Group #28363865 — subgroups

Subgroups — #28363865

Order 13952.

320 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20295050	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812626	C₂	1	4	yes	maximal, complemented
3	#20294872	C₂	1	1	yes	maximal, characteristic, complemented
4	#532×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
7	C₄⋊#11724784	—	2	2	no	—
8	#16812626	—	2	8	no	—
9	#16812606	—	2	2	no	—
10	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
12	#127×C₁₀₉	C₄	1	1	yes	characteristic, complemented
13	#124×C₁₀₉	C₄	1	2	yes	complemented
14	#121×C₁₀₉	—	2	2	no	—
15	C₄×D₄×C₁₀₉	—	2	2	no	—
16	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₀₉	—	2	2	no	—
18	#13182013	—	2	4	no	—
19	C₂×#11724784	—	2	2	no	—
20	C₄×Dic₁₀₉	—	2	2	no	—
21	C₄×Dic₁₀₉	D₄	1	4	yes	complemented
22	#41×C₁₀₉	—	2	4	no	—
23	#41×C₁₀₉	—	2	4	no	—
24	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂×D₄×C₁₀₉	—	2	2	no	—
26	C₂×D₄×C₁₀₉	—	2	2	no	—
27	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₀₉	—	2	2	no	—
29	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
30	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₁₀₉	—	2	2	no	—
32	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
33	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₁₀₉	D₄	1	2	yes	—
36	C₂³×C₁₀₉	—	2	2	no	—
37	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₁₀₉	—	2	4	no	—
39	D₄×C₁₀₉	—	2	4	no	—
40	D₄×C₁₀₉	—	4	4	no	—
41	D₄×C₁₀₉	—	4	4	no	—
42	C₂×C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₄×C₁₀₉	—	4	2	no	—
44	C₂×C₄×C₁₀₉	—	4	2	no	—
45	C₂×C₄×C₁₀₉	—	4	2	no	—
46	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	#41	1	2	yes	—
52	C₂×Dic₁₀₉	—	2	4	no	—
53	C₂×Dic₁₀₉	—	4	4	no	—
54	#11724784	—	4	2	no	—
55	C₂²×C₁₀₉	—	2	2	no	—
56	C₂²×C₁₀₉	—	4	2	no	—
57	C₂²×C₁₀₉	—	4	4	no	—
58	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₀₉	#121	1	2	yes	—
60	C₄×C₁₀₉	—	2	2	no	—
61	C₄×C₁₀₉	—	2	2	no	—
62	C₄×C₁₀₉	—	4	2	no	—
63	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
64	C₄×C₁₀₉	#124	1	1	yes	characteristic
65	C₄×C₁₀₉	#127	1	1	yes	characteristic
66	Dic₁₀₉	—	4	4	no	—
67	C₂×C₁₀₉	—	2	2	no	—
68	C₂×C₁₀₉	—	8	2	no	—
69	C₂×C₁₀₉	#390	1	1	yes	characteristic
70	C₂×C₁₀₉	#421	1	2	yes	—
71	#1963	—	109	1	no	maximal, Sylow
72	C₁₀₉	#1963	1	1	yes	characteristic, complemented, Sylow
73	#532	D₁₀₉	1	1	yes	characteristic
74	#423	—	109	1	no	—
75	C₂×#104	—	109	4	no	—
76	#376	—	109	1	no	—
77	C₂×#41	C₂×D₁₀₉	1	2	yes	—
78	C₂×M₁₆	—	109	2	no	—
79	C₄⋊Q₈	Dic₁₀₉	1	1	yes	characteristic, complemented
80	#127	Dic₁₀₉	1	1	yes	characteristic, complemented
81	#124	Dic₁₀₉	1	2	yes	complemented
82	#121	—	2	2	no	—
83	C₄×D₄	—	2	2	no	—
84	C₂×C₄²	—	109	2	no	—
85	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
86	C₄⋊C₈	—	218	2	no	—
87	#104	—	218	8	no	—
88	#95	—	218	2	no	—
89	#41	—	2	4	no	—
90	#41	—	2	4	no	—
91	C₂×Q₈	C₂×Dic₁₀₉	1	2	yes	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	C₂×Dic₁₀₉	1	2	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	218	2	no	—
97	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
98	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
99	M₁₆	—	218	4	no	—
100	C₂×C₈	—	218	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₂²⋊C₄	—	2	4	no	—
103	C₄²	—	109	4	no	—
104	C₄²	—	218	2	no	—
105	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
106	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
107	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
108	C₂³	—	2	2	no	—
109	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
110	Q₈	—	2	4	no	—
111	D₄	—	2	4	no	—
112	D₄	—	4	4	no	—
113	D₄	—	4	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	436	4	no	—
119	C₂×C₄	—	218	4	no	—
120	C₂×C₄	C₂²⋊Dic₁₀₉	1	2	yes	—
121	C₂×C₄	C₂²⋊Dic₁₀₉	1	2	yes	—
122	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
123	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
125	C₂×C₄	#13182028	1	2	yes	—
126	C₈	—	436	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	4	2	no	—
129	C₂²	—	4	4	no	—
130	C₂²	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
131	C₂²	#16812670	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	2	no	—
135	C₄	—	436	4	no	—
136	C₄	D₄×Dic₁₀₉	1	2	yes	—
137	C₄	#16812686	1	1	yes	characteristic
138	C₄	#16812695	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	8	2	no	—
141	C₂	#20294929	1	1	yes	characteristic
142	C₂	#20295043	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295050

