Group #28391726 — subgroups

Subgroups — #28391726

Order 14464.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16813614	C₂	1	1	yes	maximal, characteristic
2	C₂×#16813547	C₂	1	2	yes	maximal
3	#20298504	C₂	1	1	yes	maximal, characteristic
4	#20298361	C₂	1	2	yes	maximal
5	#336×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₂₆	C₂²	1	1	yes	characteristic
7	#16813614	—	2	4	no	—
8	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
10	C₂×#14268626	—	2	2	no	—
11	#16813547	—	2	4	no	—
12	#16813529	—	2	2	no	—
13	C₂²×C₈×C₁₁₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₂₆	—	2	2	no	—
16	C₂×Dic₂₂₆	—	2	2	no	—
17	C₂×Dic₂₂₆	—	2	1	no	—
18	C₂×Dic₂₂₆	Q₈	1	2	yes	—
19	C₂²×Dic₁₁₃	—	2	2	no	—
20	C₂²×Dic₁₁₃	—	2	1	no	—
21	C₂²×Dic₁₁₃	—	2	1	no	—
22	C₄⋊Dic₁₁₃	—	2	2	no	—
23	C₄⋊Dic₁₁₃	—	2	2	no	—
24	C₄⋊Dic₁₁₃	D₄	1	4	yes	—
25	#14268626	—	2	8	no	—
26	C₂²×C₄×C₁₁₃	D₄	1	2	yes	—
27	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₁₃	—	2	2	no	—
29	C₂×C₈×C₁₁₃	—	2	2	no	—
30	C₂×C₈×C₁₁₃	—	2	2	no	—
31	C₄²×C₁₁₃	—	2	2	no	—
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	—	2	2	no	—
36	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₁₃	#41	1	1	yes	characteristic
39	C₂×C₄×C₁₁₃	#41	1	2	yes	—
40	C₂×Dic₁₁₃	—	2	4	no	—
41	C₂×Dic₁₁₃	—	2	2	no	—
42	C₂×Dic₁₁₃	—	4	1	no	—
43	C₂×Dic₁₁₃	—	4	1	no	—
44	C₂×Dic₁₁₃	—	4	2	no	—
45	C₂×Dic₁₁₃	—	4	2	no	—
46	C₂×Dic₁₁₃	—	4	1	no	—
47	C₂×Dic₁₁₃	—	4	2	no	—
48	Dic₂₂₆	—	2	4	no	—
49	Dic₂₂₆	—	4	2	no	—
50	C₈×C₁₁₃	—	4	2	no	—
51	C₂²×C₁₁₃	#120	1	1	yes	characteristic
52	C₂²×C₁₁₃	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₁₁₃	#124	1	1	yes	characteristic
54	C₂²×C₁₁₃	#135	1	2	yes	—
55	C₂²×C₁₁₃	#137	1	2	yes	—
56	C₄×C₁₁₃	—	4	2	no	—
57	C₄×C₁₁₃	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₁₁₃	#124	1	2	yes	—
59	Dic₁₁₃	—	4	2	no	—
60	Dic₁₁₃	—	4	4	no	—
61	Dic₁₁₃	—	8	1	no	—
62	C₂×C₁₁₃	#393	1	1	yes	characteristic
63	C₂×C₁₁₃	#452	1	2	yes	—
64	C₂×C₁₁₃	#479	1	2	yes	—
65	C₂×C₁₁₃	#488	1	1	yes	characteristic
66	C₂×C₁₁₃	#491	1	1	yes	characteristic
67	#2026	—	113	1	no	maximal, Sylow
68	C₁₁₃	#2026	1	1	yes	characteristic, complemented, Sylow
69	C₂×#125	—	113	1	no	—
70	C₂×#103	—	113	2	no	—
71	#391	—	113	1	no	—
72	#340	—	113	2	no	—
73	#336	D₁₁₃	1	1	yes	characteristic
74	C₂²×Q₈	—	113	1	no	—
75	C₂²×C₈	C₂×D₁₁₃	1	2	yes	—
76	#125	—	226	4	no	—
77	C₂×C₄⋊C₄	—	113	2	no	—
78	C₂×C₄⋊C₄	—	113	1	no	—
79	C₂×C₄⋊C₄	—	226	2	no	—
80	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
81	#103	—	226	4	no	—
82	#95	—	226	2	no	—
83	C₂×Q₈	—	113	2	no	—
84	C₂×Q₈	—	226	2	no	—
85	C₂×Q₈	—	226	2	no	—
86	C₂×Q₈	—	226	1	no	—
87	C₂²×C₄	—	226	1	no	—
88	C₂²×C₄	—	226	1	no	—
89	C₂²×C₄	—	226	2	no	—
90	C₂²×C₄	D₄₅₂	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₄⋊C₄	—	113	4	no	—
96	C₄⋊C₄	—	226	8	no	—
97	C₄⋊C₄	—	226	2	no	—
98	C₄⋊C₄	—	226	2	no	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×D₄₅₂	1	1	yes	characteristic
101	Q₈	—	452	2	no	—
102	Q₈	—	226	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	452	1	no	—
107	C₂×C₄	—	452	2	no	—
108	C₂×C₄	—	452	1	no	—
109	C₂×C₄	—	452	1	no	—
110	C₂×C₄	—	452	2	no	—
111	C₂×C₄	—	452	2	no	—
112	C₂×C₄	—	226	4	no	—
113	C₂×C₄	—	226	2	no	—
114	C₂×C₄	D₄×D₁₁₃	1	2	yes	—
115	C₂×C₄	Q₈×D₁₁₃	1	1	yes	characteristic
116	C₂×C₄	#14268646	1	1	yes	characteristic
117	C₂×C₄	#14268647	1	2	yes	—
118	C₈	—	4	2	no	—
119	C₂²	#16813592	1	1	yes	characteristic
120	C₂²	#16813602	1	1	yes	characteristic
121	C₂²	#16813610	1	1	yes	characteristic
122	C₂²	#16813644	1	2	yes	—
123	C₂²	#16813655	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	904	1	no	—
126	C₄	—	452	4	no	—
127	C₄	—	452	2	no	—
128	C₄	#16813600	1	2	yes	—
129	C₄	#16813608	1	2	yes	—
130	C₂	#20298511	1	1	yes	characteristic
131	C₂	#20298762	1	2	yes	—
132	C₂	#20298906	1	2	yes	—
133	C₂	#20298953	1	1	yes	characteristic
134	C₂	#20298967	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813614

