Group #28378605 — subgroups

Subgroups — #28378605

Order 13952.

160 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 64 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	#20296047	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11724786	C₂	1	1	yes	maximal, characteristic
3	#20295890	C₂	1	2	yes	maximal, complemented
4	#20295887	C₂	1	2	yes	maximal, complemented
5	#20295076	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724786	—	2	1	no	—
7	C₄⋊#11724786	—	2	1	no	—
8	C₄⋊#11724786	C₂²	1	2	yes	—
9	C₄⋊#11724786	C₂²	1	1	yes	characteristic
10	C₄⋊#11724786	C₂²	1	1	yes	characteristic
11	C₂²⋊#11724786	—	2	1	no	—
12	#16812754	—	2	1	no	—
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	C₂×#11724786	—	2	1	no	—
19	C₂×#11724786	—	2	1	no	—
20	C₂×#11724786	—	2	2	no	—
21	C₂×#11724786	—	4	2	no	—
22	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
23	C₂²×Dic₁₀₉	Q₈	1	1	yes	characteristic
24	C₂×#11724784	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
28	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	—	2	2	no	—
33	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	—	2	2	no	—
36	C₂×Dic₁₀₉	—	2	1	no	—
37	C₂×Dic₁₀₉	—	2	1	no	—
38	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
42	#11724786	—	4	1	no	—
43	#11724786	—	4	2	no	—
44	#11724786	—	4	1	no	—
45	C₈×C₁₀₉	—	2	2	no	—
46	#11724784	—	2	2	no	—
47	C₂²×C₁₀₉	—	2	1	no	—
48	C₂²×C₁₀₉	—	2	1	no	—
49	C₂²×C₁₀₉	#108	1	2	yes	—
50	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₁₀₉	—	2	1	no	—
52	C₄×C₁₀₉	#131	1	2	yes	—
53	Dic₁₀₉	—	2	1	no	—
54	Dic₁₀₉	—	2	1	no	—
55	Dic₁₀₉	—	2	1	no	—
56	Dic₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
57	Dic₁₀₉	#137	1	1	yes	characteristic
58	C₂×C₁₀₉	—	2	2	no	—
59	C₂×C₁₀₉	#424	1	1	yes	characteristic
60	C₂×C₁₀₉	#428	1	1	yes	characteristic
61	C₂×C₁₀₉	C₂×#108	1	1	yes	characteristic
62	#1573	—	109	1	no	maximal, Sylow
63	C₁₀₉	#1573	1	1	yes	characteristic, complemented, Sylow
64	#432	—	109	1	no	—
65	#431	—	109	1	no	—
66	C₂×C₄⋊C₈	—	109	1	no	—
67	#335	—	109	2	no	—
68	#334	—	109	2	no	—
69	C₂²×C₈	—	218	1	no	—
70	C₂×C₄²	—	109	1	no	—
71	C₄⋊C₈	—	109	2	no	—
72	C₄⋊C₈	—	109	1	no	—
73	C₄⋊C₈	—	109	2	no	—
74	C₄⋊C₈	—	109	1	no	—
75	C₄⋊C₈	—	218	1	no	—
76	C₂²⋊C₈	—	109	1	no	—
77	C₂²⋊C₈	—	218	1	no	—
78	C₂²⋊C₈	#89137	1	1	yes	characteristic
79	#97	—	218	1	no	—
80	C₄×C₈	—	109	2	no	—
81	C₂²×C₄	—	109	1	no	—
82	C₂²×C₄	—	109	1	no	—
83	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
84	C₂×C₈	—	109	2	no	—
85	C₂×C₈	—	436	2	no	—
86	C₂×C₈	—	218	1	no	—
87	C₂×C₈	—	218	1	no	—
88	C₂×C₈	—	218	2	no	—
89	C₂×C₈	C₂×#89137	1	2	yes	—
90	C₄²	—	109	1	no	—
91	C₄²	—	109	1	no	—
92	C₄²	—	109	1	no	—
93	C₄²	—	109	1	no	—
94	C₂³	C₄⋊#89137	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	109	1	no	—
97	C₂×C₄	—	109	1	no	—
98	C₂×C₄	—	109	1	no	—
99	C₂×C₄	—	109	1	no	—
100	C₂×C₄	—	218	1	no	—
101	C₂×C₄	—	218	1	no	—
102	C₂×C₄	—	218	2	no	—
103	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
104	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	436	2	no	—
107	C₈	—	436	1	no	—
108	C₈	—	436	1	no	—
109	C₈	—	218	2	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	#16812769	1	2	yes	—
113	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	109	1	no	—
116	C₄	—	109	1	no	—
117	C₄	—	218	1	no	—
118	C₄	—	218	1	no	—
119	C₄	—	218	1	no	—
120	C₄	#16812786	1	2	yes	—
121	C₂	—	2	2	no	—
122	C₂	#20296032	1	1	yes	characteristic
123	C₂	#20296037	1	1	yes	characteristic
124	C₂	C₂×#16812769	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296047

C₂

yes

maximal, characteristic

C₂×C₄⋊#11724786

C₂

yes

maximal, characteristic

#20295890

C₂

yes

maximal, complemented

#20295887

C₂

yes

maximal, complemented

#20295076

C₂

yes

maximal, characteristic

C₂²×#11724786

—

C₄⋊#11724786

—

C₄⋊#11724786

C₂²

yes

—

C₄⋊#11724786

C₂²

yes

characteristic

C₄⋊#11724786

C₂²

yes

characteristic

C₂²⋊#11724786

—

#16812754

—

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

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×Q₈

yes

—

—

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#108

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

#131

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

yes

Dic₁₀₉

yes

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1573

yes

characteristic, complemented, Sylow

#432

—

109

—

#431

—

109

—

C₂×C₄⋊C₈

—

109

—

#335

—

109

—

#334

—

109

—

C₂²×C₈

—

218

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

—

218

—

C₂²⋊C₈

#89137

yes

characteristic

#97

—

218

—

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×#89137

yes

characteristic

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

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

218

—

101

C₂×C₄

—

218

—

102

C₂×C₄

—

218

—

103

C₂×C₄

C₄⋊#89137

yes

characteristic

104

C₂×C₄

C₂²×#89137

yes

characteristic

105

C₈

—

106

C₈

—

436

—

107

C₈

—

436

—

108

C₈

—

436

—

109

C₈

—

218

—

110

C₂²

—

111

C₂²

—

112

C₂²

#16812769

yes

—

113

C₂²

C₂×C₄⋊#89137

yes

characteristic

114

C₄

—

115

C₄

—

109

—

116

C₄

—

109

—

117

C₄

—

218

—

118

C₄

—

218

—

119

C₄

—

218

—

120

C₄

#16812786

yes

—

121

C₂

—

122

C₂

#20296032

yes

characteristic

123

C₂

#20296037

yes

characteristic

124

C₂

C₂×#16812769

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	#1573
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. #20295890	C₂	semidirect
4. #20295887	C₂	semidirect
63. C₁₀₉	#1573	semidirect

H ≅ G/N

type

3. #20295890

C₂

semidirect

4. #20295887

C₂

semidirect

63. C₁₀₉

#1573

semidirect

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

subgroup	index
#20296047	2
C₂×C₄⋊#11724786	2
#20295890	2
#20295887	2
#20295076	2
#1573	109

subgroup

index

109

Subgroups — #28378605

Conjugacy classes of subgroups

Complements

Maximal subgroups