Group #28373541 — subgroups

Subgroups — #28373541

Order 13952.

356 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 145 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	#20295581	C₂	1	2	yes	maximal
2	#20295429	C₂	1	2	yes	maximal, complemented
3	#20295426	C₂	1	4	yes	maximal, complemented
4	#20295424	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295061	C₂	1	4	yes	maximal
6	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
7	#498×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
9	#16812699	C₂²	1	2	yes	—
10	#16812690	C₂²	1	4	yes	—
11	#16812679	—	2	2	no	—
12	#16812675	—	2	2	no	—
13	Q₈×Dic₁₀₉	—	2	2	no	—
14	#16812664	—	2	2	no	—
15	#16812657	C₂²	1	4	yes	—
16	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
17	#16812632	C₂²	1	8	yes	—
18	#16812631	C₂²	1	4	yes	—
19	C₄×#11724784	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₀₉	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
22	#106×C₁₀₉	C₂²	1	4	yes	—
23	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×Dic₂₁₈	—	2	2	no	—
25	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
26	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
27	C₂×#11724784	C₂³	1	2	yes	—
28	C₈×D₁₀₉	Q₈	1	8	yes	complemented
29	C₄⋊Dic₁₀₉	—	2	2	no	—
30	C₄⋊Dic₁₀₉	—	2	4	no	—
31	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
32	#13182008	—	2	4	no	—
33	#13182006	—	2	4	no	—
34	C₄×Dic₁₀₉	—	2	4	no	—
35	C₄×Dic₁₀₉	D₄	1	2	yes	—
36	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₁₀₉	—	2	2	no	—
38	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
39	C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
41	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₁₀₉	—	4	4	no	—
44	C₂×C₄×C₁₀₉	—	2	4	no	—
45	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₁₀₉	—	2	4	no	—
48	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
49	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₄×D₁₀₉	—	2	4	no	—
51	C₄×D₁₀₉	C₂×Q₈	1	4	yes	—
52	Dic₂₁₈	—	4	4	no	—
53	C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
54	#11724784	C₂×Q₈	1	4	yes	—
55	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
56	C₂×D₁₀₉	—	2	2	no	—
57	C₂×D₁₀₉	C₄⋊Q₈	1	2	yes	—
58	C₄×C₁₀₉	—	4	4	no	—
59	C₄×C₁₀₉	#135	1	4	yes	—
60	C₄×C₁₀₉	C₂²×Q₈	1	2	yes	—
61	Dic₁₀₉	—	2	2	no	—
62	Dic₁₀₉	—	4	4	no	—
63	Dic₁₀₉	C₄⋊Q₈	1	2	yes	—
64	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₁₀₉	C₂×#135	1	2	yes	—
66	D₁₀₉	—	2	2	no	—
67	#3160	—	109	1	no	maximal, Sylow
68	C₁₀₉	#3160	1	1	yes	characteristic, complemented, Sylow
69	#533	—	109	2	no	—
70	#500	—	109	2	no	—
71	#499	—	109	4	no	—
72	#498	—	109	1	no	—
73	#498	D₁₀₉	1	1	yes	characteristic, complemented
74	#427	—	109	4	no	—
75	C₂×C₄×C₈	—	109	1	no	—
76	C₂²×C₈	—	109	2	no	—
77	C₄⋊Q₈	—	109	2	no	—
78	C₄⋊Q₈	C₂×D₁₀₉	1	2	yes	—
79	#125	—	109	4	no	—
80	C₂²⋊Q₈	—	218	2	no	—
81	C₂²⋊Q₈	—	218	2	no	—
82	C₄×Q₈	—	218	2	no	—
83	C₄×Q₈	—	218	2	no	—
84	#117	—	109	4	no	—
85	C₂×C₄²	—	109	1	no	—
86	#107	—	109	8	no	—
87	#106	—	109	4	no	—
88	#106	C₂×D₁₀₉	1	4	yes	—
89	C₄×C₈	—	109	2	no	—
90	C₄×C₈	—	109	1	no	—
91	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	218	2	no	—
94	C₂²×C₄	—	109	1	no	—
95	C₂²×C₄	—	109	2	no	—
96	C₂×C₈	—	109	2	no	—
97	C₂×C₈	—	109	8	no	—
98	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	109	4	no	—
101	C₄⋊C₄	—	218	4	no	—
102	C₄⋊C₄	—	218	4	no	—
103	C₄⋊C₄	—	218	2	no	—
104	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
105	C₂²⋊C₄	—	218	4	no	—
106	C₄²	—	109	2	no	—
107	C₄²	—	109	1	no	—
108	C₄²	—	218	4	no	—
109	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
110	C₂³	—	109	1	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	436	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	109	1	no	—
115	C₂×C₄	—	109	4	no	—
116	C₂×C₄	—	109	2	no	—
117	C₂×C₄	—	218	4	no	—
118	C₂×C₄	—	218	4	no	—
119	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
120	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
121	C₈	—	109	4	no	—
122	C₈	Q₈×D₁₀₉	1	4	yes	—
123	C₂²	—	109	2	no	—
124	C₂²	—	218	2	no	—
125	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
126	C₄	—	4	4	no	—
127	C₄	—	109	2	no	—
128	C₄	—	436	4	no	—
129	C₄	—	218	2	no	—
130	C₄	#16812720	1	4	yes	—
131	C₄	C₂×Q₈×D₁₀₉	1	2	yes	—
132	C₂	—	218	2	no	—
133	C₂	C₄⋊Q₈×D₁₀₉	1	1	yes	characteristic
134	C₂	C₂×#16812720	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295581

