Group #28378599 — subgroups

Subgroups — #28378599

Order 13952.

168 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 70 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	#20296029	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11724786	C₂	1	1	yes	maximal, characteristic
3	#20295888	C₂	1	2	yes	maximal, complemented
4	#20295887	C₂	1	2	yes	maximal, complemented
5	#20295076	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182038	—	2	1	no	—
7	C₂²×#11724786	—	2	1	no	—
8	#16812766	—	2	1	no	—
9	C₄⋊#11724786	—	2	1	no	—
10	C₄⋊#11724786	C₂²	1	2	yes	—
11	C₄⋊#11724786	C₂²	1	1	yes	characteristic
12	C₄⋊#11724786	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	#16812628	C₄	1	2	yes	—
15	C₂²⋊#11724784	C₄	1	1	yes	characteristic
16	C₈×Dic₁₀₉	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
18	#13182038	—	4	2	no	—
19	C₂×#11724786	—	2	2	no	—
20	C₂×#11724786	—	2	1	no	—
21	C₂×#11724786	—	2	1	no	—
22	C₂×#11724786	—	4	2	no	—
23	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
24	C₂²×Dic₁₀₉	Q₈	1	1	yes	characteristic
25	C₂×#11724784	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
28	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
29	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	—	4	1	no	—
37	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×Dic₁₀₉	SD₁₆	1	2	yes	—
40	C₂×Dic₁₀₉	SD₁₆	1	2	yes	—
41	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
43	#11724786	—	4	2	no	—
44	#11724786	—	4	1	no	—
45	#11724786	—	4	1	no	—
46	C₈×C₁₀₉	—	2	2	no	—
47	#11724784	—	2	2	no	—
48	C₂²×C₁₀₉	—	2	1	no	—
49	C₂²×C₁₀₉	—	2	1	no	—
50	C₂²×C₁₀₉	#106	1	2	yes	—
51	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₁₀₉	—	2	1	no	—
53	C₄×C₁₀₉	#131	1	2	yes	—
54	Dic₁₀₉	—	2	1	no	—
55	Dic₁₀₉	—	2	1	no	—
56	Dic₁₀₉	#106	1	2	yes	—
57	Dic₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
58	Dic₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
59	C₂×C₁₀₉	—	2	2	no	—
60	C₂×C₁₀₉	#424	1	1	yes	characteristic
61	C₂×C₁₀₉	C₂×#106	1	1	yes	characteristic
62	C₂×C₁₀₉	#430	1	1	yes	characteristic
63	#1570	—	109	1	no	maximal, Sylow
64	C₁₀₉	#1570	1	1	yes	characteristic, complemented, Sylow
65	#431	—	109	1	no	—
66	#423	—	109	1	no	—
67	C₂×C₄⋊C₈	—	109	1	no	—
68	#334	—	109	2	no	—
69	#334	—	109	2	no	—
70	C₂×M₁₆	—	218	1	no	—
71	C₂²×C₈	—	218	1	no	—
72	C₂×C₄²	—	109	1	no	—
73	C₄⋊C₈	—	109	2	no	—
74	C₄⋊C₈	—	109	2	no	—
75	C₄⋊C₈	—	109	1	no	—
76	C₄⋊C₈	—	109	1	no	—
77	C₄⋊C₈	—	218	1	no	—
78	C₄⋊C₈	—	218	1	no	—
79	C₂²⋊C₈	—	109	1	no	—
80	C₂²⋊C₈	#89137	1	1	yes	characteristic
81	C₄×C₈	—	109	2	no	—
82	C₂²×C₄	—	109	1	no	—
83	C₂²×C₄	—	109	1	no	—
84	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
85	M₁₆	—	436	2	no	—
86	C₂×C₈	—	109	2	no	—
87	C₂×C₈	—	436	2	no	—
88	C₂×C₈	—	218	2	no	—
89	C₂×C₈	—	218	1	no	—
90	C₂×C₈	—	218	1	no	—
91	C₂×C₈	C₂×#89137	1	2	yes	—
92	C₄²	—	109	1	no	—
93	C₄²	—	109	1	no	—
94	C₄²	—	109	1	no	—
95	C₄²	—	109	1	no	—
96	C₂³	C₄⋊#89137	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	109	2	no	—
99	C₂×C₄	—	109	2	no	—
100	C₂×C₄	—	109	1	no	—
101	C₂×C₄	—	109	1	no	—
102	C₂×C₄	—	109	1	no	—
103	C₂×C₄	—	109	1	no	—
104	C₂×C₄	—	436	1	no	—
105	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
106	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	436	2	no	—
109	C₈	—	436	1	no	—
110	C₈	—	436	1	no	—
111	C₈	—	218	2	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	#16812767	1	2	yes	—
115	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	109	1	no	—
118	C₄	—	109	1	no	—
119	C₄	—	109	2	no	—
120	C₄	—	218	1	no	—
121	C₄	—	218	1	no	—
122	C₄	#16812786	1	2	yes	—
123	C₂	—	2	2	no	—
124	C₂	#20296032	1	1	yes	characteristic
125	C₂	C₂×#16812767	1	1	yes	characteristic
126	C₂	#20296041	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296029

