Group #28378600 — subgroups

Subgroups — #28378600

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	#20296030	C₂	1	1	yes	maximal, characteristic
2	D₁₀₉⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20295888	C₂	1	2	yes	maximal, complemented
4	#20295887	C₂	1	2	yes	maximal, complemented
5	#20295078	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182039	—	2	1	no	—
7	C₂×D₁₀₉⋊C₈	—	2	1	no	—
8	#16812766	—	2	1	no	—
9	C₄⋊#11724786	—	2	1	no	—
10	C₄⋊#11724786	C₂²	1	1	yes	characteristic
11	C₄⋊#11724786	C₂²	1	1	yes	characteristic
12	C₄⋊#11724786	C₂²	1	2	yes	—
13	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
14	C₄⋊#11724784	C₄	1	1	yes	characteristic
15	#16812611	C₄	1	2	yes	—
16	C₈×Dic₁₀₉	C₂²	1	2	yes	—
17	C₄⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
18	#13182039	—	4	2	no	—
19	D₁₀₉⋊C₈	—	4	2	no	—
20	C₂×#11724786	—	2	2	no	—
21	C₂×#11724786	—	2	1	no	—
22	C₂×#11724786	—	2	1	no	—
23	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
25	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
26	C₂×#11724784	C₂×C₄	1	2	yes	—
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	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
39	C₄×D₁₀₉	—	2	1	no	—
40	C₄×D₁₀₉	—	2	1	no	—
41	C₄×D₁₀₉	—	4	1	no	—
42	C₄×D₁₀₉	SD₁₆	1	2	yes	—
43	C₄×D₁₀₉	SD₁₆	1	2	yes	—
44	#11724786	—	4	1	no	—
45	#11724786	—	4	1	no	—
46	#11724786	—	4	2	no	—
47	C₈×C₁₀₉	—	2	2	no	—
48	#11724784	—	2	2	no	—
49	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×D₁₀₉	—	2	1	no	—
51	C₂×D₁₀₉	—	2	1	no	—
52	C₂×D₁₀₉	#106	1	2	yes	—
53	C₄×C₁₀₉	—	2	1	no	—
54	C₄×C₁₀₉	#106	1	2	yes	—
55	C₄×C₁₀₉	#131	1	1	yes	characteristic
56	C₄×C₁₀₉	#131	1	1	yes	characteristic
57	Dic₁₀₉	—	2	1	no	—
58	Dic₁₀₉	—	2	1	no	—
59	Dic₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
60	Dic₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₁₀₉	#424	1	1	yes	characteristic
62	C₂×C₁₀₉	C₂×#106	1	1	yes	characteristic
63	C₂×C₁₀₉	#430	1	1	yes	characteristic
64	D₁₀₉	—	2	2	no	—
65	#1570	—	109	1	no	maximal, Sylow
66	C₁₀₉	#1570	1	1	yes	characteristic, complemented, Sylow
67	#431	—	109	1	no	—
68	#423	—	109	1	no	—
69	C₂×C₄⋊C₈	—	109	1	no	—
70	#334	—	109	2	no	—
71	#334	—	109	2	no	—
72	C₂×M₁₆	—	218	1	no	—
73	C₂²×C₈	—	218	1	no	—
74	C₂×C₄²	—	109	1	no	—
75	C₄⋊C₈	—	109	1	no	—
76	C₄⋊C₈	—	109	1	no	—
77	C₄⋊C₈	—	109	2	no	—
78	C₄⋊C₈	—	109	1	no	—
79	C₄⋊C₈	—	218	1	no	—
80	C₄⋊C₈	—	218	1	no	—
81	C₄⋊C₈	#89137	1	1	yes	characteristic
82	C₂²⋊C₈	—	109	2	no	—
83	C₄×C₈	—	109	2	no	—
84	C₂²×C₄	—	109	1	no	—
85	C₂²×C₄	—	109	1	no	—
86	C₂²×C₄	—	109	1	no	—
87	M₁₆	—	436	2	no	—
88	C₂×C₈	—	109	2	no	—
89	C₂×C₈	—	436	2	no	—
90	C₂×C₈	—	218	1	no	—
91	C₂×C₈	—	218	1	no	—
92	C₂×C₈	—	218	2	no	—
93	C₂×C₈	C₂×#89137	1	2	yes	—
94	C₄²	—	109	1	no	—
95	C₄²	—	109	1	no	—
96	C₄²	—	109	1	no	—
97	C₄²	C₂×#89137	1	1	yes	characteristic
98	C₂³	—	109	1	no	—
99	C₂×C₄	—	109	1	no	—
100	C₂×C₄	—	109	2	no	—
101	C₂×C₄	—	109	2	no	—
102	C₂×C₄	—	109	1	no	—
103	C₂×C₄	—	109	1	no	—
104	C₂×C₄	—	436	1	no	—
105	C₂×C₄	—	218	1	no	—
106	C₂×C₄	—	218	1	no	—
107	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
108	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
109	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	436	1	no	—
112	C₈	—	436	1	no	—
113	C₈	—	436	2	no	—
114	C₈	—	218	2	no	—
115	C₂²	—	109	2	no	—
116	C₂²	—	218	1	no	—
117	C₂²	—	218	1	no	—
118	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	109	1	no	—
121	C₄	—	109	1	no	—
122	C₄	—	218	1	no	—
123	C₄	—	218	1	no	—
124	C₄	#16812767	1	2	yes	—
125	C₄	#16812786	1	1	yes	characteristic
126	C₄	#16812787	1	1	yes	characteristic
127	C₂	—	218	2	no	—
128	C₂	#20296033	1	1	yes	characteristic
129	C₂	C₂×#16812767	1	1	yes	characteristic
130	C₂	#20296041	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296030

