Group #28378585 — subgroups

Subgroups — #28378585

Order 13952.

180 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 84 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	#20296042	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	#20295888	C₂	1	2	yes	maximal, complemented
5	M₁₆×Dic₁₀₉	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	C₄×#11724786	—	2	1	no	—
13	C₂×#13182013	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	#16812609	C₄	1	2	yes	—
16	C₈×Dic₁₀₉	C₂²	1	2	yes	—
17	C₂×M₁₆×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	#13182013	C₈	1	4	yes	complemented
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	M₁₆×C₁₀₉	—	2	2	no	—
32	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₉	—	2	1	no	—
39	C₂×Dic₁₀₉	—	2	1	no	—
40	C₂×Dic₁₀₉	—	4	1	no	—
41	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
45	#11724786	—	4	1	no	—
46	#11724786	—	4	2	no	—
47	#11724786	—	4	1	no	—
48	C₈×C₁₀₉	—	2	2	no	—
49	#11724784	C₂×C₈	1	4	yes	complemented
50	C₂²×C₁₀₉	—	2	1	no	—
51	C₂²×C₁₀₉	—	2	1	no	—
52	C₂²×C₁₀₉	C₄⋊C₈	1	2	yes	—
53	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
55	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
56	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
57	Dic₁₀₉	—	2	1	no	—
58	Dic₁₀₉	—	2	1	no	—
59	Dic₁₀₉	—	2	1	no	—
60	Dic₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	Dic₁₀₉	#137	1	1	yes	characteristic
62	C₂×C₁₀₉	—	2	2	no	—
63	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₁₀₉	#428	1	1	yes	characteristic
65	C₂×C₁₀₉	#430	1	1	yes	characteristic
66	#1565	—	109	1	no	maximal, Sylow
67	C₁₀₉	#1565	1	1	yes	characteristic, complemented, Sylow
68	#431	—	109	1	no	—
69	C₂×C₄⋊C₈	—	109	1	no	—
70	C₄×M₁₆	—	109	1	no	—
71	#335	—	109	2	no	—
72	#334	—	109	2	no	—
73	C₂×M₁₆	—	109	1	no	—
74	C₂×M₁₆	#89137	1	1	yes	characteristic
75	C₂²×C₈	—	218	1	no	—
76	C₂×C₄²	—	109	1	no	—
77	C₄⋊C₈	—	109	1	no	—
78	C₄⋊C₈	—	109	2	no	—
79	C₄⋊C₈	—	109	1	no	—
80	C₄⋊C₈	—	218	1	no	—
81	C₂²⋊C₈	—	218	1	no	—
82	#97	—	109	2	no	—
83	C₄×C₈	—	109	2	no	—
84	C₄×C₈	—	218	1	no	—
85	C₂²×C₄	—	109	1	no	—
86	C₂²×C₄	—	109	1	no	—
87	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	109	4	no	—
90	C₂×C₈	—	109	2	no	—
91	C₂×C₈	—	436	2	no	—
92	C₂×C₈	—	218	1	no	—
93	C₂×C₈	—	218	1	no	—
94	C₂×C₈	—	218	2	no	—
95	C₂×C₈	C₂×#89137	1	2	yes	—
96	C₄²	—	109	1	no	—
97	C₄²	—	109	1	no	—
98	C₄²	—	109	1	no	—
99	C₄²	—	109	1	no	—
100	C₂³	C₄⋊#89137	1	1	yes	characteristic
101	C₂×C₄	—	109	1	no	—
102	C₂×C₄	—	109	1	no	—
103	C₂×C₄	—	109	1	no	—
104	C₂×C₄	—	109	1	no	—
105	C₂×C₄	—	436	1	no	—
106	C₂×C₄	—	218	1	no	—
107	C₂×C₄	—	218	1	no	—
108	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
109	C₂×C₄	D₁₀₉⋊C₈	1	2	yes	—
110	C₂×C₄	#13182038	1	2	yes	—
111	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	109	4	no	—
114	C₈	—	436	2	no	—
115	C₈	—	436	1	no	—
116	C₈	—	436	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#16812766	1	2	yes	—
120	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
121	C₄	—	109	1	no	—
122	C₄	—	109	1	no	—
123	C₄	—	218	1	no	—
124	C₄	—	218	1	no	—
125	C₄	—	218	1	no	—
126	C₄	#16812766	1	2	yes	—
127	C₄	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
128	C₄	C₂×#13182038	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	C₂×#16812766	1	1	yes	characteristic
131	C₂	#20296037	1	1	yes	characteristic
132	C₂	#20296041	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296042

