Finite Group Databasebeta

← #28348407

Subgroups — #28348407

Order 13696.

408 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293340C211yesmaximal, characteristic, complemented
2#20293296C211yesmaximal, characteristic, complemented
3#20292972C211yesmaximal, characteristic, complemented
4#20292951C211yesmaximal, characteristic, complemented
5#20292949C211yesmaximal, characteristic, complemented
6#20292920C211yesmaximal, characteristic, complemented
7#20292863C211yesmaximal, characteristic, complemented
8#20292862C211yesmaximal, characteristic, complemented
9#20292821C211yesmaximal, characteristic, complemented
10#20292775C211yesmaximal, characteristic, complemented
11#20292772C211yesmaximal, characteristic, complemented
12#20292712C211yesmaximal, characteristic, complemented
13C8×C22⋊D107C211yesmaximal, characteristic, complemented
14#20292622C211yesmaximal, characteristic, complemented
15#416×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133805C2211yescharacteristic
18C2×D856C2211yescharacteristic
19C2×#13133803C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic
21#16812108C2211yescharacteristic
22#16812105—21no—
23#16812105C2211yescharacteristic
24#16812097—21no—
25#16812095—21no—
26#16812084C2211yescharacteristic
27#16812083—21no—
28#16812083C2211yescharacteristic
29#16812083C2211yescharacteristic
30#16812082—21no—
31#16812080—21no—
32#16812080C2211yescharacteristic
33Q8×Dic107C2211yescharacteristic
34#16812071C2211yescharacteristic
35C4×C22⋊D107—21no—
36C4×C22⋊D107C2211yescharacteristic
37#16812067C2211yescharacteristic
38#16812066—21no—
39#16812043C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812040C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43#16812033C2211yescharacteristic
44#16812031C2211yescharacteristic
45#16812031C2211yescharacteristic
46#16812030C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×#41×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic
54#117×C107C2211yescharacteristic
55#103×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59#13133819—22no—
60#13133819—41no—
61C2×Dic214—21no—
62C2×D428—21no—
63C2×D428—21no—
64C2×D428C2311yescharacteristic
65C2×D428C2311yescharacteristic
66C2×C22⋊D107—21no—
67C2×C22⋊D107—21no—
68C2×C22⋊D107—21no—
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C2×C22⋊D107D411yescharacteristic
72C2×C4×D107—21no—
73C2×C4×D107—21no—
74C2×C4×D107C2311yescharacteristic
75#13133805—22no—
76D856—41no—
77#13133803—41no—
78C2×C107⋊C8C2311yescharacteristic
79C2×C107⋊C8C2311yescharacteristic
80C8×D107—41no—
81C4⋊Dic107—21no—
82C4⋊Dic107C2311yescharacteristic
83C4⋊Dic107C2311yescharacteristic
84Dic107⋊C4—21no—
85Dic107⋊C4D411yescharacteristic
86C22⋊Dic107—21no—
87C22⋊Dic107D411yescharacteristic
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4D411yescharacteristic
94C4×Dic107—21no—
95C4×Dic107—21no—
96C4×Dic107C2311yescharacteristic
97#41×C107—22no—
98#41×C107—41no—
99C2×Q8×C107C2311yescharacteristic
100C2×D4×C107—21no—
101C2×D4×C107C2311yescharacteristic
102C22×C4×C107—21no—
