Group #28362488 — subgroups

Subgroups — #28362488

Order 13952.

144 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 55 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	#20295376	C₂	1	1	yes	maximal, characteristic
2	#20295321	C₂	1	1	yes	maximal, characteristic
3	Q₈×#11724784	C₂	1	1	yes	maximal, characteristic
4	#20294778	C₂	1	1	yes	maximal, characteristic
5	#20294774	C₂	1	1	yes	maximal, characteristic
6	#20294772	C₂	1	1	yes	maximal, characteristic
7	#475×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#16812699	C₂²	1	1	yes	characteristic
9	#16812632	—	2	1	no	—
10	C₄⋊#11724784	—	2	1	no	—
11	C₄⋊#11724784	C₂²	1	1	yes	characteristic
12	C₄⋊#11724784	C₂²	1	1	yes	characteristic
13	#16812623	—	2	1	no	—
14	#16812623	—	2	1	no	—
15	#16812623	—	2	1	no	—
16	C₄×#11724784	—	2	1	no	—
17	C₄×#11724784	C₂²	1	1	yes	characteristic
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	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	#103×C₁₀₉	—	2	1	no	—
23	C₂×Dic₂₁₈	—	2	1	no	—
24	C₂×#11724784	—	2	1	no	—
25	C₂×#11724784	—	2	1	no	—
26	C₂×#11724784	D₄	1	1	yes	characteristic
27	C₂×#11724784	D₄	1	1	yes	characteristic
28	C₄⋊Dic₁₀₉	—	2	1	no	—
29	C₄⋊Dic₁₀₉	—	2	1	no	—
30	C₂×Q₈×C₁₀₉	—	2	1	no	—
31	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₀₉	—	2	1	no	—
33	C₄⋊C₄×C₁₀₉	—	2	1	no	—
34	C₄⋊C₄×C₁₀₉	—	2	1	no	—
35	C₄⋊C₄×C₁₀₉	—	2	1	no	—
36	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
37	C₄²×C₁₀₉	—	2	1	no	—
38	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
39	Q₈×C₁₀₉	—	2	1	no	—
40	Q₈×C₁₀₉	—	4	1	no	—
41	Q₈×C₁₀₉	—	4	1	no	—
42	Q₈×C₁₀₉	Q₁₆	1	2	yes	—
43	C₂×C₄×C₁₀₉	—	2	1	no	—
44	C₂×C₄×C₁₀₉	—	2	1	no	—
45	C₂×C₄×C₁₀₉	—	4	1	no	—
46	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
49	C₂×Dic₁₀₉	—	4	1	no	—
50	Dic₂₁₈	—	8	1	no	—
51	C₈×C₁₀₉	—	4	1	no	—
52	#11724784	—	2	1	no	—
53	#11724784	—	2	1	no	—
54	#11724784	—	4	1	no	—
55	#11724784	—	4	1	no	—
56	C₂²×C₁₀₉	#121	1	1	yes	characteristic
57	C₄×C₁₀₉	—	2	2	no	—
58	C₄×C₁₀₉	—	2	1	no	—
59	C₄×C₁₀₉	—	4	1	no	—
60	C₄×C₁₀₉	—	8	1	no	—
61	C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₁₀₉	#135	1	1	yes	characteristic
63	C₄×C₁₀₉	#135	1	1	yes	characteristic
64	C₄×C₁₀₉	#137	1	1	yes	characteristic
65	Dic₁₀₉	—	8	1	no	—
66	C₂×C₁₀₉	#464	1	1	yes	characteristic
67	C₂×C₁₀₉	#467	1	1	yes	characteristic
68	C₂×C₁₀₉	#471	1	1	yes	characteristic
69	#1684	—	109	1	no	maximal, Sylow
70	C₁₀₉	#1684	1	1	yes	characteristic, complemented, Sylow
71	#487	—	109	1	no	—
72	#477	—	109	1	no	—
73	#475	D₁₀₉	1	1	yes	characteristic
74	C₈×Q₈	—	109	1	no	—
75	#335	—	109	1	no	—
76	#333	—	109	1	no	—
77	#332	—	109	1	no	—
78	C₄⋊Q₈	—	109	1	no	—
79	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
80	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
81	#107	—	218	1	no	—
82	#106	—	2	1	no	—
83	C₄⋊C₈	—	109	1	no	—
84	C₄⋊C₈	—	109	1	no	—
85	C₄⋊C₈	—	218	1	no	—
86	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
87	#103	—	2	1	no	—
88	#103	—	218	1	no	—
89	#103	—	218	1	no	—
90	#103	—	218	1	no	—
91	C₄×C₈	—	109	1	no	—
92	C₄×C₈	—	218	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	218	1	no	—
95	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	109	1	no	—
98	C₂×C₈	—	109	1	no	—
99	C₂×C₈	—	218	1	no	—
100	C₂×C₈	—	218	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	218	1	no	—
105	C₄⋊C₄	—	218	1	no	—
106	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
107	C₄²	—	2	1	no	—
108	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
109	Q₈	—	2	1	no	—
110	Q₈	—	4	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	872	1	no	—
113	Q₈	#13182019	1	2	yes	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	436	1	no	—
118	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
120	C₂×C₄	#13182028	1	1	yes	characteristic
121	C₈	—	4	1	no	—
122	C₈	—	436	1	no	—
123	C₈	—	436	1	no	—
124	C₈	—	218	1	no	—
125	C₈	—	218	1	no	—
126	C₂²	#16812671	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	872	1	no	—
132	C₄	C₂×#13182019	1	1	yes	characteristic
133	C₄	#16812719	1	1	yes	characteristic
134	C₄	#16812720	1	1	yes	characteristic
135	C₄	#16812728	1	1	yes	characteristic
136	C₂	#20295246	1	1	yes	characteristic
137	C₂	#20295265	1	1	yes	characteristic
138	C₂	#20295291	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295376

