Finite Group Databasebeta

← #28374851

Subgroups — #28374851

Order 13952.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20295734C211yesmaximal, characteristic, complemented
2#20295660C211yesmaximal, characteristic
3#20295371C211yesmaximal, characteristic, complemented
4#20295346C211yesmaximal, characteristic, complemented
5#20295342C211yesmaximal, characteristic, complemented
6#20295315C211yesmaximal, characteristic, complemented
7#20295258C211yesmaximal, characteristic, complemented
8#20295224C211yesmaximal, characteristic, complemented
9#20295176C211yesmaximal, characteristic, complemented
10#20295175C211yesmaximal, characteristic, complemented
11#20295111C211yesmaximal, characteristic, complemented
12C8×C22⋊D109C211yesmaximal, characteristic, complemented
13C2×#16812621C211yesmaximal, characteristic, complemented
14#102×D109C211yesmaximal, characteristic, complemented
15#465×C109C211yesmaximal, characteristic, complemented
16C22×D436C2211yescharacteristic
17C2×D4×D109—21no—
18C2×D4×D109C2211yescharacteristic
19C2×#13182016C2211yescharacteristic
20C2×#13182016C2211yescharacteristic, complemented
21C2×C8×D109C2211yescharacteristic, complemented
22#16812698C2211yescharacteristic, complemented
23#16812695C2211yescharacteristic
24#16812689—21no—
25#16812679—21no—
26#16812676—21no—
27#16812675C2211yescharacteristic, complemented
28#16812674C2211yescharacteristic
29#16812672—21no—
30#16812669—21no—
31#16812667—21no—
32Q8×Dic109C2211yescharacteristic, complemented
33#16812661C2211yescharacteristic
34C4×C22⋊D109C2211yescharacteristic
35#16812659—21no—
36C2×C4⋊Dic109C2211yescharacteristic, complemented
37C4⋊C4×D109C2211yescharacteristic, complemented
38#16812630C2211yescharacteristic, complemented
39#16812630C2211yescharacteristic, complemented
40#16812628C2211yescharacteristic, complemented
41#16812621C2211yescharacteristic
42#16812621C2211yescharacteristic
43#16812621C2211yescharacteristic, complemented
44#16812621C2211yescharacteristic, complemented
45#16812620C2211yescharacteristic, complemented
46#16812620C2211yescharacteristic, complemented
47#16812619C2211yescharacteristic
48#16812619C2211yescharacteristic, complemented
49C22⋊#11724784C2211yescharacteristic, complemented
50#16812611C2211yescharacteristic, complemented
51C8×Dic109C2211yescharacteristic, complemented
52C2×SD16×C109C2211yescharacteristic, complemented
53C22×C8×C109C2211yescharacteristic, complemented
54C22⋊Q8×C109C2211yescharacteristic, complemented
55#121×C109C2211yescharacteristic
56#106×C109C2211yescharacteristic, complemented
57#103×C109C2211yescharacteristic, complemented
58#102×C109C2211yescharacteristic, complemented
59C23×D109—21no—
60C23×D109—21no—
61C2×Dic218—21no—
62C2×D436—22no—
63C2×D436—21no—
64C2×D436—21no—
65C2×D436—41no—
66C2×D436C2311yescharacteristic
67C2×D436C2311yescharacteristic
68C2×C22⋊D109—21no—
69C2×C22⋊D109—21no—
70C2×C22⋊D109—21no—
71C2×C22⋊D109D411yescharacteristic
72D4×D109—22no—
73D4×D109—41no—
74D4×D109—44no—
75C22×Dic109—21no—
76C2×C4×D109—21no—
77C2×C4×D109—21no—
78C2×C4×D109C2311yescharacteristic
79#13182016—22no—
80#13182016—41no—
81C2×#11724784C2311yescharacteristic
82C2×#11724784C2311yescharacteristic
83C8×D109—41no—
84C4⋊Dic109—21no—
85C4⋊Dic109—21no—
86C4⋊Dic109C2311yescharacteristic
87C4⋊Dic109C2311yescharacteristic
88#13182008—21no—
89#13182008—21no—
90#13182008—21no—
91#13182008D411yescharacteristic
92C22⋊Dic109D411yescharacteristic
