Finite Group Databasebeta

← #28350051

Subgroups — #28350051

Order 13696.

1112 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#20293359C213yesmaximal, complemented
2#20293353C211yesmaximal, characteristic, complemented
3#20293348C213yesmaximal, complemented
4#20293329C213yesmaximal, complemented
5#20293323C213yesmaximal, complemented
6#20293296C211yesmaximal, characteristic, complemented
7#20293288C211yesmaximal, characteristic, complemented
8#20293172C213yesmaximal, complemented
9#20293163C213yesmaximal, complemented
10#20293156C213yesmaximal, complemented
11#41×Dic107C211yesmaximal, characteristic, complemented
12C4×#13133819C211yesmaximal, characteristic, complemented
13C4×#13133817C213yesmaximal, complemented
14C2×Q8×Dic107C211yesmaximal, characteristic, complemented
15#548×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133817C2213yes—
18#16812108C2213yes—
19#16812105C2213yes—
20#16812104C2216yes—
21#16812102C2216yes—
22#16812101C2213yes—
23#16812100C2213yes—
24#16812098C2213yes—
25#16812095C2216yes—
26#16812093C2216yes—
27#16812091C2216yes—
28#16812085C2213yes—
29#16812083C2213yes—
30#16812083C2213yes—
31#16812081C2216yes—
32#16812080C2213yes—
33Q8×Dic107C2213yes—
34Q8×Dic107C2211yescharacteristic
35Q8×Dic107C2211yescharacteristic
36Q8×Dic107C2213yes—
37Q8×Dic107C2211yescharacteristic
38C4×Dic214C2213yes—
39#16812074C2213yes—
40C4×D428C2213yes—
41D4×Dic107C2213yes—
42D4×Dic107C2213yes—
43D4×Dic107C2213yes—
44#16812071C2213yes—
45C4×C22⋊D107C2213yes—
46C4×C22⋊D107C2213yes—
47(C22⋊D107)⋊C4C2216yes—
48(C22⋊D107)⋊C4C2213yes—
49(C22⋊D107)⋊C4C2213yes—
50#16812068C2213yes—
51#16812067C2213yes—
52#16812067C2213yes—
53#16812065C2213yes—
54#16812065C2211yescharacteristic
55C2×C4⋊Dic107C2213yes—
56C2×C4×Dic107C2213yes—
57C2×C4×Dic107C2213yes—
58C42×D107C2213yes—
59C2×#41×C107C2211yescharacteristic
60C22×Q8×C107C2211yescharacteristic
61#124×C107C2213yes—
62C22⋊Q8×C107C2213yes—
63#121×C107C2213yes—
64C4×Q8×C107C2211yescharacteristic
65C4×D4×C107C2213yes—
66#13133819—24no—
67#13133817—212no—
68C2×Dic214C2313yes—
69C2×D428C2313yes—
70C2×C22⋊D107C2313yes—
71C2×C22⋊D107C2313yes—
72C2×C22⋊D107C2316yes—
73C22×Dic107C2313yes—
74C22×Dic107C2313yes—
75C22×Dic107C2313yes—
76C22×Dic107C2311yescharacteristic
77C2×C4×D107C2313yes—
78C2×C4×D107C2313yes—
79C2×C4×D107C2313yes—
80C4⋊Dic107C2313yes—
81C4⋊Dic107C2313yes—
82C4⋊Dic107C2316yes—
83C4⋊Dic107C2313yes—
84C4⋊Dic107C2313yes—
85Dic107⋊C4C2311yescharacteristic
86Dic107⋊C4C2313yes—
87Dic107⋊C4C2311yescharacteristic
88Dic107⋊C4C2313yes—
89Dic107⋊C4C2316yes—
90C22⋊Dic107C2311yescharacteristic
91C22⋊Dic107C2313yes—
92C22⋊Dic107C2316yes—
93(C2×D107)⋊C4C2313yes—
94(C2×D107)⋊C4C2313yes—
95(C2×D107)⋊C4C2316yes—
96(C2×D107)⋊C4C2316yes—
97C4×Dic107C2311yescharacteristic
98C4×Dic107C2313yes—
99C4×Dic107C2313yes—
100C4×Dic107C2311yescharacteristic
101C4×Dic107C2313yes—
102C4×Dic107C2313yes—
103C4×Dic107C2313yes—
104C4×Dic107C2313yes—
105C4×Dic107C2313yes—
106#41×C107—24no—
107C2×Q8×C107—24no—
108C2×Q8×C107C2313yes—
