Group #27173718 — subgroups

Subgroups — #27173718

Order 12928.

784 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 309 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×#15692214	C₂	1	2	yes	maximal
2	#20265084	C₂	1	8	yes	maximal, complemented
3	#20265040	C₂	1	4	yes	maximal
4	C₂³×Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	#15692214	—	2	8	no	—
6	C₂×C₄⋊Dic₁₀₁	C₂²	1	4	yes	—
7	C₂×C₂²⋊Dic₁₀₁	C₂²	1	4	yes	—
8	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
9	#15692162	C₂²	1	16	yes	—
10	C₂²×Q₈×C₁₀₁	—	2	8	no	—
11	C₂²×Q₈×C₁₀₁	C₄	1	8	yes	complemented
12	C₂²×Q₈×C₁₀₁	C₂²	1	4	yes	—
13	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₁	C₂³	1	8	yes	—
16	C₄⋊Dic₁₀₁	—	2	8	no	—
17	C₂²⋊Dic₁₀₁	—	2	8	no	—
18	C₄×Dic₁₀₁	—	2	8	no	—
19	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
20	C₂×Q₈×C₁₀₁	—	2	32	no	—
21	C₂×Q₈×C₁₀₁	—	2	8	no	—
22	C₂×Q₈×C₁₀₁	—	2	8	no	—
23	C₂×Q₈×C₁₀₁	C₂×C₄	1	16	yes	—
24	C₂²×C₄×C₁₀₁	—	2	8	no	—
25	C₂²×C₄×C₁₀₁	C₂×C₄	1	8	yes	—
26	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₁₀₁	D₄	1	8	yes	—
28	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
30	C₂³×C₁₀₁	—	2	4	no	—
31	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
32	C₂³×C₁₀₁	C₂×D₄	1	4	yes	—
33	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
34	Q₈×C₁₀₁	—	2	32	no	—
35	C₂×C₄×C₁₀₁	—	2	16	no	—
36	C₂×C₄×C₁₀₁	—	2	8	no	—
37	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	16	yes	—
38	C₂×C₄×C₁₀₁	C₂²×C₄	1	8	yes	—
39	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
40	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
42	C₂×Dic₁₀₁	—	2	8	no	—
43	C₂×Dic₁₀₁	—	2	16	no	—
44	C₂²×C₁₀₁	—	2	4	no	—
45	C₂²×C₁₀₁	—	2	8	no	—
46	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
47	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
48	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
49	C₂²×C₁₀₁	2₋¹⁺⁴	1	4	yes	—
50	C₄×C₁₀₁	—	2	8	no	—
51	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	8	yes	—
52	Dic₁₀₁	—	4	8	no	—
53	C₂×C₁₀₁	—	2	4	no	—
54	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₁₀₁	#519	1	2	yes	—
56	C₂×C₁₀₁	#536	1	4	yes	—
57	#2310	—	101	1	no	maximal, Sylow
58	C₁₀₁	#2310	1	1	yes	characteristic, complemented, Sylow
59	C₂³×Q₈	D₁₀₁	1	1	yes	characteristic
60	C₂×#117	—	101	2	no	—
61	#391	—	101	8	no	—
62	#381	—	101	4	no	—
63	C₂²×Q₈	—	2	8	no	—
64	C₂²×Q₈	Dic₁₀₁	1	8	yes	complemented
65	C₂²×Q₈	C₂×D₁₀₁	1	4	yes	—
66	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
67	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
68	#117	—	202	8	no	—
69	C₂×C₄⋊C₄	—	101	4	no	—
70	C₂×C₂²⋊C₄	—	101	4	no	—
71	C₂×C₄²	—	101	4	no	—
72	#95	—	101	16	no	—
73	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
74	C₂×Q₈	—	2	32	no	—
75	C₂×Q₈	—	2	8	no	—
76	C₂×Q₈	—	2	8	no	—
77	C₂×Q₈	C₂×Dic₁₀₁	1	16	yes	—
78	C₂²×C₄	—	2	8	no	—
79	C₂²×C₄	—	101	8	no	—
80	C₂²×C₄	C₂×Dic₁₀₁	1	8	yes	—
81	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	—
82	C₂²×C₄	C₂²⋊D₁₀₁	1	8	yes	—
83	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
84	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
85	C₄⋊C₄	—	202	8	no	—
86	C₂²⋊C₄	—	202	8	no	—
87	C₄²	—	202	8	no	—
88	C₂³	—	2	4	no	—
89	C₂³	C₂²×Dic₁₀₁	1	2	yes	—
90	C₂³	C₂×C₂²⋊D₁₀₁	1	4	yes	—
91	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
92	Q₈	—	2	32	no	—
93	C₂×C₄	—	2	8	no	—
94	C₂×C₄	—	2	16	no	—
95	C₂×C₄	—	202	8	no	—
96	C₂×C₄	—	202	16	no	—
97	C₂×C₄	C₂²⋊Dic₁₀₁	1	16	yes	—
98	C₂×C₄	C₂²×Dic₁₀₁	1	8	yes	—
99	C₂×C₄	C₂²×Dic₁₀₁	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
101	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
102	C₂²	—	2	8	no	—
103	C₂²	—	2	4	no	—
104	C₂²	C₂×C₂²⋊Dic₁₀₁	1	4	yes	—
105	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
106	C₂²	C₂²×C₂²⋊D₁₀₁	1	2	yes	—
107	C₂²	#15692303	1	4	yes	—
108	C₄	—	2	8	no	—
109	C₄	—	404	8	no	—
110	C₄	C₂×C₂²⋊Dic₁₀₁	1	8	yes	—
111	C₂	—	2	4	no	—
112	C₂	C₂²×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
113	C₂	#20265655	1	2	yes	—
114	C₂	#20265751	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692214

