Group #28406705 — subgroups

Subgroups — #28406705

Order 14464.

308 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 78 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×#16813685	C₂	1	4	yes	maximal
2	#20299557	C₂	1	1	yes	maximal, characteristic
3	#20299542	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×D₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14268655	C₂²	1	2	yes	—
6	C₂×C₄⋊#90673	C₂²	1	2	yes	—
7	#16813685	—	2	8	no	—
8	#16813677	—	2	2	no	—
9	(C₂×#90673)⋊C₄	—	2	2	no	—
10	C₂×Q₈×D₁₁₃	C₂²	1	2	yes	—
11	#16813622	C₄	1	1	yes	characteristic, complemented
12	#16813621	—	2	2	no	—
13	#16813620	C₄	1	2	yes	complemented
14	C₄⋊C₄×D₁₁₃	—	2	2	no	—
15	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₁₁₃	C₄	1	1	yes	characteristic, complemented
17	C₂²×#90673	—	2	2	no	—
18	#14268655	—	2	2	no	—
19	#14268655	D₄	1	4	yes	—
20	C₂×#11744753	—	2	2	no	—
21	C₄⋊#90673	—	2	4	no	—
22	C₂×Dic₂₂₆	—	2	2	no	—
23	C₂×Dic₂₂₆	C₂×C₄	1	2	yes	—
24	Q₈×D₁₁₃	—	2	4	no	—
25	Q₈×D₁₁₃	—	2	4	no	—
26	C₂×C₄×D₁₁₃	—	2	2	no	—
27	C₂×C₄×D₁₁₃	D₄	1	2	yes	—
28	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₁₃	—	2	2	no	—
30	#14268626	—	2	4	no	—
31	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₁₁₃	D₄	1	2	yes	—
33	C₂×Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
34	C₄⋊C₄×C₁₁₃	—	2	2	no	—
35	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×D₁₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×#90673	—	4	2	no	—
38	C₂×#90673	—	4	4	no	—
39	Q₈×C₁₁₃	—	2	4	no	—
40	C₂×C₄×C₁₁₃	—	2	2	no	—
41	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₁₁₃	—	2	2	no	—
44	C₂×Dic₁₁₃	C₂²⋊C₄	1	2	yes	—
45	C₂×Dic₁₁₃	C₂×D₄	1	1	yes	characteristic
46	C₄×D₁₁₃	—	2	4	no	—
47	C₄×D₁₁₃	—	4	2	no	—
48	C₄×D₁₁₃	—	4	2	no	—
49	C₄×D₁₁₃	C₂×D₄	1	2	yes	—
50	C₄×D₁₁₃	#41	1	2	yes	—
51	Dic₂₂₆	—	2	4	no	—
52	Dic₂₂₆	—	4	4	no	—
53	#11744753	—	2	4	no	—
54	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×D₁₁₃	#124	1	1	yes	characteristic
56	C₂×D₁₁₃	#127	1	1	yes	characteristic
57	C₂×D₁₁₃	#137	1	4	yes	—
58	#90673	—	8	2	no	—
59	C₄×C₁₁₃	—	2	2	no	—