C₂

yes

maximal, characteristic

C₂×C₄⋊#11724786

C₂

yes

maximal, characteristic

#20295888

C₂

yes

maximal, complemented

#20295887

C₂

yes

maximal, complemented

#20295076

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

C₄⋊#11724786

C₂²

yes

characteristic

C₄⋊#11724786

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812628

C₄

yes

—

C₂²⋊#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

—

C₂²⋊C₈×C₁₀₉

C₄

yes

—

—

—

—

—

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂²×Dic₁₀₉

Q₈

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

SD₁₆

yes

—

C₂×Dic₁₀₉

SD₁₆

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×Q₈

yes

—

—

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#106

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

#131

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#106

yes

—

Dic₁₀₉

C₂×SD₁₆

yes

characteristic

Dic₁₀₉

C₂×SD₁₆

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1570

yes

characteristic, complemented, Sylow

#431

—

109

—

#423

—

109

—

C₂×C₄⋊C₈

—

109

—

#334

—

109

—

#334

—

109

—

C₂×M₁₆

—

218

—

C₂²×C₈

—

218

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

#89137

yes

characteristic

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×#89137

yes

characteristic

M₁₆

—

436

—

C₂×C₈

—

109

—

C₂×C₈

—

436

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

C₄⋊#89137

yes

characteristic

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

109

—

103

C₂×C₄

—

109

—

104

C₂×C₄

—

436

—

105

C₂×C₄

C₄⋊#89137

yes

characteristic

106

C₂×C₄

C₂²×#89137

yes

characteristic

107

C₈

—

108

C₈

—

436

—

109

C₈

—

436

—

110

C₈

—

436

—

111

C₈

—

218

—

112

C₂²

—

113

C₂²

—

114

C₂²

#16812767

yes

—

115

C₂²

C₂×C₄⋊#89137

yes

characteristic

116

C₄

—

117

C₄

—

109

—

118

C₄

—

109

—

119

C₄

—

109

—

120

C₄

—

218

—

121

C₄

—

218

—

122

C₄

#16812786

yes

—

123

C₂

—

124

C₂

#20296032

yes

characteristic

125

C₂

C₂×#16812767

yes

characteristic

126

C₂

#20296041

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂²⋊C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₄⋊C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₂²⋊C₈	#89137
hypocenter	C₁₀₉	#1570
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₂²⋊C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₉

C₄⋊C₄

cosocle

C₄×Dic₁₀₉

C₂³

C₂²⋊C₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
3. #20295888	C₂	semidirect
4. #20295887	C₂	semidirect
64. C₁₀₉	#1570	semidirect

H ≅ G/N

type

3. #20295888

C₂

semidirect

4. #20295887

C₂

semidirect

64. C₁₀₉

#1570

semidirect

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

subgroup	index
#20296029	2
C₂×C₄⋊#11724786	2
#20295888	2
#20295887	2
#20295076	2
#1570	109

subgroup

index

109

Subgroups — #28378599

Conjugacy classes of subgroups

Complements

Maximal subgroups