Group #28362411 — subgroups

Subgroups — #28362411

Order 13952.

150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20295318	C₂	1	1	yes	maximal, characteristic
2	#20295233	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic
4	#20294777	C₂	1	1	yes	maximal, characteristic
5	#20294775	C₂	1	1	yes	maximal, characteristic
6	#20294772	C₂	1	1	yes	maximal, characteristic
7	#498×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182019	—	2	1	no	—
9	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
10	#16812632	—	2	1	no	—
11	C₄⋊#11724784	C₂²	1	1	yes	characteristic
12	C₄⋊#11724784	C₂²	1	1	yes	characteristic
13	C₄⋊#11724784	C₂²	1	1	yes	characteristic
14	#16812628	—	2	1	no	—
15	#16812623	—	2	1	no	—
16	#16812622	—	2	1	no	—
17	C₈×Dic₁₀₉	—	2	1	no	—
18	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	#106×C₁₀₉	—	2	1	no	—
21	#106×C₁₀₉	—	2	1	no	—
22	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
24	#13182019	—	4	2	no	—
25	C₂×#11724784	—	2	1	no	—
26	C₂×#11724784	—	2	1	no	—
27	C₂×#11724784	—	2	1	no	—
28	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
29	#13182008	—	2	1	no	—
30	C₄×Dic₁₀₉	—	2	1	no	—
31	C₂×Q₈×C₁₀₉	—	2	1	no	—
32	C₂×Q₈×C₁₀₉	—	2	1	no	—
33	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₁₀₉	—	2	1	no	—
36	C₄⋊C₄×C₁₀₉	—	2	1	no	—
37	C₄⋊C₄×C₁₀₉	—	2	1	no	—
38	C₄⋊C₄×C₁₀₉	—	2	1	no	—
39	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
40	Q₈×C₁₀₉	—	4	1	no	—
41	Q₈×C₁₀₉	—	4	1	no	—
42	Q₈×C₁₀₉	—	8	1	no	—
43	C₂×C₄×C₁₀₉	—	4	1	no	—
44	C₂×C₄×C₁₀₉	—	4	1	no	—
45	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
48	C₂×Dic₁₀₉	—	2	1	no	—
49	C₂×Dic₁₀₉	—	2	1	no	—
50	Dic₂₁₈	—	2	1	no	—
51	Dic₂₁₈	SD₁₆	1	2	yes	—
52	C₈×C₁₀₉	—	2	1	no	—
53	C₈×C₁₀₉	Q₁₆	1	2	yes	complemented
54	#11724784	—	4	1	no	—
55	#11724784	—	4	1	no	—
56	#11724784	—	4	1	no	—
57	C₂²×C₁₀₉	#121	1	1	yes	characteristic
58	C₄×C₁₀₉	—	2	1	no	—
59	C₄×C₁₀₉	—	8	1	no	—
60	C₄×C₁₀₉	—	8	1	no	—
61	C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₁₀₉	#135	1	1	yes	characteristic
64	C₄×C₁₀₉	#137	1	1	yes	characteristic
65	Dic₁₀₉	—	2	2	no	—
66	Dic₁₀₉	—	4	1	no	—
67	C₂×C₁₀₉	#462	1	1	yes	characteristic
68	C₂×C₁₀₉	#465	1	1	yes	characteristic
69	C₂×C₁₀₉	#471	1	1	yes	characteristic
70	#1672	—	109	1	no	maximal, Sylow
71	C₁₀₉	#1672	1	1	yes	characteristic, complemented, Sylow
72	#498	D₁₀₉	1	1	yes	characteristic
73	#477	—	109	1	no	—
74	#462	—	109	1	no	—
75	C₈×Q₈	—	109	1	no	—
76	#334	—	109	1	no	—
77	#333	—	109	1	no	—
78	#332	—	109	1	no	—
79	C₂×Q₁₆	—	218	1	no	—
80	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
81	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
82	C₄×Q₈	—	109	1	no	—
83	#107	—	218	1	no	—
84	#106	—	2	1	no	—
85	#106	—	2	1	no	—
86	C₄⋊C₈	—	109	1	no	—
87	C₄⋊C₈	—	109	1	no	—
88	C₄⋊C₈	—	109	1	no	—
89	C₄⋊C₈	—	218	1	no	—
90	#103	—	218	1	no	—
91	#103	—	218	1	no	—
92	C₄×C₈	—	218	1	no	—
93	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	109	1	no	—
97	Q₁₆	—	436	2	no	—
98	C₂×C₈	—	218	1	no	—
99	C₂×C₈	—	218	1	no	—
100	C₂×C₈	—	218	1	no	—
101	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	109	1	no	—
108	C₄⋊C₄	—	218	1	no	—
109	C₄²	—	218	1	no	—
110	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	8	1	no	—
114	Q₈	—	109	2	no	—
115	Q₈	—	218	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	218	1	no	—
119	C₂×C₄	—	218	1	no	—
120	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
122	C₂×C₄	#13182028	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	436	1	no	—
125	C₈	—	436	1	no	—
126	C₈	—	436	1	no	—
127	C₈	#13182019	1	2	yes	complemented
128	C₂²	#16812670	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	8	1	no	—
132	C₄	—	436	1	no	—
133	C₄	—	218	2	no	—
134	C₄	SD₁₆×D₁₀₉	1	1	yes	characteristic
135	C₄	C₂×#13182019	1	1	yes	characteristic
136	C₄	#16812720	1	1	yes	characteristic
137	C₄	#16812730	1	1	yes	characteristic
138	C₂	#20295233	1	1	yes	characteristic
139	C₂	#20295252	1	1	yes	characteristic
140	C₂	#20295286	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295318