60	C₄×C₁₁₃	—	4	2	no	—
61	C₄×C₁₁₃	C₄×D₄	1	2	yes	—
62	Dic₁₁₃	—	2	2	no	—
63	Dic₁₁₃	—	4	2	no	—
64	Dic₁₁₃	#121	1	2	yes	—
65	C₂×C₁₁₃	#390	1	1	yes	characteristic
66	C₂×C₁₁₃	#419	1	2	yes	—
67	D₁₁₃	#491	1	2	yes	—
68	D₁₁₃	#497	1	2	yes	—
69	#1960	—	113	1	no	maximal, Sylow
70	C₁₁₃	#1960	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	113	1	no	—
72	C₂×#103	—	113	4	no	—
73	C₂×#97	—	113	1	no	—
74	#382	—	113	1	no	—
75	C₂²×Q₈	—	113	2	no	—
76	C₂²×C₈	—	113	2	no	—
77	C₄⋊Q₈	—	113	2	no	—
78	C₄⋊Q₈	—	113	1	no	—
79	C₄⋊Q₈	—	226	2	no	—
80	C₄⋊Q₈	#90673	1	1	yes	characteristic, complemented
81	C₂×C₄⋊C₄	—	113	2	no	—
82	C₂×C₄⋊C₄	—	226	2	no	—
83	C₂×C₄²	—	113	1	no	—
84	#103	—	226	8	no	—
85	#97	—	226	2	no	—
86	#95	—	226	2	no	—
87	C₂×Q₈	—	113	2	no	—
88	C₂×Q₈	—	226	2	no	—
89	C₂×Q₈	—	226	4	no	—
90	C₂×Q₈	—	226	4	no	—
91	C₂×Q₈	C₂×#90673	1	2	yes	—
92	C₂²×C₄	—	113	1	no	—
93	C₂²×C₄	—	113	2	no	—
94	C₂²×C₄	—	226	2	no	—
95	C₂²×C₄	—	226	2	no	—
96	C₂×C₈	—	113	4	no	—
97	C₂×C₈	—	226	2	no	—
98	C₂×C₈	—	226	2	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	226	4	no	—
101	C₄⋊C₄	—	226	4	no	—
102	C₄⋊C₄	—	226	2	no	—
103	C₄²	—	113	2	no	—
104	C₄²	—	113	1	no	—
105	C₄²	C₂×#90673	1	1	yes	characteristic
106	C₂³	—	113	1	no	—
107	Q₈	—	2	4	no	—
108	Q₈	—	452	4	no	—
109	Q₈	—	226	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	113	1	no	—
112	C₂×C₄	—	113	2	no	—
113	C₂×C₄	—	113	2	no	—
114	C₂×C₄	—	113	2	no	—
115	C₂×C₄	—	452	2	no	—
116	C₂×C₄	—	452	2	no	—
117	C₂×C₄	—	452	2	no	—
118	C₂×C₄	—	452	4	no	—
119	C₂×C₄	—	226	4	no	—
120	C₂×C₄	—	226	2	no	—
121	C₂×C₄	C₂²⋊#90673	1	2	yes	—
122	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
123	C₈	—	226	4	no	—
124	C₂²	—	113	1	no	—
125	C₂²	—	113	1	no	—
126	C₂²	—	113	4	no	—
127	C₂²	C₂×C₂²⋊#90673	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	4	2	no	—
130	C₄	—	113	2	no	—
131	C₄	—	904	2	no	—
132	C₄	—	452	2	no	—
133	C₄	—	226	2	no	—
134	C₄	D₄×#90673	1	2	yes	—
135	C₂	—	113	2	no	—
136	C₂	—	113	2	no	—
137	C₂	#20299550	1	1	yes	characteristic
138	C₂	#20299591	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813685

