Group #28364316 — subgroups

Subgroups — #28364316

Order 13952.

266 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20295049	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812626	C₂	1	2	yes	maximal, complemented
3	#20295010	C₂	1	2	yes	maximal, complemented
4	#20294784	C₂	1	1	yes	maximal, characteristic, complemented
5	#575×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	1	yes	characteristic
7	C₂×#13182013	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	#16812628	—	2	2	no	—
10	#16812626	—	2	2	no	—
11	#16812626	—	2	2	no	—
12	#16812618	—	2	2	no	—
13	#16812616	—	2	2	no	—
14	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
15	#137×C₁₀₉	—	2	2	no	—
16	AGL(1,ℤ/8)×C₁₀₉	—	2	2	no	—
17	#135×C₁₀₉	—	2	2	no	—
18	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂²×Dic₁₀₉	—	2	1	no	—
20	#13182013	—	2	2	no	—
21	#13182013	—	2	2	no	—
22	#13182013	—	2	2	no	—
23	C₂×#11724784	—	2	1	no	—
24	C₂×#11724784	—	2	2	no	—
25	C₄×Dic₁₀₉	—	2	1	no	—
26	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
28	#41×C₁₀₉	—	2	2	no	—
29	#41×C₁₀₉	—	2	2	no	—
30	#41×C₁₀₉	—	2	2	no	—
31	#41×C₁₀₉	—	2	2	no	—
32	C₂×Q₈×C₁₀₉	D₄	1	2	yes	—
33	C₂×D₄×C₁₀₉	—	2	2	no	—
34	C₂×D₄×C₁₀₉	D₄	1	2	yes	—
35	C₂²×C₄×C₁₀₉	—	2	2	no	—
36	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
37	Q₁₆×C₁₀₉	—	2	2	no	—
38	SD₁₆×C₁₀₉	—	2	4	no	—
39	D₈×C₁₀₉	—	2	2	no	—
40	M₁₆×C₁₀₉	—	2	2	no	—
41	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
42	C₂³×C₁₀₉	—	2	2	no	—
43	C₂³×C₁₀₉	#41	1	1	yes	characteristic
44	Q₈×C₁₀₉	—	2	2	no	—
45	Q₈×C₁₀₉	—	2	2	no	—
46	D₄×C₁₀₉	—	2	2	no	—
47	D₄×C₁₀₉	—	2	2	no	—
48	D₄×C₁₀₉	—	4	2	no	—
49	D₄×C₁₀₉	—	4	2	no	—
50	C₂×C₄×C₁₀₉	—	2	2	no	—
51	C₂×C₄×C₁₀₉	—	4	2	no	—
52	C₂×C₄×C₁₀₉	—	4	2	no	—
53	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₀₉	#41	1	2	yes	—
57	C₂×Dic₁₀₉	—	2	1	no	—
58	C₂×Dic₁₀₉	—	2	1	no	—
59	C₂×Dic₁₀₉	—	4	2	no	—
60	C₈×C₁₀₉	—	2	2	no	—
61	#11724784	—	4	1	no	—
62	#11724784	—	4	2	no	—
63	C₂²×C₁₀₉	—	4	1	no	—
64	C₂²×C₁₀₉	—	4	2	no	—
65	C₂²×C₁₀₉	—	8	2	no	—
66	C₂²×C₁₀₉	#121	1	1	yes	characteristic
67	C₂²×C₁₀₉	#123	1	2	yes	—
68	C₄×C₁₀₉	—	4	2	no	—
69	C₄×C₁₀₉	#120	1	1	yes	characteristic
70	C₄×C₁₀₉	#121	1	2	yes	—
71	C₄×C₁₀₉	#124	1	1	yes	characteristic
72	Dic₁₀₉	—	2	2	no	—
73	Dic₁₀₉	—	4	1	no	—
74	C₂×C₁₀₉	—	2	1	no	—
75	C₂×C₁₀₉	—	2	2	no	—
76	C₂×C₁₀₉	—	8	2	no	—
77	C₂×C₁₀₉	#394	1	1	yes	characteristic
78	#2050	—	109	1	no	maximal, Sylow
79	C₁₀₉	#2050	1	1	yes	characteristic, complemented, Sylow
80	#575	D₁₀₉	1	1	yes	characteristic
81	#423	—	109	1	no	—
82	C₂×#104	—	109	2	no	—
83	#413	—	109	2	no	—
84	#338	—	109	1	no	—
85	C₂×#41	C₂×D₁₀₉	1	2	yes	—
86	#137	—	2	2	no	—
87	AGL(1,ℤ/8)	—	2	2	no	—
88	#135	—	2	2	no	—
89	C₂×M₁₆	—	109	2	no	—
90	C₂×M₁₆	—	109	1	no	—
91	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
92	C₂×C₄²	—	109	1	no	—
93	C₄⋊C₈	—	218	2	no	—
94	#104	—	218	2	no	—
95	#104	—	218	2	no	—
96	#101	—	218	2	no	—
97	#100	—	218	2	no	—
98	#41	—	2	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	#41	—	2	2	no	—
102	C₂×Q₈	C₂²⋊D₁₀₉	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	C₂²⋊D₁₀₉	1	2	yes	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	218	1	no	—
107	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
108	Q₁₆	—	2	2	no	—
109	SD₁₆	—	2	4	no	—
110	D₈	—	2	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	218	2	no	—
113	M₁₆	—	218	2	no	—
114	M₁₆	—	218	2	no	—
115	C₂×C₈	—	218	1	no	—
116	C₂×C₈	—	218	2	no	—
117	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
118	C₄²	—	109	1	no	—
119	C₄²	—	109	1	no	—
120	C₄²	—	218	1	no	—
121	C₂³	—	2	2	no	—
122	C₂³	#13182028	1	1	yes	characteristic
123	Q₈	—	2	2	no	—
124	Q₈	—	2	2	no	—
125	D₄	—	2	2	no	—
126	D₄	—	2	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	436	2	no	—
133	C₂×C₄	—	218	1	no	—
134	C₂×C₄	—	218	1	no	—
135	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
138	C₂×C₄	#13182028	1	2	yes	—
139	C₈	—	2	2	no	—
140	C₈	—	436	1	no	—
141	C₈	—	436	2	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	8	2	no	—
145	C₂²	#16812670	1	1	yes	characteristic
146	C₂²	#16812680	1	2	yes	—
147	C₄	—	4	2	no	—
148	C₄	—	436	1	no	—
149	C₄	—	218	2	no	—
150	C₄	#16812668	1	1	yes	characteristic
151	C₄	#16812671	1	2	yes	—
152	C₄	#16812686	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	2	no	—
155	C₂	—	8	2	no	—
156	C₂	#20294940	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295049

