Finite Group Databasebeta

← #28338468

Subgroups — #28338468

Order 13696.

260 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20292816C211yesmaximal, characteristic, complemented
2#20292813C211yesmaximal, characteristic, complemented
3#20292794C211yesmaximal, characteristic, complemented
4#20292420C211yesmaximal, characteristic, complemented
5#20292417C211yesmaximal, characteristic, complemented
6#20292414C211yesmaximal, characteristic, complemented
7#560×C107C211yesmaximal, characteristic, complemented
8#16812154C2211yescharacteristic
9#16812135—21no—
10#16812092—21no—
11#16812078—21no—
12#16812037—21no—
13#16812026C2211yescharacteristic
14#16812024C2211yescharacteristic
15#16812024C2211yescharacteristic
16#16812023—21no—
17#16812023—21no—
18#127×C107C2211yescharacteristic
19#126×C107—21no—
20#123×C107—21no—
21#123×C107C2211yescharacteristic
22#121×C107—21no—
23#121×C107—21no—
24#121×C107—21no—
25#120×C107—21no—
26#120×C107C2211yescharacteristic
27#13133818—21no—
28#13133817—41no—
29C2×C22⋊D107—21no—
30C2×C22⋊D107—21no—
31C2×C22⋊D107—41no—
32D4×D107—41no—
33#13133806—41no—
34#13133803—41no—
35#13133802—21no—
36Dic107⋊C4—41no—
37C22⋊Dic107—21no—
38C22⋊Dic107—21no—
39C22⋊Dic107—41no—
40(C2×D107)⋊C4—41no—
41C24×C107D411yescharacteristic
42C2×D4×C107—21no—
43C2×D4×C107—21no—
44C2×D4×C107—21no—
45C2×D4×C107—21no—
46C2×D4×C107—21no—
47C2×D4×C107D411yescharacteristic, complemented
48C2×D4×C107C2311yescharacteristic
49C22×C4×C107—21no—
50C22×C4×C107D411yescharacteristic, complemented
51C4⋊C4×C107—21no—
52C4⋊C4×C107—21no—
53C4⋊C4×C107—21no—
54C22⋊C4×C107—21no—
55C22⋊C4×C107—21no—
56C22⋊C4×C107—21no—
57C22⋊C4×C107—21no—
58C22⋊C4×C107—21no—
59C22⋊C4×C107D411yescharacteristic
60C22⋊C4×C107D411yescharacteristic, complemented
61C42×C107D411yescharacteristic, complemented
62C23×C107—21no—
63C23×C107—21no—
64C23×C107—41no—
65C23×C107—81no—
66C23×C107C2×D411yescharacteristic
67C23×C107C2×D411yescharacteristic
68C22×D107—21no—
69C22×D107—21no—
70D4×C107—41no—
71D4×C107—41no—
72D4×C107—81no—
73D4×C107—81no—
74D4×C107—81no—
75D4×C107—81no—
76D4×C107—81no—
77C2×C4×C107—21no—
78C2×C4×C107—21no—
79C2×C4×C107—21no—
80C2×C4×C107—21no—
81C2×C4×C107—41no—
82C2×C4×C107—81no—
83C2×C4×C107C2×D411yescharacteristic
84C22⋊D107—41no—
85C22⋊D107—41no—
86C22⋊D107—41no—
87C22⋊D107—82no—
88C2×Dic107—21no—
89C2×Dic107—21no—
90C2×Dic107—41no—
91C2×Dic107—41no—
92D428—21no—
93C4×D107—41no—
94Dic214—21no—
95C107⋊C8—41no—
96C22×C107—21no—
97C22×C107—21no—
98C22×C107—41no—
99C22×C107—41no—
100C22×C107—41no—
101C22×C107—41no—
102C22×C107—41no—
103C22×C107—41no—
104C22×C107—42no—
105C22×C107—41no—
106C22×C107—41no—
107C22×C107—81no—
108C22×C107#12011yescharacteristic
109C2×D107—41no—
110C2×D107—81no—
111C2×D107—81no—
112C4×C107—21no—
113C4×C107—41no—
114C4×C107—41no—
115C4×C107—41no—
116C4×C107—41no—
117Dic107—41no—
118Dic107—81no—
119Dic107—81no—
120C2×C107—21no—
121C2×C107—41no—
122C2×C107—41no—
123C2×C107—81no—
124C2×C107—81no—
125C2×C107B(4,𝔽2)11yescharacteristic