C₂

yes

maximal

#20299557

C₂

yes

maximal, characteristic

#20299542

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₁₁₃

C₂

yes

maximal, characteristic

C₂×#14268655

C₂²

yes

—

C₂×C₄⋊#90673

C₂²

yes

—

#16813685

—

#16813677

—

(C₂×#90673)⋊C₄

—

C₂×Q₈×D₁₁₃

C₂²

yes

—

#16813622

C₄

yes

characteristic, complemented

#16813621

—

#16813620

C₄

yes

complemented

C₄⋊C₄×D₁₁₃

—

C₄²×D₁₁₃

C₂²

yes

characteristic

C₄⋊Q₈×C₁₁₃

C₄

yes

characteristic, complemented

C₂²×#90673

—

#14268655

—

#14268655

D₄

yes

—

C₂×#11744753

—

C₄⋊#90673

—

C₂×Dic₂₂₆

—

C₂×Dic₂₂₆

C₂×C₄

yes

—

Q₈×D₁₁₃

—

Q₈×D₁₁₃

—

C₂×C₄×D₁₁₃

—

C₂×C₄×D₁₁₃

D₄

yes

—

C₂×C₄×D₁₁₃

C₂³

yes

characteristic

C₄⋊Dic₁₁₃

—

#14268626

—

C₄×Dic₁₁₃

C₂×C₄

yes

characteristic

C₄×Dic₁₁₃

D₄

yes

—

C₂×Q₈×C₁₁₃

C₂×C₄

yes

—

C₄⋊C₄×C₁₁₃

—

C₄²×C₁₁₃

C₂×C₄

yes

characteristic

C₂²×D₁₁₃

C₂×D₄

yes

characteristic

C₂×#90673

—

C₂×#90673

—

Q₈×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

C₂²⋊C₄

yes

—

C₂×Dic₁₁₃

C₂×D₄

yes

characteristic

C₄×D₁₁₃

—

C₄×D₁₁₃

—

C₄×D₁₁₃

—

C₄×D₁₁₃

C₂×D₄

yes

—

C₄×D₁₁₃

#41

yes

—

Dic₂₂₆

—

Dic₂₂₆

—

#11744753

—

C₂²×C₁₁₃

C₂×C₂²⋊C₄

yes

C₂×D₁₁₃

yes

C₂×D₁₁₃

yes

C₂×D₁₁₃

yes

—

#90673

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₄×D₄

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

#121

yes

—

C₂×C₁₁₃

#390

yes

characteristic

C₂×C₁₁₃

#419

yes

—

D₁₁₃

#491

yes

—

D₁₁₃

#497

yes

—

#1960

—

113

maximal, Sylow

C₁₁₃

#1960

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

113

—

C₂×#103

—

113

—

C₂×#97

—

113

—

#382

—

113

—

C₂²×Q₈

—

113

—

C₂²×C₈

—

113

—

C₄⋊Q₈

—

113

—

C₄⋊Q₈

—

113

—

C₄⋊Q₈

—

226

—

C₄⋊Q₈

#90673

yes

characteristic, complemented

C₂×C₄⋊C₄

—

113

—

C₂×C₄⋊C₄

—

226

—

C₂×C₄²

—

113

—

#103

—

226

—

#97

—

226

—

#95

—

226

—

C₂×Q₈

—

113

—

C₂×Q₈

—

226

—

C₂×Q₈

—

226

—

C₂×Q₈

—

226

—

C₂×Q₈

C₂×#90673

yes

—

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

—

226

—

C₂²×C₄

—

226

—

C₂×C₈

—

113

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₄⋊C₄

—

100

C₄⋊C₄

—

226

—

101

C₄⋊C₄

—

226

—

102

C₄⋊C₄

—

226

—

103

C₄²

—

113

—

104

C₄²

—

113

—

105

C₄²

C₂×#90673

yes

characteristic

106

C₂³

—

113

—

107

Q₈

—

108

Q₈

—

452

—

109

Q₈

—

226

—

110

C₂×C₄

—

111

C₂×C₄

—

113

—

112

C₂×C₄

—

113

—

113

C₂×C₄

—

113

—

114

C₂×C₄

—

113

—

115

C₂×C₄

—

452

—

116

C₂×C₄

—

452

—

117

C₂×C₄

—

452

—

118

C₂×C₄

—

452

—

119

C₂×C₄

—

226

—

120

C₂×C₄

—

226

—

121

C₂×C₄

C₂²⋊#90673

yes

—

122

C₂×C₄

C₂²×#90673

yes

characteristic

123

C₈

—

226

—

124

C₂²

—

113

—

125

C₂²

—

113

—

126

C₂²

—

113

—

127

C₂²

C₂×C₂²⋊#90673

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

113

—

131

C₄

—

904

—

132

C₄

—

452

—

133

C₄

—

226

—

134

C₄

D₄×#90673

yes

—

135

C₂

—

113

—

136

C₂

—

113

—

137

C₂

#20299550

yes

characteristic

138

C₂

#20299591

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#90673
commutator	C₂×C₄×C₁₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#90673
Fitting	C₄⋊Q₈×C₁₁₃	C₄
socle	C₂²×C₁₁₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₁₃	C₂×D₄
cosocle	C₂×C₄×D₁₁₃	C₂³
hypercenter	C₄⋊Q₈	#90673
hypocenter	C₁₁₃	#1960
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#90673

commutator

C₂×C₄×C₁₁₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#90673

Fitting

C₄⋊Q₈×C₁₁₃

C₄

socle

C₂²×C₁₁₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₁₃

C₂×D₄

cosocle

C₂×C₄×D₁₁₃

C₂³

C₄⋊Q₈

C₁₁₃

G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
11. #16813622	C₄	semidirect
13. #16813620	C₄	semidirect
16. C₄⋊Q₈×C₁₁₃	C₄	semidirect
70. C₁₁₃	#1960	semidirect
80. C₄⋊Q₈	#90673	semidirect

H ≅ G/N

type

11. #16813622

C₄

semidirect

13. #16813620

C₄

semidirect

16. C₄⋊Q₈×C₁₁₃

C₄

semidirect

70. C₁₁₃

#1960

semidirect

80. C₄⋊Q₈

#90673

semidirect

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

subgroup	index
C₂×#16813685	2
#20299557	2
#20299542	2
C₄⋊Q₈×D₁₁₃	2
#1960	113

subgroup

index

C₄⋊Q₈×D₁₁₃

113

Subgroups — #28406705

Conjugacy classes of subgroups

Complements

Maximal subgroups