Finite Group Databasebeta

← #28348279

Subgroups — #28348279

Order 13696.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D428C211yesmaximal, characteristic
2#20292975C211yesmaximal, characteristic, complemented
3#20292945C212yesmaximal, complemented
4#20292914C211yesmaximal, characteristic, complemented
5#20292858C212yesmaximal, complemented
6#20292827C211yesmaximal, characteristic, complemented
7#20292777C212yesmaximal, complemented
8C4×#13133805C211yesmaximal, characteristic, complemented
9D4×C107⋊C8C211yesmaximal, characteristic, complemented
10C2×#16812029C212yesmaximal, complemented
11#554×C107C211yesmaximal, characteristic, complemented
12C22×D428C2212yes—
13C2×D4×D107—21no—
14C2×#13133805C2211yescharacteristic
15C2×#13133805C2212yescomplemented
16C22×C107⋊C8C2212yescomplemented
17#16812106C2211yescharacteristic
18#16812084—21no—
19#16812083C2212yes—
20#16812079—22no—
21C4×D428C2211yescharacteristic
22#16812040C2212yescomplemented
23#16812040C2211yescharacteristic, complemented
24C4⋊(C107⋊C8)C2211yescharacteristic, complemented
25#16812033C2211yescharacteristic, complemented
26#16812029C2211yescharacteristic
27#16812029C2212yes—
28#16812029C2212yescomplemented
29#16812029C2212yescomplemented
30#16812029C2212yescomplemented
31C22⋊(C107⋊C8)C2212yescomplemented
32C4×C107⋊C8C2211yescharacteristic, complemented
33C4⋊Q8×C107C2211yescharacteristic, complemented
34#125×C107—21no—
35C22⋊Q8×C107—22no—
36C22⋊Q8×C107C2212yescomplemented
37C4×Q8×C107C2211yescharacteristic, complemented
38C4×D4×C107—21no—
39C4×D4×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2212yescomplemented
41C23×D107—22no—
42C2×D428—24no—
43C2×D428—22no—
44C2×D428—21no—
45C2×D428—21no—
46C2×D428—42no—
47C2×D428C2312yes—
48C2×D428C2311yescharacteristic
49C2×C22⋊D107—22no—
50D4×D107—44no—
51C2×C4×D107—21no—
52#13133805—22no—
53#13133805—42no—
54C2×C107⋊C8—42no—
55C2×C107⋊C8C2311yescharacteristic
56C2×C107⋊C8C2312yes—
57C2×C107⋊C8C2311yescharacteristic
58C4⋊Dic107C2311yescharacteristic
59(C2×D107)⋊C4—21no—
60(C2×D107)⋊C4—22no—
61C2×Q8×C107—21no—
62C2×Q8×C107C2311yescharacteristic
63C2×D4×C107D411yescharacteristic
64C22×C4×C107—22no—
65C22×C4×C107C2312yes—
66C4⋊C4×C107—22no—
67C4⋊C4×C107—21no—
68C4⋊C4×C107—21no—
69C4⋊C4×C107—22no—
70C4⋊C4×C107D411yescharacteristic
71C4⋊C4×C107C2311yescharacteristic
72C4⋊C4×C107C2312yes—
73C22⋊C4×C107—22no—
74C22⋊C4×C107D412yes—
75C42×C107—21no—
76C42×C107C2311yescharacteristic
77C23×C107C2×D412yes—
78C22×D107—21no—
79C22×D107—22no—
80C22×D107—44no—
81C22×D107—44no—
82C22×D107—82no—
83Q8×C107—21no—
84Q8×C107—21no—
85Q8×C107—81no—
86D4×C107SD1614yescomplemented
87C2×C4×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107—21no—
90C2×C4×C107—21no—
91C2×C4×C107—22no—
92C2×C4×C107—82no—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2×D412yes—
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2411yescharacteristic
97C22⋊D107—44no—
98C2×Dic107—21no—
99D428—22no—
100D428—21no—
101D428—22no—
102D428—44no—
103D428—42no—
104D428—81no—
105D428#4112yes—
106C4×D107—42no—
107C107⋊C8—21no—
108C107⋊C8—21no—
109C107⋊C8—22no—
110C22×C107—22no—
111C22×C107—22no—
112C22×C107C2×SD1614yes—
113C22×C107C22×D411yescharacteristic
114C2×D107—22no—
115C2×D107—41no—
116C2×D107—41no—
117C2×D107—42no—
118C2×D107—42no—
119C2×D107—42no—
120C2×D107—88no—
121C4×C107—22no—
122C4×C107—21no—
123C4×C107—41no—
124C4×C107—41no—
125C4×C107—42no—
126C4×C107C2×SD1612yes—
127C4×C107C2×#4111yescharacteristic
128C4×C107H(5,𝔽2)11yescharacteristic
129Dic107—41no—
130C2×C107—24no—
131C2×C107#54611yescharacteristic
132C2×C107C22×SD1611yescharacteristic
133C2×C107#57611yescharacteristic
134D107—42no—
135D107—82no—
136#3295—1071nomaximal, Sylow
137C107#329511yescharacteristic, complemented, Sylow
138#554D10711yescharacteristic, complemented
139D42—1071no—
140#486—1071no—
