Finite Group Databasebeta

← #28346820

Subgroups — #28346820

Order 13696.

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

normal subgroups only

This group has 404 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#20293196C211yesmaximal, characteristic
2#20292993C212yesmaximal, complemented
3#20292989C212yesmaximal, complemented
4#20292984C212yesmaximal, complemented
5#20292975C212yesmaximal, complemented
6#20292631C211yesmaximal, characteristic, complemented
7#20292626C211yesmaximal, characteristic, complemented
8#20292621C211yesmaximal, characteristic, complemented
9C2×#16812029C211yesmaximal, characteristic, complemented
10#20292583C211yesmaximal, characteristic, complemented
11#544×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×D428C2211yescharacteristic
14C2×#13133802C2211yescharacteristic
15C22×C107⋊C8C2211yescharacteristic
16#16812106C2212yes—
17#16812097C2212yes—
18#16812084—21no—
19#16812084—21no—
20#16812079—22no—
21#16812032C2211yescharacteristic, complemented
22#16812032C2211yescharacteristic, complemented
23#16812032C2212yescomplemented
24#16812029C2212yes—
25#16812029C2212yes—
26#16812029C2212yes—
27#16812029C2212yes—
28#16812029C2211yescharacteristic, complemented
29#16812029C2211yescharacteristic, complemented
30#16812029C2212yescomplemented
31#16812020C2212yescomplemented
32C4×C107⋊C8C2212yescomplemented
33C4⋊Q8×C107—21no—
34C4⋊Q8×C107C2212yescomplemented
35#125×C107—21no—
36#125×C107C2212yescomplemented
37C22⋊Q8×C107—22no—
38#117×C107C2211yescharacteristic
39#117×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41C23×D107—21no—
42#13133818—22no—
43#13133818—22no—
44C2×Dic214C2311yescharacteristic
45C2×D428—22no—
46C2×D428—22no—
47C2×D428—21no—
48C2×D428—22no—
49C2×D428—22no—
50C2×D428C2312yes—
51C2×D428C2311yescharacteristic
52C2×C22⋊D107—21no—
53C2×C4×D107—21no—
54#13133802—22no—
55C2×C107⋊C8—22no—
56C2×C107⋊C8C2311yescharacteristic
57C2×C107⋊C8C2311yescharacteristic
58C2×C107⋊C8C2312yes—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—22no—
61C2×Q8×C107—22no—
62C22×C4×C107—21no—
63C22×C4×C107C2311yescharacteristic
64C4⋊C4×C107—21no—
65C4⋊C4×C107—22no—
66C4⋊C4×C107—22no—
67C4⋊C4×C107—22no—
68C4⋊C4×C107D411yescharacteristic
69C4⋊C4×C107D411yescharacteristic
70C4⋊C4×C107C2312yes—
71C4⋊C4×C107C2311yescharacteristic
72C4⋊C4×C107C2311yescharacteristic
73C22⋊C4×C107—21no—
74C22⋊C4×C107D412yes—
75C42×C107—21no—
76C42×C107C2312yes—
77C23×C107C2×D411yescharacteristic
78C22×D107—22no—
79C22×D107—21no—
80C22×D107—42no—
81C22×D107—81no—
82C22×D107—81no—
83Q8×C107—82no—
84C2×C4×C107—21no—
85C2×C4×C107—21no—
86C2×C4×C107—22no—
87C2×C4×C107—81no—
88C2×C4×C107C2×D412yes—
89C2×C4×C107C2×D412yes—
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107#4112yes—
92C2×C4×C107#4112yes—
93C2×C4×C107C2411yescharacteristic
94C22⋊D107—42no—
95C2×Dic107—21no—
96D428—22no—
97D428—22no—
98D428—21no—
99D428—21no—
100D428—42no—
101D428—82no—
102D428—82no—
103C4×D107—42no—
104Dic214—21no—
105Dic214—21no—
106C107⋊C8—21no—
107C107⋊C8—21no—
108C107⋊C8—22no—
109C22×C107—21no—
110C22×C107—21no—
111C22×C107#12412yes—
112C22×C107C22×D411yescharacteristic
113C2×D107—41no—
114C2×D107—42no—
115C2×D107—41no—
116C2×D107—42no—
117C2×D107—41no—
118C2×D107—42no—
119C2×D107—84no—
120C4×C107—22no—
121C4×C107—22no—
122C4×C107—42no—
123C4×C107—41no—
124C4×C107—41no—
125C4×C107#12412yes—
126C4×C107C2×#4111yescharacteristic
127C4×C107C2×#4111yescharacteristic
128Dic107—41no—
129C2×C107—22no—
130C2×C107C2×#12411yescharacteristic
131C2×C107#57611yescharacteristic
132C2×C107#57711yescharacteristic
133D107—81no—
134D107—82no—
135#3136—1071nomaximal, Sylow
