Finite Group Databasebeta

← #28350559

Subgroups — #28350559

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#20293363C213yesmaximal, complemented
2#20293353C211yesmaximal, characteristic, complemented
3#20293341C213yesmaximal, complemented
4#20293329C213yesmaximal, complemented
5#20293317C213yesmaximal, complemented
6#20293284C211yesmaximal, characteristic, complemented
7#20293284C211yesmaximal, characteristic, complemented
8#20293244C213yesmaximal, complemented
9#20293231C213yesmaximal, complemented
10#20293158C213yesmaximal, complemented
11#20293157C213yesmaximal, complemented
12#20293110C211yesmaximal, characteristic, complemented
13#20293103C211yesmaximal, characteristic, complemented
14#41×Dic107C211yesmaximal, characteristic, complemented
15#550×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107C2211yescharacteristic
17#16812108C2213yes—
18#16812104C2211yescharacteristic
19#16812102C2213yes—
20#16812102C2216yes—
21#16812101C2213yes—
22#16812100C2213yes—
23#16812099C2213yes—
24#16812096C2213yes—
25#16812096C2213yes—
26#16812096C2216yes—
27#16812095C2213yes—
28#16812093C2213yes—
29#16812090C2213yes—
30#16812090C2216yes—
31#16812090C2216yes—
32#16812088C2213yes—
33#16812088C2213yes—
34#16812086C2213yes—
35#16812081C2213yes—
36#16812081C2216yes—
37#16812080C2213yes—
38#16812080C2213yes—
39Q8×Dic107C2211yescharacteristic
40Q8×Dic107C2211yescharacteristic
41Q8×Dic107C2211yescharacteristic
42C4×Dic214C2213yes—
43C4×Dic214C2213yes—
44#16812074C2213yes—
45D4×Dic107C2213yes—
46D4×Dic107C2216yes—
47D4×Dic107C2213yes—
48D4×Dic107C2213yes—
49D4×Dic107C2213yes—
50C4×C22⋊D107C2213yes—
51C4×C22⋊D107C2213yes—
52#16812068C2213yes—
53#16812068C2213yes—
54#16812067C2213yes—
55#16812066C2213yes—
56#16812065C2216yes—
57C2×C4⋊Dic107C2213yes—
58C2×C4×Dic107C2213yes—
59C2×#41×C107C2211yescharacteristic
60C2×#41×C107C2211yescharacteristic
61#127×C107C2213yes—
62#121×C107C2213yes—
63#121×C107C2213yes—
64C4×Q8×C107C2211yescharacteristic
65C4×D4×C107C2213yes—
66C2×Dic214C2313yes—
67C2×Dic214C2313yes—
68Q8×D107—24no—
69C2×C22⋊D107C2313yes—
70C2×C22⋊D107C2313yes—
71C22×Dic107C2313yes—
72C22×Dic107C2313yes—
73C22×Dic107C2316yes—
74C2×C4×D107C2313yes—
75C4⋊Dic107C2313yes—
76C4⋊Dic107C2313yes—
77C4⋊Dic107C2313yes—
78C4⋊Dic107C2311yescharacteristic
79C4⋊Dic107C2313yes—
80C4⋊Dic107C2313yes—
81Dic107⋊C4C2313yes—
82Dic107⋊C4C2316yes—
83Dic107⋊C4C2313yes—
84Dic107⋊C4C2313yes—
85Dic107⋊C4C2313yes—
86Dic107⋊C4C2311yescharacteristic
87Dic107⋊C4C2313yes—
88C22⋊Dic107C2316yes—
89C22⋊Dic107C2313yes—
90C22⋊Dic107C2316yes—
91C22⋊Dic107C2313yes—
92C22⋊Dic107C2313yes—
93C22⋊Dic107C2313yes—
94C22⋊Dic107C2316yes—
95(C2×D107)⋊C4C2311yescharacteristic
96(C2×D107)⋊C4C2313yes—
97(C2×D107)⋊C4C2311yescharacteristic
98(C2×D107)⋊C4C2311yescharacteristic
99C4×Dic107C2311yescharacteristic
100C4×Dic107C2313yes—
101C4×Dic107C2313yes—
102C4×Dic107C2313yes—
