀₁

#340

yes

—

C₂×C₁₀₁

#343

yes

characteristic

#1296

—

101

maximal, Sylow

C₁₀₁

#1296

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

101

—

#384

D₁₀₁

yes

characteristic

C₂²×C₈

—

202

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

101

—

C₄⋊C₈

—

202

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₀₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂×C₈

—

404

—

C₂×C₈

—

404

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₂₀₂

yes

characteristic

C₄²

C₄×D₁₀₁

yes

—

C₄²

D₄₀₄

yes

characteristic

C₂³

C₂²⋊Dic₁₀₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₀₁

yes

characteristic

C₂×C₄

#13108051

yes

—

C₂×C₄

#13108053

yes

—

C₂×C₄

C₄⋊Dic₁₀₁

yes

characteristic

C₂×C₄

#13108059

yes

—

C₂×C₄

#13108061

yes

—

C₂×C₄

#13108062

yes

—

C₂×C₄

#13108064

yes

—

C₈

—

404

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#15692179

yes

—

C₄

—

C₄

—

C₄

—

C₄

#15692175

yes

—

C₄

#15692178

yes

—

C₄

#15692186

yes

—

C₄

#15692188

yes

—

100

C₂

#20264921

yes

characteristic

101

C₂

#20264923

yes

—

102

C₂

#20264926

yes

characteristic

103

C₂

#20264939

yes

—

104

C₂

#20264946

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₁₀₁
commutator	C₂×C₄×C₁₀₁	C₄²
Frattini	C₂×C₄²	C₂×D₁₀₁
Fitting	#384×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₀₁	C₂²
hypercenter	#384	D₁₀₁
hypocenter	C₁₀₁	#1296
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂²⋊Dic₁₀₁

commutator

C₂×C₄×C₁₀₁

C₄²

Frattini

C₂×C₄²

C₂×D₁₀₁

Fitting

#384×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂²⋊C₄

cosocle

C₂×C₄²×C₁₀₁

C₂²

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
5. C₄⋊#11717579	C₄	semidirect
53. C₁₀₁	#1296	semidirect

H ≅ G/N

type

5. C₄⋊#11717579

C₄

semidirect

53. C₁₀₁

#1296

semidirect

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

subgroup	index
C₂×C₄⋊#11717579	2
#384×C₁₀₁	2
#1296	101

subgroup

index

C₂×C₄⋊#11717579

#384×C₁₀₁

#1296

101

Subgroups — #27169015

Conjugacy classes of subgroups

Complements

Maximal subgroups