126D107—81no—
127#2200—1071nomaximal, Sylow
128C107#220011yescharacteristic, complemented, Sylow
129#560D10711yescharacteristic, complemented
130#458—1071no—
131B(4,𝔽2)—1071no—
132#453—1071no—
133#353—1071no—
134#352—1071no—
135AΣL(1,𝔽16)—1071no—
136H(5,𝔽2)—1071no—
137AGL(1,ℤ/8)—2141no—
138#127C2×D10711yescharacteristic
139#126—21no—
140#123—21no—
141#123—2141no—
142#123C2×D10711yescharacteristic
143#121—21no—
144#121—21no—
145#121—21no—
146#120—21no—
147#120—2141no—
148#120C2×D10711yescharacteristic
149#104—2141no—
150#100—1071no—
151#99—1071no—
152#99—1071no—
153#99—2141no—
154#99—2141no—
155C24C22⋊D10711yescharacteristic
156#41—4281no—
157#41—2141no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—4281no—
164C2×D4—4281no—
165C2×D4—2141no—
166C2×D4—2141no—
167C2×D4C22⋊D10711yescharacteristic, complemented
168C2×D4C22×D10711yescharacteristic
169C22×C4—21no—
170C22×C4C22⋊D10711yescharacteristic, complemented
171SD16—4281no—
172D8—4281no—
173M16—2141no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—4281no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—4281no—
184C22⋊C4—4281no—
185C22⋊C4—2141no—
186C22⋊C4—2141no—
187C22⋊C4C22⋊D10711yescharacteristic
188C22⋊C4C22⋊D10711yescharacteristic, complemented
189C42C22⋊D10711yescharacteristic, complemented
190C23—21no—
191C23—21no—
192C23—41no—
193C23—81no—
194C23—2141no—
195C23—2141no—
196C23C2×C22⋊D10711yescharacteristic
197C23C2×C22⋊D10711yescharacteristic
198Q8—2141no—
199D4—41no—
200D4—41no—
201D4—81no—
202D4—81no—
203D4—81no—
204D4—81no—
205D4—81no—
206D4—8562no—
207D4—4281no—
208D4—4281no—
209D4—4281no—
210D4—2141no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—81no—
217C2×C4—4281no—
218C2×C4—4281no—
219C2×C4—4281no—
220C2×C4—2141no—
221C2×C4—2141no—
222C2×C4C2×C22⋊D10711yescharacteristic
223C8—4281no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22—81no—
236C22—8561no—
237C22—8561no—
238C22—4281no—
239C22#1681207811yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—41no—
245C4—8561no—
246C4—8561no—
247C4—4281no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—81no—
253C2—8561no—
254C2#2029281411yescharacteristic
subgroupG/H
centerC2#20292814
commutator, cosocleC2×D4×C107C23
FrattiniC2×D4C22×D107
Fitting#560×C107C2
socleC2×C107B(4,𝔽2)
hypercenter#560D107
hypocenterC107#2200
LCSG ▷ C2×D4×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #560

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20292816C2semidirect
2. #20292813C2semidirect
3. #20292794C2semidirect
4. #20292420C2semidirect
5. #20292417C2semidirect
6. #20292414C2semidirect
7. #560×C107C2semidirect
47. C2×D4×C107D4semidirect
50. C22×C4×C107D4semidirect
60. C22⋊C4×C107D4semidirect
61. C42×C107D4semidirect
128. C107#2200semidirect
129. #560D107semidirect
167. C2×D4C22⋊D107semidirect
170. C22×C4C22⋊D107semidirect
188. C22⋊C4C22⋊D107semidirect
189. C42C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202928162
#202928132
#202927942
#202924202
#202924172
#202924142
#560×C1072
#2200107