93#13182006—21no—
94#13182006—21no—
95#13182006—21no—
96#13182006D411yescharacteristic
97C4×Dic109—21no—
98C4×Dic109C2311yescharacteristic
99C2×Q8×C109C2311yescharacteristic
100C2×D4×C109—21no—
101C2×D4×C109C2311yescharacteristic
102C22×C4×C109C2311yescharacteristic
103SD16×C109—41no—
104C2×C8×C109—41no—
105C2×C8×C109C2311yescharacteristic
106C2×C8×C109C2311yescharacteristic
107C4⋊C4×C109—21no—
108C4⋊C4×C109C2311yescharacteristic
109C4⋊C4×C109C2311yescharacteristic
110C22⋊C4×C109—21no—
111C22⋊C4×C109—21no—
112C23×C109—21no—
113C23×C109C2×D411yescharacteristic
114C22×D109—21no—
115C22×D109—21no—
116C22×D109—42no—
117C22×D109—42no—
118C22×D109—42no—
119C22×D109—42no—
120C22×D109—81no—
121C22×D109—81no—
122C22×D109C2×D411yescharacteristic
123Q8×C109—21no—
124Q8×C109—21no—
125D4×C109—21no—
126D4×C109—21no—
127D4×C109—42no—
128C2×C4×C109—21no—
129C2×C4×C109—21no—
130C2×C4×C109—21no—
131C2×C4×C109—21no—
132C2×C4×C109—21no—
133C2×C4×C109C2×D411yescharacteristic
134C2×C4×C109C2411yescharacteristic
135C22⋊D109—42no—
136C22⋊D109—42no—
137C22⋊D109—81no—
138C22⋊D109SD1614yescomplemented
139C2×Dic109—21no—
140C2×Dic109—21no—
141C2×Dic109—81no—
142C2×Dic109C2×D411yescharacteristic
143C2×Dic109C2×D411yescharacteristic
144C2×Dic109C2×D411yescharacteristic
145D436—21no—
146D436—21no—
147D436—21no—
148D436—42no—
149D436—42no—
150D436#4112yes—
151C4×D109—21no—
152C4×D109—21no—
153C4×D109—42no—
154C4×D109—81no—
155Dic218—81no—
156C8×C109—21no—
157C8×C109—21no—
158#11724784—21no—
159#11724784—21no—
160C22×C109—21no—
161C22×C109—21no—
162C22×C109—41no—
163C22×C109—41no—
164C22×C109—41no—
165C22×C109C2×SD1612yes—
166C22×C109C22×D411yescharacteristic
167C2×D109—21no—
168C2×D109—21no—
169C2×D109—22no—
170C2×D109—41no—
171C2×D109—41no—
172C2×D109—41no—
173C2×D109—41no—
174C2×D109—41no—
175C2×D109—84no—
176C2×D109—84no—
177C2×D109C2×SD1612yes—
178C4×C109—21no—
179C4×C109—41no—
180C4×C109—41no—
181C4×C109—41no—
182C4×C109C2×#4111yescharacteristic
183C4×C109H(5,𝔽2)11yescharacteristic
184Dic109—21no—
185Dic109—21no—
186Dic109—41no—
187Dic109—41no—
188Dic109C2×SD1612yes—
189C2×C109—22no—
190C2×C109—81no—
191C2×C109#54611yescharacteristic
192C2×C109C22×SD1611yescharacteristic
193C2×C109#57611yescharacteristic
194D109—22no—
195D109—42no—
196D109—81no—
197#3295—1091nomaximal, Sylow
198C109#329511yescharacteristic, complemented, Sylow
199#554—1091no—
200D42—1091no—
201#486—1091no—
202#481—1091no—
203#481—1091no—
204#476—1091no—
205#465—1091no—
206#465D10911yescharacteristic, complemented
207#460—1091no—
208#450—1091no—
209#450—1091no—
210C4×SD16—1091no—
211C8×D4—1091no—
212C2×#102—1091no—
213C2×#102—1091no—
214C22×D4—1091no—
215C22×D4—1091no—
216C22×D4—2181no—
217C2×SD16—1091no—
218C2×SD16—1091no—
219C2×SD16C2×D10911yescharacteristic, complemented
220C22×C8—1091no—
221C22×C8C2×D10911yescharacteristic, complemented
222C4⋊Q8—1091no—
223#127—1091no—
224#125—2181no—
225C22⋊Q8—1091no—
226C22⋊Q8—2181no—
227C22⋊Q8—2181no—
228C22⋊Q8C2×D10911yescharacteristic, complemented
229#121—1091no—
230#121—2181no—
231#121C2×D10911yescharacteristic
232#120—2181no—
233#120—2181no—
234C4×Q8—1091no—
235C4×D4—1091no—
236C4×D4—1091no—
237C4×D4—2181no—
238C2×C4⋊C4—1091no—
239C2×C4⋊C4—1091no—
240#106—1091no—
241#106—1091no—
242#106C2×D10911yescharacteristic, complemented
243C4⋊C8—1091no—
244#103C2×D10911yescharacteristic, complemented
245#102—1091no—
246#102—1091no—
247#102—1091no—
248#102—1091no—
249#102—1091no—
