Group #28363760 — subgroups

Subgroups — #28363760

Order 13952.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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×#16812631	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812631	C₂	1	1	yes	maximal, characteristic
3	#20294924	C₂	1	2	yes	maximal
4	#20294789	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
8	C₂×#13182008	—	2	2	no	—
9	#16812631	—	2	2	no	—
10	#16812631	C₄	1	4	yes	complemented
11	#16812606	—	2	2	no	—
12	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₀₉	—	2	2	no	—
16	C₂²×Dic₁₀₉	—	2	2	no	—
17	C₂²×Dic₁₀₉	—	2	2	no	—
18	C₄⋊Dic₁₀₉	—	2	2	no	—
19	C₄⋊Dic₁₀₉	—	2	2	no	—
20	C₄⋊Dic₁₀₉	—	2	2	no	—
21	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
22	#13182008	—	4	4	no	—
23	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
24	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₀₉	—	2	1	no	—
26	C₂×C₈×C₁₀₉	—	2	1	no	—
27	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
29	C₂×C₈×C₁₀₉	Q₈	1	2	yes	—
30	C₂×C₈×C₁₀₉	Q₈	1	2	yes	—
31	C₄²×C₁₀₉	—	2	1	no	—
32	C₄²×C₁₀₉	—	2	1	no	—
33	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	—	2	2	no	—
35	C₂×C₄×C₁₀₉	—	2	2	no	—
36	C₂×C₄×C₁₀₉	SD₁₆	1	4	yes	—
37	C₂×C₄×C₁₀₉	C₂²×C₄	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×C₄×C₁₀₉	#41	1	1	yes	characteristic
43	C₂×Dic₁₀₉	—	2	4	no	—
44	C₂×Dic₁₀₉	—	4	2	no	—
45	C₂×Dic₁₀₉	—	4	2	no	—
46	C₂×Dic₁₀₉	—	4	2	no	—
47	C₂×Dic₁₀₉	—	4	2	no	—
48	C₂×Dic₁₀₉	—	4	2	no	—
49	C₈×C₁₀₉	—	2	2	no	—
50	C₈×C₁₀₉	C₄⋊C₄	1	4	yes	complemented
51	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
52	C₂²×C₁₀₉	#121	1	1	yes	characteristic
53	C₂²×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₀₉	C₂×SD₁₆	1	2	yes	—
55	C₂²×C₁₀₉	#135	1	2	yes	—
56	C₄×C₁₀₉	—	2	4	no	—
57	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
59	C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
60	C₄×C₁₀₉	#125	1	1	yes	characteristic
61	Dic₁₀₉	—	4	4	no	—
62	Dic₁₀₉	—	8	2	no	—
63	C₂×C₁₀₉	#389	1	1	yes	characteristic
64	C₂×C₁₀₉	C₄×SD₁₆	1	2	yes	—
65	C₂×C₁₀₉	#465	1	2	yes	—
66	C₂×C₁₀₉	#498	1	1	yes	characteristic
67	C₂×C₁₀₉	#499	1	1	yes	characteristic
68	#1941	—	109	1	no	maximal, Sylow
69	C₁₀₉	#1941	1	1	yes	characteristic, complemented, Sylow
70	C₂×#106	—	109	1	no	—
71	C₂×#106	—	109	1	no	—
72	C₂×C₄×C₈	D₁₀₉	1	1	yes	characteristic
73	#389	—	109	2	no	—
74	#340	—	109	2	no	—
75	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	109	2	no	—
78	C₂×C₄⋊C₄	—	109	2	no	—
79	C₂×C₄⋊C₄	—	218	2	no	—
80	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
81	#106	—	109	4	no	—
82	#106	—	218	2	no	—
83	C₄×C₈	—	2	2	no	—
84	#95	—	218	2	no	—
85	C₂²×C₄	—	218	2	no	—
86	C₂²×C₄	—	218	2	no	—
87	C₂²×C₄	D₄₃₆	1	2	yes	—
88	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	Dic₂₁₈	1	2	yes	—
92	C₂×C₈	Dic₂₁₈	1	2	yes	—
93	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
94	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
95	C₄⋊C₄	—	109	4	no	—
96	C₄⋊C₄	—	436	4	no	—
97	C₄⋊C₄	—	218	2	no	—
98	C₄⋊C₄	—	218	2	no	—
99	C₄⋊C₄	—	218	2	no	—
100	C₄²	—	2	1	no	—
101	C₄²	—	2	1	no	—
102	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	436	2	no	—
106	C₂×C₄	—	436	2	no	—
107	C₂×C₄	—	436	2	no	—
108	C₂×C₄	—	436	2	no	—
109	C₂×C₄	—	436	2	no	—
110	C₂×C₄	—	218	4	no	—
111	C₂×C₄	#13182018	1	4	yes	—
112	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
113	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
115	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
116	C₂×C₄	#13182029	1	1	yes	characteristic
117	C₂×C₄	#13182029	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	#13182008	1	4	yes	complemented
120	C₂²	C₄×D₄₃₆	1	1	yes	characteristic
121	C₂²	#16812673	1	1	yes	characteristic
122	C₂²	#16812699	1	1	yes	characteristic
123	C₂²	C₂×#13182018	1	2	yes	—
124	C₂²	#16812721	1	2	yes	—
125	C₄	—	2	4	no	—
126	C₄	—	872	2	no	—
127	C₄	—	436	4	no	—
128	C₄	C₂×#13182008	1	1	yes	characteristic
129	C₄	C₄×Dic₂₁₈	1	1	yes	characteristic
130	C₄	#16812678	1	1	yes	characteristic
131	C₄	#16812691	1	1	yes	characteristic
132	C₂	#20294924	1	1	yes	characteristic
133	C₂	C₄×#13182018	1	2	yes	—
134	C₂	#20295255	1	2	yes	—
135	C₂	#20295424	1	1	yes	characteristic
136	C₂	#20295428	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812631

