Group #28378598 — subgroups

Subgroups — #28378598

Order 13952.

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

normal subgroups only

This group has 166 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	#20296048	C₂	1	1	yes	maximal, characteristic
2	#20296030	C₂	1	1	yes	maximal, characteristic
3	#20295890	C₂	1	2	yes	maximal, complemented
4	#20295888	C₂	1	2	yes	maximal, complemented
5	#20294978	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182039	—	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	#16812756	—	2	1	no	—
12	#16812754	—	2	1	no	—
13	C₂×#13182012	C₄	1	2	yes	—
14	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
15	#16812610	C₄	1	1	yes	characteristic
16	C₈×Dic₁₀₉	C₂²	1	2	yes	—
17	#97×C₁₀₉	C₄	1	1	yes	characteristic
18	#13182039	—	4	2	no	—
19	C₂×#11724786	—	2	1	no	—
20	C₂×#11724786	—	2	1	no	—
21	C₂×#11724786	—	2	2	no	—
22	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
24	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
25	#13182012	—	2	4	no	—
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	2	no	—
40	C₄×D₁₀₉	—	2	1	no	—
41	C₄×D₁₀₉	—	2	1	no	—
42	C₄×D₁₀₉	—	4	1	no	—
43	#11724786	—	4	1	no	—
44	#11724786	—	4	2	no	—
45	#11724786	—	4	1	no	—
46	C₈×C₁₀₉	—	2	2	no	—
47	#11724784	M₁₆	1	4	yes	complemented
48	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×D₁₀₉	—	2	1	no	—
50	C₂×D₁₀₉	—	2	1	no	—
51	C₂×D₁₀₉	—	2	1	no	—
52	C₄×C₁₀₉	—	2	1	no	—
53	C₄×C₁₀₉	—	2	1	no	—
54	C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
55	Dic₁₀₉	—	2	1	no	—
56	Dic₁₀₉	—	2	1	no	—
57	Dic₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
58	Dic₁₀₉	#137	1	1	yes	characteristic
59	C₂×C₁₀₉	#423	1	1	yes	characteristic
60	C₂×C₁₀₉	#428	1	1	yes	characteristic
61	C₂×C₁₀₉	#430	1	1	yes	characteristic
62	D₁₀₉	—	4	1	no	—
63	#1569	—	109	1	no	maximal, Sylow
64	C₁₀₉	#1569	1	1	yes	characteristic, complemented, Sylow
65	#432	—	109	1	no	—
66	#423	—	109	1	no	—
67	C₄×M₁₆	—	109	1	no	—
68	#335	—	109	2	no	—
69	#334	—	109	2	no	—
70	C₂×M₁₆	—	109	2	no	—
71	C₂×M₁₆	—	218	1	no	—
72	C₂×C₄²	—	109	1	no	—
73	C₄⋊C₈	—	109	1	no	—
74	C₄⋊C₈	—	109	2	no	—
75	C₄⋊C₈	—	109	1	no	—
76	C₄⋊C₈	—	218	1	no	—
77	C₂²⋊C₈	—	218	1	no	—
78	#97	—	109	1	no	—
79	#97	—	218	1	no	—
80	#97	#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	—	109	1	no	—
85	M₁₆	—	436	2	no	—
86	M₁₆	—	218	4	no	—
87	C₂×C₈	—	109	2	no	—
88	C₂×C₈	—	218	1	no	—
89	C₂×C₈	—	218	1	no	—
90	C₂×C₈	—	218	2	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	C₂×#89137	1	1	yes	characteristic
96	C₂³	—	109	1	no	—
97	C₂×C₄	—	109	1	no	—
98	C₂×C₄	—	109	1	no	—
99	C₂×C₄	—	109	1	no	—
100	C₂×C₄	—	436	1	no	—
101	C₂×C₄	—	218	1	no	—
102	C₂×C₄	—	218	1	no	—
103	C₂×C₄	—	218	2	no	—
104	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
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	—	109	4	no	—
109	C₈	—	436	2	no	—
110	C₈	—	436	1	no	—
111	C₈	—	436	1	no	—
112	C₂²	—	218	1	no	—
113	C₂²	—	218	1	no	—
114	C₂²	—	218	1	no	—
115	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	109	1	no	—
119	C₄	—	109	1	no	—
120	C₄	—	218	1	no	—
121	C₄	—	218	1	no	—
122	C₄	D₁₀₉⋊M₁₆	1	2	yes	—
123	C₂	—	436	1	no	—
124	C₂	#20296031	1	1	yes	characteristic
125	C₂	#20296037	1	1	yes	characteristic
126	C₂	#20296041	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296048

C₂

yes

maximal, characteristic

#20296030

C₂

yes

maximal, characteristic

#20295890

C₂

yes

maximal, complemented

#20295888

C₂

yes

maximal, complemented

#20294978

C₂

yes

maximal, characteristic

C₂×#13182039

—

C₄⋊#11724786

—

C₄⋊#11724786

C₂²

yes

—

C₄⋊#11724786

C₂²

yes

characteristic

C₄⋊#11724786

C₂²

yes

—

—

C₄

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

#16812610

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

—

#97×C₁₀₉

C₄

yes

—

—

—

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

Q₈

yes

characteristic

#13182012

—

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

—

#11724786

—

#11724786

—

#11724786

—

C₈×C₁₀₉

—

#11724784

M₁₆

yes

complemented

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

yes

Dic₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1569

—

109

maximal, Sylow

C₁₀₉

#1569

yes

characteristic, complemented, Sylow

#432

—

109

—

#423

—

109

—

C₄×M₁₆

—

109

—

#335

—

109

—

#334

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

218

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₂²⋊C₈

—

218

—

#97

—

109

—

#97

—

218

—

#97

#89137

yes

characteristic

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

436

—

M₁₆

—

218

—

C₂×C₈

—

109

—

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

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

436

—

101

C₂×C₄

—

218

—

102

C₂×C₄

—

218

—

103

C₂×C₄

—

218

—

104

C₂×C₄

C₄⋊#89137

yes

characteristic

105

C₂×C₄

C₄⋊#89137

yes

characteristic

106

C₂×C₄

C₂²×#89137

yes

characteristic

107

C₈

—

108

C₈

—

109

—

109

C₈

—

436

—

110

C₈

—

436

—

111

C₈

—

436

—

112

C₂²

—

218

—

113

C₂²

—

218

—

114

C₂²

—

218

—

115

C₂²

C₂×C₄⋊#89137

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

109

—

119

C₄

—

109

—

120

C₄

—

218

—

121

C₄

—

218

—

122

C₄

D₁₀₉⋊M₁₆

yes

—

123

C₂

—

436

—

124

C₂

#20296031

yes

characteristic

125

C₂

#20296037

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	#97×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂²×D₁₀₉	C₄⋊C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	#97	#89137
hypocenter	C₁₀₉	#1569
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

#97×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂²×D₁₀₉

C₄⋊C₄

cosocle

C₄×Dic₁₀₉

C₂³

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
3. #20295890	C₂	semidirect
4. #20295888	C₂	semidirect
47. #11724784	M₁₆	semidirect
64. C₁₀₉	#1569	semidirect

H ≅ G/N

type

3. #20295890

C₂

semidirect

4. #20295888

C₂

semidirect

47. #11724784

M₁₆

semidirect

64. C₁₀₉

#1569

semidirect

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

subgroup	index
#20296048	2
#20296030	2
#20295890	2
#20295888	2
#20294978	2
#1569	109

subgroup

index

109

Subgroups — #28378598

Conjugacy classes of subgroups

Complements

Maximal subgroups