Finite Group Databasebeta

← #28350571

Subgroups — #28350571

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#20293359C213yesmaximal, complemented
2#20293356C211yesmaximal, characteristic, complemented
3#20293342C213yesmaximal, complemented
4#20293324C213yesmaximal, complemented
5#20293311C213yesmaximal, complemented
6#20293296C211yesmaximal, characteristic, complemented
7#20293288C211yesmaximal, characteristic, complemented
8#20293246C213yesmaximal, complemented
9#20293230C213yesmaximal, complemented
10#20293163C213yesmaximal, complemented
11#20293157C213yesmaximal, complemented
12#20293109C211yesmaximal, characteristic, complemented
13#20293105C211yesmaximal, characteristic, complemented
14#41×Dic107C211yesmaximal, characteristic, complemented
15#548×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17#16812109C2213yes—
18#16812105C2213yes—
19#16812104C2213yes—
20#16812102C2211yescharacteristic
21#16812102C2216yes—
22#16812100C2213yes—
23#16812100C2213yes—
24#16812099C2213yes—
25#16812098C2213yes—
26#16812095C2213yes—
27#16812095C2216yes—
28#16812093C2213yes—
29#16812093C2216yes—
30#16812091C2216yes—
31#16812090C2213yes—
32#16812086C2213yes—
33#16812085C2213yes—
34#16812083C2213yes—
35#16812083C2213yes—
36#16812081C2213yes—
37#16812080C2216yes—
38#16812080C2213yes—
39Q8×Dic107C2213yes—
40Q8×Dic107C2211yescharacteristic
41Q8×Dic107C2211yescharacteristic
42Q8×Dic107C2213yes—
43Q8×Dic107C2211yescharacteristic
44C4×Dic214C2213yes—
45C4×D428C2213yes—
46D4×Dic107C2213yes—
47D4×Dic107C2213yes—
48#16812071C2213yes—
49C4×C22⋊D107C2213yes—
50C4×C22⋊D107C2213yes—
51(C22⋊D107)⋊C4C2216yes—
52#16812068C2213yes—
53#16812068C2213yes—
54#16812068C2213yes—
55#16812067C2216yes—
56#16812066C2213yes—
57C4⋊C4×D107C2213yes—
58C2×C4×Dic107C2213yes—
59C2×#41×C107C2211yescharacteristic
60C22×Q8×C107C2211yescharacteristic
61#124×C107C2213yes—
62C22⋊Q8×C107C2213yes—
63#121×C107C2213yes—
64C4×Q8×C107C2211yescharacteristic
65C4×D4×C107C2213yes—
66#13133819—24no—
67C2×Dic214C2313yes—
68C2×D428C2313yes—
69C2×C22⋊D107C2316yes—
70C2×C22⋊D107C2313yes—
71C2×C22⋊D107C2313yes—
72C22×Dic107C2313yes—
73C22×Dic107C2313yes—
74C2×C4×D107C2313yes—
75C2×C4×D107C2316yes—
76C4⋊Dic107C2313yes—
77C4⋊Dic107C2316yes—
78C4⋊Dic107C2313yes—
79C4⋊Dic107C2313yes—
80C4⋊Dic107C2313yes—
81C4⋊Dic107C2313yes—
82C4⋊Dic107C2311yescharacteristic
83Dic107⋊C4C2313yes—
84Dic107⋊C4C2311yescharacteristic
85Dic107⋊C4C2313yes—
86Dic107⋊C4C2311yescharacteristic
87Dic107⋊C4C2313yes—
88Dic107⋊C4C2313yes—
89C22⋊Dic107C2313yes—
90C22⋊Dic107C2311yescharacteristic
91C22⋊Dic107C2316yes—
92C22⋊Dic107C2311yescharacteristic
93C22⋊Dic107C2313yes—
94(C2×D107)⋊C4C2313yes—
95(C2×D107)⋊C4C2313yes—
96(C2×D107)⋊C4C2313yes—
97(C2×D107)⋊C4C2316yes—
98(C2×D107)⋊C4C2313yes—
99C4×Dic107C2311yescharacteristic
