Group #28399508 — subgroups

Subgroups — #28399508

Order 14464.

364 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 197 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	#20299155	C₂	1	1	yes	maximal, characteristic
2	#20298728	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₂₂₆	C₂	1	4	yes	maximal, complemented
4	#20298624	C₂	1	2	yes	maximal
5	#20298622	C₂	1	1	yes	maximal, characteristic
6	#20298553	C₂	1	2	yes	maximal
7	C₂×C₄×C₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#14268631	C₂²	1	2	yes	—
9	#16813622	C₄	1	2	yes	—
10	#16813614	C₄	1	2	yes	—
11	#16813601	C₄	1	4	yes	—
12	C₄×Dic₂₂₆	C₂²	1	4	yes	—
13	#16813581	C₂²	1	2	yes	—
14	C₄⋊#11744751	C₂²	1	4	yes	—
15	#16813551	C₂²	1	8	yes	—
16	#16813532	C₂²	1	4	yes	—
17	C₈×Dic₁₁₃	C₂²	1	4	yes	—
18	C₂²×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₁₁₃	C₂²	1	2	yes	—
22	C₄×C₈×C₁₁₃	C₂²	1	2	yes	—
23	C₂×Dic₂₂₆	C₂×C₄	1	4	yes	—
24	#14268631	—	2	4	no	—
25	C₂×#11744751	C₂³	1	4	yes	—
26	C₄⋊Dic₁₁₃	C₂×C₄	1	4	yes	—
27	C₄⋊Dic₁₁₃	C₂×C₄	1	4	yes	—
28	#14268626	C₂×C₄	1	8	yes	—
29	C₂²⋊Dic₁₁₃	C₂×C₄	1	4	yes	—
30	C₄×Dic₁₁₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₁₃	—	2	2	no	—
34	C₂×C₈×C₁₁₃	Q₈	1	4	yes	complemented
35	C₂×C₈×C₁₁₃	C₂³	1	2	yes	—
36	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
38	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₁₁₃	C₂³	1	2	yes	—
41	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₁₃	—	2	4	no	—
43	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₁₃	#41	1	2	yes	—
48	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₁₁₃	C₂²×C₄	1	8	yes	—
50	Dic₂₂₆	—	2	8	no	—
51	C₈×C₁₁₃	—	2	2	no	—
52	C₈×C₁₁₃	C₂×Q₈	1	2	yes	—
53	C₈×C₁₁₃	C₂×Q₈	1	2	yes	—
54	#11744751	—	2	4	no	—
55	C₂²×C₁₁₃	—	2	2	no	—
56	C₂²×C₁₁₃	C₄×Q₈	1	2	yes	—
57	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₁₁₃	—	2	2	no	—
59	C₄×C₁₁₃	C₄×Q₈	1	2	yes	—
60	C₄×C₁₁₃	#131	1	4	yes	—
61	C₄×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
63	Dic₁₁₃	—	2	8	no	—
64	C₂×C₁₁₃	—	2	2	no	—
65	C₂×C₁₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₁₁₃	C₂×#131	1	2	yes	—
67	#2960	—	113	1	no	maximal, Sylow
68	C₁₁₃	#2960	1	1	yes	characteristic, complemented, Sylow
69	#533	—	113	1	no	—
70	#446	—	113	4	no	—
71	C₈×Q₈	—	113	4	no	—
72	#424	—	113	2	no	—
73	#423	—	113	1	no	—
74	#405	—	113	2	no	—
75	C₂×C₄×C₈	D₁₁₃	1	1	yes	characteristic
76	C₂×M₁₆	—	113	2	no	—
77	C₂²×C₈	C₂×D₁₁₃	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₁₁₃	1	1	yes	characteristic
79	C₄⋊Q₈	—	113	2	no	—
80	#125	—	113	2	no	—
81	C₂²⋊Q₈	—	113	4	no	—
82	C₄×Q₈	—	113	4	no	—
83	#117	—	113	2	no	—
84	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
85	C₄⋊C₈	—	113	8	no	—
86	C₄⋊C₈	—	113	4	no	—
87	#97	—	113	4	no	—
88	C₄×C₈	—	113	4	no	—
89	C₄×C₈	C₂×D₁₁₃	1	2	yes	—
90	C₄×C₈	C₂×D₁₁₃	1	2	yes	—
91	C₂×Q₈	—	113	4	no	—
92	C₂²×C₄	C₄×D₁₁₃	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
94	M₁₆	—	226	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	113	4	no	—
97	C₂×C₈	Dic₂₂₆	1	4	yes	complemented
98	C₂×C₈	C₂²×D₁₁₃	1	2	yes	—
99	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
100	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
101	C₄⋊C₄	—	113	4	no	—
102	C₄⋊C₄	—	113	8	no	—
103	C₄⋊C₄	—	113	4	no	—
104	C₂²⋊C₄	—	113	4	no	—
105	C₄²	—	113	4	no	—
106	C₄²	C₄×D₁₁₃	1	1	yes	characteristic
107	C₄²	C₄×D₁₁₃	1	1	yes	characteristic
108	C₄²	C₂²×D₁₁₃	1	2	yes	—
109	C₂³	C₂×C₄×D₁₁₃	1	1	yes	characteristic
110	Q₈	—	226	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	113	8	no	—
113	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
116	C₂×C₄	C₂×Dic₂₂₆	1	2	yes	—
117	C₂×C₄	#14268647	1	2	yes	—
118	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	226	4	no	—
121	C₈	C₂×Dic₂₂₆	1	2	yes	—
122	C₈	C₂×Dic₂₂₆	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	C₄×Dic₂₂₆	1	2	yes	—
125	C₂²	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	226	8	no	—
128	C₄	C₄×Dic₂₂₆	1	2	yes	—
129	C₄	#16813628	1	4	yes	—
130	C₄	C₂²×Dic₂₂₆	1	1	yes	characteristic
131	C₄	C₂×#14268647	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×C₄×Dic₂₂₆	1	1	yes	characteristic
134	C₂	C₂×#16813628	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20299155

