Group #28377883 — subgroups

Subgroups — #28377883

Order 13952.

164 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 46 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	#20296046	C₂	1	1	yes	maximal, characteristic
2	#20296046	C₂	1	1	yes	maximal, characteristic
3	C₄×#13182008	C₂	1	1	yes	maximal, characteristic
4	#16812766	—	2	1	no	—
5	#16812766	C₄	1	2	yes	complemented
6	C₂²⋊#11724786	—	2	1	no	—
7	C₂²⋊#11724786	—	2	1	no	—
8	#16812754	—	2	1	no	—
9	#16812754	C₄	1	2	yes	complemented
10	C₂×#13182008	—	2	1	no	—
11	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	#16812606	—	2	1	no	—
14	C₂×C₄²×C₁₀₉	C₄	1	1	yes	characteristic
15	C₂×#11724786	—	2	2	no	—
16	C₂×#11724786	—	2	2	no	—
17	C₂²×Dic₁₀₉	—	2	1	no	—
18	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
19	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
20	#13182008	—	2	4	no	—
21	C₄×Dic₁₀₉	—	4	1	no	—
22	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
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	C₂²×C₄×C₁₀₉	—	2	1	no	—
27	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₀₉	—	4	1	no	—
29	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	—	2	1	no	—
31	C₂×C₄×C₁₀₉	—	2	1	no	—
32	C₂×C₄×C₁₀₉	—	2	2	no	—
33	C₂×C₄×C₁₀₉	—	4	1	no	—
34	C₂×C₄×C₁₀₉	—	4	1	no	—
35	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×Dic₁₀₉	—	2	2	no	—
38	C₂×Dic₁₀₉	—	2	2	no	—
39	C₂×Dic₁₀₉	—	4	1	no	—
40	C₂×Dic₁₀₉	—	4	1	no	—
41	C₂×Dic₁₀₉	—	4	1	no	—
42	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
46	#11724786	—	4	2	no	—
47	#11724786	—	4	2	no	—
48	C₂²×C₁₀₉	—	2	1	no	—
49	C₂²×C₁₀₉	—	2	1	no	—
50	C₂²×C₁₀₉	#95	1	1	yes	characteristic
51	C₂²×C₁₀₉	#100	1	1	yes	characteristic
52	C₂²×C₁₀₉	#101	1	1	yes	characteristic
53	C₄×C₁₀₉	—	2	1	no	—
54	C₄×C₁₀₉	—	2	1	no	—
55	C₄×C₁₀₉	—	4	2	no	—
56	Dic₁₀₉	—	2	1	no	—
57	Dic₁₀₉	—	2	1	no	—
58	Dic₁₀₉	—	2	1	no	—
59	Dic₁₀₉	—	4	2	no	—
60	Dic₁₀₉	#104	1	2	yes	—
61	C₂×C₁₀₉	—	2	1	no	—
62	C₂×C₁₀₉	—	2	1	no	—
63	C₂×C₁₀₉	H(3,ℤ/4)	1	1	yes	characteristic
64	C₂×C₁₀₉	#339	1	1	yes	characteristic
65	C₂×C₁₀₉	#343	1	1	yes	characteristic
66	#1285	—	109	1	no	maximal, Sylow
67	C₁₀₉	#1285	1	1	yes	characteristic, complemented, Sylow
68	#432	—	109	1	no	—
69	#432	—	109	1	no	—
70	C₄×C₄⋊C₄	—	109	1	no	—
71	C₂×C₄⋊C₄	—	218	1	no	—
72	C₂×C₄²	—	109	1	no	—
73	C₂×C₄²	—	109	1	no	—
74	C₂×C₄²	#89137	1	1	yes	characteristic
75	C₄⋊C₈	—	109	2	no	—
76	C₄⋊C₈	—	218	1	no	—
77	C₂²⋊C₈	—	218	1	no	—
78	C₂²⋊C₈	—	218	1	no	—
79	#97	—	109	2	no	—
80	#97	—	218	1	no	—
81	#95	—	218	1	no	—
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	109	1	no	—
84	C₂²×C₄	—	109	1	no	—
85	C₂²×C₄	—	218	1	no	—
86	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
87	C₂×C₈	—	218	2	no	—
88	C₂×C₈	—	218	2	no	—
89	C₄⋊C₄	—	218	4	no	—
90	C₄²	—	4	1	no	—
91	C₄²	—	109	1	no	—
92	C₄²	—	109	1	no	—
93	C₄²	—	109	1	no	—
94	C₄²	—	109	1	no	—
95	C₄²	—	436	1	no	—
96	C₂³	C₂²⋊#89137	1	1	yes	characteristic
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	109	1	no	—
103	C₂×C₄	—	109	1	no	—
104	C₂×C₄	—	109	1	no	—
105	C₂×C₄	—	109	1	no	—
106	C₂×C₄	—	436	1	no	—
107	C₂×C₄	—	436	1	no	—
108	C₂×C₄	—	436	1	no	—
109	C₂×C₄	—	218	2	no	—
110	C₂×C₄	—	218	2	no	—
111	C₂×C₄	C₄×#89137	1	1	yes	characteristic
112	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
113	C₈	—	436	2	no	—
114	C₈	—	436	2	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
118	C₂²	#16812759	1	1	yes	characteristic
119	C₂²	#16812760	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	—	4	2	no	—
123	C₄	—	109	2	no	—
124	C₄	—	436	2	no	—
125	C₄	—	218	1	no	—
126	C₄	—	218	1	no	—
127	C₄	—	218	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#20295893	1	1	yes	characteristic
131	C₂	#20295897	1	1	yes	characteristic
132	C₂	#20295906	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296046