100C4×Dic107C2313yes—
101C4×Dic107C2313yes—
102C4×Dic107C2313yes—
103C4×Dic107C2316yes—
104C4×Dic107C2313yes—
105#41×C107—24no—
106C2×Q8×C107—24no—
107C2×Q8×C107C2313yes—
108C2×Q8×C107C2311yescharacteristic
109C2×D4×C107C2313yes—
110C2×D4×C107C2313yes—
111C22×C4×C107C2313yes—
112C22×C4×C107C2313yes—
113C4⋊C4×C107C2313yes—
114C4⋊C4×C107C2313yes—
115C22⋊C4×C107C2316yes—
116C22⋊C4×C107C2313yes—
117C22⋊C4×C107C2311yescharacteristic
118C42×C107C2313yes—
119C23×C107C2413yes—
120C23×C107C2411yescharacteristic
121C22×D107C2413yes—
122Q8×C107—26no—
123Q8×C107#4114yes—
124D4×C107—26no—
125D4×C107—43no—
126C2×C4×C107—26no—
127C2×C4×C107—26no—
128C2×C4×C107C2411yescharacteristic
129C2×C4×C107C2413yes—
130C2×C4×C107C2413yes—
131C2×C4×C107C2411yescharacteristic
132C2×C4×C107C2413yes—
133C22⋊D107—43no—
134C22⋊D107—46no—
135C22⋊D107—43no—
136C2×Dic107—43no—
137C2×Dic107—43no—
138C2×Dic107C2411yescharacteristic
139C2×Dic107C2411yescharacteristic
140C2×Dic107C2413yes—
141C2×Dic107C2413yes—
142C2×Dic107C2411yescharacteristic
143C2×Dic107C2413yes—
144C2×Dic107C2411yescharacteristic
145D428—26no—
146C4×D107—26no—
147C4×D107—46no—
148Dic214—43no—
149C22×C107—21no—
150C22×C107—23no—
151C22×C107—23no—
152C22×C107—23no—
153C22×C107—21no—
154C22×C107—21no—
155C22×C107C2511yescharacteristic
156C2×D107—23no—
157C2×D107—23no—
158C2×D107—23no—
159C4×C107—21no—
160C4×C107—23no—
161C4×C107—21no—
162C4×C107—23no—
163C4×C107C2×#4116yes—
164Dic107—21no—
165Dic107—23no—
166Dic107—23no—
167Dic107—23no—
168Dic107—21no—
169Dic107—21no—
170Dic107—21no—
171C2×C107—41no—
172C2×C107—43no—
173C2×C107C22×#4111yescharacteristic
174C2×C107#58511yescharacteristic
175C2×C107#58511yescharacteristic
176D107—43no—
177#3524—1071nomaximal, Sylow
178C107#352411yescharacteristic, complemented, Sylow
179#559—1073no—
180#557—1071no—
181#555—1073no—
182#553—1073no—
183#552—1073no—
184#550—1071no—
185#548—1071no—
186#548D10711yescharacteristic, complemented
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×#41C2×D10711yescharacteristic
196C22×Q8C2×D10711yescharacteristic
197C4⋊Q8—1073no—
198#127—1073no—
199#126—1073no—
200#126—1071no—
201#126—1076no—
202#125—1073no—
203#125—1073no—
204#125—1073no—
205#124—1073no—
206#124—1073no—
207#124—1076no—
208#124C2×D10713yes—
209#123—1073no—
210#123—1076no—
211#123—1073no—
212#123—1076no—
213C22⋊Q8—1073no—
214C22⋊Q8—1073no—
215C22⋊Q8C2×D10713yes—
216#121—1073no—
217#121—1076no—
218#121—1073no—
219#121—1073no—
220#121—1073no—
221#121C2×D10713yes—
222C4×Q8—1071no—
223C4×Q8—1071no—
224C4×Q8—1073no—
225C4×Q8—1071no—
226C4×Q8—1073no—
227C4×Q8—1073no—
228C4×Q8C2×D10711yescharacteristic
229C4×D4—1073no—
230C4×D4—1073no—
231C4×D4—1073no—
232C4×D4—1073no—
233C4×D4—1073no—
234C4×D4—1076no—
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—2144no—
