Group #28364576 — subgroups

Subgroups — #28364576

Order 13952.

254 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 91 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×#16812630	C₂	1	2	yes	maximal
2	C₂×#16812622	C₂	1	2	yes	maximal
3	#20294932	C₂	1	1	yes	maximal, characteristic
4	#20294781	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic
7	C₂²×#11724784	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
9	#16812630	—	2	4	no	—
10	#16812622	—	2	4	no	—
11	#16812606	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₁₈	—	2	2	no	—
16	C₂×Dic₂₁₈	—	2	2	no	—
17	C₂×Dic₂₁₈	—	2	1	no	—
18	C₂×Dic₂₁₈	Q₈	1	2	yes	—
19	C₂²×Dic₁₀₉	—	2	1	no	—
20	C₂²×Dic₁₀₉	—	2	1	no	—
21	C₂×#11724784	—	2	2	no	—
22	C₂×#11724784	—	2	2	no	—
23	C₂×#11724784	D₄	1	4	yes	—
24	C₄⋊Dic₁₀₉	—	2	2	no	—
25	C₂²×C₄×C₁₀₉	—	2	2	no	—
26	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
27	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₉	—	2	2	no	—
29	C₄⋊C₄×C₁₀₉	—	2	2	no	—
30	C₄⋊C₄×C₁₀₉	—	4	4	no	—
31	C₄²×C₁₀₉	—	2	2	no	—
32	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄×C₁₀₉	—	2	2	no	—
35	C₂×C₄×C₁₀₉	—	4	2	no	—
36	C₂×C₄×C₁₀₉	—	4	4	no	—
37	C₂×C₄×C₁₀₉	SD₁₆	1	4	yes	—
38	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	#41	1	2	yes	—
41	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
42	C₂×Dic₁₀₉	—	2	2	no	—
43	C₂×Dic₁₀₉	—	4	1	no	—
44	C₂×Dic₁₀₉	—	4	1	no	—
45	C₂×Dic₁₀₉	—	4	1	no	—
46	C₂×Dic₁₀₉	—	4	2	no	—
47	Dic₂₁₈	—	2	4	no	—
48	Dic₂₁₈	—	4	2	no	—
49	#11724784	—	2	4	no	—
50	C₂²×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₀₉	#123	1	1	yes	characteristic
52	C₂²×C₁₀₉	#127	1	1	yes	characteristic
53	C₂²×C₁₀₉	C₂×SD₁₆	1	2	yes	—
54	C₂²×C₁₀₉	#137	1	2	yes	—
55	C₄×C₁₀₉	—	2	4	no	—
56	C₄×C₁₀₉	—	8	2	no	—
57	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₁₀₉	#123	1	2	yes	—
59	Dic₁₀₉	—	4	2	no	—
60	Dic₁₀₉	—	8	1	no	—
61	C₂×C₁₀₉	#399	1	1	yes	characteristic
62	C₂×C₁₀₉	#475	1	2	yes	—
63	C₂×C₁₀₉	#483	1	2	yes	—
64	C₂×C₁₀₉	#492	1	1	yes	characteristic
65	C₂×C₁₀₉	#497	1	1	yes	characteristic
66	#2096	—	109	1	no	maximal, Sylow
67	C₁₀₉	#2096	1	1	yes	characteristic, complemented, Sylow
68	C₂×#106	—	109	2	no	—
69	C₂×#103	—	109	2	no	—
70	#391	—	109	1	no	—
71	#384	D₁₀₉	1	1	yes	characteristic
72	#336	—	109	1	no	—
73	C₂²×Q₈	—	109	1	no	—
74	C₂²×C₈	—	109	2	no	—
75	C₂×C₄⋊C₄	—	2	2	no	—
76	C₂×C₄⋊C₄	—	109	1	no	—
77	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
78	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
79	#106	—	218	4	no	—
80	#103	—	218	4	no	—
81	#95	—	218	2	no	—
82	C₂×Q₈	—	109	2	no	—
83	C₂×Q₈	—	218	2	no	—
84	C₂×Q₈	—	218	2	no	—
85	C₂×Q₈	—	218	1	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	218	1	no	—
88	C₂²×C₄	—	218	1	no	—
89	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
91	C₂×C₈	—	109	4	no	—
92	C₂×C₈	—	218	2	no	—
93	C₂×C₈	—	218	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	4	4	no	—
97	C₄⋊C₄	—	218	2	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
100	Q₈	—	436	2	no	—
101	Q₈	—	218	4	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	436	1	no	—
107	C₂×C₄	—	436	2	no	—
108	C₂×C₄	—	436	1	no	—
109	C₂×C₄	—	436	1	no	—
110	C₂×C₄	—	218	2	no	—
111	C₂×C₄	#13182017	1	4	yes	—
112	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
113	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
114	C₂×C₄	#13182029	1	2	yes	—
115	C₂×C₄	#13182030	1	1	yes	characteristic
116	C₈	—	218	4	no	—
117	C₂²	#16812675	1	1	yes	characteristic
118	C₂²	#16812682	1	1	yes	characteristic
119	C₂²	#16812695	1	1	yes	characteristic
120	C₂²	C₂×#13182017	1	2	yes	—
121	C₂²	#16812730	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	8	2	no	—
124	C₄	—	872	1	no	—
125	C₄	—	436	2	no	—
126	C₄	#16812677	1	2	yes	—
127	C₄	#16812684	1	2	yes	—
128	C₂	#20294961	1	1	yes	characteristic
129	C₂	#20295308	1	2	yes	—
130	C₂	#20295356	1	2	yes	—
131	C₂	#20295398	1	1	yes	characteristic
132	C₂	#20295418	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812630

