Group #27384474 — subgroups

Subgroups — C₂×#20274869 #27384474

Order 13568.

1230 conjugacy classes of subgroups; only the 651 normal subgroups are shown.

normal subgroups only

This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 651 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	C₂²×C₄⋊#88322	C₂	1	6	yes	maximal
2	#20274869	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	C₂²	1	4	yes	—
5	C₂×C₄⋊#88322	—	2	24	no	—
6	C₂×C₄⋊#88322	C₂²	1	24	yes	—
7	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
8	C₅₃×#376	C₄	1	4	yes	—
9	C₅₃×#376	C₂²	1	4	yes	—
10	C₂×#95×C₅₃	C₄	1	4	yes	—
11	C₂²×#88322	—	2	24	no	—
12	C₂²×#88322	C₂³	1	8	yes	—
13	C₄⋊#88322	—	2	48	no	—
14	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
15	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
17	C₂×C₄²×C₅₃	D₄	1	12	yes	—
18	C₂×C₄²×C₅₃	Q₈	1	12	yes	—
19	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
20	#95×C₅₃	C₂×C₄	1	16	yes	—
21	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	—	2	24	no	—
23	C₂²×C₄×C₅₃	C₄⋊C₄	1	24	yes	—
24	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
25	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
27	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
28	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
29	C₄²×C₅₃	C₄⋊C₄	1	24	yes	—
30	C₄²×C₅₃	C₂×D₄	1	12	yes	—
31	C₄²×C₅₃	C₂×Q₈	1	12	yes	—
32	C₂×#88322	—	2	24	no	—
33	C₂×#88322	—	4	16	no	—
34	C₂³×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
35	C₂³×C₅₃	#131	1	8	yes	—
36	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	—	2	24	no	—
38	C₂×C₄×C₅₃	—	2	48	no	—
39	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	24	yes	—
40	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
41	C₂×C₄×C₅₃	#127	1	4	yes	—
42	C₂×C₄×C₅₃	C₄⋊Q₈	1	12	yes	—
43	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
44	C₂×C₄×C₅₃	C₂²×Q₈	1	3	yes	—
45	#88322	—	4	8	no	—
46	C₂×#1843	—	53	1	no	maximal, Sylow
47	C₂²×C₅₃	#384	1	4	yes	—
48	C₂²×C₅₃	#424	1	24	yes	—
49	C₂²×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
50	C₂²×C₅₃	C₂×#131	1	4	yes	—
51	C₄×C₅₃	—	2	24	no	—
52	C₄×C₅₃	#384	1	4	yes	—
53	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
54	C₄×C₅₃	C₂×C₄⋊Q₈	1	3	yes	—
55	C₂²×C₄⋊C₈	—	53	6	no	—
56	C₂×#376	D₅₃	1	1	yes	characteristic
57	#1843	—	53	8	no	—
58	C₂×C₅₃	#1843	1	8	yes	complemented
59	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
60	C₂×C₅₃	C₂×#424	1	6	yes	—
61	C₂³×C₈	—	53	4	no	—
62	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
63	C₂×C₄⋊C₈	—	53	24	no	—
64	C₂×C₄⋊C₈	—	106	24	no	—
65	#376	Dic₅₃	1	4	yes	—
66	#376	C₂×D₅₃	1	4	yes	—
67	C₂×#95	Dic₅₃	1	4	yes	—
68	C₅₃	C₂×#1843	1	1	yes	characteristic, complemented, Sylow
69	C₂³×C₄	C₂×Dic₅₃	1	6	yes	—
70	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
71	C₂²×C₈	—	53	8	no	—
72	C₂²×C₈	—	106	24	no	—
73	C₂×C₄²	Dic₁₀₆	1	12	yes	—
74	C₂×C₄²	D₂₁₂	1	12	yes	—
75	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
76	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
77	C₄⋊C₈	—	106	48	no	—
78	#95	C₂×Dic₅₃	1	16	yes	—
79	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
80	C₂²×C₄	—	2	24	no	—
81	C₂²×C₄	C₄⋊Dic₅₃	1	24	yes	—
82	C₂²×C₄	C₂×Dic₁₀₆	1	6	yes	—
83	C₂²×C₄	C₂×D₂₁₂	1	6	yes	—
84	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
85	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
86	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
87	C₂×C₈	—	106	24	no	—
88	C₂×C₈	—	212	16	no	—
89	C₄²	C₄⋊Dic₅₃	1	24	yes	—
90	C₄²	C₂×Dic₁₀₆	1	12	yes	—
91	C₄²	C₂×D₂₁₂	1	12	yes	—
92	C₂³	C₂×C₄⋊Dic₅₃	1	6	yes	—
93	C₂³	#13133224	1	8	yes	—
94	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
95	C₂×C₄	—	2	24	no	—
96	C₂×C₄	—	2	48	no	—
97	C₂×C₄	C₂×C₄⋊Dic₅₃	1	24	yes	—
98	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
99	C₂×C₄	#13133214	1	4	yes	—
100	C₂×C₄	#13133217	1	12	yes	—
101	C₂×C₄	C₂²×D₂₁₂	1	3	yes	—
102	C₂×C₄	C₂²×Dic₁₀₆	1	3	yes	—
103	C₈	—	212	8	no	—
104	C₂²	#16810493	1	4	yes	—
105	C₂²	#16810650	1	24	yes	—
106	C₂²	C₂²×C₄⋊Dic₅₃	1	3	yes	—
107	C₂²	C₂×#13133224	1	4	yes	—
108	C₄	—	2	24	no	—
109	C₄	#16810493	1	4	yes	—
110	C₄	C₂×#13133214	1	1	yes	characteristic
111	C₄	C₂×#13133217	1	3	yes	—
112	C₂	#20274869	1	8	yes	direct product factor
113	C₂	C₂×#16810493	1	1	yes	characteristic
114	C₂	C₂×#16810650	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊#88322

