Finite Group Databasebeta

← #28347972

Subgroups — #28347972

Order 13696.

494 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×#16812140C211yesmaximal, characteristic, complemented
2C22×#13133805C211yesmaximal, characteristic, complemented
3#20293273C211yesmaximal, characteristic
4#20293019C211yesmaximal, characteristic, complemented
5#20292875C211yesmaximal, characteristic, complemented
6#20292866C211yesmaximal, characteristic, complemented
7#20292849C211yesmaximal, characteristic, complemented
8#20292829C211yesmaximal, characteristic, complemented
9#20292787C211yesmaximal, characteristic, complemented
10#20292783C211yesmaximal, characteristic, complemented
11#20292777C211yesmaximal, characteristic, complemented
12#20292766C211yesmaximal, characteristic, complemented
13#20292725C211yesmaximal, characteristic, complemented
14#20292699C211yesmaximal, characteristic, complemented
15D4×Q8×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C22×D428C2211yescharacteristic
18#16812140—24no—
19C2×#13133808C2211yescharacteristic, complemented
20C2×#13133808C2211yescharacteristic, complemented
21C2×#13133808C2211yescharacteristic, complemented
22C2×#13133805—24no—
23C2×#13133805C2211yescharacteristic
24C2×#13133805C2211yescharacteristic
25C2×#13133805C2211yescharacteristic
26C2×#13133805C2211yescharacteristic
27C2×#13133805C2211yescharacteristic
28C2×#13133805C2211yescharacteristic
29C2×#13133802C2211yescharacteristic, complemented
30C22×C107⋊C8C2211yescharacteristic
31#16812097C2211yescharacteristic
32#16812094—21no—
33#16812088C2211yescharacteristic
34#16812083C2211yescharacteristic
35#16812082—21no—
36#16812077—21no—
37C4×D428C2211yescharacteristic
38C22⋊C4×D107—21no—
39#16812042C2211yescharacteristic, complemented
40C4⋊(C107⋊C8)C2211yescharacteristic, complemented
41#16812033C2211yescharacteristic, complemented
42#16812033C2211yescharacteristic, complemented
43#16812033C2211yescharacteristic, complemented
44#16812032C2211yescharacteristic, complemented
45#16812029C2211yescharacteristic
46#16812029C2211yescharacteristic
47C22⋊(C107⋊C8)C2211yescharacteristic
48C22⋊(C107⋊C8)C2211yescharacteristic, complemented
49#16812020C2211yescharacteristic, complemented
50C22×Q8×C107C2211yescharacteristic
51C22×Q8×C107C2211yescharacteristic, complemented
52C4⋊Q8×C107—21no—
53C4⋊Q8×C107C2211yescharacteristic, complemented
54C22⋊Q8×C107—21no—
55C22⋊Q8×C107—21no—
56C22⋊Q8×C107C2211yescharacteristic
57C22⋊Q8×C107C2211yescharacteristic, complemented
58C4×Q8×C107C2211yescharacteristic, complemented
59C4×D4×C107—21no—
60C4×D4×C107C2211yescharacteristic
61C23×D107—21no—
62#13133818—22no—
63#13133818—41no—
64C2×Dic214C2311yescharacteristic
65C2×D428—22no—
66C2×D428—21no—
67C2×D428—41no—
68C2×D428C2311yescharacteristic
69C2×D428C2311yescharacteristic
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107—21no—
74#13133808—22no—
75#13133808—22no—
76#13133808—41no—
77#13133805—22no—
78#13133805—22no—
79#13133805—22no—
80#13133805—22no—
81#13133805—41no—
82#13133805D414yescomplemented
83#13133802—22no—
84C2×C107⋊C8—22no—
85C2×C107⋊C8C2311yescharacteristic
86C2×C107⋊C8C2311yescharacteristic
87C2×C107⋊C8C2311yescharacteristic
88C2×C107⋊C8C2311yescharacteristic
89C4⋊Dic107C2311yescharacteristic
90Dic107⋊C4—21no—
91C22⋊Dic107—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96C2×Q8×C107—21no—
97C2×Q8×C107—21no—
98C2×Q8×C107—22no—
99C2×Q8×C107—22no—
100C2×Q8×C107—41no—
101C2×Q8×C107—41no—
102C2×Q8×C107C2311yescharacteristic
103C2×Q8×C107C2311yescharacteristic
104C2×Q8×C107C2311yescharacteristic
105C2×D4×C107D411yescharacteristic
106C22×C4×C107—21no—
107C22×C4×C107—21no—
