Finite Group Databasebeta

← #28350573

Subgroups — #28350573

Order 13696.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293362C213yesmaximal, complemented
2#20293343C213yesmaximal, complemented
3#20293330C213yesmaximal, complemented
4#20293315C213yesmaximal, complemented
5#20293299C211yesmaximal, characteristic, complemented
6#20293289C211yesmaximal, characteristic, complemented
7#20293289C211yesmaximal, characteristic, complemented
8#20293251C213yesmaximal, complemented
9#20293233C213yesmaximal, complemented
10#20293164C213yesmaximal, complemented
11#20293162C213yesmaximal, complemented
12#20293109C211yesmaximal, characteristic, complemented
13#20293103C211yesmaximal, characteristic, complemented
14C4×#13133819C211yesmaximal, characteristic, complemented
15#557×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133819C2211yescharacteristic
18C2×Q8×D107C2211yescharacteristic
19#16812106C2213yes—
20#16812104C2213yes—
21#16812104C2211yescharacteristic
22#16812104C2216yes—
23#16812100C2213yes—
24#16812098C2213yes—
25#16812098C2216yes—
26#16812097C2213yes—
27#16812097C2213yes—
28#16812094C2216yes—
29#16812094C2213yes—
30#16812094C2213yes—
31#16812094C2216yes—
32#16812089C2213yes—
33#16812089C2213yes—
34#16812085C2213yes—
35#16812084C2213yes—
36#16812084C2213yes—
37#16812084C2213yes—
38#16812084C2213yes—
39#16812084C2216yes—
40#16812084C2213yes—
41Q8×Dic107C2211yescharacteristic
42C4×Dic214C2213yes—
43#16812074C2213yes—
44C4×D428C2213yes—
45C4×D428C2213yes—
46C4×D428C2213yes—
47C4×D428C2213yes—
48C4×D428C2213yes—
49#16812071C2213yes—
50#16812071C2216yes—
51#16812071C2213yes—
52#16812067C2216yes—
53#16812067C2213yes—
54#16812067C2213yes—
55#16812066C2213yes—
56#16812066C2213yes—
57C4⋊C4×D107C2213yes—
58C42×D107C2213yes—
59C4⋊Q8×C107C2213yes—
60#125×C107C2213yes—
61#125×C107C2213yes—
62C4×Q8×C107C2213yes—
63C4×Q8×C107C2211yescharacteristic
64C4×Q8×C107C2211yescharacteristic
65C4×Q8×C107C2211yescharacteristic
66#13133819—24no—
67#13133819—24no—
68C2×Dic214C2313yes—
69Q8×D107—24no—
70C2×D428C2313yes—
71C2×D428C2313yes—
72C2×D428C2316yes—
73C2×D428C2313yes—
74C2×D428C2313yes—
75C2×D428C2313yes—
76C2×C4×D107C2313yes—
77C2×C4×D107C2313yes—
78C2×C4×D107C2313yes—
79C2×C4×D107C2313yes—
80C2×C4×D107C2313yes—
81C2×C4×D107C2316yes—
82C4⋊Dic107C2313yes—
83C4⋊Dic107C2313yes—
84C4⋊Dic107C2311yescharacteristic
85C4⋊Dic107C2313yes—
86Dic107⋊C4C2313yes—
87Dic107⋊C4C2311yescharacteristic
88Dic107⋊C4C2313yes—
89Dic107⋊C4C2313yes—
90(C2×D107)⋊C4C2311yescharacteristic
91(C2×D107)⋊C4C2313yes—
92(C2×D107)⋊C4C2313yes—
93(C2×D107)⋊C4C2313yes—
94(C2×D107)⋊C4C2313yes—
95(C2×D107)⋊C4C2316yes—
96(C2×D107)⋊C4C2316yes—
97(C2×D107)⋊C4C2313yes—
98(C2×D107)⋊C4C2316yes—
99(C2×D107)⋊C4C2311yescharacteristic
100(C2×D107)⋊C4C2311yescharacteristic
101(C2×D107)⋊C4C2313yes—
102(C2×D107)⋊C4C2313yes—
103C4×Dic107C2311yescharacteristic
