Group #27181473 — subgroups

Subgroups — C₂×#20265547 #27181473

Order 12928.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15692255	C₂	1	1	yes	maximal, characteristic
2	#20265547	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15692178	C₂	1	4	yes	maximal
4	C₂×#15692166	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
7	C₂²×#11717579	C₂²	1	2	yes	—
8	#15692255	—	2	2	no	—
9	#15692255	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₁₀₁	—	2	2	no	—
11	#15692178	C₂²	1	16	yes	—
12	#15692166	C₂²	1	4	yes	—
13	#125×C₁₀₁	—	2	2	no	—
14	#125×C₁₀₁	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
16	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₀₂	—	2	8	no	—
19	C₂×Dic₂₀₂	—	2	2	no	—
20	C₂×Dic₂₀₂	C₂³	1	4	yes	—
21	C₂²×Dic₁₀₁	—	2	2	no	—
22	C₂×#11717579	—	2	4	no	—
23	C₂×#11717579	C₂³	1	4	yes	—
24	C₄⋊Dic₁₀₁	—	2	8	no	—
25	C₂²×C₄×C₁₀₁	—	2	2	no	—
26	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
27	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₁	—	2	8	no	—
29	C₄⋊C₄×C₁₀₁	—	2	2	no	—
30	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
31	C₄²×C₁₀₁	D₄	1	2	yes	—
32	C₄²×C₁₀₁	C₂³	1	2	yes	—
33	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₁	—	2	4	no	—
35	C₂×C₄×C₁₀₁	—	2	4	no	—
36	C₂×C₄×C₁₀₁	—	4	4	no	—
37	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
38	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	#41	1	4	yes	—
40	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
41	C₂×Dic₁₀₁	—	2	4	no	—
42	C₂×Dic₁₀₁	—	4	4	no	—
43	Dic₂₀₂	—	2	8	no	—
44	Dic₂₀₂	—	4	4	no	—
45	#11717579	—	2	4	no	—
46	C₂²×C₁₀₁	#124	1	2	yes	—
47	C₂²×C₁₀₁	#137	1	4	yes	—
48	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₁₀₁	—	2	4	no	—
50	C₄×C₁₀₁	—	4	4	no	—
51	C₄×C₁₀₁	#124	1	2	yes	—
52	C₄×C₁₀₁	C₂×#41	1	2	yes	—
53	Dic₁₀₁	—	4	4	no	—
54	C₂×C₁₀₁	#491	1	4	yes	complemented
55	C₂×C₁₀₁	C₂×#124	1	1	yes	characteristic
56	C₂×C₁₀₁	C₂×#137	1	2	yes	—
57	C₂×#491	—	101	1	no	maximal, Sylow
58	C₁₀₁	C₂×#491	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₄⋊Q₈	—	101	1	no	—
60	C₂×#125	D₁₀₁	1	1	yes	characteristic
61	#491	—	101	8	no	—
62	C₂×#103	—	101	4	no	—
63	C₂×#97	—	101	1	no	—
64	C₂²×Q₈	—	101	2	no	—
65	C₂²×C₈	—	101	2	no	—
66	C₄⋊Q₈	—	101	4	no	—
67	C₄⋊Q₈	—	202	2	no	—
68	#125	—	2	2	no	—
69	#125	C₂×D₁₀₁	1	4	yes	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	202	2	no	—
72	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
73	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
74	#103	—	101	16	no	—
75	#97	—	101	4	no	—
76	C₂×Q₈	—	101	4	no	—
77	C₂×Q₈	—	202	8	no	—
78	C₂×Q₈	—	202	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	202	2	no	—
81	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
82	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
83	C₂×C₈	—	101	4	no	—
84	C₂×C₈	—	202	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	202	8	no	—
88	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
89	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
90	C₄²	C₂²×D₁₀₁	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
92	Q₈	—	404	4	no	—
93	Q₈	—	202	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	404	4	no	—
98	C₂×C₄	—	202	4	no	—
99	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
101	C₂×C₄	#13108075	1	4	yes	—
102	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
103	C₈	—	202	4	no	—
104	C₂²	#15692244	1	2	yes	—
105	C₂²	#15692285	1	4	yes	—
106	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	4	4	no	—
109	C₄	—	404	4	no	—
110	C₄	#15692244	1	2	yes	—
111	C₄	C₂×#13108075	1	2	yes	—
112	C₂	#20265547	1	4	yes	direct product factor
113	C₂	C₂×#15692244	1	1	yes	characteristic
114	C₂	C₂×#15692285	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692255