103C4×Dic107C2316yes—
104C4×Dic107C2313yes—
105#41×C107—24no—
106#41×C107—24no—
107C2×Q8×C107C2311yescharacteristic
108C2×D4×C107C2313yes—
109C2×D4×C107C2313yes—
110C2×D4×C107C2316yes—
111C2×D4×C107C2313yes—
112C22×C4×C107C2313yes—
113C22×C4×C107C2313yes—
114C4⋊C4×C107C2313yes—
115C4⋊C4×C107C2311yescharacteristic
116C22⋊C4×C107C2313yes—
117C22⋊C4×C107C2313yes—
118C42×C107C2313yes—
119C23×C107C2413yes—
120C23×C107C2413yes—
121C22×D107C2411yescharacteristic
122Q8×C107#4114yes—
123D4×C107—26no—
124D4×C107—26no—
125D4×C107—46no—
126D4×C107—43no—
127C2×C4×C107—26no—
128C2×C4×C107—26no—
129C2×C4×C107C2411yescharacteristic
130C2×C4×C107C2413yes—
131C2×C4×C107C2411yescharacteristic
132C2×C4×C107C2411yescharacteristic
133C2×C4×C107C2413yes—
134C22⋊D107—43no—
135C22⋊D107—43no—
136C2×Dic107—43no—
137C2×Dic107—43no—
138C2×Dic107—46no—
139C2×Dic107C2411yescharacteristic
140C2×Dic107C2411yescharacteristic
141C2×Dic107C2413yes—
142C2×Dic107C2411yescharacteristic
143C2×Dic107C2413yes—
144C2×Dic107C2413yes—
145C2×Dic107C2413yes—
146C4×D107—26no—
147Dic214—26no—
148Dic214—43no—
149C22×C107—23no—
150C22×C107—23no—
151C22×C107—23no—
152C22×C107—23no—
153C22×C107—23no—
154C22×C107—23no—
155C22×C107C2511yescharacteristic
156C2×D107—21no—
157C2×D107—21no—
158C2×D107—21no—
159C4×C107—21no—
160C4×C107—21no—
161C4×C107—23no—
162C4×C107—21no—
163C4×C107C2×#4116yes—
164Dic107—23no—
165Dic107—21no—
166Dic107—23no—
167Dic107—21no—
168Dic107—21no—
169Dic107—23no—
170Dic107—23no—
171C2×C107—43no—
172C2×C107—43no—
173C2×C107C22×#4111yescharacteristic
174C2×C107#58511yescharacteristic
175C2×C107#58511yescharacteristic
176D107—41no—
177#3524—1071nomaximal, Sylow
178C107#352411yescharacteristic, complemented, Sylow
179#559—1073no—
180#557—1071no—
181#555—1073no—
182#553—1073no—
183#552—1073no—
184#550D10711yescharacteristic, complemented
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×#41C2×D10711yescharacteristic
195C2×#41C2×D10711yescharacteristic
196C22×Q8—1071no—
197C4⋊Q8—1073no—
198#127C2×D10713yes—
199#126—1071no—
200#126—1073no—
201#126—1076no—
202#125—1073no—
203#125—1073no—
204#125—1073no—
205#124—1073no—
206#124—1073no—
207#124—1076no—
208#124—1073no—
209#123—1076no—
210#123—1073no—
211#123—1076no—
212#123—1073no—
213C22⋊Q8—1073no—
214C22⋊Q8—1073no—
215C22⋊Q8—1073no—
216#121—1073no—
217#121—1073no—
218#121—1073no—
219#121—1076no—
220#121C2×D10713yes—
221#121C2×D10713yes—
222C4×Q8—1071no—
223C4×Q8—1071no—
224C4×Q8—1071no—
225C4×Q8—1073no—
226C4×Q8—1073no—
227C4×Q8—1073no—
228C4×Q8C2×D10711yescharacteristic
229C4×D4—1073no—
230C4×D4—1073no—
231C4×D4—1073no—
232C4×D4—1073no—
233C4×D4—1076no—
234C4×D4—1073no—
235C4×D4—1073no—
236C4×D4C2×D10713yes—
237#117—1073no—
238#117—1073no—
239#117—1073no—
240#117—1076no—
241#117—1073no—
242C2×C4⋊C4—1073no—
243C2×C42—1073no—
244#41—24no—
245#41—24no—
246C2×Q8—1073no—
247C2×Q8—1073no—