C₂

yes

maximal, characteristic

#20295233

C₂

yes

maximal, characteristic

C₈×Dic₂₁₈

C₂

yes

maximal, characteristic

#20294777

C₂

yes

maximal, characteristic

#20294775

C₂

yes

maximal, characteristic

#20294772

C₂

yes

maximal, characteristic

#498×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182019

—

C₄×Dic₂₁₈

C₂²

yes

characteristic

#16812632

—

C₄⋊#11724784

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

—

—

—

C₈×Dic₁₀₉

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

#106×C₁₀₉

—

#106×C₁₀₉

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

D₄

yes

—

—

—

—

C₄⋊Dic₁₀₉

D₄

yes

characteristic

#13182008

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

SD₁₆

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

Q₁₆

yes

—

—

—

C₂²×C₁₀₉

#121

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₄×C₁₀₉

C₂×Q₁₆

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1672

yes

characteristic, complemented, Sylow

#498

D₁₀₉

yes

characteristic

#477

—

109

—

#462

—

109

—

C₈×Q₈

—

109

—

#334

—

109

—

#333

—

109

—

#332

—

109

—

C₂×Q₁₆

—

218

—

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

C₄×Q₈

—

109

—

#107

—

218

—

#106

—

#106

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

#103

—

218

—

#103

—

218

—

C₄×C₈

—

218

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

109

—

Q₁₆

—

436

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

100

C₂×C₈

—

218

—

101

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

102

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

109

—

108

C₄⋊C₄

—

218

—

109

C₄²

—

218

—

110

C₄²

C₂²×D₁₀₉

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

109

—

115

Q₈

—

218

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

218

—

119

C₂×C₄

—

218

—

120

C₂×C₄

D₄×D₁₀₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

122

C₂×C₄

#13182028

yes

characteristic

123

C₈

—

124

C₈

—

436

—

125

C₈

—

436

—

126

C₈

—

436

—

127

C₈

#13182019

yes

complemented

128

C₂²

#16812670

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

436

—

133

C₄

—

218

—

134

C₄

SD₁₆×D₁₀₉

yes

characteristic

135

C₄

C₂×#13182019

yes

characteristic

136

C₄

#16812720

yes

characteristic

137

C₄

#16812730

yes

characteristic

138

C₂

#20295233

yes

characteristic

139

C₂

#20295252

yes

characteristic

140

C₂

#20295286

yes

characteristic

	subgroup	G/H
center	C₂²	#16812670
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#498×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#121
hypercenter	#498	D₁₀₉
hypocenter	C₁₀₉	#1672
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#16812670

commutator, cosocle

C₄²×C₁₀₉

C₂³

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#498×C₁₀₉

C₂

C₂²×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
53. C₈×C₁₀₉	Q₁₆	semidirect
71. C₁₀₉	#1672	semidirect
127. C₈	#13182019	semidirect

H ≅ G/N

type

53. C₈×C₁₀₉

Q₁₆

semidirect

71. C₁₀₉

#1672

semidirect

127. C₈

#13182019

semidirect

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

subgroup	index
#20295318	2
#20295233	2
C₈×Dic₂₁₈	2
#20294777	2
#20294775	2
#20294772	2
#498×C₁₀₉	2
#1672	109

subgroup

index

C₈×Dic₂₁₈

#498×C₁₀₉

109

Subgroups — #28362411

Conjugacy classes of subgroups

Complements

Maximal subgroups