Finite Group Databasebeta

← #28347049

Subgroups — #28347049

Order 13696.

408 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293172C211yesmaximal, characteristic, complemented
2#20293163C211yesmaximal, characteristic, complemented
3#20293035C211yesmaximal, characteristic, complemented
4#20293013C211yesmaximal, characteristic, complemented
5#20292990C211yesmaximal, characteristic, complemented
6#20292988C211yesmaximal, characteristic, complemented
7#20292880C211yesmaximal, characteristic, complemented
8#20292878C211yesmaximal, characteristic, complemented
9#20292875C211yesmaximal, characteristic, complemented
10SD16×Dic107C211yesmaximal, characteristic, complemented
11#20292712C211yesmaximal, characteristic, complemented
12#20292709C211yesmaximal, characteristic, complemented
13#20292707C211yesmaximal, characteristic, complemented
14M16×Dic107C211yesmaximal, characteristic, complemented
15#468×C107C211yesmaximal, characteristic, complemented
16C2×#13133817—21no—
17C2×#13133805C2211yescharacteristic
18C2×D856C2211yescharacteristic
19C2×#13133803C2211yescharacteristic
20C2×#13133802C2211yescharacteristic
21#16812108C2211yescharacteristic
22#16812105C2211yescharacteristic
23#16812104—21no—
24#16812100C2211yescharacteristic
25#16812098C2211yescharacteristic
26#16812095—21no—
27#16812093—21no—
28#16812083C2211yescharacteristic
29#16812083C2211yescharacteristic
30#16812081—21no—
31Q8×Dic107C2211yescharacteristic
32D4×Dic107C2211yescharacteristic
33#16812071C2211yescharacteristic
34#16812071C2211yescharacteristic
35(C22⋊D107)⋊C4—21no—
36(C22⋊D107)⋊C4—21no—
37#16812067—21no—
38C2×C4×Dic107C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812041C2211yescharacteristic
41#16812040C2211yescharacteristic
42#16812033C2211yescharacteristic
43#16812031C2211yescharacteristic
44#16812031C2211yescharacteristic
45#16812030C2211yescharacteristic
46#16812029C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812019C2211yescharacteristic
49#16812019C2211yescharacteristic
50C8×Dic107C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×SD16×C107C2211yescharacteristic
53C2×M16×C107C2211yescharacteristic
54C22⋊Q8×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56#107×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59#13133817—44no—
60C2×Dic214—21no—
61C2×D428C2311yescharacteristic
62C2×D428C2311yescharacteristic
63C2×C22⋊D107—21no—
64C2×C22⋊D107—21no—
65C2×C22⋊D107—21no—
66C2×C22⋊D107—21no—
67C22×Dic107—21no—
68C22×Dic107D411yescharacteristic
69C22×Dic107D411yescharacteristic
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72#13133805—22no—
73D856—22no—
74#13133803—22no—
75#13133802—22no—
76C2×C107⋊C8C2311yescharacteristic
77C2×C107⋊C8C2311yescharacteristic
78C4⋊Dic107—21no—
79C4⋊Dic107—21no—
80C4⋊Dic107C2311yescharacteristic
81C4⋊Dic107C2311yescharacteristic
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84C22⋊Dic107—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91C4×Dic107—21no—
92C4×Dic107D411yescharacteristic
93C4×Dic107D411yescharacteristic
94C4×Dic107C2311yescharacteristic
95C4×Dic107C2311yescharacteristic
96C2×Q8×C107C2311yescharacteristic
97C2×D4×C107—21no—
98C2×D4×C107C2311yescharacteristic
99C22×C4×C107C2311yescharacteristic
100SD16×C107—22no—
101M16×C107—22no—
102C2×C8×C107C2311yescharacteristic
103C2×C8×C107C2311yescharacteristic
104C4⋊C4×C107—21no—
105C4⋊C4×C107C2311yescharacteristic
106C4⋊C4×C107C2311yescharacteristic
107C22⋊C4×C107—21no—
108C22⋊C4×C107—21no—
109C23×C107—21no—
110C23×C107C2×D411yescharacteristic
111C22×D107—21no—
112C22×D107—21no—
113Q8×C107—21no—
114Q8×C107—21no—
115D4×C107—21no—
116D4×C107—21no—
117D4×C107—42no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2411yescharacteristic
125C22⋊D107—42no—
126C22⋊D107—42no—
127C22⋊D107—42no—
128C22⋊D107—42no—
129C2×Dic107—22no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—41no—
133C2×Dic107—42no—
134C2×Dic107C2×D411yescharacteristic
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137C2×Dic107C2×D411yescharacteristic
138D428—21no—
139D428—21no—
140D428—21no—
141D428—21no—
142C4×D107—42no—
143C4×D107—42no—
144Dic214—42no—
145C8×C107—21no—
146C8×C107#4112yescomplemented
147C107⋊C8—21no—
148C107⋊C8#4112yescomplemented
149C22×C107—21no—
150C22×C107—21no—
151C22×C107—21no—
152C22×C107—41no—
153C22×C107—41no—
154C22×C107—41no—
155C22×C107C22×D411yescharacteristic
156C2×D107—41no—
157C2×D107—41no—
158C2×D107—41no—
159C2×D107—41no—
160C2×D107—41no—
161C2×D107—41no—
162C4×C107—21no—
163C4×C107—41no—
164C4×C107—41no—
165C4×C107—41no—
166C4×C107C2×#4111yescharacteristic
167C4×C107C2×#4111yescharacteristic