109C2×Q8×C107C2311yescharacteristic
110C2×D4×C107C2313yes—
111C2×D4×C107C2313yes—
112C22×C4×C107C2313yes—
113C22×C4×C107C2313yes—
114C4⋊C4×C107C2313yes—
115C4⋊C4×C107C2313yes—
116C22⋊C4×C107C2311yescharacteristic
117C22⋊C4×C107C2313yes—
118C22⋊C4×C107C2316yes—
119C42×C107C2313yes—
120C23×C107C2413yes—
121C23×C107C2411yescharacteristic
122C22×D107C2413yes—
123Q8×C107—26no—
124Q8×C107#4114yescomplemented
125D4×C107—26no—
126D4×C107—26no—
127C2×C4×C107—26no—
128C2×C4×C107—26no—
129C2×C4×C107C2411yescharacteristic
130C2×C4×C107C2413yes—
131C2×C4×C107C2413yes—
132C2×C4×C107C2411yescharacteristic
133C2×C4×C107C2413yes—
134C22⋊D107—26no—
135C22⋊D107—26no—
136C22⋊D107—46no—
137C2×Dic107—26no—
138C2×Dic107—43no—
139C2×Dic107—43no—
140C2×Dic107#4114yescomplemented
141C2×Dic107C2411yescharacteristic
142C2×Dic107C2413yes—
143C2×Dic107C2411yescharacteristic
144C2×Dic107C2413yes—
145C2×Dic107C2413yes—
146C2×Dic107C2411yescharacteristic
147C2×Dic107C2411yescharacteristic
148D428—26no—
149C4×D107—26no—
150C4×D107—26no—
151C4×D107—43no—
152Dic214—26no—
153C22×C107—21no—
154C22×C107—21no—
155C22×C107—23no—
156C22×C107—23no—
157C22×C107—23no—
158C22×C107C2×#4112yes—
159C22×C107C2511yescharacteristic
160C2×D107—23no—
161C2×D107—23no—
162C2×D107—23no—
163C4×C107—23no—
164C4×C107—21no—
165C4×C107—23no—
166C4×C107—21no—
167C4×C107C2×#4116yes—
168Dic107—23no—
169Dic107—21no—
170Dic107—23no—
171Dic107—21no—
172Dic107—23no—
173Dic107C2×#4112yes—
174Dic107C2×#4112yes—
175C2×C107—22no—
176C2×C107—43no—
177C2×C107C22×#4111yescharacteristic
178C2×C107C22×#4111yescharacteristic
179C2×C107#58511yescharacteristic
180D107—43no—
181#3480—1071nomaximal, Sylow
182C107#348011yescharacteristic, complemented, Sylow
183#559—1073no—
184#557—1071no—
185#556—1073no—
186#553—1073no—
187#553—1073no—
188#550—1071no—
189#548—1071no—
190#548D10711yescharacteristic, complemented
191#532—1073no—
192#529—1073no—
193#529—1073no—
194C4×#41—1071no—
195C4×#41—1071no—
196C4×#41—1073no—
197C2×C4×Q8—1071no—
198C2×#41—1071no—
199C2×#41—1073no—
200C2×#41C2×D10711yescharacteristic
201C22×Q8C2×D10711yescharacteristic
202C4⋊Q8—1073no—
203#127—1073no—
204#126—1076no—
205#126—1076no—
206#125—1073no—
207#125—1073no—
208#124—1073no—
209#124—1076no—
210#124C2×D10713yes—
211#123—1076no—
212#123—1076no—
213C22⋊Q8—1073no—
214C22⋊Q8C2×D10713yes—
215#121—1073no—
216#121—1076no—
217#121—1073no—
218#121—1073no—
219#121C2×D10713yes—
220C4×Q8—1071no—
221C4×Q8—1073no—
222C4×Q8—1071no—
223C4×Q8—1073no—
224C4×Q8—1071no—
225C4×Q8—1073no—
226C4×Q8—1073no—
227C4×Q8C2×D10711yescharacteristic
228C4×D4—1073no—
229C4×D4—1073no—
230C4×D4—1073no—
231C4×D4—1073no—
232C4×D4—1073no—
233C4×D4—1073no—
234C4×D4—1073no—
235C4×D4—1073no—
236C4×D4—1076no—
237C4×D4—1073no—
238C4×D4C2×D10713yes—
239#117—1073no—
240#117—1073no—
241#117—1073no—
242#117—1071no—
243#117—1073no—
244C2×C4⋊C4—1073no—
245C2×C42—1073no—
246C2×C42—1073no—
247C2×C42—1073no—
248#41—24no—
249#41—21412no—
250#41—2144no—
251C2×Q8—24no—
252C2×Q8—1073no—
253C2×Q8C22×D10713yes—