104C4×Dic107C2313yes—
105C4×Dic107C2313yes—
106C4×Dic107C2316yes—
107C2×Q8×C107C2313yes—
108C2×Q8×C107C2311yescharacteristic
109C4⋊C4×C107C2313yes—
110C4⋊C4×C107C2313yes—
111C4⋊C4×C107C2313yes—
112C4⋊C4×C107C2316yes—
113C4⋊C4×C107C2313yes—
114C4⋊C4×C107C2313yes—
115C4⋊C4×C107C2311yescharacteristic
116C42×C107C2313yes—
117C42×C107C2313yes—
118C42×C107C2313yes—
119C22×D107C2411yescharacteristic
120C22×D107C2413yes—
121C22×D107C2413yes—
122Q8×C107—43no—
123Q8×C107#4114yes—
124C2×C4×C107C2411yescharacteristic
125C2×C4×C107C2411yescharacteristic
126C2×C4×C107C2411yescharacteristic
127C2×C4×C107C2413yes—
128C2×C4×C107C2413yes—
129C2×C4×C107C2413yes—
130C2×C4×C107C2413yes—
131C2×Dic107C2411yescharacteristic
132C2×Dic107C2413yes—
133C2×Dic107C2411yescharacteristic
134C2×Dic107C2413yes—
135C2×Dic107C2411yescharacteristic
136D428—26no—
137D428—26no—
138D428—43no—
139D428—43no—
140D428—43no—
141D428—46no—
142C4×D107—26no—
143C4×D107—26no—
144C4×D107—26no—
145C4×D107—43no—
146C4×D107—46no—
147C4×D107—43no—
148Dic214—26no—
149C22×C107C2511yescharacteristic
150C2×D107—21no—
151C2×D107—23no—
152C2×D107—23no—
153C2×D107—23no—
154C2×D107—23no—
155C2×D107—23no—
156C2×D107—21no—
157C2×D107—23no—
158C2×D107—21no—
159C4×C107—21no—
160C4×C107—23no—
161C4×C107—23no—
162C4×C107—21no—
163C4×C107—23no—
164C4×C107—21no—
165C4×C107C2×#4116yes—
166Dic107—23no—
167Dic107—21no—
168Dic107—21no—
169Dic107—21no—
170Dic107—23no—
171C2×C107C22×#4111yescharacteristic
172C2×C107#58511yescharacteristic
173C2×C107#58511yescharacteristic
174D107—41no—
175D107—43no—
176D107—43no—
177#3524—1071nomaximal, Sylow
178C107#352411yescharacteristic, complemented, Sylow
179#559—1073no—
180#557D10711yescharacteristic, complemented
181#555—1073no—
182#553—1073no—
183#552—1073no—
184#550—1071no—
185#548—1071no—
186#548—1071no—
187#542—1073no—
188#540—1073no—
189#529—1073no—
190#529—1073no—
191#520—1071no—
192#519—1071no—
193C4×#41—1071no—
194C2×#41—1071no—
195C2×#41—1071no—
196C22×Q8—1071no—
197C4⋊Q8C2×D10713yes—
198#127—1073no—
199#126—1071no—
200#126—1073no—
201#126—1076no—
202#125—1073no—
203#125C2×D10713yes—
204#125C2×D10713yes—
205#124—1073no—
206#124—1076no—
207#124—1073no—
208#124—1073no—
209#123—1076no—
210#123—1073no—
211#123—1073no—
212#123—1076no—
213C22⋊Q8—1073no—
214C22⋊Q8—1073no—
215C22⋊Q8—1073no—
216#121—1073no—
217#121—1073no—
218#121—1073no—
219#121—1073no—
220#121—1076no—
221#121—1073no—
222C4×Q8—1071no—
223C4×Q8—1073no—
224C4×Q8—1073no—
225C4×Q8C2×D10711yescharacteristic
226C4×Q8C2×D10711yescharacteristic
227C4×Q8C2×D10711yescharacteristic
228C4×Q8C2×D10713yes—
229C4×D4—1073no—
230C4×D4—1073no—
231C4×D4—1076no—
232C4×D4—1073no—
233C4×D4—1073no—
234C4×D4—1073no—
235C4×D4—1073no—
236C4×D4—1073no—
237#117—1073no—
238#117—1073no—
239#117—1076no—
240#117—1073no—
241#117—1073no—
242C2×C4⋊C4—1073no—
243C2×C42—1073no—
244#41—2144no—
245#41—2144no—