108C22×C4×C107C2311yescharacteristic
109C22×C4×C107C2311yescharacteristic
110C4⋊C4×C107—21no—
111C4⋊C4×C107—21no—
112C4⋊C4×C107—21no—
113C4⋊C4×C107—21no—
114C4⋊C4×C107—21no—
115C4⋊C4×C107D411yescharacteristic
116C4⋊C4×C107C2311yescharacteristic
117C22⋊C4×C107—21no—
118C22⋊C4×C107—21no—
119C22⋊C4×C107D411yescharacteristic
120C22⋊C4×C107D411yescharacteristic
121C42×C107—21no—
122C42×C107C2311yescharacteristic
123C23×C107C2×D411yescharacteristic
124C23×C107C2×D411yescharacteristic
125C22×D107—21no—
126C22×D107—21no—
127C22×D107—42no—
128C22×D107—42no—
129C22×D107—81no—
130Q8×C107—21no—
131Q8×C107—21no—
132Q8×C107—21no—
133Q8×C107—21no—
134Q8×C107—21no—
135Q8×C107—42no—
136Q8×C107—42no—
137Q8×C107C2×D412yes—
138D4×C107—22no—
139C2×C4×C107—21no—
140C2×C4×C107—21no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—21no—
145C2×C4×C107—21no—
146C2×C4×C107—21no—
147C2×C4×C107—42no—
148C2×C4×C107—42no—
149C2×C4×C107C2×D411yescharacteristic
150C2×C4×C107C2×D411yescharacteristic
151C2×C4×C107C2×D411yescharacteristic
152C2×C4×C107C2×D411yescharacteristic
153C2×C4×C107C2411yescharacteristic
154C22⋊D107—42no—
155C22⋊D107—81no—
156C2×Dic107—21no—
157C2×Dic107—21no—
158D428—21no—
159D428—21no—
160D428—21no—
161D428—42no—
162D428C2×D412yes—
163C4×D107—42no—
164C4×D107—81no—
165Dic214—21no—
166Dic214—21no—
167C107⋊C8—21no—
168C107⋊C8—21no—
169C107⋊C8—21no—
170C107⋊C8C2×D412yes—
171C22×C107—21no—
172C22×C107—21no—
173C22×C107—21no—
174C22×C107—21no—
175C22×C107—21no—
176C22×C107#13712yescomplemented
177C22×C107C22×D411yescharacteristic
178C2×D107—22no—
179C2×D107—41no—
180C2×D107—41no—
181C2×D107—41no—
182C2×D107—41no—
183C2×D107—41no—
184C2×D107—84no—
185C4×C107—21no—
186C4×C107—21no—
187C4×C107—22no—
188C4×C107—21no—
189C4×C107—21no—
190C4×C107—41no—
191C4×C107—41no—
192C4×C107—41no—
193C4×C107C22×D411yescharacteristic
194C4×C107H(5,𝔽2)11yescharacteristic
195Dic107—41no—
196Dic107—41no—
197C2×C107—22no—
198C2×C107—41no—
199C2×C107D4211yescharacteristic
200C2×C107C2×#13711yescharacteristic
201C2×C107#57711yescharacteristic
202D107—42no—
203D107—81no—
204#3266—1071nomaximal, Sylow
205C107#326611yescharacteristic, complemented, Sylow
206C2×#137—1071no—
207C22×SD16—1071no—
208D4×Q8D10711yescharacteristic, complemented
209#546—1071no—
210#497—1071no—
211#468—1071no—
212#467—1071no—
213#464—1071no—
214#461—1071no—
215#452—1071no—
216#451—1071no—
217#450—1071no—
218#448—1071no—
219#440—1071no—
220#435—1071no—
221C2×#41—1071no—
222C22×Q8C2×D10711yescharacteristic
223C22×Q8C2×D10711yescharacteristic, complemented
224C22×D4—1071no—
225#137—2144no—
226C2×Q16—1071no—
227C2×Q16—1071no—
228C2×Q16—1071no—
229C2×SD16—1071no—
230C2×SD16—1071no—
231C2×SD16—1071no—
232C2×SD16—1071no—
233C2×SD16—1071no—
234C2×SD16—1071no—
235C2×SD16—2144no—
236C2×M16—1071no—
237C22×C8—1071no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D10711yescharacteristic, complemented
240#124—1071no—
241#123—2141no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—1071no—
245C22⋊Q8C2×D10711yescharacteristic
246C22⋊Q8C2×D10711yescharacteristic, complemented
247#121—1071no—
248#121—2141no—
249#120—2141no—
250C4×Q8C2×D10711yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—1071no—
253C4×D4C2×D10711yescharacteristic
254C2×C22⋊C4—2141no—
255#107—1071no—
256C4⋊C8—1071no—
257#103—1071no—
258#103—1071no—
259#103—1071no—
260#103—1071no—
261#102—1071no—
262#102—1071no—
263C22⋊C8—1071no—
264C22⋊C8—1071no—
265#97—1071no—
266C24—2141no—
267#41—4281no—
268#41—2142no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—1071no—
276C2×Q8C22×D10711yescharacteristic
277C2×Q8C22×D10711yescharacteristic