168Dic107—21no—
169Dic107—21no—
170Dic107—21no—
171Dic107—41no—
172Dic107—41no—
173Dic107AGL(1,ℤ/8)12yes—
174C2×C107—41no—
175C2×C107—81no—
176C2×C107#53211yescharacteristic
177C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
178C2×C107#57511yescharacteristic
179D107—81no—
180D107—81no—
181#3171—1071nomaximal, Sylow
182C107#317111yescharacteristic, complemented, Sylow
183#532—1071no—
184#529—1071no—
185#501—1071no—
186#496—1071no—
187#490—1071no—
188#489—1071no—
189#469—1071no—
190#469—1071no—
191#468—1071no—
192#468D10711yescharacteristic, complemented
193C4×SD16—1071no—
194C4×SD16—1071no—
195C4×D8—1071no—
196C4×D8—1071no—
197C4×M16—1071no—
198C2×#41—2141no—
199C2×SD16—1071no—
200C2×SD16C2×D10711yescharacteristic
201C2×D8—1071no—
202C2×D8—1071no—
203C2×M16—1071no—
204C2×M16C2×D10711yescharacteristic
205C4⋊Q8—1071no—
206#127—1071no—
207#126—2141no—
208#125—1071no—
209#124—1071no—
210#124—2141no—
211#123—2141no—
212C22⋊Q8C2×D10711yescharacteristic
213#121—1071no—
214#121—1071no—
215#121—2141no—
216#121C2×D10711yescharacteristic
217C4×Q8—1071no—
218C4×D4—1071no—
219C4×D4—1071no—
220C4×D4—1071no—
221C4×D4—2141no—
222C4×D4—2141no—
223#117—2141no—
224C2×C42—1071no—
225#107—1071no—
226#107C2×D10711yescharacteristic
227#106—1071no—
228#106—1071no—
229#103—1071no—
230#103C2×D10711yescharacteristic
231#102—1071no—
232#102—1071no—
233#102—1071no—
234#102—1071no—
235#102—1071no—
236#102C2×D10711yescharacteristic
237#97—1071no—
238#97—1071no—
239C4×C8—1071no—
240C4×C8—1071no—
241#41—4284no—
242C2×Q8—2141no—
243C2×Q8C22×D10711yescharacteristic
244C2×D4—21no—
245C2×D4—1071no—
246C2×D4—1071no—
247C2×D4—2141no—
248C2×D4—2141no—
249C2×D4—2141no—
250C2×D4—2141no—
251C2×D4C22×D10711yescharacteristic
252C22×C4—1071no—
253C22×C4—1071no—
254C22×C4—2141no—
255C22×C4—2141no—
256C22×C4—2141no—
257C22×C4C22×D10711yescharacteristic
258SD16—22no—
259SD16—2142no—
260D8—2142no—
261D8—2142no—
262M16—22no—
263M16—2142no—
264C2×C8—1071no—
265C2×C8—1071no—
266C2×C8C22×D10711yescharacteristic
267C2×C8C22×D10711yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—1071no—
270C4⋊C4—1071no—
271C4⋊C4—2141no—
272C4⋊C4—2141no—
273C4⋊C4—2141no—
274C4⋊C4—2141no—
275C4⋊C4C22×D10711yescharacteristic
276C4⋊C4C22×D10711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2141no—
280C22⋊C4—2141no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C22⋊C4—2141no—
284C22⋊C4—2141no—
285C22⋊C4—2141no—
286C42—1071no—
287C42—1071no—
288C42—1071no—
289C42—1071no—
290C42—2141no—
291C23—21no—
292C23—2141no—
293C23—2141no—
294C23D4×D10711yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—4282no—
298D4—21no—
299D4—21no—
300D4—42no—
301D4—4282no—
302D4—4282no—
303D4—4282no—
304D4—4282no—
305D4—2141no—
306D4—2141no—
307D4—2141no—
308D4—2141no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—1071no—
315C2×C4—1071no—
316C2×C4—1071no—
317C2×C4—1071no—
318C2×C4—4282no—
319C2×C4—4282no—
320C2×C4—4282no—
321C2×C4—4281no—
322C2×C4—2142no—
323C2×C4—2141no—
324C2×C4—2141no—
325C2×C4D4×D10711yescharacteristic
326C2×C4C23×D10711yescharacteristic
327C8—21no—
328C8—1072no—
329C8—2141no—
330C8#1313381712yescomplemented
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—4281no—
338C22—4281no—
339C22—4281no—
340C22—4281no—
341C22—4281no—
342C22—4281no—
343C22C2×D4×D10711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—1072no—
349C4—4281no—
350C4—4281no—
351C4—2141no—
352C4—2141no—
353C4—2141no—
354C4#41×D10711yescharacteristic
355C4C2×#1313381711yescharacteristic
356C2—41no—
357C2—81no—
358C2—8561no—
359C2—8561no—
360C2#2029317211yescharacteristic
361C2AGL(1,ℤ/8)×D10711yescharacteristic
362C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#468×C107C2
socleC22×C107C22×D4
hypercenter#468D107
hypocenterC107#3171
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #20293172C2semidirect
2. #20293163C2semidirect
3. #20293035C2semidirect
4. #20293013C2semidirect
5. #20292990C2semidirect
6. #20292988C2semidirect
7. #20292880C2semidirect
8. #20292878C2semidirect
9. #20292875C2semidirect
10. SD16×Dic107C2semidirect
11. #20292712C2semidirect
12. #20292709C2semidirect
13. #20292707C2semidirect
14. M16×Dic107C2semidirect
15. #468×C107C2semidirect
146. C8×C107#41semidirect
148. C107⋊C8#41semidirect
182. C107#3171semidirect
192. #468D107semidirect
330. C8#13133817semidirect

Maximal subgroups

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

subgroupindex
#202931722
#202931632
#202930352
#202930132
#202929902
#202929882
#202928802
#202928782
#202928752
SD16×Dic1072
#202927122
#202927092
#202927072
M16×Dic1072
#468×C1072
#3171107