C₂

yes

maximal, characteristic

D₁₀₉⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

#20295888

C₂

yes

maximal, complemented

#20295887

C₂

yes

maximal, complemented

#20295078

C₂

yes

maximal, characteristic

C₂×#13182039

—

C₂×D₁₀₉⋊C₈

—

#16812766

—

C₄⋊#11724786

—

C₄⋊#11724786

C₂²

yes

characteristic

C₄⋊#11724786

C₂²

yes

characteristic

C₄⋊#11724786

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊#11724784

C₄

yes

characteristic

#16812611

C₄

yes

—

C₈×Dic₁₀₉

C₂²

yes

—

C₄⋊C₈×C₁₀₉

C₄

yes

characteristic

#13182039

—

D₁₀₉⋊C₈

—

C₂×#11724786

—

C₂×#11724786

—

C₂×#11724786

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

Q₈

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×Q₈

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

SD₁₆

yes

—

C₄×D₁₀₉

SD₁₆

yes

—

#11724786

—

#11724786

—

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#106

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#106

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×SD₁₆

yes

characteristic

Dic₁₀₉

C₂×SD₁₆

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1570

—

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

—

C₄⋊C₈

#89137

yes

characteristic

C₂²⋊C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

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

C₂×#89137

yes

characteristic

C₂³

—

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

—

218

—

106

C₂×C₄

—

218

—

107

C₂×C₄

C₄⋊#89137

yes

characteristic

108

C₂×C₄

C₄⋊#89137

yes

characteristic

109

C₂×C₄

C₂²×#89137

yes

characteristic

110

C₈

—

111

C₈

—

436

—

112

C₈

—

436

—

113

C₈

—

436

—

114

C₈

—

218

—

115

C₂²

—

109

—

116

C₂²

—

218

—

117

C₂²

—

218

—

118

C₂²

C₂×C₄⋊#89137

yes

characteristic

119

C₄

—

120

C₄

—

109

—

121

C₄

—

109

—

122

C₄

—

218

—

123

C₄

—

218

—

124

C₄

#16812767

yes

—

125

C₄

#16812786

yes

characteristic

126

C₄

#16812787

yes

characteristic

127

C₂

—

218

—

128

C₂

#20296033

yes

characteristic

129

C₂

C₂×#16812767

yes

characteristic

130

C₂

#20296041

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄⋊C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂²×D₁₀₉	C₄⋊C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₄⋊C₈	#89137
hypocenter	C₁₀₉	#1570
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄⋊C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂²×D₁₀₉

C₄⋊C₄

cosocle

C₄×Dic₁₀₉

C₂³

C₄⋊C₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

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

H ≅ G/N

type

3. #20295888

C₂

semidirect

4. #20295887

C₂

semidirect

66. C₁₀₉

#1570

semidirect

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

subgroup	index
#20296030	2
D₁₀₉⋊(C₄⋊C₈)	2
#20295888	2
#20295887	2
#20295078	2
#1570	109

subgroup

index

D₁₀₉⋊(C₄⋊C₈)

109

Subgroups — #28378600

Conjugacy classes of subgroups

Complements

Maximal subgroups