C₂

yes

maximal

C₂×#16812622

C₂

yes

maximal

#20294932

C₂

yes

maximal, characteristic

#20294781

C₂

yes

maximal, characteristic

#384×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

characteristic

C₂²×#11724784

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

—

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

Q₈

yes

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

D₄

yes

—

C₄⋊Dic₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724784

—

C₂²×C₁₀₉

C₂²⋊Q₈

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

C₂×SD₁₆

yes

—

C₂²×C₁₀₉

#137

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

#123

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#399

yes

characteristic

C₂×C₁₀₉

#475

yes

—

C₂×C₁₀₉

#483

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2096

yes

characteristic, complemented, Sylow

C₂×#106

—

109

—

C₂×#103

—

109

—

#391

—

109

—

#384

D₁₀₉

yes

characteristic

#336

—

109

—

C₂²×Q₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#106

—

218

—

#103

—

218

—

#95

—

218

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂²×C₄

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₄²

—

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

100

Q₈

—

436

—

101

Q₈

—

218

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

436

—

107

C₂×C₄

—

436

—

108

C₂×C₄

—

436

—

109

C₂×C₄

—

436

—

110

C₂×C₄

—

218

—

111

C₂×C₄

#13182017

yes

—

112

C₂×C₄

D₄×D₁₀₉

yes

—

113

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

114

C₂×C₄

#13182029

yes

—

115

C₂×C₄

#13182030

yes

characteristic

116

C₈

—

218

—

117

C₂²

#16812675

yes

characteristic

118

C₂²

#16812682

yes

characteristic

119

C₂²

#16812695

yes

characteristic

120

C₂²

C₂×#13182017

yes

—

121

C₂²

#16812730

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

872

—

125

C₄

—

436

—

126

C₄

#16812677

yes

—

127

C₄

#16812684

yes

—

128

C₂

#20294961

yes

characteristic

129

C₂

#20295308

yes

—

130

C₂

#20295356

yes

—

131

C₂

#20295398

yes

characteristic

132

C₂

#20295418

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator, cosocle	C₂²×C₄×C₁₀₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#384×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	#384	D₁₀₉
hypocenter	C₁₀₉	#2096
LCS	G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator, cosocle

C₂²×C₄×C₁₀₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#384×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
67. C₁₀₉	#2096	semidirect

H ≅ G/N

type

67. C₁₀₉

#2096

semidirect

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

subgroup	index
C₂×#16812630	2
C₂×#16812622	2
#20294932	2
#20294781	2
#384×C₁₀₉	2
#2096	109

subgroup

index

#384×C₁₀₉

109

Subgroups — #28364576

Conjugacy classes of subgroups

Complements

Maximal subgroups