Finite Group Databasebeta

← #28345995

Subgroups — #28345995

Order 13696.

504 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812133C211yesmaximal, characteristic, complemented
2C2×#16812128C211yesmaximal, characteristic, complemented
3#20293190C211yesmaximal, characteristic, complemented
4#20293106C211yesmaximal, characteristic, complemented
5#20292850C211yesmaximal, characteristic, complemented
6#20292848C211yesmaximal, characteristic, complemented
7#20292842C211yesmaximal, characteristic, complemented
8#20292840C211yesmaximal, characteristic, complemented
9#20292826C211yesmaximal, characteristic, complemented
10#20292824C211yesmaximal, characteristic, complemented
11#20292820C211yesmaximal, characteristic, complemented
12#20292652C211yesmaximal, characteristic, complemented
13#20292625C211yesmaximal, characteristic, complemented
14#102×D107C211yesmaximal, characteristic, complemented
15#459×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×#13133817C2211yescharacteristic
18C2×D4×D107C2211yescharacteristic
19#16812133—24no—
20#16812128—24no—
21C2×Dic428C2211yescharacteristic, complemented
22C2×#13133807C2211yescharacteristic
23C2×#13133806C2211yescharacteristic
24C2×#13133806C2211yescharacteristic, complemented
25C2×#13133806C2211yescharacteristic, complemented
26C2×#13133803C2211yescharacteristic
27C2×#13133801C2211yescharacteristic
28C2×C8×D107C2211yescharacteristic
29#16812105C2211yescharacteristic
30#16812097C2211yescharacteristic
31#16812096C2211yescharacteristic
32#16812081—21no—
33#16812081—21no—
34#16812077—21no—
35#16812077—21no—
36C4×Dic214C2211yescharacteristic, complemented
37#16812074C2211yescharacteristic, complemented
38D4×Dic107C2211yescharacteristic, complemented
39C4×C22⋊D107—21no—
40(C22⋊D107)⋊C4—21no—
41#16812066C2211yescharacteristic
42#16812065—21no—
43C2×Dic107⋊C4—21no—
44C4⋊C4×D107C2211yescharacteristic
45C4⋊(C107⋊C8)C2211yescharacteristic, complemented
46Dic107⋊C8C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812032C2211yescharacteristic, complemented
50#16812031C2211yescharacteristic
51#16812030C2211yescharacteristic
52#16812030C2211yescharacteristic, complemented
53#16812029C2211yescharacteristic
54C2×D8×C107C2211yescharacteristic
55C2×D8×C107C2211yescharacteristic, complemented
56#127×C107C2211yescharacteristic
57C4×D4×C107C2211yescharacteristic, complemented
58C4⋊C8×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133817—22no—
63#13133817—22no—
64#13133817—22no—
65#13133817D414yescomplemented
66C2×Dic214C2311yescharacteristic
67C2×Dic214C2311yescharacteristic
68C2×D428C2311yescharacteristic
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74D4×D107—22no—
75D4×D107—22no—
76C22×Dic107—21no—
77C22×Dic107—21no—
78C22×Dic107—21no—
79C2×C4×D107—21no—
80C2×C4×D107C2311yescharacteristic
81Dic428—22no—
82#13133807—22no—
83#13133806—22no—
84#13133806—22no—
85#13133806—22no—
86#13133803—22no—
87#13133801—22no—
88C2×C107⋊C8C2311yescharacteristic
89C2×C107⋊C8C2311yescharacteristic
90C8×D107—22no—
91C4⋊Dic107C2311yescharacteristic
92Dic107⋊C4—21no—
93Dic107⋊C4—21no—
94Dic107⋊C4—21no—
95Dic107⋊C4D411yescharacteristic
96Dic107⋊C4D411yescharacteristic
97C22⋊Dic107—21no—
98C22⋊Dic107—21no—
99C22⋊Dic107—21no—
100(C2×D107)⋊C4—21no—
101(C2×D107)⋊C4—21no—
102(C2×D107)⋊C4D411yescharacteristic
103(C2×D107)⋊C4D411yescharacteristic
104C4×Dic107—21no—
105C4×Dic107—21no—
106C4×Dic107C2311yescharacteristic
107C2×D4×C107—21no—
108C2×D4×C107—21no—
109C2×D4×C107C2311yescharacteristic
110C2×D4×C107C2311yescharacteristic
111C2×D4×C107C2311yescharacteristic
