Group #28389298 — subgroups

Subgroups — #28389298

Order 14464.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₄⋊Q₈×D₁₁₃	C₂	1	1	yes	maximal, characteristic
2	#20298654	C₂	1	2	yes	maximal
3	#20298347	C₂	1	1	yes	maximal, characteristic, complemented
4	#20298346	C₂	1	2	yes	maximal, complemented
5	#333×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₁₁₃	C₄	1	2	yes	—
7	#16813622	C₂²	1	1	yes	characteristic
8	#16813621	—	2	2	no	—
9	#16813620	C₄	1	2	yes	—
10	C₄⋊C₄×D₁₁₃	—	2	2	no	—
11	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
12	C₄⋊#11744751	C₂²	1	2	yes	—
13	#16813534	—	2	2	no	—
14	#16813532	—	2	2	no	—
15	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	—
17	C₂×Dic₂₂₆	—	2	2	no	—
18	C₂×Dic₂₂₆	C₂×C₄	1	2	yes	—
19	Q₈×D₁₁₃	—	4	4	no	—
20	C₂×C₄×D₁₁₃	—	2	2	no	—
21	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₁₁₃	D₄	1	2	yes	—
23	C₂×#11744751	—	2	2	no	—
24	C₄⋊Dic₁₁₃	—	2	2	no	—
25	#14268626	—	2	4	no	—
26	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₁₃	D₄	1	2	yes	—
28	C₂×Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₁₃	—	2	2	no	—
30	C₄⋊C₄×C₁₁₃	—	2	2	no	—
31	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
32	C₂²×D₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
33	Q₈×C₁₁₃	—	4	2	no	—
34	C₂×C₄×C₁₁₃	—	2	2	no	—
35	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
37	C₂×Dic₁₁₃	—	2	2	no	—
38	C₂×Dic₁₁₃	C₂²⋊C₄	1	2	yes	—
39	C₂×Dic₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×D₁₁₃	—	2	2	no	—
41	C₄×D₁₁₃	—	2	2	no	—
42	C₄×D₁₁₃	—	2	2	no	—
43	C₄×D₁₁₃	—	4	4	no	—
44	Dic₂₂₆	—	4	4	no	—
45	Dic₂₂₆	—	4	2	no	—
46	C₈×C₁₁₃	—	4	2	no	—
47	#11744751	—	4	2	no	—
48	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₁₁₃	—	2	2	no	—
50	C₂×D₁₁₃	#100	1	2	yes	—
51	C₄×C₁₁₃	—	2	1	no	—
52	C₄×C₁₁₃	—	4	2	no	—
53	C₄×C₁₁₃	#137	1	2	yes	—
54	C₄×C₁₁₃	#137	1	2	yes	—
55	Dic₁₁₃	—	2	1	no	—
56	Dic₁₁₃	—	2	2	no	—
57	Dic₁₁₃	—	4	2	no	—
58	C₂×C₁₁₃	C₂×#100	1	1	yes	characteristic
59	C₂×C₁₁₃	#419	1	2	yes	—
60	D₁₁₃	—	2	2	no	—
61	#1558	—	113	1	no	maximal, Sylow
62	C₁₁₃	#1558	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₄⋊Q₈	—	113	1	no	—
64	#432	—	113	2	no	—
65	#333	—	113	1	no	—
66	#333	—	113	2	no	—
67	#333	D₁₁₃	1	1	yes	characteristic, complemented
68	C₂²×Q₈	—	113	2	no	—
69	C₄⋊Q₈	—	113	2	no	—
70	C₄⋊Q₈	—	113	1	no	—
71	C₄⋊Q₈	—	226	2	no	—
72	C₄⋊Q₈	C₂×D₁₁₃	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	226	2	no	—
74	C₂×C₄²	—	113	1	no	—
75	C₄⋊C₈	—	113	2	no	—
76	C₄⋊C₈	C₂×D₁₁₃	1	2	yes	—
77	C₂²⋊C₈	—	226	2	no	—
78	#97	—	226	2	no	—
79	C₂×Q₈	—	113	2	no	—
80	C₂×Q₈	—	452	4	no	—
81	C₂×Q₈	—	226	2	no	—
82	C₂×Q₈	C₄×D₁₁₃	1	2	yes	—
83	C₂²×C₄	—	113	1	no	—
84	C₂²×C₄	—	113	2	no	—
85	C₂²×C₄	—	226	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	226	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	226	4	no	—
90	C₄⋊C₄	—	226	2	no	—
91	C₄²	—	113	1	no	—
92	C₄²	—	113	2	no	—
93	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
94	C₂³	—	113	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	452	2	no	—
97	Q₈	—	452	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	113	2	no	—
100	C₂×C₄	—	113	1	no	—
101	C₂×C₄	—	452	4	no	—
102	C₂×C₄	—	226	2	no	—
103	C₂×C₄	—	226	2	no	—
104	C₂×C₄	—	226	2	no	—
105	C₂×C₄	—	226	2	no	—
106	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
107	C₂×C₄	D₄×D₁₁₃	1	2	yes	—
108	C₈	—	4	2	no	—
109	C₈	—	452	2	no	—
110	C₂²	—	113	2	no	—
111	C₂²	—	226	2	no	—
112	C₂²	C₂²⋊C₄×D₁₁₃	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	452	2	no	—
116	C₄	—	226	1	no	—
117	C₄	—	226	2	no	—
118	C₄	#16813651	1	2	yes	—
119	C₄	#16813654	1	2	yes	—
120	C₂	—	226	2	no	—
121	C₂	#100×D₁₁₃	1	1	yes	characteristic
122	C₂	#20298605	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₁₁₃