C₂

yes

maximal, characteristic

#20295321

C₂

yes

maximal, characteristic

Q₈×#11724784

C₂

yes

maximal, characteristic

#20294778

C₂

yes

maximal, characteristic

#20294774

C₂

yes

maximal, characteristic

#20294772

C₂

yes

maximal, characteristic

#475×C₁₀₉

C₂

yes

maximal, characteristic

#16812699

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

C₄×Q₈×C₁₀₉

C₂²

yes

characteristic

#106×C₁₀₉

—

C₄⋊C₈×C₁₀₉

C₂²

yes

characteristic

#103×C₁₀₉

—

C₂×Dic₂₁₈

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

D₄

yes

characteristic

C₂×#11724784

D₄

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

Q₁₆

yes

—

C₂×C₄×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

—

Dic₂₁₈

—

C₈×C₁₀₉

—

—

—

—

—

C₂²×C₁₀₉

#121

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×Q₁₆

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1684

yes

characteristic, complemented, Sylow

#487

—

109

—

#477

—

109

—

#475

D₁₀₉

yes

characteristic

C₈×Q₈

—

109

—

#335

—

109

—

#333

—

109

—

#332

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

C₄×Q₈

C₂×D₁₀₉

yes

characteristic

#107

—

218

—

#106

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

C₂×D₁₀₉

yes

characteristic

#103

—

#103

—

218

—

#103

—

218

—

#103

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₂×Q₈

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

100

C₂×C₈

—

218

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

218

—

105

C₄⋊C₄

—

218

—

106

C₄⋊C₄

C₂²⋊D₁₀₉

yes

characteristic

107

C₄²

—

108

C₄²

C₂²×D₁₀₉

yes

characteristic

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

872

—

113

Q₈

#13182019

yes

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

436

—

118

C₂×C₄

D₄×D₁₀₉

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

120

C₂×C₄

#13182028

yes

characteristic

121

C₈

—

122

C₈

—

436

—

123

C₈

—

436

—

124

C₈

—

218

—

125

C₈

—

218

—

126

C₂²

#16812671

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

872

—

132

C₄

C₂×#13182019

yes

characteristic

133

C₄

#16812719

yes

characteristic

134

C₄

#16812720

yes

characteristic

135

C₄

#16812728

yes

characteristic

136

C₂

#20295246

yes

characteristic

137

C₂

#20295265

yes

characteristic

138

C₂

#20295291

yes

characteristic

	subgroup	G/H
center	C₂²	#16812671
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#475×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#121
hypercenter	#475	D₁₀₉
hypocenter	C₁₀₉	#1684
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#16812671

commutator, cosocle

C₄²×C₁₀₉

C₂³

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#475×C₁₀₉

C₂

C₂²×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
70. C₁₀₉	#1684	semidirect

H ≅ G/N

type

70. C₁₀₉

#1684

semidirect

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

subgroup	index
#20295376	2
#20295321	2
Q₈×#11724784	2
#20294778	2
#20294774	2
#20294772	2
#475×C₁₀₉	2
#1684	109

subgroup

index

#475×C₁₀₉

109

Subgroups — #28362488

Conjugacy classes of subgroups

Complements

Maximal subgroups