C₂

yes

maximal, characteristic

C₂×#16812631

C₂

yes

maximal, characteristic

#20294924

C₂

yes

maximal

#20294789

C₂

yes

maximal

C₂×C₄×C₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182008

—

#16812631

—

#16812631

C₄

yes

complemented

#16812606

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

Q₈

yes

—

C₂×C₈×C₁₀₉

Q₈

yes

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₄⋊C₄

yes

complemented

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₂²×C₁₀₉

#121

yes

characteristic

C₂²×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₉

C₂×SD₁₆

yes

—

C₂²×C₁₀₉

#135

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

C₄×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₀₉

#125

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#389

yes

characteristic

C₂×C₁₀₉

C₄×SD₁₆

yes

—

C₂×C₁₀₉

#465

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1941

yes

characteristic, complemented, Sylow

C₂×#106

—

109

—

C₂×#106

—

109

—

C₂×C₄×C₈

D₁₀₉

yes

characteristic

#389

—

109

—

#340

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#106

—

109

—

#106

—

218

—

C₄×C₈

—

#95

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

D₄₃₆

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₂×C₈

C₂²⋊D₁₀₉

yes

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

100

C₄²

—

101

C₄²

—

102

C₂³

C₂×D₄₃₆

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

436

—

106

C₂×C₄

—

436

—

107

C₂×C₄

—

436

—

108

C₂×C₄

—

436

—

109

C₂×C₄

—

436

—

110

C₂×C₄

—

218

—

111

C₂×C₄

#13182018

yes

—

112

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

113

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

114

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

115

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

116

C₂×C₄

#13182029

yes

characteristic

117

C₂×C₄

#13182029

yes

characteristic

118

C₈

—

119

C₈

#13182008

yes

complemented

120

C₂²

C₄×D₄₃₆

yes

characteristic

121

C₂²

#16812673

yes

characteristic

122

C₂²

#16812699

yes

characteristic

123

C₂²

C₂×#13182018

yes

—

124

C₂²

#16812721

yes

—

125

C₄

—

126

C₄

—

872

—

127

C₄

—

436

—

128

C₄

C₂×#13182008

yes

characteristic

129

C₄

C₄×Dic₂₁₈

yes

characteristic

130

C₄

#16812678

yes

characteristic

131

C₄

#16812691

yes

characteristic

132

C₂

#20294924

yes

characteristic

133

C₂

C₄×#13182018

yes

—

134

C₂

#20295255

yes

—

135

C₂

#20295424

yes

characteristic

136

C₂

#20295428

yes

characteristic

	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	#1941
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

#1941

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
10. #16812631	C₄	semidirect
50. C₈×C₁₀₉	C₄⋊C₄	semidirect
69. C₁₀₉	#1941	semidirect
119. C₈	#13182008	semidirect

H ≅ G/N

type

10. #16812631

C₄

semidirect

50. C₈×C₁₀₉

C₄⋊C₄

semidirect

69. C₁₀₉

#1941

semidirect

119. C₈

#13182008

semidirect

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

subgroup	index
C₂×#16812631	2
#20294924	2
#20294789	2
C₂×C₄×C₈×C₁₀₉	2
#1941	109

subgroup

index

C₂×C₄×C₈×C₁₀₉

109

Subgroups — #28363760

Conjugacy classes of subgroups

Complements

Maximal subgroups