Group #28391546 — subgroups

Subgroups — #28391546

Order 14464.

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

normal subgroups only

This group has 276 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×#16813622	C₂	1	1	yes	maximal, characteristic
2	C₂×#16813533	C₂	1	1	yes	maximal, characteristic
3	#20298360	C₂	1	4	yes	maximal
4	#382×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₂₆	—	2	1	no	—
6	C₂²×#11744751	C₂²	1	2	yes	—
7	#16813622	—	2	2	no	—
8	#16813622	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₁₃	—	2	1	no	—
10	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
11	#16813533	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
14	#95×C₁₁₃	—	2	1	no	—
15	#95×C₁₁₃	—	2	1	no	—
16	C₂×Dic₂₂₆	—	2	4	no	—
17	C₂×Dic₂₂₆	—	4	4	no	—
18	C₂²×Dic₁₁₃	—	2	2	no	—
19	C₂×#11744751	—	2	2	no	—
20	C₂×#11744751	—	2	2	no	—
21	C₂×#11744751	Q₈	1	4	yes	complemented
22	C₄⋊Dic₁₁₃	—	2	4	no	—
23	C₄⋊Dic₁₁₃	—	2	2	no	—
24	C₄⋊Dic₁₁₃	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₁₃	—	2	2	no	—
26	C₂²×C₄×C₁₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₁₃	—	2	4	no	—
30	C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
31	C₄²×C₁₁₃	D₄	1	2	yes	—
32	C₂³×C₁₁₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₁₃	—	2	2	no	—
34	C₂×C₄×C₁₁₃	—	2	2	no	—
35	C₂×C₄×C₁₁₃	—	4	2	no	—
36	C₂×C₄×C₁₁₃	—	4	2	no	—
37	C₂×C₄×C₁₁₃	—	4	2	no	—
38	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₁₁₃	#41	1	2	yes	—
44	C₂×Dic₁₁₃	—	2	4	no	—
45	C₂×Dic₁₁₃	—	4	4	no	—
46	Dic₂₂₆	—	4	8	no	—
47	#11744751	—	2	4	no	—
48	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₁₃	#124	1	1	yes	characteristic
50	C₂²×C₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₁₃	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₁₁₃	#137	1	2	yes	—
53	C₄×C₁₁₃	—	2	2	no	—
54	C₄×C₁₁₃	—	2	2	no	—
55	C₄×C₁₁₃	—	8	2	no	—
56	C₄×C₁₁₃	C₄×Q₈	1	2	yes	—
57	C₄×C₁₁₃	C₂²⋊Q₈	1	2	yes	—
58	Dic₁₁₃	—	4	4	no	—
59	C₂×C₁₁₃	#391	1	1	yes	characteristic
60	C₂×C₁₁₃	#418	1	1	yes	characteristic
61	C₂×C₁₁₃	#419	1	1	yes	characteristic
62	C₂×C₁₁₃	#489	1	2	yes	—
63	C₂×C₁₁₃	#501	1	2	yes	—
64	#1993	—	113	1	no	maximal, Sylow
65	C₁₁₃	#1993	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄⋊Q₈	—	113	1	no	—
67	C₂×#97	—	113	1	no	—
68	#382	D₁₁₃	1	1	yes	characteristic
69	#340	—	113	4	no	—
70	C₂²×Q₈	—	226	1	no	—
71	C₂²×C₈	—	113	2	no	—
72	C₄⋊Q₈	—	113	4	no	—
73	C₄⋊Q₈	—	226	2	no	—
74	C₂×C₄⋊C₄	—	113	2	no	—
75	C₂×C₄⋊C₄	—	226	1	no	—
76	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
77	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
78	#97	—	226	2	no	—
79	#95	—	2	1	no	—
80	#95	—	2	1	no	—
81	C₂×Q₈	—	452	4	no	—
82	C₂×Q₈	—	226	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	226	2	no	—
85	C₂²×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
88	C₂×C₈	—	113	4	no	—
89	C₂×C₈	—	226	2	no	—
90	C₂×C₈	—	226	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	113	4	no	—
93	C₄⋊C₄	—	226	4	no	—
94	C₄⋊C₄	—	226	2	no	—
95	C₄²	C₄×D₁₁₃	1	2	yes	—
96	C₄²	D₄₅₂	1	2	yes	—
97	C₂³	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
98	Q₈	—	452	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	452	4	no	—
105	C₂×C₄	—	226	4	no	—
106	C₂×C₄	#14268624	1	2	yes	—
107	C₂×C₄	#14268624	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
109	C₂×C₄	C₂×D₄₅₂	1	1	yes	characteristic
110	C₂×C₄	Q₈×D₁₁₃	1	2	yes	—
111	C₂×C₄	#14268646	1	2	yes	—
112	C₈	—	226	4	no	—
113	C₂²	C₂×#14268624	1	1	yes	characteristic
114	C₂²	#16813609	1	1	yes	characteristic
115	C₂²	#16813620	1	1	yes	characteristic
116	C₂²	#16813647	1	2	yes	—
117	C₂²	#16813652	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	8	2	no	—
121	C₄	—	452	4	no	—
122	C₄	#16813587	1	2	yes	—
123	C₄	#16813602	1	2	yes	—
124	C₂	#20298503	1	1	yes	characteristic
125	C₂	#20298602	1	1	yes	characteristic
126	C₂	#20298606	1	1	yes	characteristic
127	C₂	#20298957	1	2	yes	—
128	C₂	#20299008	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813622