C₂

yes

maximal

#20274869

C₂

yes

maximal, direct product factor

C₂×C₅₃×#376

C₂

yes

maximal, characteristic

C₂³×#88322

C₂²

yes

—

C₂×C₄⋊#88322

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂²×C₄²×C₅₃

C₂²

yes

—

C₅₃×#376

C₄

yes

—

C₅₃×#376

C₂²

yes

—

C₂×#95×C₅₃

C₄

yes

—

C₂²×#88322

—

C₂²×#88322

C₂³

yes

—

C₄⋊#88322

—

C₂³×C₄×C₅₃

C₂×C₄

yes

—

C₂³×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

—

#95×C₅₃

C₂×C₄

yes

—

C₂⁴×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×Q₈

yes

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

—

C₂³×C₅₃

#131

yes

—

C₂³×C₅₃

C₂³×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

#127

yes

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

—

C₂×C₄×C₅₃

C₂²×Q₈

yes

—

—

—

C₂²×C₅₃

yes

—

C₂²×C₅₃

#424

yes

—

C₂²×C₅₃

C₂²×C₄⋊C₄

yes

—

C₂²×C₅₃

C₂×#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

#384

yes

—

C₄×C₅₃

C₂×#127

yes

characteristic

C₄×C₅₃

C₂×C₄⋊Q₈

yes

—

C₂²×C₄⋊C₈

—

C₂×#376

D₅₃

yes

characteristic

#1843

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂³×C₈

—

C₂²×C₄²

C₂×D₅₃

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

106

—

#376

Dic₅₃

yes

—

#376

C₂×D₅₃

yes

—

C₂×#95

Dic₅₃

yes

—

C₅₃

C₂×#1843

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×Dic₅₃

yes

—

C₂³×C₄

C₂²×D₅₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂×C₄²

Dic₁₀₆

yes

—

C₂×C₄²

D₂₁₂

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

—

C₄⋊C₈

—

106

—

#95

C₂×Dic₅₃

yes

—

C₂⁴

C₂²×Dic₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄⋊Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₁₀₆

yes

—

C₂²×C₄

C₂×D₂₁₂

yes

—

C₂²×C₄

C₂²×Dic₅₃

yes

—

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

C₂²×C₄

C₂³×D₅₃

yes

characteristic

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₄²

C₄⋊Dic₅₃

yes

—

C₄²

C₂×Dic₁₀₆

yes

—

C₄²

C₂×D₂₁₂

yes

—

C₂³

C₂×C₄⋊Dic₅₃

yes

—

C₂³

#13133224

yes

—

C₂³

C₂³×Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

—

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

—

C₂×C₄

#13133214

yes

—

100

C₂×C₄

#13133217

yes

—

101

C₂×C₄

C₂²×D₂₁₂

yes

—

102

C₂×C₄

C₂²×Dic₁₀₆

yes

—

103

C₈

—

212

—

104

C₂²

#16810493

yes

—

105

C₂²

#16810650

yes

—

106

C₂²

C₂²×C₄⋊Dic₅₃

yes

—

107

C₂²

C₂×#13133224

yes

—

108

C₄

—

109

C₄

#16810493

yes

—

110

C₄

C₂×#13133214

yes

characteristic

111

C₄

C₂×#13133217

yes

—

112

C₂

#20274869

yes

direct product factor

113

C₂

C₂×#16810493

yes

characteristic

114

C₂

C₂×#16810650

yes

—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#376	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#376	D₅₃
hypocenter	C₅₃	C₂×#1843
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

subgroup

G/H

center

C₂³×C₄

C₂²×D₅₃

commutator

C₂³×C₅₃

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₅₃

Fitting

C₂×C₅₃×#376

C₂

socle, prime radical

C₂⁴×C₅₃

C₂²×C₄

cosocle

C₂²×C₄×C₅₃

C₂⁴

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³×C₄ ◁ C₂×#376

N	H ≅ G/N	type
2. #20274869	C₂	direct (N × H)
58. C₂×C₅₃	#1843	semidirect
68. C₅₃	C₂×#1843	semidirect
112. C₂	#20274869	direct (N × H)

H ≅ G/N

type

2. #20274869

C₂

direct (N × H)

58. C₂×C₅₃

#1843

semidirect

68. C₅₃

C₂×#1843

semidirect

112. C₂

#20274869

direct (N × H)

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

subgroup	index
C₂²×C₄⋊#88322	2
#20274869	2
C₂×C₅₃×#376	2
C₂×#1843	53

subgroup

index

C₂²×C₄⋊#88322

#20274869

C₂×C₅₃×#376

C₂×#1843

Subgroups — C₂×#20274869 #27384474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20274869 #27384474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274869 #27384474