248C2×Q8—2144no—
249C2×Q8C22×D10711yescharacteristic
250C2×D4—1073no—
251C2×D4—1073no—
252C2×D4C22×D10713yes—
253C2×D4C22×D10713yes—
254C2×D4C22×D10716yes—
255C2×D4C22×D10713yes—
256C22×C4—1073no—
257C22×C4—1073no—
258C22×C4—1076no—
259C22×C4—1073no—
260C22×C4C22×D10713yes—
261C22×C4C22×D10713yes—
262C4⋊C4—1073no—
263C4⋊C4—1073no—
264C4⋊C4—1073no—
265C4⋊C4—1071no—
266C4⋊C4—1073no—
267C4⋊C4—1073no—
268C4⋊C4—1073no—
269C4⋊C4—1073no—
270C4⋊C4—1076no—
271C4⋊C4—1073no—
272C4⋊C4—1073no—
273C4⋊C4—1071no—
274C4⋊C4—1073no—
275C4⋊C4C22×D10713yes—
276C4⋊C4C22×D10711yescharacteristic
277C22⋊C4—1073no—
278C22⋊C4—1076no—
279C22⋊C4—1073no—
280C22⋊C4—1076no—
281C22⋊C4—1073no—
282C22⋊C4—1076no—
283C22⋊C4—1073no—
284C22⋊C4—1071no—
285C22⋊C4—1073no—
286C22⋊C4—1071no—
287C22⋊C4—1071no—
288C22⋊C4C22×D10713yes—
289C22⋊C4C22×D10713yes—
290C42—1071no—
291C42—1073no—
292C42—1073no—
293C42—1073no—
294C42—1076no—
295C42—1073no—
296C42C22×D10713yes—
297C23—1071no—
298C23C23×D10713yes—
299C23C23×D10713yes—
300Q8—4283no—
301Q8—2146no—
302Q8#1313381714yes—
303D4—26no—
304D4—26no—
305D4—46no—
306D4—43no—
307D4—4283no—
308D4—4283no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—1071no—
312C2×C4—1071no—
313C2×C4—1073no—
314C2×C4—1071no—
315C2×C4—1073no—
316C2×C4—1073no—
317C2×C4—1073no—
318C2×C4—4283no—
319C2×C4—4283no—
320C2×C4—4286no—
321C2×C4—2146no—
322C2×C4C23×D10711yescharacteristic
323C2×C4C23×D10713yes—
324C2×C4C23×D10711yescharacteristic
325C2×C4C23×D10711yescharacteristic
326C2×C4C23×D10713yes—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—23no—
331C22—23no—
332C22—23no—
333C22—2141no—
334C22—2141no—
335C22—2141no—
336C22C24×D10711yescharacteristic
337C4—21no—
338C4—21no—
339C4—23no—
340C4—21no—
341C4—2143no—
342C4—2141no—
343C4—2143no—
344C4—2141no—
345C4—2141no—
346C4—2143no—
347C4—2143no—
348C4C2×#1313381716yes—
349C2—43no—
350C2—43no—
351C2—4281no—
352C2C22×#1313381711yescharacteristic
353C2#2029346211yescharacteristic
354C2#2029346311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D107
commutator, socle, cosocleC22×C107C25
Fitting#550×C107C2
hypercenter#550D107
hypocenterC107#3524
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #20293363C2semidirect
2. #20293353C2semidirect
3. #20293341C2semidirect
4. #20293329C2semidirect
5. #20293317C2semidirect
6. #20293284C2semidirect
7. #20293284C2semidirect
8. #20293244C2semidirect
9. #20293231C2semidirect
10. #20293158C2semidirect
11. #20293157C2semidirect
12. #20293110C2semidirect
13. #20293103C2semidirect
14. #41×Dic107C2semidirect
15. #550×C107C2semidirect
178. C107#3524semidirect
184. #550D107semidirect

Maximal subgroups

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

subgroupindex
#202933632
#202933532
#202933412
#202933292
#202933172
#202932842
#202932442
#202932312
#202931582
#202931572
#202931102
#202931032
#41×Dic1072
#550×C1072
#3524107