Finite Group Databasebeta

← #28346419

Subgroups — #28346419

Order 13696.

418 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 133 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#20293191C211yesmaximal, characteristic
2#20292951C212yesmaximal, complemented
3#20292951C212yesmaximal, complemented
4#20292945C212yesmaximal, complemented
5#20292940C212yesmaximal, complemented
6#20292667C211yesmaximal, characteristic, complemented
7#20292663C211yesmaximal, characteristic, complemented
8#20292631C211yesmaximal, characteristic, complemented
9C2×#16812029C211yesmaximal, characteristic, complemented
10#20292594C211yesmaximal, characteristic, complemented
11#520×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13C2×#13133802C2211yescharacteristic
14C22×C107⋊C8C2211yescharacteristic
15#16812105—21no—
16#16812098—21no—
17#16812083C2212yes—
18#16812083C2212yes—
19#16812077—22no—
20#16812068C2211yescharacteristic
21#16812042C2211yescharacteristic, complemented
22#16812042C2211yescharacteristic, complemented
23#16812042C2212yescomplemented
24#16812040C2211yescharacteristic, complemented
25#16812040C2211yescharacteristic, complemented
26#16812040C2212yescomplemented
27#16812029C2212yes—
28#16812029C2212yes—
29#16812029C2212yes—
30#16812029C2212yes—
31C22⋊(C107⋊C8)C2212yescomplemented
32C22⋊(C107⋊C8)C2212yescomplemented
33C2×#41×C107C2211yescharacteristic
34C4×Q8×C107—21no—
35C4×D4×C107—21no—
36C4×D4×C107—22no—
37#117×C107C2211yescharacteristic
38#117×C107C2212yescomplemented
39C2×C4⋊C4×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2212yescomplemented
41C23×D107—21no—
42C2×D428—22no—
43C2×D428—22no—
44C2×D428—21no—
45C2×D428C2312yes—
46C2×C22⋊D107—22no—
47C2×C22⋊D107—22no—
48#13133802—22no—
49C2×C107⋊C8—22no—
50C2×C107⋊C8C2311yescharacteristic
51C2×C107⋊C8C2312yes—
52C2×C107⋊C8C2311yescharacteristic
53C4⋊Dic107C2311yescharacteristic
54C4⋊Dic107C2311yescharacteristic
55C22⋊Dic107—21no—
56(C2×D107)⋊C4—22no—
57(C2×D107)⋊C4—22no—
58C4×Dic107—21no—
59#41×C107—22no—
60#41×C107—22no—
61C2×Q8×C107D411yescharacteristic
62C2×D4×C107D412yes—
63C2×D4×C107D411yescharacteristic
64C22×C4×C107—21no—
65C22×C4×C107—22no—
66C22×C4×C107C2312yes—
67C22×C4×C107C2311yescharacteristic
68C4⋊C4×C107—21no—
69C4⋊C4×C107—22no—
70C4⋊C4×C107C2312yes—
71C4⋊C4×C107C2311yescharacteristic
72C4⋊C4×C107C2311yescharacteristic
73C22⋊C4×C107—21no—
74C22⋊C4×C107—22no—
75C42×C107—21no—
76C42×C107—22no—
77C23×C107C2×D412yes—
78C23×C107C2×D411yescharacteristic
79C22×D107—22no—
80C22×D107—42no—
81C22×D107—81no—
82C22×D107—81no—
83Q8×C107—22no—
84D4×C107—24no—
85D4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107—21no—
89C2×C4×C107—21no—
90C2×C4×C107—22no—
91C2×C4×C107—81no—
92C2×C4×C107—82no—
93C2×C4×C107C2×D412yes—
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107#4112yes—
96C2×C4×C107#4112yes—
97C2×C4×C107C2411yescharacteristic
98C22⋊D107—82no—
99C22⋊D107—82no—
100C2×Dic107—21no—
101C2×Dic107—21no—
102D428—22no—
103D428—22no—
104D428—42no—
105C107⋊C8—21no—
106C107⋊C8—21no—
107C107⋊C8—22no—
108C22×C107—21no—
109C22×C107—21no—
110C22×C107—22no—
111C22×C107—22no—
112C22×C107—22no—
113C22×C107#12312yes—
114C22×C107C22×D411yescharacteristic
115C2×D107—42no—
116C2×D107—42no—
117C2×D107—42no—
118C2×D107—84no—
119C4×C107—22no—
120C4×C107—41no—
121C4×C107—42no—
122C4×C107—41no—
123C4×C107#12312yes—
124C4×C107C2×#4111yescharacteristic
125C4×C107C2×#4111yescharacteristic
126Dic107—41no—
127Dic107—41no—
128C2×C107—22no—
129C2×C107—42no—
130C2×C107C2×#12311yescharacteristic
131C2×C107#57611yescharacteristic
132C2×C107#57711yescharacteristic
133D107—82no—
