Group #28377882 — subgroups

Subgroups — #28377882

Order 13952.

164 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 44 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	#20296049	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#89137	C₂	1	1	yes	maximal, characteristic
3	#20295081	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#89137	—	2	1	no	—
5	C₂×C₄×#89137	C₄	1	2	yes	—
6	#16812766	C₄	1	2	yes	complemented
7	#16812756	—	2	1	no	—
8	#16812754	—	2	1	no	—
9	(C₂×#89137)⋊C₄	—	2	1	no	—
10	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
11	#16812628	—	2	1	no	—
12	#16812611	—	2	1	no	—
13	#16812610	C₄	1	1	yes	characteristic, complemented
14	#97×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂²×#89137	—	2	2	no	—
16	C₂×#11724786	—	2	2	no	—
17	C₄⋊#89137	—	2	4	no	—
18	C₄×#89137	—	4	2	no	—
19	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
21	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
22	C₂×#11724784	—	2	1	no	—
23	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
25	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
26	C₂×C₈×C₁₀₉	—	2	1	no	—
27	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×#89137	—	2	4	no	—
30	C₂×#89137	—	4	2	no	—
31	C₂×#89137	—	4	2	no	—
32	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₄×D₁₀₉	—	2	1	no	—
39	C₄×D₁₀₉	—	2	1	no	—
40	C₄×D₁₀₉	—	2	2	no	—
41	C₄×D₁₀₉	—	4	1	no	—
42	#11724786	—	4	2	no	—
43	C₈×C₁₀₉	—	4	1	no	—
44	#11724784	—	4	1	no	—
45	C₂²×C₁₀₉	#95	1	1	yes	characteristic
46	C₂×D₁₀₉	—	2	1	no	—
47	C₂×D₁₀₉	—	2	1	no	—
48	C₂×D₁₀₉	#100	1	1	yes	characteristic
49	C₂×D₁₀₉	#101	1	1	yes	characteristic
50	#89137	—	4	4	no	—
51	C₄×C₁₀₉	—	2	1	no	—
52	C₄×C₁₀₉	—	2	1	no	—
53	C₄×C₁₀₉	—	2	1	no	—
54	Dic₁₀₉	—	2	1	no	—
55	Dic₁₀₉	—	2	1	no	—
56	Dic₁₀₉	#104	1	2	yes	—
57	C₂×C₁₀₉	H(3,ℤ/4)	1	1	yes	characteristic
58	C₂×C₁₀₉	#339	1	1	yes	characteristic
59	C₂×C₁₀₉	#343	1	1	yes	characteristic
60	D₁₀₉	—	2	1	no	—
61	D₁₀₉	—	2	1	no	—
62	#1285	—	109	1	no	maximal, Sylow
63	C₁₀₉	#1285	1	1	yes	characteristic, complemented, Sylow
64	#432	—	109	1	no	—
65	#432	—	109	1	no	—
66	C₄×C₄⋊C₄	—	109	1	no	—
67	C₂×C₄⋊C₄	—	218	1	no	—
68	C₂×C₄²	—	109	1	no	—
69	C₂×C₄²	—	109	2	no	—
70	C₄⋊C₈	—	109	2	no	—
71	C₄⋊C₈	—	218	1	no	—
72	C₂²⋊C₈	—	218	1	no	—
73	C₂²⋊C₈	—	218	1	no	—
74	#97	—	109	1	no	—
75	#97	—	218	1	no	—
76	#97	#89137	1	1	yes	characteristic, complemented
77	#95	—	218	1	no	—
78	C₂²×C₄	—	109	1	no	—
79	C₂²×C₄	—	109	1	no	—
80	C₂²×C₄	—	109	1	no	—
81	C₂²×C₄	—	218	2	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	218	2	no	—
84	C₂×C₈	—	218	1	no	—
85	C₄⋊C₄	—	218	4	no	—
86	C₄²	—	109	1	no	—
87	C₄²	—	109	1	no	—
88	C₄²	—	109	1	no	—
89	C₄²	—	436	2	no	—
90	C₄²	C₂×#89137	1	1	yes	characteristic
91	C₂³	—	109	1	no	—
92	C₂×C₄	—	109	1	no	—
93	C₂×C₄	—	109	1	no	—
94	C₂×C₄	—	109	1	no	—
95	C₂×C₄	—	436	1	no	—
96	C₂×C₄	—	436	2	no	—
97	C₂×C₄	—	436	2	no	—
98	C₂×C₄	—	218	2	no	—
99	C₂×C₄	—	218	1	no	—
100	C₂×C₄	—	218	1	no	—
101	C₂×C₄	—	218	4	no	—
102	C₂×C₄	C₄×#89137	1	1	yes	characteristic
103	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
104	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
105	C₈	—	4	1	no	—
106	C₈	—	436	1	no	—
107	C₈	—	436	2	no	—
108	C₂²	—	109	1	no	—
109	C₂²	—	109	1	no	—
110	C₂²	—	218	1	no	—
111	C₂²	—	218	1	no	—
112	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	1	no	—
116	C₄	—	109	2	no	—
117	C₄	—	436	4	no	—
118	C₄	—	218	1	no	—
119	C₄	—	218	1	no	—
120	C₂	—	218	1	no	—
121	C₂	—	218	1	no	—
122	C₂	#20295893	1	1	yes	characteristic
123	C₂	#20295898	1	1	yes	characteristic
124	C₂	#20295905	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296049

