Group #28363813 — subgroups

Subgroups — #28363813

Order 13952.

196 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20295068	C₂	1	2	yes	maximal
2	C₂×#16812631	C₂	1	1	yes	maximal, characteristic
3	#20294981	C₂	1	1	yes	maximal, characteristic
4	#20294794	C₂	1	2	yes	maximal
5	C₂×#108×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	1	yes	characteristic
7	#16812658	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
9	#16812633	—	2	2	no	—
10	#16812631	—	2	2	no	—
11	#16812631	—	2	1	no	—
12	#16812631	—	2	1	no	—
13	#16812609	—	2	1	no	—
14	C₈×Dic₁₀₉	—	2	1	no	—
15	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	#108×C₁₀₉	C₄	1	4	yes	complemented
18	C₂²×Dic₁₀₉	—	2	2	no	—
19	#13182013	—	2	1	no	—
20	#13182013	Q₈	1	2	yes	—
21	C₂×#11724784	—	2	1	no	—
22	C₄⋊Dic₁₀₉	—	2	2	no	—
23	C₄⋊Dic₁₀₉	—	2	2	no	—
24	C₄⋊Dic₁₀₉	D₄	1	2	yes	—
25	C₂²⋊Dic₁₀₉	—	2	1	no	—
26	C₄×Dic₁₀₉	—	2	1	no	—
27	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₀₉	—	2	2	no	—
29	M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
30	C₂×C₈×C₁₀₉	—	2	2	no	—
31	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₀₉	Q₈	1	2	yes	—
35	C₂³×C₁₀₉	#41	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
42	C₂×Dic₁₀₉	—	2	2	no	—
43	C₂×Dic₁₀₉	—	4	2	no	—
44	C₂×Dic₁₀₉	—	8	2	no	—
45	C₈×C₁₀₉	—	2	4	no	—
46	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
47	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
48	#11724784	—	2	2	no	—
49	C₂²×C₁₀₉	—	2	2	no	—
50	C₂²×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
51	C₂²×C₁₀₉	#121	1	1	yes	characteristic
52	C₂²×C₁₀₉	#125	1	1	yes	characteristic
53	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
55	C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
56	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
57	Dic₁₀₉	—	4	2	no	—
58	Dic₁₀₉	—	8	2	no	—
59	C₂×C₁₀₉	—	2	1	no	—
60	C₂×C₁₀₉	—	2	1	no	—
61	C₂×C₁₀₉	#389	1	1	yes	characteristic
62	C₂×C₁₀₉	#471	1	2	yes	—
63	#1951	—	109	1	no	maximal, Sylow
64	C₁₀₉	#1951	1	1	yes	characteristic, complemented, Sylow
65	C₂×#108	D₁₀₉	1	1	yes	characteristic
66	#428	—	109	2	no	—
67	C₂×#106	—	109	1	no	—
68	#405	—	109	1	no	—
69	#343	—	109	2	no	—
70	C₂×M₁₆	—	109	1	no	—
71	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
72	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
73	#117	—	109	1	no	—
74	C₂×C₄⋊C₄	—	109	2	no	—
75	#108	Dic₁₀₉	1	4	yes	complemented
76	#107	—	218	2	no	—
77	#106	—	218	2	no	—
78	#106	—	218	1	no	—
79	#106	—	218	1	no	—
80	#97	—	218	1	no	—
81	C₄×C₈	—	218	1	no	—
82	C₂²×C₄	—	218	2	no	—
83	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	109	2	no	—
86	M₁₆	—	218	1	no	—
87	M₁₆	C₂×Dic₁₀₉	1	4	yes	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	218	1	no	—
90	C₂×C₈	Dic₂₁₈	1	2	yes	—
91	C₂×C₈	D₄₃₆	1	1	yes	characteristic
92	C₂×C₈	D₄₃₆	1	1	yes	characteristic
93	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
94	C₄⋊C₄	—	109	2	no	—
95	C₄⋊C₄	—	218	2	no	—
96	C₄⋊C₄	—	218	2	no	—
97	C₂²⋊C₄	—	218	1	no	—
98	C₄²	—	218	1	no	—
99	C₂³	#13182030	1	1	yes	characteristic
100	C₂×C₄	—	872	2	no	—
101	C₂×C₄	—	436	2	no	—
102	C₂×C₄	—	218	2	no	—
103	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
104	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
105	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
108	C₂×C₄	#13182028	1	1	yes	characteristic
109	C₈	—	2	4	no	—
110	C₈	—	218	2	no	—
111	C₈	C₄⋊Dic₁₀₉	1	2	yes	—
112	C₈	C₄⋊Dic₁₀₉	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	Q₈×Dic₁₀₉	1	1	yes	characteristic
115	C₂²	#16812674	1	1	yes	characteristic
116	C₂²	#16812690	1	1	yes	characteristic
117	C₄	—	872	2	no	—
118	C₄	—	436	2	no	—
119	C₄	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
120	C₄	D₄×Dic₁₀₉	1	1	yes	characteristic
121	C₄	#16812679	1	1	yes	characteristic
122	C₄	#16812698	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#20294925	1	1	yes	characteristic
126	C₂	#20295290	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295068