C₂

yes

maximal, characteristic

#20298654

C₂

yes

maximal

#20298347

C₂

yes

maximal, characteristic, complemented

#20298346

C₂

yes

maximal, complemented

#333×C₁₁₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₁₁₃

C₄

yes

—

#16813622

C₂²

yes

characteristic

#16813621

—

#16813620

C₄

yes

—

C₄⋊C₄×D₁₁₃

—

C₄²×D₁₁₃

C₂²

yes

characteristic

C₄⋊#11744751

C₂²

yes

—

#16813534

—

#16813532

—

C₄⋊Q₈×C₁₁₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₁₃

C₂²

yes

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

C₂×C₄

yes

—

Q₈×D₁₁₃

—

C₂×C₄×D₁₁₃

—

C₂×C₄×D₁₁₃

C₂×C₄

yes

characteristic

C₂×C₄×D₁₁₃

D₄

yes

—

C₂×#11744751

—

C₄⋊Dic₁₁₃

—

#14268626

—

C₄×Dic₁₁₃

C₂×C₄

yes

characteristic

C₄×Dic₁₁₃

D₄

yes

—

C₂×Q₈×C₁₁₃

C₂×C₄

yes

—

C₂×C₈×C₁₁₃

—

C₄⋊C₄×C₁₁₃

—

C₄²×C₁₁₃

C₂³

yes

characteristic

C₂²×D₁₁₃

C₂²⋊C₄

yes

characteristic

Q₈×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×D₄

yes

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

C₂²⋊C₄

yes

—

C₂×Dic₁₁₃

C₂²⋊C₄

yes

characteristic

C₄×D₁₁₃

—

C₄×D₁₁₃

—

C₄×D₁₁₃

—

C₄×D₁₁₃

—

Dic₂₂₆

—

Dic₂₂₆

—

C₈×C₁₁₃

—

#11744751

—

C₂²×C₁₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₁₃

—

C₂×D₁₁₃

#100

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

#137

yes

—

C₄×C₁₁₃

#137

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

C₂×#100

yes

characteristic

C₂×C₁₁₃

#419

yes

—

D₁₁₃

—

#1558

—

113

maximal, Sylow

C₁₁₃

#1558

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

113

—

#432

—

113

—

#333

—

113

—

#333

—

113

—

#333

D₁₁₃

yes

characteristic, complemented

C₂²×Q₈

—

113

—

C₄⋊Q₈

—

113

—

C₄⋊Q₈

—

113

—

C₄⋊Q₈

—

226

—

C₄⋊Q₈

C₂×D₁₁₃

yes

characteristic

C₂×C₄⋊C₄

—

226

—

C₂×C₄²

—

113

—

C₄⋊C₈

—

113

—

C₄⋊C₈

C₂×D₁₁₃

yes

—

C₂²⋊C₈

—

226

—

#97

—

226

—

C₂×Q₈

—

113

—

C₂×Q₈

—

452

—

C₂×Q₈

—

226

—

C₂×Q₈

C₄×D₁₁₃

yes

—