246C2×Q8—24no—
247C2×Q8—1073no—
248C2×Q8C22×D10713yes—
249C2×Q8C22×D10711yescharacteristic
250C2×D4—1073no—
251C2×D4—1076no—
252C2×D4—1073no—
253C2×D4—1073no—
254C2×D4C22×D10713yes—
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—1071no—
265C4⋊C4—1073no—
266C4⋊C4—1073no—
267C4⋊C4—1073no—
268C4⋊C4—1073no—
269C4⋊C4—1076no—
270C4⋊C4—1071no—
271C4⋊C4—1073no—
272C4⋊C4—1073no—
273C4⋊C4—1071no—
274C4⋊C4—1073no—
275C4⋊C4C22×D10713yes—
276C4⋊C4C22×D10713yes—
277C22⋊C4—1071no—
278C22⋊C4—1073no—
279C22⋊C4—1076no—
280C22⋊C4—1071no—
281C22⋊C4—1073no—
282C22⋊C4—1073no—
283C22⋊C4—1073no—
284C22⋊C4—1073no—
285C22⋊C4—1076no—
286C22⋊C4—1073no—
287C22⋊C4C22×D10716yes—
288C22⋊C4C22×D10711yescharacteristic
289C22⋊C4C22×D10713yes—
290C42—1071no—
291C42—1073no—
292C42—1073no—
293C42—1073no—
294C42—1076no—
295C42—1073no—
296C42C22×D10713yes—
297C23—1073no—
298C23C23×D10711yescharacteristic
299C23C23×D10713yes—
300Q8—26no—
301Q8—4283no—
302Q8#1313381714yes—
303D4—26no—
304D4—43no—
305D4—4283no—
306D4—4283no—
307D4—4286no—
308D4—2146no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—1071no—
312C2×C4—1073no—
313C2×C4—1071no—
314C2×C4—1073no—
315C2×C4—1071no—
316C2×C4—1073no—
317C2×C4—1071no—
318C2×C4—4286no—
319C2×C4—4283no—
320C2×C4—4283no—
321C2×C4—2146no—
322C2×C4C23×D10711yescharacteristic
323C2×C4C23×D10713yes—
324C2×C4C23×D10711yescharacteristic
325C2×C4C23×D10713yes—
326C2×C4C23×D10713yes—
327C22—21no—
328C22—23no—
329C22—23no—
330C22—21no—
331C22—23no—
332C22—21no—
333C22—2143no—
334C22—2143no—
335C22—2143no—
336C22C24×D10711yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—2141no—
342C4—2143no—
343C4—2141no—
344C4—2141no—
345C4—2141no—
346C4—2143no—
347C4—2143no—
348C4C2×#1313381716yes—
349C2—41no—
350C2—43no—
351C2—4283no—
352C2C22×#1313381711yescharacteristic
353C2#2029346311yescharacteristic
354C2#2029346411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D107
commutator, socle, cosocleC22×C107C25
Fitting#548×C107C2
hypercenter#548D107
hypocenterC107#3524
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #20293359C2semidirect
2. #20293356C2semidirect
3. #20293342C2semidirect
4. #20293324C2semidirect
5. #20293311C2semidirect
6. #20293296C2semidirect
7. #20293288C2semidirect
8. #20293246C2semidirect
9. #20293230C2semidirect
10. #20293163C2semidirect
11. #20293157C2semidirect
12. #20293109C2semidirect
13. #20293105C2semidirect
14. #41×Dic107C2semidirect
15. #548×C107C2semidirect
178. C107#3524semidirect
186. #548D107semidirect

Maximal subgroups

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

subgroupindex
#202933592
#202933562
#202933422
#202933242
#202933112
#202932962
#202932882
#202932462
#202932302
#202931632
#202931572
#202931092
#202931052
#41×Dic1072
#548×C1072
#3524107