C₂

yes

maximal

#20265084

C₂

yes

maximal, complemented

#20265040

C₂

yes

maximal

C₂³×Q₈×C₁₀₁

C₂

yes

maximal, characteristic

#15692214

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₁

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

#15692162

C₂²

yes

—

C₂²×Q₈×C₁₀₁

—

C₂²×Q₈×C₁₀₁

C₄

yes

complemented

C₂²×Q₈×C₁₀₁

C₂²

yes

—

C₂³×C₄×C₁₀₁

C₂²

yes

—

C₂³×C₄×C₁₀₁

C₂²

yes

characteristic

C₂²×Dic₁₀₁

C₂³

yes

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂⁴×C₁₀₁

C₂³

yes

characteristic

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂²×C₄

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

—

C₂³×C₁₀₁

C₂⁴

yes

characteristic

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₀₁

C₂³×C₄

yes

characteristic

C₂²×C₁₀₁

C₂²×D₄

yes

—

C₂²×C₁₀₁

2₋¹⁺⁴

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×C₂²⋊C₄

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₁₀₁

#519

yes

—

C₂×C₁₀₁

#536

yes

—

#2310

—

101

maximal, Sylow

C₁₀₁

#2310

yes

characteristic, complemented, Sylow

C₂³×Q₈

D₁₀₁

yes

characteristic

C₂×#117

—

101

—

#391

—

101

—

#381

—

101

—

C₂²×Q₈

—

C₂²×Q₈

Dic₁₀₁

yes

complemented

C₂²×Q₈

C₂×D₁₀₁

yes

—

C₂³×C₄

C₂×D₁₀₁

yes

—

C₂³×C₄

C₂×D₁₀₁

yes

characteristic

#117

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₄²

—

101

—

#95

—

101

—

C₂⁴

C₂²×D₁₀₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₀₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

—

C₄⋊C₄

—

202

—

C₂²⋊C₄

—

202

—

C₄²

—

202

—

C₂³

—

C₂³

C₂²×Dic₁₀₁

yes

—

C₂³

C₂×C₂²⋊D₁₀₁

yes

—

C₂³

C₂³×D₁₀₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

C₂×C₄

C₂²×Dic₁₀₁

yes

—

C₂×C₄

C₂²×Dic₁₀₁

yes

—

100

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

101

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×C₂²⋊Dic₁₀₁

yes

—

105

C₂²

C₂³×Dic₁₀₁

yes

characteristic

106

C₂²

C₂²×C₂²⋊D₁₀₁

yes

—

107

C₂²

#15692303

yes

—

108

C₄

—

109

C₄

—

404

—

110

C₄

C₂×C₂²⋊Dic₁₀₁

yes

—

111

C₂

—

112

C₂

C₂²×C₂²⋊Dic₁₀₁

yes

characteristic

113

C₂

#20265655

yes

—

114

C₂

#20265751

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂³×Q₈×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂⁴×C₁₀₁	C₂³
hypercenter	C₂³×Q₈	D₁₀₁
hypocenter	C₁₀₁	#2310
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₁

commutator

C₂²×C₁₀₁

C₂³×C₄

Fitting

C₂³×Q₈×C₁₀₁

C₂

socle, cosocle

C₂³×C₁₀₁

C₂⁴

prime radical

C₂⁴×C₁₀₁

C₂³

hypercenter

C₂³×Q₈

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂³ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. #20265084	C₂	semidirect
11. C₂²×Q₈×C₁₀₁	C₄	semidirect
58. C₁₀₁	#2310	semidirect
64. C₂²×Q₈	Dic₁₀₁	semidirect

H ≅ G/N

type

2. #20265084

C₂

semidirect

11. C₂²×Q₈×C₁₀₁

C₄

semidirect

58. C₁₀₁

#2310

semidirect

64. C₂²×Q₈

Dic₁₀₁

semidirect

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

subgroup	index
C₂×#15692214	2
#20265084	2
#20265040	2
C₂³×Q₈×C₁₀₁	2
#2310	101

subgroup

index

C₂³×Q₈×C₁₀₁

101

Subgroups — #27173718

Conjugacy classes of subgroups

Complements

Maximal subgroups