C₂

yes

maximal, characteristic

#20296046

C₂

yes

maximal, characteristic

C₄×#13182008

C₂

yes

maximal, characteristic

#16812766

—

#16812766

C₄

yes

—

—

—

C₄

yes

complemented

C₂×#13182008

—

C₂×C₄×Dic₁₀₉

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812606

—

C₂×C₄²×C₁₀₉

C₄

yes

characteristic

C₂×#11724786

—

C₂×#11724786

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂²×Dic₁₀₉

D₄

yes

characteristic

#13182008

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

#11724786

—

#11724786

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#104

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

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

—

C₂×C₄²

#89137

yes

characteristic

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₂²⋊C₈

—

218

—

C₂²⋊C₈

—

218

—

#97

—

109

—

#97

—

218

—

#95

—

218

—

C₂²×C₄

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂×#89137

yes

characteristic

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

218

—

C₄²

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

436

—

C₂³

C₂²⋊#89137

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

109

—

103

C₂×C₄

—

109

—

104

C₂×C₄

—

109

—

105

C₂×C₄

—

109

—

106

C₂×C₄

—

436

—

107

C₂×C₄

—

436

—

108

C₂×C₄

—

436

—

109

C₂×C₄

—

218

—

110

C₂×C₄

—

218

—

111

C₂×C₄

C₄×#89137

yes

characteristic

112

C₂×C₄

C₄⋊#89137

yes

characteristic

113

C₈

—

436

—

114

C₈

—

436

—

115

C₂²

—

116

C₂²

—

117

C₂²

(C₂×#89137)⋊C₄

yes

characteristic

118

C₂²

#16812759

yes

characteristic

119

C₂²

#16812760

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

109

—

124

C₄

—

436

—

125

C₄

—

218

—

126

C₄

—

218

—

127

C₄

—

218

—

128

C₂

—

129

C₂

—

130

C₂

#20295893

yes

characteristic

131

C₂

#20295897

yes

characteristic

132

C₂

#20295906

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#89137)⋊C₄
commutator	C₂×C₄×C₁₀₉	C₄²
Frattini	C₂²×C₄	C₂×#89137
Fitting	C₂×C₄²×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#95
prime radical	C₂³×C₁₀₉	C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₀₉	C₂²
hypercenter	C₂×C₄²	#89137
hypocenter	C₁₀₉	#1285
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂²

(C₂×#89137)⋊C₄

commutator

C₂×C₄×C₁₀₉

C₄²

Frattini

C₂²×C₄

C₂×#89137

Fitting

C₂×C₄²×C₁₀₉

C₄

socle

C₂²×C₁₀₉

#95

prime radical

C₂³×C₁₀₉

C₂²⋊C₄

cosocle

C₂×C₄×Dic₁₀₉

C₂²

C₂×C₄²

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
5. #16812766	C₄	semidirect
9. #16812754	C₄	semidirect
67. C₁₀₉	#1285	semidirect

H ≅ G/N

type

5. #16812766

C₄

semidirect

9. #16812754

C₄

semidirect

67. C₁₀₉

#1285

semidirect

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

subgroup	index
#20296046	2
C₄×#13182008	2
#1285	109

subgroup

index

#20296046

C₄×#13182008

#1285

109

Subgroups — #28377883

Conjugacy classes of subgroups

Complements

Maximal subgroups