C₂

yes

maximal, characteristic

#20265547

C₂

yes

maximal, direct product factor

C₂×#15692178

C₂

yes

maximal

C₂×#15692166

C₂

yes

maximal, characteristic

C₂×#125×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

C₂²

yes

—

C₂²×#11717579

C₂²

yes

—

#15692255

—

#15692255

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

—

#15692178

C₂²

yes

—

#15692166

C₂²

yes

—

#125×C₁₀₁

—

#125×C₁₀₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

C₂³

yes

—

C₂²×Dic₁₀₁

—

C₂×#11717579

—

C₂×#11717579

C₂³

yes

—

C₄⋊Dic₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

C₂³

yes

—

C₄²×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

C₂³

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

#11717579

—

C₂²×C₁₀₁

#124

yes

—

C₂²×C₁₀₁

#137

yes

—

C₂²×C₁₀₁

C₂²×D₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#124

yes

—

C₄×C₁₀₁

C₂×#41

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

C₂×#491

—

101

maximal, Sylow

C₁₀₁

C₂×#491

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

101

—

C₂×#125

D₁₀₁

yes

characteristic

#491

—

101

—

C₂×#103

—

101

—

C₂×#97

—

101

—

C₂²×Q₈

—

101

—

C₂²×C₈

—

101

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

—

202

—

#125

—

#125

C₂×D₁₀₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#103

—

101

—

#97

—

101

—

C₂×Q₈

—

101

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂²×C₄

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

202

—

C₄⋊C₄

C₂²×D₁₀₁

yes

—

C₄²

C₂²⋊D₁₀₁

yes

—

C₄²

C₂²×D₁₀₁

yes

—

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

Q₈

—

404

—

Q₈

—

202

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

404

—

C₂×C₄

—

202

—

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

100

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

101

C₂×C₄

#13108075

yes

—

102

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

103

C₈

—

202

—

104

C₂²

#15692244

yes

—

105

C₂²

#15692285

yes

—

106

C₂²

C₂²×C₂²⋊D₁₀₁

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

404

—

110

C₄

#15692244

yes

—

111

C₄

C₂×#13108075

yes

—

112

C₂

#20265547

yes

direct product factor

113

C₂

C₂×#15692244

yes

characteristic

114

C₂

C₂×#15692285

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₂×#125×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂×D₄
hypercenter	C₂×#125	D₁₀₁
hypocenter	C₁₀₁	C₂×#491
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator, cosocle

C₂×C₄×C₁₀₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₁

Fitting

C₂×#125×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂×D₄

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #20265547	C₂	direct (N × H)
54. C₂×C₁₀₁	#491	semidirect
58. C₁₀₁	C₂×#491	semidirect
112. C₂	#20265547	direct (N × H)

H ≅ G/N

type

2. #20265547

C₂

direct (N × H)

54. C₂×C₁₀₁

#491

semidirect

58. C₁₀₁

C₂×#491

semidirect

112. C₂

#20265547

direct (N × H)

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

subgroup	index
C₂×#15692255	2
#20265547	2
C₂×#15692178	2
C₂×#15692166	2
C₂×#125×C₁₀₁	2
C₂×#491	101

subgroup

index

C₂×#125×C₁₀₁

101

Subgroups — C₂×#20265547 #27181473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20265547 #27181473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265547 #27181473