C₂

yes

maximal, characteristic

C₂×C₄⋊#11724786

C₂

yes

maximal, characteristic

#20295890

C₂

yes

maximal, complemented

#20295888

C₂

yes

maximal, complemented

M₁₆×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724786

—

C₄⋊#11724786

—

C₄⋊#11724786

C₂²

yes

—

C₄⋊#11724786

C₂²

yes

characteristic

C₄⋊#11724786

C₂²

yes

—

—

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812609

C₄

yes

—

C₈×Dic₁₀₉

C₂²

yes

—

C₂×M₁₆×C₁₀₉

C₄

yes

—

—

—

—

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂²×Dic₁₀₉

Q₈

yes

characteristic

#13182013

C₈

yes

complemented

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

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

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

C₂×C₈

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄⋊C₈

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄⋊C₈

yes

—

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

yes

Dic₁₀₉

yes

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄⋊C₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1565

yes

characteristic, complemented, Sylow

#431

—

109

—

C₂×C₄⋊C₈

—

109

—

C₄×M₁₆

—

109

—

#335

—

109

—

#334

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

#89137

yes

characteristic

C₂²×C₈

—

218

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₂²⋊C₈

—

218

—

#97

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×#89137

yes

characteristic

M₁₆

—

M₁₆

—

109

—

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

—

100

C₂³

C₄⋊#89137

yes

characteristic

101

C₂×C₄

—

109

—

102

C₂×C₄

—

109

—

103

C₂×C₄

—

109

—

104

C₂×C₄

—

109

—

105

C₂×C₄

—

436

—

106

C₂×C₄

—

218

—

107

C₂×C₄

—

218

—

108

C₂×C₄

C₄⋊#89137

yes

characteristic

109

C₂×C₄

D₁₀₉⋊C₈

yes

—

110

C₂×C₄

#13182038

yes

—

111

C₂×C₄

C₂²×#89137

yes

characteristic

112

C₈

—

113

C₈

—

109

—

114

C₈

—

436

—

115

C₈

—

436

—

116

C₈

—

436

—

117

C₂²

—

118

C₂²

—

119

C₂²

#16812766

yes

—

120

C₂²

C₂×C₄⋊#89137

yes

characteristic

121

C₄

—

109

—

122

C₄

—

109

—

123

C₄

—

218

—

124

C₄

—

218

—

125

C₄

—

218

—

126

C₄

#16812766

yes

—

127

C₄

C₂×D₁₀₉⋊C₈

yes

characteristic

128

C₄

C₂×#13182038

yes

characteristic

129

C₂

—

130

C₂

C₂×#16812766

yes

characteristic

131

C₂

#20296037

yes

characteristic

132

C₂

#20296041

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#89137
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂×M₁₆×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₄⋊C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₂×M₁₆	#89137
hypocenter	C₁₀₉	#1565
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₂²

C₂×C₄⋊#89137

commutator

C₄×C₁₀₉

C₂²×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₂×M₁₆×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₉

C₄⋊C₄

cosocle

C₄×Dic₁₀₉

C₂³

C₂×M₁₆

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
3. #20295890	C₂	semidirect
4. #20295888	C₂	semidirect
24. #13182013	C₈	semidirect
49. #11724784	C₂×C₈	semidirect
67. C₁₀₉	#1565	semidirect

H ≅ G/N

type

3. #20295890

C₂

semidirect

4. #20295888

C₂

semidirect

24. #13182013

C₈

semidirect

49. #11724784

C₂×C₈

semidirect

67. C₁₀₉

#1565

semidirect

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

subgroup	index
#20296042	2
C₂×C₄⋊#11724786	2
#20295890	2
#20295888	2
M₁₆×Dic₁₀₉	2
#1565	109

subgroup

index

M₁₆×Dic₁₀₉

109

Subgroups — #28378585

Conjugacy classes of subgroups

Complements

Maximal subgroups