112C22×C4×C107—21no—
113D8×C107—22no—
114D8×C107—22no—
115C2×C8×C107C2311yescharacteristic
116C2×C8×C107C2311yescharacteristic
117C4⋊C4×C107C2311yescharacteristic
118C22⋊C4×C107—21no—
119C42×C107C2311yescharacteristic
120C23×C107—21no—
121C23×C107—21no—
122C23×C107—21no—
123C22×D107—21no—
124C22×D107—42no—
125C22×D107—81no—
126C22×D107—81no—
127C22×D107C2×D411yescharacteristic
128D4×C107—21no—
129D4×C107—21no—
130D4×C107—21no—
131D4×C107—21no—
132D4×C107—21no—
133D4×C107—81no—
134D4×C107—81no—
135D4×C107C2×D412yes—
136C2×C4×C107—21no—
137C2×C4×C107—81no—
138C2×C4×C107C2×D411yescharacteristic
139C2×C4×C107C2×D411yescharacteristic
140C2×C4×C107C2411yescharacteristic
141C22⋊D107—24no—
142C22⋊D107—42no—
143C22⋊D107—42no—
144C22⋊D107—81no—
145C22⋊D107—81no—
146C2×Dic107—21no—
147C2×Dic107—24no—
148C2×Dic107—21no—
149C2×Dic107—21no—
150C2×Dic107—42no—
151C2×Dic107—81no—
152C2×Dic107C2×D411yescharacteristic
153C2×Dic107C2×D411yescharacteristic
154C2×Dic107C2×D411yescharacteristic
155D428—21no—
156D428—21no—
157C4×D107—81no—
158C4×D107C2×D412yes—
159C4×D107#4112yes—
160Dic214—21no—
161Dic214—21no—
162Dic214—21no—
163Dic214C2×D412yes—
164C8×C107—21no—
165C8×C107—21no—
166C107⋊C8—21no—
167C107⋊C8—21no—
168C22×C107—22no—
169C22×C107—41no—
170C22×C107—41no—
171C22×C107—41no—
172C22×C107—41no—
173C22×C107—41no—
174C22×C107—41no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107C22×D411yescharacteristic
178C2×D107—21no—
179C2×D107—21no—
180C2×D107—41no—
181C2×D107—41no—
182C2×D107—41no—
183C2×D107—84no—
184C2×D107#12112yes—
185C4×C107—21no—
186C4×C107—21no—
187C4×C107—41no—
188C4×C107C22×D411yescharacteristic
189C4×C107C2×#4111yescharacteristic
190Dic107—22no—
191Dic107—21no—
192Dic107—21no—
193Dic107—41no—
194Dic107—41no—
195Dic107#12112yes—
196C2×C107—42no—
197C2×C107—81no—
198C2×C107—81no—
199C2×C107C2×#12111yescharacteristic
200C2×C107#57611yescharacteristic
201C2×C107#57711yescharacteristic
202D107—22no—
203D107—81no—
204#3045—1071nomaximal, Sylow
205C107#304511yescharacteristic, complemented, Sylow
206C2×AGL(1,ℤ/8)—1071no—
207C2×#135—1071no—
208#535—1071no—
209#520—1071no—
210#464—1071no—
211#464—1071no—
212#463—1071no—
213#463—1071no—
214#460—1071no—
215#460—1071no—
216#459—1071no—
217#459D10711yescharacteristic, complemented
218#424—1071no—
219#417—1071no—
220C2×#102—1071no—
221C2×#41—1071no—
222C2×#41—1071no—
223C22×D4—1071no—
224AGL(1,ℤ/8)—2144no—
225#135—2144no—
226C2×Q16—1071no—
227C2×SD16—1071no—
228C2×SD16—1071no—
229C2×SD16—1071no—
230C2×SD16—1071no—
231C2×D8—1071no—
232C2×D8C2×D10711yescharacteristic
233C2×D8C2×D10711yescharacteristic, complemented
234C2×M16—1071no—
235C22×C8—1071no—
236#127—1071no—
237#127C2×D10711yescharacteristic
238#124—1071no—
239#124—1071no—
240#121—2141no—
241#121—2141no—
242#120—2141no—
243#120—2141no—
244C4×Q8—1071no—
245C4×Q8—1071no—
246C4×D4—1071no—
247C4×D4—2141no—
248C4×D4—2141no—
249C4×D4C2×D10711yescharacteristic, complemented
250#117—1071no—
251#117—2141no—
252C2×C4⋊C4—1071no—
253C2×C4⋊C4—2141no—
254C4⋊C8—1071no—
255C4⋊C8—1071no—
256C4⋊C8—1071no—
257C4⋊C8C2×D10711yescharacteristic, complemented
258#103—1071no—
259#103—1071no—
260#102—1071no—
261#102—1071no—
262#102—1071no—
263#102—1071no—
264#102C2×D10711yescharacteristic
265#102C2×D10711yescharacteristic, complemented
266C24—2141no—
267#41—1074no—
268#41—2142no—
269#41—2142no—
270#41—2142no—
271C2×Q8—1071no—
272C2×Q8—1071no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—1071no—
276C2×D4—2142no—
277C2×D4—2142no—
278C2×D4—2141no—
279C2×D4—2141no—
