Group #28363979 — subgroups

Subgroups — #28363979

Order 13952.

292 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 121 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×#16812699	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20294789	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₈	—	2	1	no	—
6	#16812699	—	2	2	no	—
7	#16812699	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
9	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₀₉	—	2	1	no	—
12	#16812606	—	2	1	no	—
13	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₀₉	—	2	2	no	—
16	C₂×Dic₂₁₈	—	2	4	no	—
17	C₂×Dic₂₁₈	—	4	4	no	—
18	C₂²×Dic₁₀₉	—	2	2	no	—
19	C₂²×Dic₁₀₉	—	2	2	no	—
20	C₄⋊Dic₁₀₉	—	2	4	no	—
21	C₄⋊Dic₁₀₉	—	2	2	no	—
22	C₄⋊Dic₁₀₉	—	2	2	no	—
23	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
24	C₄⋊Dic₁₀₉	Q₈	1	4	yes	complemented
25	C₄×Dic₁₀₉	—	4	2	no	—
26	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	—	2	2	no	—
30	C₂×C₈×C₁₀₉	—	2	4	no	—
31	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₄²×C₁₀₉	D₄	1	2	yes	—
33	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	—	2	2	no	—
35	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₉	SD₁₆	1	2	yes	—
38	C₂×C₄×C₁₀₉	Q₁₆	1	2	yes	—
39	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₁₀₉	#41	1	2	yes	—
43	C₂×Dic₁₀₉	—	2	4	no	—
44	C₂×Dic₁₀₉	—	2	4	no	—
45	C₂×Dic₁₀₉	—	4	2	no	—
46	C₂×Dic₁₀₉	—	4	2	no	—
47	C₂×Dic₁₀₉	—	4	4	no	—
48	Dic₂₁₈	—	4	8	no	—
49	C₈×C₁₀₉	—	4	2	no	—
50	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
52	C₂²×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
54	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₀₉	—	2	2	no	—
56	C₄×C₁₀₉	#103	1	4	yes	—
57	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
58	C₄×C₁₀₉	#124	1	1	yes	characteristic
59	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₁₀₉	—	4	4	no	—
61	Dic₁₀₉	—	4	4	no	—
62	C₂×C₁₀₉	#391	1	1	yes	characteristic
63	C₂×C₁₀₉	C₂×#103	1	1	yes	characteristic
64	C₂×C₁₀₉	#418	1	1	yes	characteristic
65	C₂×C₁₀₉	#476	1	2	yes	—
66	C₂×C₁₀₉	#477	1	2	yes	—
67	#1987	—	109	1	no	maximal, Sylow
68	C₁₀₉	#1987	1	1	yes	characteristic, complemented, Sylow
69	C₂×C₄⋊Q₈	—	109	1	no	—
70	C₂×C₄⋊C₈	D₁₀₉	1	1	yes	characteristic
71	C₄×C₄⋊C₄	—	109	1	no	—
72	#340	—	109	4	no	—
73	C₂²×Q₈	—	218	1	no	—
74	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
75	C₄⋊Q₈	—	109	4	no	—
76	C₄⋊Q₈	—	218	2	no	—
77	C₂×C₄⋊C₄	—	109	2	no	—
78	C₂×C₄⋊C₄	—	109	2	no	—
79	C₂×C₄⋊C₄	—	218	1	no	—
80	C₂×C₄²	—	218	1	no	—
81	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
82	C₄⋊C₈	—	2	2	no	—
83	#95	—	218	1	no	—
84	C₂×Q₈	—	436	4	no	—
85	C₂×Q₈	—	218	4	no	—
86	C₂²×C₄	—	218	2	no	—
87	C₂²×C₄	—	218	2	no	—
88	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
89	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	4	no	—
93	C₄⋊C₄	—	109	4	no	—
94	C₄⋊C₄	—	109	4	no	—
95	C₄⋊C₄	—	218	4	no	—
96	C₄⋊C₄	—	218	2	no	—
97	C₄⋊C₄	—	218	2	no	—
98	C₄²	—	436	2	no	—
99	C₄²	C₄×D₁₀₉	1	2	yes	—
100	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
101	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
102	Q₈	—	436	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	436	2	no	—
105	C₂×C₄	—	436	4	no	—
106	C₂×C₄	—	436	2	no	—
107	C₂×C₄	—	218	4	no	—
108	C₂×C₄	—	218	4	no	—
109	C₂×C₄	#13182006	1	2	yes	—
110	C₂×C₄	#13182006	1	2	yes	—
111	C₂×C₄	#13182018	1	2	yes	—
112	C₂×C₄	Dic₄₃₆	1	2	yes	—
113	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
116	C₂×C₄	#13182028	1	2	yes	—
117	C₈	—	4	2	no	—
118	C₂²	C₂×#13182006	1	1	yes	characteristic
119	C₂²	#16812679	1	2	yes	—
120	C₂²	C₂×#13182018	1	1	yes	characteristic
121	C₂²	C₂×Dic₄₃₆	1	1	yes	characteristic
122	C₂²	#16812727	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	436	4	no	—
125	C₄	—	436	4	no	—
126	C₄	#16812624	1	4	yes	—
127	C₄	#16812664	1	2	yes	—
128	C₄	#16812686	1	1	yes	characteristic
129	C₄	#16812697	1	1	yes	characteristic
130	C₂	#20294931	1	1	yes	characteristic
131	C₂	C₂×#16812624	1	1	yes	characteristic
132	C₂	#20295032	1	1	yes	characteristic
133	C₂	#20295317	1	2	yes	—
134	C₂	#20295322	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₀₉