136C107#313611yescharacteristic, complemented, Sylow
137#544D10711yescharacteristic, complemented
138#535—1071no—
139#490—1072no—
140#489—1072no—
141#488—1072no—
142#486—1072no—
143#418—1071no—
144#417—1071no—
145#416—1071no—
146C2×#102—1071no—
147#405—1071no—
148C2×#41—1071no—
149C22×D4—1071no—
150C2×M16—1071no—
151C22×C8—1071no—
152C4⋊Q8—21no—
153C4⋊Q8C2×D10712yescomplemented
154#127—1072no—
155#125—21no—
156#125C2×D10712yescomplemented
157#124—1072no—
158C22⋊Q8—22no—
159#121—2141no—
160#121—2141no—
161#120—2142no—
162#117C2×D10711yescharacteristic
163#117C2×D10711yescharacteristic
164C2×C4⋊C4C2×D10711yescharacteristic
165#103—1071no—
166#103—1071no—
167#103—1072no—
168#102—1072no—
169#102—1072no—
170#102—1071no—
171#102—1071no—
172#102—1072no—
173#102—1072no—
174#102—1072no—
175#97—1072no—
176C4×C8—1072no—
177C24—2141no—
178#41—2142no—
179#41—2142no—
180C2×Q8—22no—
181C2×Q8—1071no—
182C2×D4—1072no—
183C2×D4—1071no—
184C2×D4—2141no—
185C2×D4—2142no—
186C2×D4—2142no—
187C2×D4—2141no—
188C2×D4—2142no—
189C2×D4—2142no—
190C22×C4—21no—
191C22×C4—2141no—
192C22×C4C22×D10711yescharacteristic
193M16—2142no—
194C2×C8—1071no—
195C2×C8—1072no—
196C2×C8—1071no—
197C2×C8—2142no—
198C4⋊C4—21no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4C22⋊D10711yescharacteristic
203C4⋊C4C22⋊D10711yescharacteristic
204C4⋊C4C22×D10712yes—
205C4⋊C4C22×D10711yescharacteristic
206C4⋊C4C22×D10711yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—2142no—
209C22⋊C4—2142no—
210C22⋊C4C22⋊D10712yes—
211C42—21no—
212C42C22×D10712yes—
213C23—8561no—
214C23—8561no—
215C23—4282no—
216C23—2142no—
217C23—2141no—
218C23C2×C22⋊D10711yescharacteristic
219Q8—82no—
220Q8—2141no—
221Q8—2141no—
222D4—8562no—
223D4—8562no—
224D4—4282no—
225D4—4282no—
226D4—2142no—
227D4—2142no—
228D4—2141no—
229D4—2141no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—81no—
234C2×C4—4282no—
235C2×C4—2141no—
236C2×C4C2×C22⋊D10712yes—
237C2×C4C2×C22⋊D10712yes—
238C2×C4C2×C22⋊D10711yescharacteristic
239C2×C4#1313381912yes—
240C2×C4#1313381912yes—
241C2×C4C23×D10711yescharacteristic
242C8—2141no—
243C8—2141no—
244C8—2142no—
245C22—21no—
246C22—21no—
247C22—8564no—
248C22—4281no—
249C22—4282no—
250C22—4281no—
251C22—4282no—
252C22—4281no—
253C22—4282no—
254C22#1681209812yes—
255C22C22×C22⋊D10711yescharacteristic
256C4—22no—
257C4—22no—
258C4—42no—
259C4—41no—
260C4—41no—
261C4—4281no—
262C4#1681209812yes—
263C4C2×#1313381911yescharacteristic
264C4C2×#1313381911yescharacteristic
265C2—22no—
266C2—8562no—
267C2—8561no—
268C2C2×#1681209811yescharacteristic
269C2#2029343011yescharacteristic
270C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#544×C107C2
socleC22×C107C22×D4
prime radical#20293196C2
hypercenter#544D107
hypocenterC107#3136
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #544

Complements

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

NH ≅ G/Ntype
2. #20292993C2semidirect
3. #20292989C2semidirect
4. #20292984C2semidirect
5. #20292975C2semidirect
6. #20292631C2semidirect
7. #20292626C2semidirect
8. #20292621C2semidirect
9. C2×#16812029C2semidirect
10. #20292583C2semidirect
11. #544×C107C2semidirect
21. #16812032C22semidirect
22. #16812032C22semidirect
23. #16812032C22semidirect
28. #16812029C22semidirect
29. #16812029C22semidirect
30. #16812029C22semidirect
31. #16812020C22semidirect
32. C4×C107⋊C8C22semidirect
34. C4⋊Q8×C107C22semidirect
36. #125×C107C22semidirect
136. C107#3136semidirect
137. #544D107semidirect
153. C4⋊Q8C2×D107semidirect
156. #125C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931962
#202929932
#202929892
#202929842
#202929752
#202926312
#202926262
#202926212
C2×#168120292
#202925832
#544×C1072
#3136107