Group #28374426 — subgroups

Subgroups — #28374426

Order 13952.

342 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 125 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	#20295583	C₂	1	2	yes	maximal
2	#20295312	C₂	1	4	yes	maximal, complemented
3	#20295284	C₂	1	4	yes	maximal
4	#20295237	C₂	1	4	yes	maximal, complemented
5	#423×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₃₆	C₂²	1	4	yes	—
7	#16812699	C₂²	1	2	yes	—
8	#16812699	C₂²	1	2	yes	—
9	#16812691	—	2	2	no	—
10	#16812678	—	2	2	no	—
11	#16812678	C₂²	1	4	yes	—
12	C₄×Dic₂₁₈	—	2	2	no	—
13	C₄×Dic₂₁₈	C₂²	1	4	yes	—
14	#16812658	—	2	2	no	—
15	#16812631	C₂²	1	4	yes	—
16	#16812624	C₂²	1	8	yes	—
17	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	—
20	C₂×Dic₂₁₈	—	2	2	no	—
21	C₂×Dic₂₁₈	C₂³	1	4	yes	—
22	Dic₄₃₆	—	4	4	no	—
23	C₄⋊Dic₁₀₉	—	2	2	no	—
24	C₄⋊Dic₁₀₉	—	2	2	no	—
25	C₄⋊Dic₁₀₉	—	2	2	no	—
26	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
27	#13182008	—	2	4	no	—
28	#13182008	—	2	4	no	—
29	C₂²⋊Dic₁₀₉	—	2	4	no	—
30	C₄×Dic₁₀₉	—	2	4	no	—
31	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
32	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₀₉	—	4	2	no	—
34	C₂×C₈×C₁₀₉	C₂³	1	4	yes	—
35	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
36	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
37	C₄²×C₁₀₉	C₂³	1	2	yes	—
38	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	—	2	4	no	—
40	C₂×C₄×C₁₀₉	—	2	1	no	—
41	C₂×C₄×C₁₀₉	—	2	1	no	—
42	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₁₀₉	—	2	4	no	—
47	C₂×Dic₁₀₉	—	2	4	no	—
48	Dic₂₁₈	—	2	4	no	—
49	Dic₂₁₈	—	2	4	no	—
50	Dic₂₁₈	—	4	4	no	—
51	C₈×C₁₀₉	—	2	4	no	—
52	C₂²×C₁₀₉	—	2	1	no	—
53	C₂²×C₁₀₉	—	2	2	no	—
54	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₁₀₉	—	2	1	no	—
56	C₄×C₁₀₉	—	2	2	no	—
57	C₄×C₁₀₉	#137	1	4	yes	—
58	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
59	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
60	Dic₁₀₉	—	4	4	no	—
61	Dic₁₀₉	—	4	4	no	—
62	C₂×C₁₀₉	—	4	1	no	—
63	C₂×C₁₀₉	#537	1	1	yes	characteristic
64	C₂×C₁₀₉	C₂×#137	1	2	yes	—
65	#3255	—	109	1	no	maximal, Sylow
66	C₁₀₉	#3255	1	1	yes	characteristic, complemented, Sylow
67	#533	—	109	2	no	—
68	#475	—	109	4	no	—
69	#470	—	109	4	no	—
70	#462	—	109	4	no	—
71	#423	D₁₀₉	1	1	yes	characteristic
72	C₂×Q₁₆	—	109	4	no	—
73	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
74	C₄⋊Q₈	—	109	2	no	—
75	C₄⋊Q₈	—	109	2	no	—
76	#125	—	218	2	no	—
77	C₂²⋊Q₈	—	109	4	no	—
78	C₂²⋊Q₈	—	218	2	no	—
79	C₄×Q₈	—	109	4	no	—
80	C₄×Q₈	—	218	2	no	—
81	#117	—	218	2	no	—
82	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
83	#106	—	109	4	no	—
84	C₄⋊C₈	C₂×D₁₀₉	1	4	yes	—
85	#103	—	109	8	no	—
86	C₂×Q₈	—	109	4	no	—
87	C₂×Q₈	—	218	2	no	—
88	C₂²×C₄	D₄₃₆	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
90	Q₁₆	—	436	4	no	—
91	M₁₆	—	4	2	no	—
92	C₂×C₈	C₂²×D₁₀₉	1	4	yes	—
93	C₄⋊C₄	—	109	4	no	—
94	C₄⋊C₄	—	218	4	no	—
95	C₄⋊C₄	—	218	4	no	—
96	C₄⋊C₄	—	218	2	no	—
97	C₄⋊C₄	—	218	2	no	—
98	C₄⋊C₄	—	218	2	no	—
99	C₂²⋊C₄	—	218	4	no	—
100	C₄²	—	218	4	no	—
101	C₄²	D₄₃₆	1	1	yes	characteristic
102	C₄²	D₄₃₆	1	1	yes	characteristic
103	C₄²	C₂²×D₁₀₉	1	2	yes	—
104	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
105	Q₈	—	436	4	no	—
106	Q₈	—	218	4	no	—
107	Q₈	—	218	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	218	4	no	—
112	C₂×C₄	—	218	4	no	—
113	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
114	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
115	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
116	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	436	4	no	—
124	C₄	—	436	4	no	—
125	C₄	#16812732	1	4	yes	—
126	C₄	#16812744	1	1	yes	characteristic
127	C₄	#16812747	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#20295607	1	1	yes	characteristic
130	C₂	C₂×#16812732	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295583