254C2×Q8C22×D10711yescharacteristic
255C2×D4—1076no—
256C2×D4—1073no—
257C2×D4—1073no—
258C2×D4—1073no—
259C2×D4C22×D10713yes—
260C2×D4C22×D10713yes—
261C22×C4—1071no—
262C22×C4—1073no—
263C22×C4—1073no—
264C22×C4—1073no—
265C22×C4—1073no—
266C22×C4—1073no—
267C22×C4—1073no—
268C22×C4C22×D10713yes—
269C22×C4C22×D10713yes—
270C4⋊C4—1073no—
271C4⋊C4—1073no—
272C4⋊C4—1076no—
273C4⋊C4—1071no—
274C4⋊C4—1071no—
275C4⋊C4—1073no—
276C4⋊C4—1073no—
277C4⋊C4—1073no—
278C4⋊C4—1073no—
279C4⋊C4—1076no—
280C4⋊C4C22×D10713yes—
281C4⋊C4C22×D10713yes—
282C22⋊C4—1071no—
283C22⋊C4—1073no—
284C22⋊C4—1076no—
285C22⋊C4—1073no—
286C22⋊C4—1076no—
287C22⋊C4—1073no—
288C22⋊C4—1076no—
289C22⋊C4C22×D10711yescharacteristic
290C22⋊C4C22×D10713yes—
291C22⋊C4C22×D10716yes—
292C42—1071no—
293C42—1071no—
294C42—1073no—
295C42—1073no—
296C42—1073no—
297C42—1073no—
298C42—1073no—
299C42—1073no—
300C42—1073no—
301C42C22×D10713yes—
302C23—1073no—
303C23C23×D10713yes—
304C23C23×D10711yescharacteristic
305Q8—26no—
306Q8—2146no—
307Q8#1313381714yescomplemented
308D4—26no—
309D4—26no—
310D4—4286no—
311D4—2146no—
312D4—2146no—
313D4—2146no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—1074no—
317C2×C4—1071no—
318C2×C4—1071no—
319C2×C4—1071no—
320C2×C4—1071no—
321C2×C4—1073no—
322C2×C4—1073no—
323C2×C4—1073no—
324C2×C4—4283no—
325C2×C4—4283no—
326C2×C4—4283no—
327C2×C4—2146no—
328C2×C4—2146no—
329C2×C4—2146no—
330C2×C4C23×D10711yescharacteristic
331C2×C4C23×D10713yes—
332C2×C4C23×D10711yescharacteristic
333C2×C4C23×D10713yes—
334C2×C4C23×D10713yes—
335C22—21no—
336C22—21no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—2143no—
341C22—2143no—
342C22—2143no—
343C22#41×D10712yes—
344C22C24×D10711yescharacteristic
345C4—23no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—1072no—
350C4—1072no—
351C4—2143no—
352C4—2141no—
353C4—2143no—
354C4—2141no—
355C4—2143no—
356C4C2×#1313381716yes—
357C2—22no—
358C2—43no—
359C2—4283no—
360C2C2×#41×D10711yescharacteristic
361C2C22×#1313381711yescharacteristic
362C2#2029346411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D107
commutator, socle, cosocleC22×C107C25
Fitting#548×C107C2
hypercenter#548D107
hypocenterC107#3480
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #20293359C2semidirect
2. #20293353C2semidirect
3. #20293348C2semidirect
4. #20293329C2semidirect
5. #20293323C2semidirect
6. #20293296C2semidirect
7. #20293288C2semidirect
8. #20293172C2semidirect
9. #20293163C2semidirect
10. #20293156C2semidirect
11. #41×Dic107C2semidirect
12. C4×#13133819C2semidirect
13. C4×#13133817C2semidirect
14. C2×Q8×Dic107C2semidirect
15. #548×C107C2semidirect
124. Q8×C107#41semidirect
140. C2×Dic107#41semidirect
182. C107#3480semidirect
190. #548D107semidirect
307. Q8#13133817semidirect

Maximal subgroups

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

subgroupindex
#202933592
#202933532
#202933482
#202933292
#202933232
#202932962
#202932882
#202931722
#202931632
#202931562
#41×Dic1072
C4×#131338192
C4×#131338172
C2×Q8×Dic1072
#548×C1072
#3480107