134#3096—1071nomaximal, Sylow
135C107#309611yescharacteristic, complemented, Sylow
136#535—1071no—
137#520D10711yescharacteristic, complemented
138#482—1072no—
139#482—1072no—
140#481—1072no—
141#480—1072no—
142#428—1071no—
143#427—1071no—
144#418—1071no—
145C2×#102—1071no—
146#408—1071no—
147C2×#41C2×D10711yescharacteristic
148C22×D4—1071no—
149C2×M16—1071no—
150C22×C8—1071no—
151#127—2141no—
152#124—2141no—
153#121—1072no—
154#121—1072no—
155#120—2142no—
156C4×Q8—21no—
157C4×D4—21no—
158C4×D4—22no—
159#117—1071no—
160#117C2×D10711yescharacteristic
161#117C2×D10712yescomplemented
162C2×C4⋊C4C2×D10711yescharacteristic
163C2×C4⋊C4C2×D10712yescomplemented
164#107—1071no—
165#107—1071no—
166#107—1072no—
167#106—1071no—
168#106—1071no—
169#106—1072no—
170#102—1072no—
171#102—1072no—
172#102—1072no—
173#102—1072no—
174C22⋊C8—1072no—
175C22⋊C8—1072no—
176C24—2141no—
177#41—22no—
178#41—22no—
179C2×Q8C22⋊D10711yescharacteristic
180C2×D4—1072no—
181C2×D4—2142no—
182C2×D4—2142no—
183C2×D4—2141no—
184C2×D4—2142no—
185C2×D4—2142no—
186C2×D4C22⋊D10712yes—
187C2×D4C22⋊D10711yescharacteristic
188C22×C4—21no—
189C22×C4—22no—
190C22×C4C22×D10712yes—
191C22×C4C22×D10711yescharacteristic
192M16—2142no—
193C2×C8—1071no—
194C2×C8—1072no—
195C2×C8—1071no—
196C2×C8—2142no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—1071no—
200C4⋊C4—1071no—
201C4⋊C4C22×D10712yes—
202C4⋊C4C22×D10711yescharacteristic
203C4⋊C4C22×D10711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—2141no—
207C22⋊C4—2142no—
208C22⋊C4—2142no—
209C42—21no—
210C42—22no—
211C42—2141no—
212C23—8561no—
213C23—8561no—
214C23—4282no—
215C23—2142no—
216C23C2×C22⋊D10712yes—
217C23C2×C22⋊D10711yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—8562no—
222D4—8562no—
223D4—4282no—
224D4—2142no—
225D4—2142no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—2141no—
234C2×C4—2141no—
235C2×C4C2×C22⋊D10712yes—
236C2×C4C2×C22⋊D10711yescharacteristic
237C2×C4#1313381812yes—
238C2×C4#1313381812yes—
239C2×C4C23×D10711yescharacteristic
240C8—2141no—
241C8—2141no—
242C8—2142no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—8564no—
249C22—4282no—
250C22—4282no—
251C22—4282no—
252C22#1681209212yes—
253C22C22×C22⋊D10711yescharacteristic
254C4—22no—
255C4—41no—
256C4—42no—
257C4—41no—
258C4—4281no—
259C4—4281no—
260C4#1681209212yes—
261C4C2×#1313381811yescharacteristic
262C4C2×#1313381811yescharacteristic
263C2—22no—
264C2—42no—
265C2—8562no—
266C2C2×#1681209211yescharacteristic
267C2#2029343011yescharacteristic
268C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#520×C107C2
socleC22×C107C22×D4
prime radical#20293191C2
hypercenter#520D107
hypocenterC107#3096
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×#41 ◁ #520

Complements

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

NH ≅ G/Ntype
2. #20292951C2semidirect
3. #20292951C2semidirect
4. #20292945C2semidirect
5. #20292940C2semidirect
6. #20292667C2semidirect
7. #20292663C2semidirect
8. #20292631C2semidirect
9. C2×#16812029C2semidirect
10. #20292594C2semidirect
11. #520×C107C2semidirect
21. #16812042C22semidirect
22. #16812042C22semidirect
23. #16812042C22semidirect
24. #16812040C22semidirect
25. #16812040C22semidirect
26. #16812040C22semidirect
31. C22⋊(C107⋊C8)C22semidirect
32. C22⋊(C107⋊C8)C22semidirect
38. #117×C107C22semidirect
40. C2×C4⋊C4×C107C22semidirect
135. C107#3096semidirect
137. #520D107semidirect
161. #117C2×D107semidirect
163. C2×C4⋊C4C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931912
#202929512
#202929452
#202929402
#202926672
#202926632
#202926312
C2×#168120292
#202925942
#520×C1072
#3096107