C₂

yes

maximal

#20295429

C₂

yes

maximal, complemented

#20295426

C₂

yes

maximal, complemented

#20295424

C₂

yes

maximal, characteristic, complemented

#20295061

C₂

yes

maximal

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

#498×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₀₉

C₂²

yes

—

#16812699

C₂²

yes

—

#16812690

C₂²

yes

—

#16812679

—

#16812675

—

Q₈×Dic₁₀₉

—

#16812664

—

#16812657

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

#16812632

C₂²

yes

—

#16812631

C₂²

yes

—

C₄×#11724784

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

#106×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×C₄×D₁₀₉

D₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₂×#11724784

C₂³

yes

—

C₈×D₁₀₉

Q₈

yes

complemented

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

#13182006

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×Q₈

yes

—

Dic₂₁₈

—

C₈×C₁₀₉

C₂×Q₈

yes

—

#11724784

C₂×Q₈

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄⋊Q₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#135

yes

—

C₄×C₁₀₉

C₂²×Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₄⋊Q₈

yes

—

C₂×C₁₀₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₉

C₂×#135

yes

—

D₁₀₉

—

#3160

—

109

maximal, Sylow

C₁₀₉

#3160

yes

characteristic, complemented, Sylow

#533

—

109

—

#500

—

109

—

#499

—

109

—

#498

—

109

—

#498

D₁₀₉

yes

characteristic, complemented

#427

—

109

—

C₂×C₄×C₈

—

109

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

—

#125

—

109

—

C₂²⋊Q₈

—

218

—

C₂²⋊Q₈

—

218

—

C₄×Q₈

—

218

—

C₄×Q₈

—

218

—

#117

—

109

—

C₂×C₄²

—

109

—

#107

—

109

—

#106

—

109

—

#106

C₂×D₁₀₉

yes

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

109

—

101

C₄⋊C₄

—

218

—

102

C₄⋊C₄

—

218

—

103

C₄⋊C₄

—

218

—

104

C₄⋊C₄

C₂²×D₁₀₉

yes

—

105

C₂²⋊C₄

—

218

—

106

C₄²

—

109

—

107

C₄²

—

109

—

108

C₄²

—

218

—

109

C₄²

C₂²×D₁₀₉

yes

characteristic

110

C₂³

—

109

—

111

Q₈

—

112

Q₈

—

436

—

113

C₂×C₄

—

114

C₂×C₄

—

109

—

115

C₂×C₄

—

109

—

116

C₂×C₄

—

109

—

117

C₂×C₄

—

218

—

118

C₂×C₄

—

218

—

119

C₂×C₄

D₄×D₁₀₉

yes

—

120

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

121

C₈

—

109

—

122

C₈

Q₈×D₁₀₉

yes

—

123

C₂²

—

109

—

124

C₂²

—

218

—

125

C₂²

C₂×D₄×D₁₀₉

yes

characteristic

126

C₄

—

127

C₄

—

109

—

128

C₄

—

436

—

129

C₄

—

218

—

130

C₄

#16812720

yes

—

131

C₄

C₂×Q₈×D₁₀₉

yes

—

132

C₂

—

218

—

133

C₂

C₄⋊Q₈×D₁₀₉

yes

characteristic

134

C₂

C₂×#16812720

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#498×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂²×D₁₀₉	C₂×D₄
hypercenter	#498	D₁₀₉
hypocenter	C₁₀₉	#3160
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#498×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

prime radical

C₂²×D₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #20295429	C₂	semidirect
3. #20295426	C₂	semidirect
4. #20295424	C₂	semidirect
7. #498×C₁₀₉	C₂	semidirect
28. C₈×D₁₀₉	Q₈	semidirect
68. C₁₀₉	#3160	semidirect
73. #498	D₁₀₉	semidirect

H ≅ G/N

type

2. #20295429

C₂

semidirect

3. #20295426

C₂

semidirect

4. #20295424

C₂

semidirect

7. #498×C₁₀₉

C₂

semidirect

28. C₈×D₁₀₉

Q₈

semidirect

68. C₁₀₉

#3160

semidirect

73. #498

D₁₀₉

semidirect

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

subgroup	index
#20295581	2
#20295429	2
#20295426	2
#20295424	2
#20295061	2
C₄×C₈×D₁₀₉	2
#498×C₁₀₉	2
#3160	109

subgroup

index

C₄×C₈×D₁₀₉

#498×C₁₀₉

#3160

109

Subgroups — #28373541

Conjugacy classes of subgroups

Complements

Maximal subgroups