246C2×Q8—1073no—
247C2×Q8—2144no—
248C2×Q8C22×D10713yes—
249C2×Q8C22×D10711yescharacteristic
250C2×D4—1073no—
251C2×D4—1073no—
252C2×D4—1076no—
253C2×D4—1073no—
254C2×D4—1073no—
255C2×D4—1073no—
256C22×C4—1073no—
257C22×C4—1073no—
258C22×C4—1076no—
259C22×C4—1073no—
260C22×C4—1073no—
261C22×C4—1073no—
262C4⋊C4—1073no—
263C4⋊C4—1073no—
264C4⋊C4—1071no—
265C4⋊C4—1073no—
266C4⋊C4—1073no—
267C4⋊C4—1073no—
268C4⋊C4—1071no—
269C4⋊C4—1073no—
270C4⋊C4C22×D10713yes—
271C4⋊C4C22×D10713yes—
272C4⋊C4C22×D10713yes—
273C4⋊C4C22×D10716yes—
274C4⋊C4C22×D10713yes—
275C4⋊C4C22×D10711yescharacteristic
276C4⋊C4C22×D10713yes—
277C22⋊C4—1071no—
278C22⋊C4—1073no—
279C22⋊C4—1073no—
280C22⋊C4—1073no—
281C22⋊C4—1076no—
282C22⋊C4—1073no—
283C22⋊C4—1076no—
284C22⋊C4—1071no—
285C22⋊C4—1073no—
286C22⋊C4—1071no—
287C22⋊C4—1073no—
288C22⋊C4—1076no—
289C22⋊C4—1073no—
290C42—1071no—
291C42—1073no—
292C42—1073no—
293C42—1076no—
294C42C22×D10713yes—
295C42C22×D10713yes—
296C42C22×D10713yes—
297C23—1071no—
298C23—1073no—
299C23—1073no—
300Q8—43no—
301Q8—2146no—
302Q8#1313381914yes—
303D4—4283no—
304D4—4286no—
305D4—4283no—
306D4—4283no—
307D4—2146no—
308D4—2146no—
309C2×C4—1071no—
310C2×C4—1071no—
311C2×C4—1073no—
312C2×C4—1073no—
313C2×C4—1071no—
314C2×C4—4283no—
315C2×C4—4283no—
316C2×C4—4286no—
317C2×C4—2146no—
318C2×C4—2146no—
319C2×C4—2146no—
320C2×C4C23×D10713yes—
321C2×C4C23×D10711yescharacteristic
322C2×C4C23×D10713yes—
323C2×C4C23×D10711yescharacteristic
324C2×C4C23×D10711yescharacteristic
325C2×C4C23×D10713yes—
326C2×C4C23×D10713yes—
327C22—2141no—
328C22—2143no—
329C22—2143no—
330C22—2143no—
331C22—2141no—
332C22—2143no—
333C22—2143no—
334C22—2143no—
335C22—2141no—
336C22C24×D10711yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—2141no—
344C4—2141no—
345C4—2143no—
346C4—2141no—
347C4—2143no—
348C4C2×#1313381916yes—
349C2—4281no—
350C2—4283no—
351C2—4283no—
352C2C22×#1313381911yescharacteristic
353C2#2029346311yescharacteristic
354C2#2029346411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D107
commutator, socle, cosocleC22×C107C25
Fitting#557×C107C2
hypercenter#557D107
hypocenterC107#3524
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #20293362C2semidirect
2. #20293343C2semidirect
3. #20293330C2semidirect
4. #20293315C2semidirect
5. #20293299C2semidirect
6. #20293289C2semidirect
7. #20293289C2semidirect
8. #20293251C2semidirect
9. #20293233C2semidirect
10. #20293164C2semidirect
11. #20293162C2semidirect
12. #20293109C2semidirect
13. #20293103C2semidirect
14. C4×#13133819C2semidirect
15. #557×C107C2semidirect
178. C107#3524semidirect
180. #557D107semidirect

Maximal subgroups

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

subgroupindex
#202933622
#202933432
#202933302
#202933152
#202932992
#202932892
#202932512
#202932332
#202931642
#202931622
#202931092
#202931032
C4×#131338192
#557×C1072
#3524107