C₂

yes

maximal, characteristic, complemented

C₂×#16812626

C₂

yes

maximal, complemented

#20295010

C₂

yes

maximal, complemented

#20294784

C₂

yes

maximal, characteristic, complemented

#575×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

characteristic

C₂×#13182013

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

—

—

—

—

C₂×#41×C₁₀₉

C₂²

yes

—

#137×C₁₀₉

—

AGL(1,ℤ/8)×C₁₀₉

—

#135×C₁₀₉

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

—

—

—

—

—

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

#41×C₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

Q₁₆×C₁₀₉

—

SD₁₆×C₁₀₉

—

D₈×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

#41

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#121

yes

characteristic

C₂²×C₁₀₉

#123

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#120

yes

characteristic

C₄×C₁₀₉

#121

yes

—

C₄×C₁₀₉

#124

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#394

yes

characteristic

#2050

—

109

maximal, Sylow

C₁₀₉

#2050

yes

characteristic, complemented, Sylow

#575

D₁₀₉

yes

characteristic

#423

—

109

—

C₂×#104

—

109

—

#413

—

109

—

#338

—

109

—

C₂×#41

C₂×D₁₀₉

yes

—

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

C₂×C₄²

—

109

—

C₄⋊C₈

—

218

—

#104

—

218

—

#104

—

218

—

#101

—

218

—

#100

—

218

—

#41

—

#41

—

100

#41

—

101

#41

—

102

C₂×Q₈

C₂²⋊D₁₀₉

yes

—

103

C₂×D₄

—

104

C₂×D₄

C₂²⋊D₁₀₉

yes

—

105

C₂²×C₄

—

106

C₂²×C₄

—

218

—

107

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

108

Q₁₆

—

109

SD₁₆

—

110

D₈

—

111

M₁₆

—

112

M₁₆

—

218

—

113

M₁₆

—

218

—

114

M₁₆

—

218

—

115

C₂×C₈

—

218

—

116

C₂×C₈

—

218

—

117

C₂×C₈

C₂²⋊D₁₀₉

yes

—

118

C₄²

—

109

—

119

C₄²

—

109

—

120

C₄²

—

218

—

121

C₂³

—

122

C₂³

#13182028

yes

characteristic

123

Q₈

—

124

Q₈

—

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

436

—

133

C₂×C₄

—

218

—

134

C₂×C₄

—

218

—

135

C₂×C₄

D₄×D₁₀₉

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

137

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

138

C₂×C₄

#13182028

yes

—

139

C₈

—

140

C₈

—

436

—

141

C₈

—

436

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#16812670

yes

characteristic

146

C₂²

#16812680

yes

—

147

C₄

—

148

C₄

—

436

—

149

C₄

—

218

—

150

C₄

#16812668

yes

characteristic

151

C₄

#16812671

yes

—

152

C₄

#16812686

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

#20294940

yes

characteristic

	subgroup	G/H
center	C₄	#16812686
commutator, cosocle	C₂²×C₄×C₁₀₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting, prime radical	#575×C₁₀₉	C₂
socle	C₂×C₁₀₉	#394
hypercenter	#575	D₁₀₉
hypocenter	C₁₀₉	#2050
LCS	G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

subgroup

G/H

center

C₄

#16812686

commutator, cosocle

C₂²×C₄×C₁₀₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting, prime radical

#575×C₁₀₉

C₂

C₂×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

N	H ≅ G/N	type
1. #20295049	C₂	semidirect
2. C₂×#16812626	C₂	semidirect
3. #20295010	C₂	semidirect
4. #20294784	C₂	semidirect
79. C₁₀₉	#2050	semidirect

H ≅ G/N

type

1. #20295049

C₂

semidirect

2. C₂×#16812626

C₂

semidirect

3. #20295010

C₂

semidirect

4. #20294784

C₂

semidirect

79. C₁₀₉

#2050

semidirect

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

subgroup	index
#20295049	2
C₂×#16812626	2
#20295010	2
#20294784	2
#575×C₁₀₉	2
#2050	109

subgroup

index

#575×C₁₀₉

109

Subgroups — #28364316

Conjugacy classes of subgroups

Complements

Maximal subgroups