C₂

yes

maximal, characteristic

C₄×C₄⋊#89137

C₂

yes

maximal, characteristic

#20295081

C₂

yes

maximal, characteristic

C₂×C₄⋊#89137

—

C₂×C₄×#89137

C₄

yes

—

#16812766

C₄

yes

—

—

—

C₄²×D₁₀₉

C₂²

yes

—

—

C₄

yes

characteristic, complemented

#97×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

C₂×#11724786

—

C₄⋊#89137

—

C₄×#89137

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×#11724784

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂²⋊C₄

yes

—

—

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

#95

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#104

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

D₁₀₉

—

#1285

—

109

maximal, Sylow

C₁₀₉

#1285

yes

characteristic, complemented, Sylow

#432

—

109

—

#432

—

109

—

C₄×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₂²⋊C₈

—

218

—

C₂²⋊C₈

—

218

—

#97

—

109

—

#97

—

218

—

#97

#89137

yes

characteristic, complemented

#95

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

436

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

100

C₂×C₄

—

218

—

101

C₂×C₄

—

218

—

102

C₂×C₄

C₄×#89137

yes

characteristic

103

C₂×C₄

C₂²⋊#89137

yes

characteristic

104

C₂×C₄

C₄⋊#89137

yes

characteristic

105

C₈

—

106

C₈

—

436

—

107

C₈

—

436

—

108

C₂²

—

109

—

109

C₂²

—

109

—

110

C₂²

—

218

—

111

C₂²

—

218

—

112

C₂²

(C₂×#89137)⋊C₄

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

109

—

117

C₄

—

436

—

118

C₄

—

218

—

119

C₄

—

218

—

120

C₂

—

218

—

121

C₂

—

218

—

122

C₂

#20295893

yes

characteristic

123

C₂

#20295898

yes

characteristic

124

C₂

#20295905

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#89137)⋊C₄
commutator	C₂×C₄×C₁₀₉	C₄²
Frattini	C₄²	C₂×#89137
Fitting	#97×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#95
prime radical	C₂²×D₁₀₉	C₂²⋊C₄
cosocle	C₄²×D₁₀₉	C₂²
hypercenter	#97	#89137
hypocenter	C₁₀₉	#1285
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

(C₂×#89137)⋊C₄

commutator

C₂×C₄×C₁₀₉

C₄²

Frattini

C₄²

C₂×#89137

Fitting

#97×C₁₀₉

C₄

socle

C₂²×C₁₀₉

#95

prime radical

C₂²×D₁₀₉

C₂²⋊C₄

cosocle

C₄²×D₁₀₉

C₂²

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
6. #16812766	C₄	semidirect
13. #16812610	C₄	semidirect
14. #97×C₁₀₉	C₄	semidirect
63. C₁₀₉	#1285	semidirect
76. #97	#89137	semidirect

H ≅ G/N

type

6. #16812766

C₄

semidirect

13. #16812610

C₄

semidirect

14. #97×C₁₀₉

C₄

semidirect

63. C₁₀₉

#1285

semidirect

76. #97

#89137

semidirect

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

subgroup	index
#20296049	2
C₄×C₄⋊#89137	2
#20295081	2
#1285	109

subgroup

index

#20296049

C₄×C₄⋊#89137

#20295081

#1285

109

Subgroups — #28377882

Conjugacy classes of subgroups

Complements

Maximal subgroups