103C22×C4×C107C2311yescharacteristic
104C2×C8×C107—41no—
105C2×C8×C107C2311yescharacteristic
106C2×C8×C107C2311yescharacteristic
107C4⋊C4×C107C2311yescharacteristic
108C4⋊C4×C107C2311yescharacteristic
109C22⋊C4×C107—21no—
110C42×C107—21no—
111C23×C107—21no—
112C23×C107C2×D411yescharacteristic
113C22×D107—21no—
114C22×D107—21no—
115C22×D107C2×D411yescharacteristic
116Q8×C107—21no—
117Q8×C107#4112yes—
118D4×C107—21no—
119D4×C107—21no—
120D4×C107—42no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—42no—
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2411yescharacteristic
129C22⋊D107—22no—
130C22⋊D107—42no—
131C22⋊D107—81no—
132C22⋊D107—81no—
133C22⋊D107—81no—
134C22⋊D107—81no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107C2×D411yescharacteristic
138C2×Dic107C2×D411yescharacteristic
139C2×Dic107C2×D411yescharacteristic
140D428—21no—
141D428—21no—
142D428—21no—
143D428—21no—
144D428—42no—
145D428—81no—
146C4×D107—21no—
147C4×D107—21no—
148C4×D107—42no—
149C4×D107—42no—
150Dic214—81no—
151C8×C107—21no—
152C8×C107—21no—
153C107⋊C8—21no—
154C107⋊C8—21no—
155C22×C107—21no—
156C22×C107—21no—
157C22×C107—21no—
158C22×C107—41no—
159C22×C107—41no—
160C22×C107—41no—
161C22×C107C22×D411yescharacteristic
162C2×D107—21no—
163C2×D107—21no—
164C2×D107—21no—
165C2×D107—41no—
166C2×D107—41no—
167C2×D107—41no—
168C2×D107—41no—
169C2×D107—41no—
170C2×D107—41no—
171C4×C107—22no—
172C4×C107—21no—
173C4×C107—41no—
174C4×C107—41no—
175C4×C107C2×#4111yescharacteristic
176C4×C107H(5,𝔽2)11yescharacteristic
177Dic107—21no—
178Dic107—21no—
179Dic107—41no—
180Dic107—41no—
181Dic107#13512yes—
182C2×C107—41no—
183C2×C107—81no—
184C2×C107#54611yescharacteristic
185C2×C107C2×#13511yescharacteristic
186C2×C107#57611yescharacteristic
187D107—41no—
188D107—81no—
189D107—81no—
190#3305—1071nomaximal, Sylow
191C107#330511yescharacteristic, complemented, Sylow
192#555—1071no—
193#550—1071no—
194#486—1071no—
195#482—1071no—
196#482—1071no—
197#477—1071no—
198#466—1071no—
199#466—1071no—
200#459—1071no—
201#449—1071no—
202#449—1071no—
203C4×SD16—1071no—
204C8×D4—1071no—
205#416—1071no—
206#416D10711yescharacteristic, complemented
207C2×#41—1071no—
208C2×#41C2×D10711yescharacteristic
209C2×SD16—1071no—
210C2×D8—1071no—
211C2×D8—1071no—
212C22×C8—1071no—
213C22×C8C2×D10711yescharacteristic
214C4⋊Q8—1071no—
215#127—1071no—
216#127—2141no—
217#124—2141no—
218#124—2141no—
219#121—1071no—
220#121—1071no—
221#121—1071no—
222#121—1071no—
223#121—2141no—
224#121—2141no—
225#121—2141no—
226C4×Q8—1071no—
227C4×D4—1071no—
228C4×D4—1071no—
229C4×D4—2141no—
230#117—1071no—
231#117—2141no—
232#117C2×D10711yescharacteristic
233#107—1071no—
234#107—1071no—
235#106—1071no—
236C4⋊C8—1071no—
237#103—1071no—
238#103C2×D10711yescharacteristic
239#103C2×D10711yescharacteristic
240#102—1071no—
241#102—1071no—
242#102—1071no—
243#102—1071no—
244#102—1071no—
245#102C2×D10711yescharacteristic
246#102C2×D10711yescharacteristic
247C22⋊C8—1071no—
248C22⋊C8—1071no—
249C4×C8—1071no—
250#41—22no—
251#41—41no—
252#41—4281no—
253#41—2142no—
254C2×Q8—2141no—