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

—

226

—

C₂×C₈

—

C₂×C₈

—

226

—

C₄⋊C₄

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₄²

—

113

—

C₄²

—

113

—

C₄²

C₂²×D₁₁₃

yes

characteristic

C₂³

—

113

—

Q₈

—

Q₈

—

452

—

Q₈

—

452

—

C₂×C₄

—

C₂×C₄

—

113

—

100

C₂×C₄

—

113

—

101

C₂×C₄

—

452

—

102

C₂×C₄

—

226

—

103

C₂×C₄

—

226

—

104

C₂×C₄

—

226

—

105

C₂×C₄

—

226

—

106

C₂×C₄

C₂×C₄×D₁₁₃

yes

characteristic

107

C₂×C₄

D₄×D₁₁₃

yes

—

108

C₈

—

109

C₈

—

452

—

110

C₂²

—

113

—

111

C₂²

—

226

—

112

C₂²

C₂²⋊C₄×D₁₁₃

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

452

—

116

C₄

—

226

—

117

C₄

—

226

—

118

C₄

#16813651

yes

—

119

C₄

#16813654

yes

—

120

C₂

—

226

—

121

C₂

#100×D₁₁₃

yes

characteristic

122

C₂

#20298605

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₁₃
commutator	C₂×C₄×C₁₁₃	C₂²×C₄
Frattini	C₄²	C₂²×D₁₁₃
Fitting	#333×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₁₃	C₂²⋊C₄
cosocle	C₄²×C₁₁₃	C₂³
hypercenter	#333	D₁₁₃
hypocenter	C₁₁₃	#1558
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₁₃

commutator

C₂×C₄×C₁₁₃

C₂²×C₄

Frattini

C₄²

C₂²×D₁₁₃

Fitting

#333×C₁₁₃

C₂

socle

C₂²×C₁₁₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₁₃

C₂²⋊C₄

cosocle

C₄²×C₁₁₃

C₂³

D₁₁₃

C₁₁₃

G ▷ C₂×C₄×C₁₁₃ ▷ C₂²×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
3. #20298347	C₂	semidirect
4. #20298346	C₂	semidirect
5. #333×C₁₁₃	C₂	semidirect
62. C₁₁₃	#1558	semidirect
67. #333	D₁₁₃	semidirect

H ≅ G/N

type

3. #20298347

C₂

semidirect

4. #20298346

C₂

semidirect

5. #333×C₁₁₃

C₂

semidirect

62. C₁₁₃

#1558

semidirect

67. #333

D₁₁₃

semidirect

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

subgroup	index
C₄⋊Q₈×D₁₁₃	2
#20298654	2
#20298347	2
#20298346	2
#333×C₁₁₃	2
#1558	113

subgroup

index

C₄⋊Q₈×D₁₁₃

#333×C₁₁₃

113

Subgroups — #28389298

Conjugacy classes of subgroups

Complements

Maximal subgroups