C₂

yes

maximal, characteristic

#20294789

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

—

#16812699

—

#16812699

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

—

#16812606

—

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₄⋊Dic₁₀₉

Q₈

yes

complemented

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

Q₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

—

C₂²×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×Q₁₆

yes

characteristic

C₂²×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#103

yes

—

C₄×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

#124

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

C₂×C₁₀₉

#477

yes

—

#1987

—

109

maximal, Sylow

C₁₀₉

#1987

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

C₂×C₄⋊C₈

D₁₀₉

yes

characteristic

C₄×C₄⋊C₄

—

109

—

#340

—

109

—

C₂²×Q₈

—

218

—

C₂²×C₈

C₂×D₁₀₉

yes

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

#95

—

218

—

C₂×Q₈

—

436

—

C₂×Q₈

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

D₄₃₆

yes

characteristic

C₂²×C₄

D₄₃₆

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

—

436

—

C₄²

C₄×D₁₀₉

yes

—

100

C₄²

C₂²⋊D₁₀₉

yes

—

101

C₂³

C₂×D₄₃₆

yes

characteristic

102

Q₈

—

436

—

103

C₂×C₄

—

104

C₂×C₄

—

436

—

105

C₂×C₄

—

436

—

106

C₂×C₄

—

436

—

107

C₂×C₄

—

218

—

108

C₂×C₄

—

218

—

109

C₂×C₄

#13182006

yes

—

110

C₂×C₄

#13182006

yes

—

111

C₂×C₄

#13182018

yes

—

112

C₂×C₄

Dic₄₃₆

yes

—

113

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

115

C₂×C₄

Q₈×D₁₀₉

yes

—

116

C₂×C₄

#13182028

yes

—

117

C₈

—

118

C₂²

C₂×#13182006

yes

characteristic

119

C₂²

#16812679

yes

—

120

C₂²

C₂×#13182018

yes

characteristic

121

C₂²

C₂×Dic₄₃₆

yes

characteristic

122

C₂²

#16812727

yes

—

123

C₄

—

124

C₄

—

436

—

125

C₄

—

436

—

126

C₄

#16812624

yes

—

127

C₄

#16812664

yes

—

128

C₄

#16812686

yes

characteristic

129

C₄

#16812697

yes

characteristic

130

C₂

#20294931

yes

characteristic

131

C₂

C₂×#16812624

yes

characteristic

132

C₂

#20295032

yes

characteristic

133

C₂

#20295317

yes

—

134

C₂

#20295322

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₃₆
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×C₄⋊C₈×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₀₉
hypocenter	C₁₀₉	#1987
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₄₃₆

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×C₄⋊C₈×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₀₉

hypocenter

C₁₀₉

#1987

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
7. #16812699	C₄	semidirect
24. C₄⋊Dic₁₀₉	Q₈	semidirect
68. C₁₀₉	#1987	semidirect

H ≅ G/N

type

7. #16812699

C₄

semidirect

24. C₄⋊Dic₁₀₉

Q₈

semidirect

68. C₁₀₉

#1987

semidirect

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

subgroup	index
C₂×#16812699	2
C₄×C₄⋊Dic₁₀₉	2
#20294789	2
C₂×C₄⋊C₈×C₁₀₉	2
#1987	109

subgroup

index

C₂×#16812699

C₄×C₄⋊Dic₁₀₉

#20294789

C₂×C₄⋊C₈×C₁₀₉

#1987

109

Subgroups — #28363979

Conjugacy classes of subgroups

Complements

Maximal subgroups