Group #27171251 — subgroups

Subgroups — #27171251

Order 12928.

216 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 101 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	1	yes	maximal, characteristic
2	#20265023	C₂	1	1	yes	maximal, characteristic
3	#20264931	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11717579	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
7	C₄⋊#11717579	—	2	2	no	—
8	#15692162	C₄	1	2	yes	—
9	#15692162	C₄	1	2	yes	—
10	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₀₁	—	2	2	no	—
13	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
14	C₂×#11717579	—	2	2	no	—
15	C₂×#11717579	—	2	4	no	—
16	C₄×Dic₁₀₁	—	2	2	no	—
17	C₄×Dic₁₀₁	—	2	2	no	—
18	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₁₀₁	—	2	2	no	—
22	C₂×C₈×C₁₀₁	—	2	4	no	—
23	C₄²×C₁₀₁	D₄	1	2	yes	—
24	C₄²×C₁₀₁	Q₈	1	2	yes	—
25	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₀₁	—	2	2	no	—
27	C₂×C₄×C₁₀₁	—	2	2	no	—
28	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₁₀₁	#41	1	2	yes	—
33	C₂×C₄×C₁₀₁	#41	1	2	yes	—
34	C₂×Dic₁₀₁	—	2	4	no	—
35	C₂×Dic₁₀₁	—	2	4	no	—
36	C₂×Dic₁₀₁	—	2	4	no	—
37	C₈×C₁₀₁	—	4	2	no	—
38	#11717579	—	4	2	no	—
39	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₀₁	#117	1	2	yes	—
41	C₂²×C₁₀₁	#131	1	2	yes	—
42	C₂²×C₁₀₁	#131	1	2	yes	—
43	C₄×C₁₀₁	—	2	2	no	—
44	C₄×C₁₀₁	—	2	2	no	—
45	C₄×C₁₀₁	#124	1	1	yes	characteristic
46	C₄×C₁₀₁	#124	1	1	yes	characteristic
47	C₄×C₁₀₁	#125	1	2	yes	—
48	Dic₁₀₁	—	4	4	no	—
49	C₂×C₁₀₁	#382	1	1	yes	characteristic
50	C₂×C₁₀₁	#424	1	1	yes	characteristic
51	C₂×C₁₀₁	#424	1	1	yes	characteristic
52	C₂×C₁₀₁	#433	1	4	yes	—
53	#1832	—	101	1	no	maximal, Sylow
54	C₁₀₁	#1832	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₄⋊C₈	—	101	1	no	—
56	C₂×C₄⋊C₈	D₁₀₁	1	1	yes	characteristic
57	#376	—	101	1	no	—
58	#336	—	101	4	no	—
59	C₂²×C₈	—	101	2	no	—
60	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
61	C₂×C₄²	—	101	2	no	—
62	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	202	2	no	—
65	#95	—	101	2	no	—
66	#95	—	101	2	no	—
67	C₂²×C₄	—	101	4	no	—
68	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
69	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	202	2	no	—
74	C₂×C₈	—	202	4	no	—
75	C₄²	—	202	2	no	—
76	C₄²	—	202	2	no	—
77	C₄²	Dic₂₀₂	1	2	yes	—
78	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
79	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	202	4	no	—
83	C₂×C₄	—	202	4	no	—
84	C₂×C₄	—	202	4	no	—
85	C₂×C₄	#13108053	1	2	yes	—
86	C₂×C₄	#13108053	1	2	yes	—
87	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₂₀₂	1	1	yes	characteristic
89	C₂×C₄	#13108073	1	2	yes	—
90	C₂×C₄	#13108075	1	2	yes	—
91	C₈	—	4	2	no	—
92	C₈	—	404	2	no	—
93	C₂²	C₂×#13108053	1	1	yes	characteristic
94	C₂²	#15692213	1	2	yes	—
95	C₂²	#15692261	1	2	yes	—
96	C₂²	#15692263	1	2	yes	—
97	C₄	—	2	2	no	—
98	C₄	—	2	2	no	—
99	C₄	—	404	4	no	—
100	C₄	#15692242	1	1	yes	characteristic
101	C₄	#15692244	1	1	yes	characteristic
102	C₄	#15692246	1	2	yes	—
103	C₂	#20265042	1	1	yes	characteristic
104	C₂	#20265203	1	1	yes	characteristic
105	C₂	#20265205	1	1	yes	characteristic
106	C₂	#20265237	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11717579

C₂

yes

maximal, characteristic

#20265023

C₂

yes

maximal, characteristic

#20264931

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

C₄⋊#11717579

—

#15692162

C₄

yes

—

#15692162

C₄

yes

—

C₂²×C₈×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₁

—

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂×#11717579

—

C₂×#11717579

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₄²×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

Q₈

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₁

#117

yes

—

C₂²×C₁₀₁

#131

yes

—

C₂²×C₁₀₁

#131

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

#1832

—

101

maximal, Sylow

C₁₀₁

#1832

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

101

—

C₂×C₄⋊C₈

D₁₀₁

yes

characteristic

#376

—

101

—

#336

—

101

—

C₂²×C₈

—

101

—

C₂²×C₈

C₂×D₁₀₁

yes

—

C₂×C₄²

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

202

—

#95

—

101

—

#95

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

C₄×D₁₀₁

yes

characteristic

C₂²×C₄

C₄×D₁₀₁

yes

characteristic

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄²

—

202

—

C₄²

—

202

—

C₄²

Dic₂₀₂

yes

—

C₄²

C₂²⋊D₁₀₁

yes

—

C₂³

C₂×C₄×D₁₀₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

#13108053

yes

—

C₂×C₄

#13108053

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

C₂×C₄

C₂×Dic₂₀₂

yes

characteristic

C₂×C₄

#13108073

yes

—

C₂×C₄

#13108075

yes

—

C₈

—

C₈

—

404

—

C₂²

C₂×#13108053

yes

characteristic

C₂²

#15692213

yes

—

C₂²

#15692261

yes

—

C₂²

#15692263

yes

—

C₄

—

C₄

—

C₄

—

404

—

100

C₄

#15692242

yes

characteristic

101

C₄

#15692244

yes

characteristic

102

C₄

#15692246

yes

—

103

C₂

#20265042

yes

characteristic

104

C₂

#20265203

yes

characteristic

105

C₂

#20265205

yes

characteristic

106

C₂

#20265237

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₁
commutator, socle, prime radical	C₂³×C₁₀₁	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₀₁	C₂
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₀₁
hypocenter	C₁₀₁	#1832
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₁

commutator, socle, prime radical

C₂³×C₁₀₁

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₀₁

C₂

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
54. C₁₀₁	#1832	semidirect

H ≅ G/N

type

54. C₁₀₁

#1832

semidirect

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

subgroup	index
C₂×C₄⋊#11717579	2
#20265023	2
#20264931	2
C₂×C₄⋊C₈×C₁₀₁	2
#1832	101

subgroup

index

C₂×C₄⋊C₈×C₁₀₁

101

Subgroups — #27171251

Conjugacy classes of subgroups

Complements

Maximal subgroups