141#481—1072no—
142#476—1071no—
143#465—1072no—
144#460—1071no—
145#450—1072no—
146C4×SD16—1071no—
147C8×D4—1071no—
148C2×#102—1072no—
149C22×D4—1072no—
150C22×D4—2141no—
151C2×SD16—1072no—
152C2×SD16—1071no—
153C22×C8—1072no—
154C4⋊Q8C2×D10711yescharacteristic, complemented
155#127—1071no—
156#125—21no—
157C22⋊Q8—22no—
158C22⋊Q8C2×D10712yescomplemented
159#121—1072no—
160#121—2141no—
161#120—2142no—
162C4×Q8C2×D10711yescharacteristic, complemented
163C4×D4—21no—
164C4×D4—1071no—
165C4×D4C2×D10711yescharacteristic
166C2×C4⋊C4C2×D10712yescomplemented
167#106—1071no—
168#106—1072no—
169C4⋊C8—1071no—
170#103—1071no—
171#102—1071no—
172#102—1072no—
173#102—1072no—
174#102—1072no—
175#102—1072no—
176C22⋊C8—1072no—
177C4×C8—1071no—
178C24—2142no—
179C2×Q8—21no—
180C2×Q8C22×D10711yescharacteristic
181C2×D4—1071no—
182C2×D4—1072no—
183C2×D4—4282no—
184C2×D4—4284no—
185C2×D4—2144no—
186C2×D4—2142no—
187C2×D4—2141no—
188C2×D4—2142no—
189C2×D4—2141no—
190C2×D4C22⋊D10711yescharacteristic
191C22×C4—22no—
192C22×C4—2141no—
193C22×C4C22×D10712yes—
194SD16—4282no—
195SD16—2142no—
196C2×C8—1071no—
197C2×C8—1071no—
198C2×C8—1072no—
199C2×C8—4282no—
200C4⋊C4—22no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—22no—
204C4⋊C4—1071no—
205C4⋊C4C22⋊D10711yescharacteristic
206C4⋊C4C22×D10711yescharacteristic
207C4⋊C4C22×D10712yes—
208C22⋊C4—22no—
209C22⋊C4—2141no—
210C22⋊C4—2142no—
211C22⋊C4C22⋊D10712yes—
212C42—21no—
213C42C22×D10711yescharacteristic
214C23—8562no—
215C23—4284no—
216C23—4284no—
217C23—2142no—
218C23—2141no—
219C23C2×C22⋊D10712yes—
220Q8—21no—
221Q8—21no—
222Q8—81no—
223D4—8561no—
224D4—1072no—
225D4—4284no—
226D4—4284no—
227D4—4282no—
228D4—2142no—
229D4—2142no—
230D4—2141no—
231D4#1313380514yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—82no—
238C2×C4—4282no—
239C2×C4—2141no—
240C2×C4C2×C22⋊D10711yescharacteristic
241C2×C4C2×C22⋊D10712yes—
242C2×C4C2×C22⋊D10711yescharacteristic
243C2×C4C23×D10711yescharacteristic
244C8—2141no—
245C8—2141no—
246C8—2142no—
247C22—22no—
248C22—22no—
249C22—8568no—
250C22—4282no—
251C22—4282no—
252C22—4282no—
253C22—4281no—
254C22—4281no—
255C22—2142no—
256C22C2×#1313380514yes—
257C22C22×C22⋊D10711yescharacteristic
258C4—21no—
259C4—22no—
260C4—41no—
261C4—42no—
262C4—41no—
263C4—4281no—
264C4C2×#1313380512yes—
265C4#41×D10711yescharacteristic
266C4#1681215511yescharacteristic
267C2—24no—
268C2—8562no—
269C2—4282no—
270C2#2029326511yescharacteristic
271C2C22×#1313380511yescharacteristic
272C2#2029343011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#554×C107C2
socleC22×C107C22×D4
prime radicalD4×D428C2
hypercenter#554D107
hypocenterC107#3295
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #20292975C2semidirect
3. #20292945C2semidirect
4. #20292914C2semidirect
5. #20292858C2semidirect
6. #20292827C2semidirect
7. #20292777C2semidirect
8. C4×#13133805C2semidirect
9. D4×C107⋊C8C2semidirect
10. C2×#16812029C2semidirect
11. #554×C107C2semidirect
15. C2×#13133805C22semidirect
16. C22×C107⋊C8C22semidirect
22. #16812040C22semidirect
23. #16812040C22semidirect
24. C4⋊(C107⋊C8)C22semidirect
25. #16812033C22semidirect
28. #16812029C22semidirect
29. #16812029C22semidirect
30. #16812029C22semidirect
31. C22⋊(C107⋊C8)C22semidirect
32. C4×C107⋊C8C22semidirect
33. C4⋊Q8×C107C22semidirect
36. C22⋊Q8×C107C22semidirect
37. C4×Q8×C107C22semidirect
40. C2×C4⋊C4×C107C22semidirect
86. D4×C107SD16semidirect
137. C107#3295semidirect
138. #554D107semidirect
154. C4⋊Q8C2×D107semidirect
158. C22⋊Q8C2×D107semidirect
162. C4×Q8C2×D107semidirect
166. C2×C4⋊C4C2×D107semidirect
231. D4#13133805semidirect

Maximal subgroups

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

subgroupindex
D4×D4282
#202929752
#202929452
#202929142
#202928582
#202928272
#202927772
C4×#131338052
D4×C107⋊C82
C2×#168120292
#554×C1072
#3295107