C₂

yes

maximal, characteristic

C₂×#16813533

C₂

yes

maximal, characteristic

#20298360

C₂

yes

maximal

#382×C₁₁₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₂₆

—

C₂²×#11744751

C₂²

yes

—

#16813622

—

#16813622

C₄

yes

complemented

C₂×C₄⋊Dic₁₁₃

—

C₂×C₄⋊Dic₁₁₃

C₂²

yes

—

#16813533

—

C₂×C₄⋊C₄×C₁₁₃

C₂²

yes

—

C₂×C₄²×C₁₁₃

C₂²

yes

characteristic

#95×C₁₁₃

—

#95×C₁₁₃

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

—

C₂²×Dic₁₁₃

—

C₂×#11744751

—

C₂×#11744751

—

C₂×#11744751

Q₈

yes

complemented

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₁₃

—

C₂²×C₄×C₁₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₁₃

—

C₄²×C₁₁₃

C₂×C₄

yes

—

C₄²×C₁₁₃

D₄

yes

—

C₂³×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₁₃

#41

yes

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

Dic₂₂₆

—

#11744751

—

C₂²×C₁₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₁₃

#124

yes

characteristic

C₂²×C₁₁₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₁₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₁₃

#137

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₄×Q₈

yes

—

C₄×C₁₁₃

C₂²⋊Q₈

yes

—

Dic₁₁₃

—

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

C₂×C₁₁₃

#501

yes

—

#1993

—

113

maximal, Sylow

C₁₁₃

#1993

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

113

—

C₂×#97

—

113

—

#382

D₁₁₃

yes

characteristic

#340

—

113

—

C₂²×Q₈

—

226

—

C₂²×C₈

—

113

—

C₄⋊Q₈

—

113

—

C₄⋊Q₈

—

226

—

C₂×C₄⋊C₄

—

113

—

C₂×C₄⋊C₄

—

226

—

C₂×C₄⋊C₄

C₂×D₁₁₃

yes

—

C₂×C₄²

C₂×D₁₁₃

yes

characteristic

#97

—

226

—

#95

—

#95

—

C₂×Q₈

—

452

—

C₂×Q₈

—

226

—

C₂²×C₄

—

C₂²×C₄

—

226

—

C₂²×C₄

C₂²⋊D₁₁₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₈

—

113

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₄⋊C₄

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₄²

C₄×D₁₁₃

yes

—

C₄²

D₄₅₂

yes

—

C₂³

C₂×C₂²⋊D₁₁₃

yes

characteristic

Q₈

—

452

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

452

—

105

C₂×C₄

—

226

—

106

C₂×C₄

#14268624

yes

—

107

C₂×C₄

#14268624

yes

—

108

C₂×C₄

C₂×C₄×D₁₁₃

yes

characteristic

109

C₂×C₄

C₂×D₄₅₂

yes

characteristic

110

C₂×C₄

Q₈×D₁₁₃

yes

—

111

C₂×C₄

#14268646

yes

—

112

C₈

—

226

—

113

C₂²

C₂×#14268624

yes

characteristic

114

C₂²

#16813609

yes

characteristic

115

C₂²

#16813620

yes

characteristic

116

C₂²

#16813647

yes

—

117

C₂²

#16813652

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

452

—

122

C₄

#16813587

yes

—

123

C₄

#16813602

yes

—

124

C₂

#20298503

yes

characteristic

125

C₂

#20298602

yes

characteristic

126

C₂

#20298606

yes

characteristic

127

C₂

#20298957

yes

—

128

C₂

#20299008

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₁₃
commutator	C₂×C₄×C₁₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₁₃
Fitting	#382×C₁₁₃	C₂
socle, prime radical	C₂³×C₁₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₁₃	C₂³
hypercenter	#382	D₁₁₃
hypocenter	C₁₁₃	#1993
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₁₃

commutator

C₂×C₄×C₁₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₁₃

Fitting

#382×C₁₁₃

C₂

socle, prime radical

C₂³×C₁₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₁₃

C₂³

D₁₁₃

C₁₁₃

G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
8. #16813622	C₄	semidirect
21. C₂×#11744751	Q₈	semidirect
65. C₁₁₃	#1993	semidirect

H ≅ G/N

type

8. #16813622

C₄

semidirect

21. C₂×#11744751

Q₈

semidirect

65. C₁₁₃

#1993

semidirect

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

subgroup	index
C₂×#16813622	2
C₂×#16813533	2
#20298360	2
#382×C₁₁₃	2
#1993	113

subgroup

index

#382×C₁₁₃

113

Subgroups — #28391546

Conjugacy classes of subgroups

Complements

Maximal subgroups