C₂

yes

maximal

C₂×#16812631

C₂

yes

maximal, characteristic

#20294981

C₂

yes

maximal, characteristic

#20294794

C₂

yes

maximal

C₂×#108×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

characteristic

#16812658

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

—

—

—

—

—

C₈×Dic₁₀₉

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#108×C₁₀₉

C₄

yes

complemented

C₂²×Dic₁₀₉

—

#13182013

—

#13182013

Q₈

yes

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

D₄

yes

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

Q₈

yes

—

C₂³×C₁₀₉

#41

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×Q₈

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#389

yes

characteristic

C₂×C₁₀₉

#471

yes

—

#1951

—

109

maximal, Sylow

C₁₀₉

#1951

yes

characteristic, complemented, Sylow

C₂×#108

D₁₀₉

yes

characteristic

#428

—

109

—

C₂×#106

—

109

—

#405

—

109

—

#343

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

109

—

C₂×C₄⋊C₄

—

109

—

#108

Dic₁₀₉

yes

complemented

#107

—

218

—

#106

—

218

—

#106

—

218

—

#106

—

218

—

#97

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

C₂×Dic₁₀₉

yes

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

D₄₃₆

yes

characteristic

C₂×C₈

D₄₃₆

yes

characteristic

C₂×C₈

C₂×Dic₁₀₉

yes

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

218

—

C₂³

#13182030

yes

characteristic

100

C₂×C₄

—

872

—

101

C₂×C₄

—

436

—

102

C₂×C₄

—

218

—

103

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

104

C₂×C₄

D₄×D₁₀₉

yes

characteristic

105

C₂×C₄

C₂×D₄₃₆

yes

characteristic

106

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

107

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

108

C₂×C₄

#13182028

yes

characteristic

109

C₈

—

110

C₈

—

218

—

111

C₈

C₄⋊Dic₁₀₉

yes

—

112

C₈

C₄⋊Dic₁₀₉

yes

—

113

C₂²

—

114

C₂²

Q₈×Dic₁₀₉

yes

characteristic

115

C₂²

#16812674

yes

characteristic

116

C₂²

#16812690

yes

characteristic

117

C₄

—

872

—

118

C₄

—

436

—

119

C₄

C₂×C₄⋊Dic₁₀₉

yes

characteristic

120

C₄

D₄×Dic₁₀₉

yes

characteristic

121

C₄

#16812679

yes

characteristic

122

C₄

#16812698

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#20294925

yes

characteristic

126

C₂

#20295290

yes

—

	subgroup	G/H
center	C₂²	#16812674
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#108×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#121
prime radical	C₂³×C₁₀₉	#41
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#108	D₁₀₉
hypocenter	C₁₀₉	#1951
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#16812674

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×#108×C₁₀₉

C₂

C₂²×C₁₀₉

C₂³×C₁₀₉

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
17. #108×C₁₀₉	C₄	semidirect
64. C₁₀₉	#1951	semidirect
75. #108	Dic₁₀₉	semidirect

H ≅ G/N

type

17. #108×C₁₀₉

C₄

semidirect

64. C₁₀₉

#1951

semidirect

75. #108

Dic₁₀₉

semidirect

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

subgroup	index
#20295068	2
C₂×#16812631	2
#20294981	2
#20294794	2
C₂×#108×C₁₀₉	2
#1951	109

subgroup

index

C₂×#108×C₁₀₉

109

Subgroups — #28363813

Conjugacy classes of subgroups

Complements

Maximal subgroups