C₂

yes

maximal, characteristic

#20298728

C₂

yes

maximal, complemented

C₈×Dic₂₂₆

C₂

yes

maximal, complemented

#20298624

C₂

yes

maximal

#20298622

C₂

yes

maximal, characteristic

#20298553

C₂

yes

maximal

C₂×C₄×C₈×C₁₁₃

C₂

yes

maximal, characteristic

C₂×#14268631

C₂²

yes

—

#16813622

C₄

yes

—

#16813614

C₄

yes

—

#16813601

C₄

yes

—

C₄×Dic₂₂₆

C₂²

yes

—

#16813581

C₂²

yes

—

C₄⋊#11744751

C₂²

yes

—

#16813551

C₂²

yes

—

#16813532

C₂²

yes

—

C₈×Dic₁₁₃

C₂²

yes

—

C₂²×C₈×C₁₁₃

C₂²

yes

characteristic

C₂²×C₈×C₁₁₃

C₂²

yes

characteristic

C₂×C₄²×C₁₁₃

C₂²

yes

characteristic

C₄×C₈×C₁₁₃

C₂²

yes

—

C₄×C₈×C₁₁₃

C₂²

yes

—

C₂×Dic₂₂₆

C₂×C₄

yes

—

#14268631

—

C₂×#11744751

C₂³

yes

—

C₄⋊Dic₁₁₃

C₂×C₄

yes

—

C₄⋊Dic₁₁₃

C₂×C₄

yes

—

#14268626

C₂×C₄

yes

—

C₂²⋊Dic₁₁₃

C₂×C₄

yes

—

C₄×Dic₁₁₃

C₂³

yes

—

C₂²×C₄×C₁₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×C₈×C₁₁₃

—

C₂×C₈×C₁₁₃

Q₈

yes

complemented

C₂×C₈×C₁₁₃

C₂³

yes

—

C₂×C₈×C₁₁₃

C₂³

yes

characteristic

C₂×C₈×C₁₁₃

C₂³

yes

characteristic

C₄²×C₁₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₁₃

C₂×C₄

yes

characteristic

C₄²×C₁₁₃

C₂³

yes

—

C₂³×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₁₃

#41

yes

—

C₂×C₄×C₁₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₁₃

C₂²×C₄

yes

—

Dic₂₂₆

—

C₈×C₁₁₃

—

C₈×C₁₁₃

C₂×Q₈

yes

—

C₈×C₁₁₃

C₂×Q₈

yes

—

#11744751

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₄×Q₈

yes

—

C₂²×C₁₁₃

C₂³×C₄

yes

characteristic

C₄×C₁₁₃

—

C₄×C₁₁₃

C₄×Q₈

yes

—

C₄×C₁₁₃

#131

yes

—

C₄×C₁₁₃

C₂²×Q₈

yes

characteristic

C₄×C₁₁₃

C₂×#41

yes

characteristic

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₁₃

C₂×#131

yes

—

#2960

—

113

maximal, Sylow

C₁₁₃

#2960

yes

characteristic, complemented, Sylow

#533

—

113

—

#446

—

113

—

C₈×Q₈

—

113

—

#424

—

113

—

#423

—

113

—

#405

—

113

—

C₂×C₄×C₈

D₁₁₃

yes

characteristic

C₂×M₁₆

—

113

—

C₂²×C₈

C₂×D₁₁₃

yes

characteristic

C₂²×C₈

C₂×D₁₁₃

yes

characteristic

C₄⋊Q₈

—

113

—

#125

—

113

—

C₂²⋊Q₈

—

113

—

C₄×Q₈

—

113

—

#117

—

113

—

C₂×C₄²

C₂×D₁₁₃

yes

characteristic

C₄⋊C₈

—

113

—

C₄⋊C₈

—

113

—

#97

—

113

—

C₄×C₈

—

113

—

C₄×C₈

C₂×D₁₁₃

yes

—

C₄×C₈

C₂×D₁₁₃

yes

—

C₂×Q₈

—