278C2×Q8C22×D10711yescharacteristic
279C2×D4—1071no—
280C2×D4—1071no—
281C2×D4—4281no—
282C2×D4—2142no—
283C2×D4—2141no—
284C2×D4—2141no—
285C2×D4—2141no—
286C2×D4C22⋊D10711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—2141no—
290C22×C4—2141no—
291C22×C4C22×D10711yescharacteristic
292C22×C4C22×D10711yescharacteristic
293Q16—4281no—
294Q16—2142no—
295Q16—2142no—
296SD16—1074no—
297SD16—4281no—
298SD16—2142no—
299SD16—2142no—
300SD16—2142no—
301SD16—2142no—
302M16—2142no—
303C2×C8—1071no—
304C2×C8—1071no—
305C2×C8—1071no—
306C2×C8—1071no—
307C2×C8—2142no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—1071no—
314C4⋊C4—2141no—
315C4⋊C4C22⋊D10711yescharacteristic
316C4⋊C4C22×D10711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—2141no—
320C22⋊C4—2141no—
321C22⋊C4—2141no—
322C22⋊C4—2141no—
323C22⋊C4—2141no—
324C22⋊C4C22⋊D10711yescharacteristic
325C22⋊C4C22⋊D10711yescharacteristic
326C42—21no—
327C42C22×D10711yescharacteristic
328C23—8561no—
329C23—4282no—
330C23—4282no—
331C23—2141no—
332C23—2141no—
333C23C2×C22⋊D10711yescharacteristic
334C23C2×C22⋊D10711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—2141no—
343Q8—2141no—
344Q8D4×D10712yes—
345D4—22no—
346D4—8561no—
347D4—1072no—
348D4—4282no—
349D4—4282no—
350D4—2141no—
351D4—2141no—
352D4—2141no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—8561no—
364C2×C4—4282no—
365C2×C4—2141no—
366C2×C4—2141no—
367C2×C4C2×C22⋊D10711yescharacteristic
368C2×C4C2×C22⋊D10711yescharacteristic
369C2×C4C2×C22⋊D10711yescharacteristic
370C2×C4C2×C22⋊D10711yescharacteristic
371C2×C4C23×D10711yescharacteristic
372C8—1072no—
373C8—2141no—
374C8—2141no—
375C8—2141no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—8564no—
382C22—4281no—
383C22—4281no—
384C22—4281no—
385C22—4281no—
386C22—4281no—
387C22—2142no—
388C22#1681214012yescomplemented
389C22C22×C22⋊D10711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—4281no—
399C4—4281no—
400C4C2×D4×D10711yescharacteristic
401C4#1681215411yescharacteristic
402C2—22no—
403C2—41no—
404C2—8561no—
405C2—4282no—
406C2D4×C22⋊D10711yescharacteristic
407C2C2×#1681214011yescharacteristic
408C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingD4×Q8×C107C2
socleC22×C107C22×D4
prime radical#20293273C2
hypercenterD4×Q8D107
hypocenterC107#3266
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16812140C2semidirect
2. C22×#13133805C2semidirect
4. #20293019C2semidirect
5. #20292875C2semidirect
6. #20292866C2semidirect
7. #20292849C2semidirect
8. #20292829C2semidirect
9. #20292787C2semidirect
10. #20292783C2semidirect
11. #20292777C2semidirect
12. #20292766C2semidirect
13. #20292725C2semidirect
14. #20292699C2semidirect
15. D4×Q8×C107C2semidirect
19. C2×#13133808C22semidirect
20. C2×#13133808C22semidirect
21. C2×#13133808C22semidirect
29. C2×#13133802C22semidirect
39. #16812042C22semidirect
40. C4⋊(C107⋊C8)C22semidirect
41. #16812033C22semidirect
42. #16812033C22semidirect
43. #16812033C22semidirect
44. #16812032C22semidirect
48. C22⋊(C107⋊C8)C22semidirect
49. #16812020C22semidirect
51. C22×Q8×C107C22semidirect
53. C4⋊Q8×C107C22semidirect
57. C22⋊Q8×C107C22semidirect
58. C4×Q8×C107C22semidirect
82. #13133805D4semidirect
176. C22×C107#137semidirect
205. C107#3266semidirect
208. D4×Q8D107semidirect
223. C22×Q8C2×D107semidirect
239. C4⋊Q8C2×D107semidirect
246. C22⋊Q8C2×D107semidirect
250. C4×Q8C2×D107semidirect
388. C22#16812140semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
C22×#131338052
#202932732
#202930192
#202928752
#202928662
#202928492
#202928292
#202927872
#202927832
#202927772
#202927662
#202927252
#202926992
D4×Q8×C1072
#3266107