Group #28361849 — subgroups

Subgroups — #28361849

Order 13952.

160 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₈×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic
2	C₄×#13182019	C₂	1	1	yes	maximal, characteristic
3	#20294776	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#11724784	C₂	1	1	yes	maximal, characteristic
5	#20294756	C₂	1	1	yes	maximal, characteristic
6	#20294748	C₂	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182019	C₄	1	1	yes	characteristic
9	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
10	#16812632	C₄	1	1	yes	characteristic
11	C₄⋊#11724784	C₂²	1	1	yes	characteristic
12	#16812628	—	2	1	no	—
13	#16812623	C₄	1	1	yes	characteristic
14	#16812622	C₄	1	1	yes	characteristic
15	C₄×#11724784	C₂²	1	1	yes	characteristic
16	C₄×#11724784	C₂²	1	1	yes	characteristic
17	C₈×Dic₁₀₉	—	2	1	no	—
18	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₁₀₉	—	2	1	no	—
20	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₁₀₉	—	2	1	no	—
22	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
24	#13182019	C₈	1	4	yes	complemented
25	C₂×#11724784	—	2	1	no	—
26	C₂×#11724784	—	2	1	no	—
27	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
28	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
29	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
30	#13182008	—	2	1	no	—
31	C₄×Dic₁₀₉	—	2	1	no	—
32	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₀₉	—	2	1	no	—
34	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₁₀₉	—	2	1	no	—
37	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₁₀₉	—	2	1	no	—
39	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
40	Q₈×C₁₀₉	—	2	1	no	—
41	Q₈×C₁₀₉	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₁₀₉	—	2	1	no	—
43	C₂×C₄×C₁₀₉	—	2	1	no	—
44	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
47	C₂×Dic₁₀₉	—	2	1	no	—
48	C₂×Dic₁₀₉	—	2	1	no	—
49	Dic₂₁₈	—	2	1	no	—
50	Dic₂₁₈	C₂×C₈	1	2	yes	—
51	C₈×C₁₀₉	—	2	1	no	—
52	C₈×C₁₀₉	—	2	1	no	—
53	C₈×C₁₀₉	—	4	1	no	—
54	#11724784	—	2	1	no	—
55	#11724784	—	4	1	no	—
56	#11724784	—	4	1	no	—
57	#11724784	C₂×C₈	1	2	yes	—
58	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
59	C₄×C₁₀₉	—	2	2	no	—
60	C₄×C₁₀₉	—	2	1	no	—
61	C₄×C₁₀₉	—	4	1	no	—
62	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
63	C₄×C₁₀₉	#131	1	1	yes	characteristic
64	C₄×C₁₀₉	#137	1	1	yes	characteristic
65	C₄×C₁₀₉	#137	1	1	yes	characteristic
66	Dic₁₀₉	—	2	2	no	—
67	Dic₁₀₉	—	4	1	no	—
68	C₂×C₁₀₉	C₈×D₄	1	1	yes	characteristic
69	C₂×C₁₀₉	#441	1	1	yes	characteristic
70	C₂×C₁₀₉	#444	1	1	yes	characteristic
71	#1579	—	109	1	no	maximal, Sylow
72	C₁₀₉	#1579	1	1	yes	characteristic, complemented, Sylow
73	C₈×Q₈	—	109	1	no	—
74	C₈×Q₈	D₁₀₉	1	1	yes	characteristic
75	C₄×Q₁₆	—	109	1	no	—
76	#334	—	109	1	no	—
77	Q₈⋊C₈	—	109	1	no	—
78	Q₈⋊C₈	—	109	1	no	—
79	#322	—	109	1	no	—
80	C₂×Q₁₆	—	109	1	no	—
81	C₄×Q₈	—	109	1	no	—
82	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
83	#107	—	109	1	no	—
84	C₄⋊C₈	—	2	1	no	—
85	C₄⋊C₈	—	109	1	no	—
86	C₄⋊C₈	—	218	1	no	—
87	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
88	#103	—	109	1	no	—
89	#103	—	109	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	—	109	1	no	—
92	C₄×C₈	—	109	1	no	—
93	C₄×C₈	—	218	1	no	—
94	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
95	C₂×Q₈	—	109	1	no	—
96	C₂×Q₈	C₄×D₁₀₉	1	1	yes	characteristic
97	Q₁₆	—	109	4	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	109	1	no	—
100	C₂×C₈	—	109	1	no	—
101	C₂×C₈	—	218	1	no	—
102	C₂×C₈	—	218	1	no	—
103	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
104	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	109	1	no	—
107	C₄⋊C₄	—	218	1	no	—
108	C₄⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	—	218	1	no	—
111	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	109	2	no	—
114	Q₈	—	218	1	no	—
115	Q₈	C₈×D₁₀₉	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	218	1	no	—
119	C₂×C₄	—	218	1	no	—
120	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
122	C₂×C₄	#13182029	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	1	no	—
125	C₈	—	4	1	no	—
126	C₈	—	109	2	no	—
127	C₈	—	436	1	no	—
128	C₈	—	436	1	no	—
129	C₈	—	218	1	no	—
130	C₂²	C₄×C₂²⋊D₁₀₉	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	436	1	no	—
135	C₄	—	218	2	no	—
136	C₄	C₂×C₈×D₁₀₉	1	1	yes	characteristic
137	C₄	#16812705	1	1	yes	characteristic
138	C₄	#16812729	1	1	yes	characteristic
139	C₄	#16812730	1	1	yes	characteristic
140	C₂	C₈×C₂²⋊D₁₀₉	1	1	yes	characteristic
141	C₂	#20295131	1	1	yes	characteristic
142	C₂	#20295146	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₂₁₈