280C2×D4—2141no—
281C2×D4—2141no—
282C2×D4—2141no—
283C2×D4C22×D10711yescharacteristic
284C2×D4C22×D10711yescharacteristic
285C2×D4C22×D10711yescharacteristic
286C22×C4—21no—
287C22×C4—1071no—
288C22×C4—2141no—
289C22×C4—2141no—
290C22×C4—2141no—
291C22×C4—2141no—
292Q16—2142no—
293SD16—2142no—
294SD16—2142no—
295SD16—2142no—
296SD16—2142no—
297D8—22no—
298D8—22no—
299D8—2142no—
300M16—2142no—
301C2×C8—1071no—
302C2×C8—1071no—
303C2×C8—2142no—
304C2×C8C22×D10711yescharacteristic
305C2×C8C22×D10711yescharacteristic
306C4⋊C4—1071no—
307C4⋊C4—1071no—
308C4⋊C4—1071no—
309C4⋊C4—2141no—
310C4⋊C4—2141no—
311C4⋊C4—2141no—
312C4⋊C4C22×D10711yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—1071no—
315C22⋊C4—1071no—
316C22⋊C4—2141no—
317C22⋊C4—2141no—
318C22⋊C4—2141no—
319C22⋊C4—2141no—
320C22⋊C4—2141no—
321C42—1071no—
322C42—2141no—
323C42—2141no—
324C42C22×D10711yescharacteristic
325C23—21no—
326C23—21no—
327C23—21no—
328C23—8561no—
329C23—8561no—
330C23—1071no—
331C23—4282no—
332C23—2141no—
333Q8—1072no—
334Q8—2141no—
335Q8—2141no—
336Q8—2141no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—21no—
342D4—81no—
343D4—81no—
344D4—8561no—
345D4—8561no—
346D4—4282no—
347D4—4282no—
348D4—2144no—
349D4—2141no—
350D4—2141no—
351D4D4×D10712yes—
352C2×C4—21no—
353C2×C4—81no—
354C2×C4—8561no—
355C2×C4—8561no—
356C2×C4—1071no—
357C2×C4—1072no—
358C2×C4—1072no—
359C2×C4—1071no—
360C2×C4—1071no—
361C2×C4—4282no—
362C2×C4—2144no—
363C2×C4—2141no—
364C2×C4—2141no—
365C2×C4—2141no—
366C2×C4D4×D10711yescharacteristic
367C2×C4D4×D10711yescharacteristic
368C2×C4C23×D10711yescharacteristic
369C8—21no—
370C8—21no—
371C8—2141no—
372C8—2141no—
373C22—22no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—8564no—
383C22—1072no—
384C22—4281no—
385C22—4281no—
386C22—4281no—
387C22—2141no—
388C22—2141no—
389C22C2×D4×D10711yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—1072no—
394C4—4281no—
395C4—4281no—
396C4—2141no—
397C4—2141no—
398C4—2142no—
399C4C2×D4×D10711yescharacteristic
400C4#41×D10711yescharacteristic
401C2—42no—
402C2—81no—
403C2—81no—
404C2—8561no—
405C2—2142no—
406C2#121×D10711yescharacteristic
407C2#2029342811yescharacteristic
408C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#459×C107C2
socleC22×C107C22×D4
hypercenter#459D107
hypocenterC107#3045
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. C2×#16812128C2semidirect
3. #20293190C2semidirect
4. #20293106C2semidirect
5. #20292850C2semidirect
6. #20292848C2semidirect
7. #20292842C2semidirect
8. #20292840C2semidirect
9. #20292826C2semidirect
10. #20292824C2semidirect
11. #20292820C2semidirect
12. #20292652C2semidirect
13. #20292625C2semidirect
14. #102×D107C2semidirect
15. #459×C107C2semidirect
21. C2×Dic428C22semidirect
24. C2×#13133806C22semidirect
25. C2×#13133806C22semidirect
36. C4×Dic214C22semidirect
37. #16812074C22semidirect
38. D4×Dic107C22semidirect
45. C4⋊(C107⋊C8)C22semidirect
46. Dic107⋊C8C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812034C22semidirect
49. #16812032C22semidirect
52. #16812030C22semidirect
55. C2×D8×C107C22semidirect
57. C4×D4×C107C22semidirect
58. C4⋊C8×C107C22semidirect
60. #102×C107C22semidirect
65. #13133817D4semidirect
205. C107#3045semidirect
217. #459D107semidirect
233. C2×D8C2×D107semidirect
249. C4×D4C2×D107semidirect
257. C4⋊C8C2×D107semidirect
265. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
C2×#168121282
#202931902
#202931062
#202928502
#202928482
#202928422
#202928402
#202928262
#202928242
#202928202
#202926522
#202926252
#102×D1072
#459×C1072
#3045107