255C2×Q8C22×D10711yescharacteristic
256C2×D4—21no—
257C2×D4—1071no—
258C2×D4—1071no—
259C2×D4—1071no—
260C2×D4—2141no—
261C2×D4—2141no—
262C2×D4—2141no—
263C2×D4—2141no—
264C2×D4—2141no—
265C2×D4—2141no—
266C2×D4—2141no—
267C2×D4C22×D10711yescharacteristic
268C22×C4—21no—
269C22×C4—1071no—
270C22×C4—2141no—
271C22×C4—2141no—
272C22×C4C22×D10711yescharacteristic
273SD16—2142no—
274D8—4281no—
275D8—4281no—
276C2×C8—41no—
277C2×C8—1071no—
278C2×C8—1071no—
279C2×C8—4281no—
280C2×C8C22×D10711yescharacteristic
281C2×C8C22×D10711yescharacteristic
282C4⋊C4—1071no—
283C4⋊C4—1071no—
284C4⋊C4—1071no—
285C4⋊C4—2141no—
286C4⋊C4—2141no—
287C4⋊C4C22×D10711yescharacteristic
288C4⋊C4C22×D10711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—1071no—
291C22⋊C4—1071no—
292C22⋊C4—2141no—
293C22⋊C4—2141no—
294C22⋊C4—2141no—
295C22⋊C4—2141no—
296C22⋊C4—2141no—
297C22⋊C4—2141no—
298C42—21no—
299C42—1071no—
300C42—2141no—
301C42—2141no—
302C23—21no—
303C23—1071no—
304C23—2141no—
305C23—2141no—
306C23D4×D10711yescharacteristic
307Q8—21no—
308Q8—8561no—
309Q8#1313381812yes—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—8561no—
314D4—8561no—
315D4—8561no—
316D4—8561no—
317D4—8561no—
318D4—4282no—
319D4—4282no—
320D4—2142no—
321D4—2141no—
322D4—2141no—
323D4—2141no—
324D4—2141no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—1071no—
332C2×C4—1071no—
333C2×C4—1071no—
334C2×C4—4282no—
335C2×C4—4282no—
336C2×C4—2141no—
337C2×C4—2141no—
338C2×C4—2141no—
339C2×C4—2141no—
340C2×C4D4×D10711yescharacteristic
341C2×C4C23×D10711yescharacteristic
342C8—21no—
343C8—21no—
344C8—2141no—
345C8—2141no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—4281no—
353C22—4281no—
354C22—4281no—
355C22—4281no—
356C22—4281no—
357C22—4281no—
358C22—2141no—
359C22—2141no—
360C22—2141no—
361C22C2×D4×D10711yescharacteristic
362C4—21no—
363C4—22no—
364C4—41no—
365C4—41no—
366C4—1072no—
367C4—4281no—
368C4—4281no—
369C4—2141no—
370C4—2141no—
371C4C2×#1313381811yescharacteristic
372C4#1681215411yescharacteristic
373C2—41no—
374C2—81no—
375C2—8561no—
376C2—8561no—
377C2—4281no—
378C2#2029326711yescharacteristic
379C2#135×D10711yescharacteristic
380C2#2029342911yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#416×C107C2
socleC22×C107C22×D4
hypercenter#416D107
hypocenterC107#3305
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #20293340C2semidirect
2. #20293296C2semidirect
3. #20292972C2semidirect
4. #20292951C2semidirect
5. #20292949C2semidirect
6. #20292920C2semidirect
7. #20292863C2semidirect
8. #20292862C2semidirect
9. #20292821C2semidirect
10. #20292775C2semidirect
11. #20292772C2semidirect
12. #20292712C2semidirect
13. C8×C22⋊D107C2semidirect
14. #20292622C2semidirect
15. #416×C107C2semidirect
191. C107#3305semidirect
206. #416D107semidirect

Maximal subgroups

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

subgroupindex
#202933402
#202932962
#202929722
#202929512
#202929492
#202929202
#202928632
#202928622
#202928212
#202927752
#202927722
#202927122
C8×C22⋊D1072
#202926222
#416×C1072
#3305107