C₂

yes

maximal, characteristic

C₄×#13182019

C₂

yes

maximal, characteristic

#20294776

C₂

yes

maximal, characteristic

Q₈⋊#11724784

C₂

yes

maximal, characteristic

#20294756

C₂

yes

maximal, characteristic

#20294748

C₂

yes

maximal, characteristic

C₈×Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182019

C₄

yes

characteristic

C₄×Dic₂₁₈

C₂²

yes

characteristic

#16812632

C₄

yes

characteristic

C₄⋊#11724784

C₂²

yes

characteristic

#16812628

—

#16812623

C₄

yes

characteristic

#16812622

C₄

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₈×Dic₁₀₉

—

C₄×Q₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

C₂×C₄

yes

characteristic

#13182019

C₈

yes

—

—

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₄⋊Dic₁₀₉

C₂×C₄

yes

characteristic

#13182008

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

C₂×C₈

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

—

—

—

C₂×C₈

yes

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₈

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

C₈×D₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1579

yes

characteristic, complemented, Sylow

C₈×Q₈

—

109

—

C₈×Q₈

D₁₀₉

yes

characteristic

C₄×Q₁₆

—

109

—

#334

—

109

—

Q₈⋊C₈

—

109

—

Q₈⋊C₈

—

109

—

#322

—

109

—

C₂×Q₁₆

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

C₂×D₁₀₉

yes

characteristic

#107

—

109

—

C₄⋊C₈

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

C₂×D₁₀₉

yes

characteristic

#103

—

109

—

#103

—

109

—

C₄×C₈

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

109

—

C₂×Q₈

C₄×D₁₀₉

yes

characteristic

Q₁₆

—

109

—

C₂×C₈

—

C₂×C₈

—

109

—

100

C₂×C₈

—

109

—

101

C₂×C₈

—

218

—

102

C₂×C₈

—

218

—

103

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

104

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

109

—

107

C₄⋊C₄

—

218

—

108

C₄⋊C₄

C₄×D₁₀₉

yes

characteristic

109

C₄²

—

110

C₄²

—

218

—

111

C₄²

C₂²×D₁₀₉

yes

characteristic

112

Q₈

—

113

Q₈

—

109

—

114

Q₈

—

218

—

115

Q₈

C₈×D₁₀₉

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

218

—

119

C₂×C₄

—

218

—

120

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

122

C₂×C₄

#13182029

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

109

—

127

C₈

—

436

—

128

C₈

—

436

—

129

C₈

—

218

—

130

C₂²

C₄×C₂²⋊D₁₀₉

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

436

—

135

C₄

—

218

—

136

C₄

C₂×C₈×D₁₀₉

yes

characteristic

137

C₄

#16812705

yes

characteristic

138

C₄

#16812729

yes

characteristic

139

C₄

#16812730

yes

characteristic

140

C₂

C₈×C₂²⋊D₁₀₉

yes

characteristic

141

C₂

#20295131

yes

characteristic

142

C₂

#20295146

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₄²	C₂²×D₁₀₉
Fitting	C₈×Q₈×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₄×D₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	C₈×Q₈	D₁₀₉
hypocenter	C₁₀₉	#1579
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₉

commutator

C₄×C₁₀₉

C₂²×C₈

Frattini

C₄²

C₂²×D₁₀₉

Fitting

C₈×Q₈×C₁₀₉

C₂

socle, prime radical

C₂²×C₁₀₉

C₄×D₄

cosocle

C₄²×C₁₀₉

C₂³

hypercenter

C₈×Q₈

D₁₀₉

hypocenter

C₁₀₉

#1579

LCS

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₈

N	H ≅ G/N	type
24. #13182019	C₈	semidirect
72. C₁₀₉	#1579	semidirect

H ≅ G/N

type

24. #13182019

C₈

semidirect

72. C₁₀₉

#1579

semidirect

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

subgroup	index
C₈×Dic₂₁₈	2
C₄×#13182019	2
#20294776	2
Q₈⋊#11724784	2
#20294756	2
#20294748	2
C₈×Q₈×C₁₀₉	2
#1579	109

subgroup

index

C₈×Dic₂₁₈

C₈×Q₈×C₁₀₉

109

Subgroups — #28361849

Conjugacy classes of subgroups

Complements

Maximal subgroups