C₂

yes

maximal

#20295312

C₂

yes

maximal, complemented

#20295284

C₂

yes

maximal

#20295237

C₂

yes

maximal, complemented

#423×C₁₀₉

C₂

yes

maximal, characteristic

C₂×Dic₄₃₆

C₂²

yes

—

#16812699

C₂²

yes

—

#16812699

C₂²

yes

—

#16812691

—

#16812678

—

#16812678

C₂²

yes

—

C₄×Dic₂₁₈

—

C₄×Dic₂₁₈

C₂²

yes

—

#16812658

—

#16812631

C₂²

yes

—

#16812624

C₂²

yes

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂³

yes

—

Dic₄₃₆

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

#13182008

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#137

yes

—

C₄×C₁₀₉

H(5,𝔽₂)

yes

characteristic

C₄×C₁₀₉

2₋¹⁺⁴

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#537

yes

characteristic

C₂×C₁₀₉

C₂×#137

yes

—

#3255

—

109

maximal, Sylow

C₁₀₉

#3255

yes

characteristic, complemented, Sylow

#533

—

109

—

#475

—

109

—

#470

—

109

—

#462

—

109

—

#423

D₁₀₉

yes

characteristic

C₂×Q₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

#125

—

218

—

C₂²⋊Q₈

—

109

—

C₂²⋊Q₈

—

218

—

C₄×Q₈

—

109

—

C₄×Q₈

—

218

—

#117

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#106

—

109

—

C₄⋊C₈

C₂×D₁₀₉

yes

—

#103

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂²×C₄

D₄₃₆

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

Q₁₆

—

436

—

M₁₆

—

C₂×C₈

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

100

C₄²

—

218

—

101

C₄²

D₄₃₆

yes

characteristic

102

C₄²

D₄₃₆

yes

characteristic

103

C₄²

C₂²×D₁₀₉

yes

—

104

C₂³

C₂×D₄₃₆

yes

characteristic

105

Q₈

—

436

—

106

Q₈

—

218

—

107

Q₈

—

218

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

218

—

112

C₂×C₄

—

218

—

113

C₂×C₄

C₂×D₄₃₆

yes

characteristic

114

C₂×C₄

C₂×D₄₃₆

yes

—

115

C₂×C₄

C₂×D₄₃₆

yes

—

116

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂²×D₄₃₆

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

436

—

124

C₄

—

436

—

125

C₄

#16812732

yes

—

126

C₄

#16812744

yes

characteristic

127

C₄

#16812747

yes

characteristic

128

C₂

—

129

C₂

#20295607

yes

characteristic

130

C₂

C₂×#16812732

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₄₃₆
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#423×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	#423	D₁₀₉
hypocenter	C₁₀₉	#3255
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₄₃₆

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#423×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

prime radical

C₂³×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. #20295312	C₂	semidirect
4. #20295237	C₂	semidirect
66. C₁₀₉	#3255	semidirect

H ≅ G/N

type

2. #20295312

C₂

semidirect

4. #20295237

C₂

semidirect

66. C₁₀₉

#3255

semidirect

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

subgroup	index
#20295583	2
#20295312	2
#20295284	2
#20295237	2
#423×C₁₀₉	2
#3255	109

subgroup

index

#423×C₁₀₉

109

Subgroups — #28374426

Conjugacy classes of subgroups

Complements

Maximal subgroups