C₂

yes

maximal, characteristic

C₂×#16813547

C₂

yes

maximal

#20298504

C₂

yes

maximal, characteristic

#20298361

C₂

yes

maximal

#336×C₁₁₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₂₆

C₂²

yes

characteristic

#16813614

—

C₂×C₄⋊Dic₁₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₁₃

C₂²

yes

—

C₂×#14268626

—

#16813547

—

#16813529

—

C₂²×C₈×C₁₁₃

C₂²

yes

—

C₂×C₄²×C₁₁₃

C₂²

yes

characteristic

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

Q₈

yes

—

C₂²×Dic₁₁₃

—

C₂²×Dic₁₁₃

—

C₂²×Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

D₄

yes

—

#14268626

—

C₂²×C₄×C₁₁₃

D₄

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×C₈×C₁₁₃

—

C₂×C₈×C₁₁₃

—

C₂×C₈×C₁₁₃

—

C₄²×C₁₁₃

—

C₂³×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₁₃

C₂×Q₈

yes

C₂×C₄×C₁₁₃

yes

C₂×C₄×C₁₁₃

yes

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

Dic₂₂₆

—

Dic₂₂₆

—

C₈×C₁₁₃

—

C₂²×C₁₁₃

#120

yes

characteristic

C₂²×C₁₁₃

C₂²⋊Q₈

yes

C₂²×C₁₁₃

yes

C₂²×C₁₁₃

yes

—

C₂²×C₁₁₃

#137

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₂²⋊Q₈

yes

—

C₄×C₁₁₃

#124

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

#393

yes

characteristic

C₂×C₁₁₃

#452

yes

—

C₂×C₁₁₃

#479

yes

—

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

113

maximal, Sylow

C₁₁₃

#2026

yes

characteristic, complemented, Sylow

C₂×#125

—

113

—

C₂×#103

—

113

—

#391

—

113

—

#340

—

113

—

#336

D₁₁₃

yes

characteristic

C₂²×Q₈

—

113

—

C₂²×C₈

C₂×D₁₁₃

yes

—

#125

—

226

—

C₂×C₄⋊C₄

—

113

—

C₂×C₄⋊C₄

—

113

—

C₂×C₄⋊C₄

—

226

—

C₂×C₄²

C₂×D₁₁₃

yes

characteristic

#103

—

226

—

#95

—

226

—

C₂×Q₈

—

113

—

C₂×Q₈

—

226

—

C₂×Q₈

—

226

—

C₂×Q₈

—

226

—

C₂²×C₄

—

226

—

C₂²×C₄

—

226

—

C₂²×C₄

—

226

—

C₂²×C₄

D₄₅₂

yes

—

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₄²

—

100

C₂³

C₂×D₄₅₂

yes

characteristic

101

Q₈

—

452

—

102

Q₈

—

226

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

452

—

107

C₂×C₄

—

452

—

108

C₂×C₄

—

452

—

109

C₂×C₄

—

452

—

110

C₂×C₄

—

452

—

111

C₂×C₄

—

452

—

112

C₂×C₄

—

226

—

113

C₂×C₄

—

226

—

114

C₂×C₄

D₄×D₁₁₃

yes

—

115

C₂×C₄

Q₈×D₁₁₃

yes

characteristic

116

C₂×C₄

#14268646

yes

characteristic

117

C₂×C₄

#14268647

yes

—

118

C₈

—

119

C₂²

#16813592

yes

characteristic

120

C₂²

#16813602

yes

characteristic

121

C₂²

#16813610

yes

characteristic

122

C₂²

#16813644

yes

—

123

C₂²

#16813655

yes

—

124

C₄

—

125

C₄

—

904

—

126

C₄

—

452

—

127

C₄

—

452

—

128

C₄

#16813600

yes

—

129

C₄

#16813608

yes

—

130

C₂

#20298511

yes

characteristic

131

C₂

#20298762

yes

—

132

C₂

#20298906

yes

—

133

C₂

#20298953

yes

characteristic

134

C₂

#20298967

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₅₂
commutator, cosocle	C₂²×C₄×C₁₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₁₃
Fitting	#336×C₁₁₃	C₂
socle, prime radical	C₂³×C₁₁₃	C₂×D₄
hypercenter	#336	D₁₁₃
hypocenter	C₁₁₃	#2026
LCS	G ▷ C₂²×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₄₅₂

commutator, cosocle

C₂²×C₄×C₁₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₁₃

Fitting

#336×C₁₁₃

C₂

socle, prime radical

C₂³×C₁₁₃

C₂×D₄

D₁₁₃

C₁₁₃

G ▷ C₂²×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
68. C₁₁₃	#2026	semidirect

H ≅ G/N

type

68. C₁₁₃

#2026

semidirect

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

subgroup	index
C₂×#16813614	2
C₂×#16813547	2
#20298504	2
#20298361	2
#336×C₁₁₃	2
#2026	113

subgroup

index

#336×C₁₁₃

113

Subgroups — #28391726

Conjugacy classes of subgroups

Complements

Maximal subgroups