113

—

C₂²×C₄

C₄×D₁₁₃

yes

—

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

M₁₆

—

226

—

C₂×C₈

—

C₂×C₈

—

113

—

C₂×C₈

Dic₂₂₆

yes

complemented

C₂×C₈

C₂²×D₁₁₃

yes

—

C₂×C₈

C₂²×D₁₁₃

yes

characteristic

100

C₂×C₈

C₂²×D₁₁₃

yes

characteristic

101

C₄⋊C₄

—

113

—

102

C₄⋊C₄

—

113

—

103

C₄⋊C₄

—

113

—

104

C₂²⋊C₄

—

113

—

105

C₄²

—

113

—

106

C₄²

C₄×D₁₁₃

yes

characteristic

107

C₄²

C₄×D₁₁₃

yes

characteristic

108

C₄²

C₂²×D₁₁₃

yes

—

109

C₂³

C₂×C₄×D₁₁₃

yes

characteristic

110

Q₈

—

226

—

111

C₂×C₄

—

112

C₂×C₄

—

113

—

113

C₂×C₄

C₂×C₄×D₁₁₃

yes

—

114

C₂×C₄

C₂×C₄×D₁₁₃

yes

characteristic

115

C₂×C₄

C₂×C₄×D₁₁₃

yes

—

116

C₂×C₄

C₂×Dic₂₂₆

yes

—

117

C₂×C₄

#14268647

yes

—

118

C₂×C₄

C₂³×D₁₁₃

yes

characteristic

119

C₈

—

120

C₈

—

226

—

121

C₈

C₂×Dic₂₂₆

yes

—

122

C₈

C₂×Dic₂₂₆

yes

—

123

C₂²

—

124

C₂²

C₄×Dic₂₂₆

yes

—

125

C₂²

C₂²×C₄×D₁₁₃

yes

characteristic

126

C₄

—

127

C₄

—

226

—

128

C₄

C₄×Dic₂₂₆

yes

—

129

C₄

#16813628

yes

—

130

C₄

C₂²×Dic₂₂₆

yes

characteristic

131

C₄

C₂×#14268647

yes

characteristic

132

C₂

—

133

C₂

C₂×C₄×Dic₂₂₆

yes

characteristic

134

C₂

C₂×#16813628

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₁₃
Fitting	C₂×C₄×C₈×C₁₁₃	C₂
prime radical	C₂³×C₁₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₁₃
hypocenter	C₁₁₃	#2960
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂²×D₁₁₃

commutator, socle

C₂²×C₁₁₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₁₃

Fitting

C₂×C₄×C₈×C₁₁₃

C₂

prime radical

C₂³×C₁₁₃

C₂²×C₄

cosocle

C₂×C₄×C₁₁₃

C₂⁴

hypercenter

C₂×C₄×C₈

D₁₁₃

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #20298728	C₂	semidirect
3. C₈×Dic₂₂₆	C₂	semidirect
34. C₂×C₈×C₁₁₃	Q₈	semidirect
68. C₁₁₃	#2960	semidirect
97. C₂×C₈	Dic₂₂₆	semidirect

H ≅ G/N

type

2. #20298728

C₂

semidirect

3. C₈×Dic₂₂₆

C₂

semidirect

34. C₂×C₈×C₁₁₃

Q₈

semidirect

68. C₁₁₃

#2960

semidirect

97. C₂×C₈

Dic₂₂₆

semidirect

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

subgroup	index
#20299155	2
#20298728	2
C₈×Dic₂₂₆	2
#20298624	2
#20298622	2
#20298553	2
C₂×C₄×C₈×C₁₁₃	2
#2960	113

subgroup

index

C₈×Dic₂₂₆

C₂×C₄×C₈×C₁₁₃

113

Subgroups — #28399508

Conjugacy classes of subgroups

Complements

Maximal subgroups