C₂

yes

maximal, characteristic, complemented

C₂×#16812626

C₂

yes

maximal, complemented

#20294872

C₂

yes

maximal, characteristic, complemented

#532×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₄⋊#11724784

—

#16812626

—

#16812606

—

C₂×#41×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₄

yes

characteristic, complemented

#127×C₁₀₉

C₄

yes

characteristic, complemented

#124×C₁₀₉

C₄

yes

complemented

#121×C₁₀₉

—

C₄×D₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

—

#13182013

—

C₂×#11724784

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

complemented

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

#121

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#390

yes

characteristic

C₂×C₁₀₉

#421

yes

—

#1963

—

109

maximal, Sylow

C₁₀₉

#1963

yes

characteristic, complemented, Sylow

#532

D₁₀₉

yes

characteristic

#423

—

109

—

C₂×#104

—

109

—

#376

—

109

—

C₂×#41

C₂×D₁₀₉

yes

—

C₂×M₁₆

—

109

—

C₄⋊Q₈

Dic₁₀₉

yes

characteristic, complemented

#127

Dic₁₀₉

yes

characteristic, complemented

#124

Dic₁₀₉

yes

complemented

#121

—

C₄×D₄

—

C₂×C₄²

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

218

—

#104

—

218

—

#95

—

218

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₁₀₉

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

218

—

100

C₂×C₈

—

218

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

109

—

104

C₄²

—

218

—

105

C₄²

C₂×Dic₁₀₉

yes

characteristic

106

C₄²

C₂×Dic₁₀₉

yes

characteristic

107

C₄²

C₂²⋊D₁₀₉

yes

—

108

C₂³

—

109

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

436

—

119

C₂×C₄

—

218

—

120

C₂×C₄

C₂²⋊Dic₁₀₉

yes

—

121

C₂×C₄

C₂²⋊Dic₁₀₉

yes

—

122

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

123

C₂×C₄

D₄×D₁₀₉

yes

—

124

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

125

C₂×C₄

#13182028

yes

—

126

C₈

—

436

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

131

C₂²

#16812670

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

436

—

136

C₄

D₄×Dic₁₀₉

yes

—

137

C₄

#16812686

yes

characteristic

138

C₄

#16812695

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#20294929

yes

characteristic

142

C₂

#20295043

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting, prime radical	#532×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#532	D₁₀₉
hypocenter	C₁₀₉	#1963
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting, prime radical

#532×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #20295050	C₂	semidirect
2. C₂×#16812626	C₂	semidirect
3. #20294872	C₂	semidirect
11. C₄⋊Q₈×C₁₀₉	C₄	semidirect
12. #127×C₁₀₉	C₄	semidirect
13. #124×C₁₀₉	C₄	semidirect
21. C₄×Dic₁₀₉	D₄	semidirect
72. C₁₀₉	#1963	semidirect
79. C₄⋊Q₈	Dic₁₀₉	semidirect
80. #127	Dic₁₀₉	semidirect
81. #124	Dic₁₀₉	semidirect

H ≅ G/N

type

1. #20295050

C₂

semidirect

2. C₂×#16812626

C₂

semidirect

3. #20294872

C₂

semidirect

11. C₄⋊Q₈×C₁₀₉

C₄

semidirect

12. #127×C₁₀₉

C₄

semidirect

13. #124×C₁₀₉

C₄

semidirect

21. C₄×Dic₁₀₉

D₄

semidirect

72. C₁₀₉

#1963

semidirect

79. C₄⋊Q₈

Dic₁₀₉

semidirect

80. #127

Dic₁₀₉

semidirect

81. #124

Dic₁₀₉

semidirect

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

subgroup	index
#20295050	2
C₂×#16812626	2
#20294872	2
#532×C₁₀₉	2
#1963	109

subgroup

index

#532×C₁₀₉

109

Subgroups — #28363865

Conjugacy classes of subgroups

Complements

Maximal subgroups