250#102—1091no—
251#102—1091no—
252#102—1091no—
253#102C2×D10911yescharacteristic, complemented
254C22⋊C8—1091no—
255C22⋊C8—1091no—
256C4×C8—1091no—
257C24—2181no—
258C24—2181no—
259C2×Q8—2181no—
260C2×Q8C22×D10911yescharacteristic
261C2×D4—21no—
262C2×D4—1091no—
263C2×D4—1091no—
264C2×D4—1091no—
265C2×D4—4361no—
266C2×D4—4364no—
267C2×D4—4361no—
268C2×D4—2182no—
269C2×D4—2181no—
270C2×D4—2181no—
271C2×D4—2181no—
272C2×D4—2181no—
273C2×D4—2182no—
274C2×D4—2181no—
275C2×D4C22×D10911yescharacteristic
276C22×C4—1091no—
277C22×C4—2181no—
278C22×C4—2181no—
279C22×C4—2181no—
280C22×C4C22×D10911yescharacteristic
281SD16—41no—
282SD16—4361no—
283SD16—2182no—
284C2×C8—41no—
285C2×C8—1091no—
286C2×C8—1091no—
287C2×C8—4361no—
288C2×C8C22×D10911yescharacteristic
289C2×C8C22×D10911yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—1091no—
292C4⋊C4—1091no—
293C4⋊C4—1091no—
294C4⋊C4—2181no—
295C4⋊C4—2181no—
296C4⋊C4—2181no—
297C4⋊C4—2181no—
298C4⋊C4—2181no—
299C4⋊C4C22×D10911yescharacteristic
300C4⋊C4C22×D10911yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—1091no—
304C22⋊C4—1091no—
305C22⋊C4—2181no—
306C22⋊C4—2181no—
307C22⋊C4—2181no—
308C42—1091no—
309C42—2181no—
310C23—21no—
311C23—8721no—
312C23—8721no—
313C23—1091no—
314C23—4362no—
315C23—4362no—
316C23—4362no—
317C23—4362no—
318C23—2181no—
319C23—2181no—
320C23D4×D10911yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—8721no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—8721no—
328D4—1094no—
329D4—1092no—
330D4—4362no—
331D4—4362no—
332D4—4362no—
333D4—4362no—
334D4—2181no—
335D4—2181no—
336D4—2181no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—8721no—
343C2×C4—8721no—
344C2×C4—1091no—
345C2×C4—1091no—
346C2×C4—1091no—
347C2×C4—4362no—
348C2×C4—2181no—
349C2×C4—2181no—
350C2×C4—2181no—
351C2×C4—2181no—
352C2×C4D4×D10911yescharacteristic
353C2×C4C23×D10911yescharacteristic
354C8—21no—
355C8—21no—
356C8—2181no—
357C8—2181no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—8724no—
364C22—8724no—
365C22—1092no—
366C22—4361no—
367C22—4361no—
368C22—4361no—
369C22—4361no—
370C22—4361no—
371C22—2182no—
372C22—2181no—
373C22—2181no—
374C22SD16×D10912yes—
375C22C2×D4×D10911yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—1092no—
381C4—4361no—
382C4—4361no—
383C4—2181no—
384C4—2181no—
385C4#41×D10911yescharacteristic
386C4#1681274411yescharacteristic
387C2—22no—
388C2—81no—
389C2—8721no—
390C2—4362no—
391C2—2182no—
392C2#2029566511yescharacteristic
393C2C2×SD16×D10911yescharacteristic
394C2#2029582811yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#465×C109C2
socleC22×C109C22×D4
prime radical#20295660C2
hypercenter#465D109
hypocenterC109#3295
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #20295734C2semidirect
3. #20295371C2semidirect
4. #20295346C2semidirect
5. #20295342C2semidirect
6. #20295315C2semidirect
7. #20295258C2semidirect
8. #20295224C2semidirect
9. #20295176C2semidirect
10. #20295175C2semidirect
11. #20295111C2semidirect
12. C8×C22⋊D109C2semidirect
13. C2×#16812621C2semidirect
14. #102×D109C2semidirect
15. #465×C109C2semidirect
20. C2×#13182016C22semidirect
21. C2×C8×D109C22semidirect
22. #16812698C22semidirect
27. #16812675C22semidirect
32. Q8×Dic109C22semidirect
36. C2×C4⋊Dic109C22semidirect
37. C4⋊C4×D109C22semidirect
38. #16812630C22semidirect
39. #16812630C22semidirect
40. #16812628C22semidirect
43. #16812621C22semidirect
44. #16812621C22semidirect
45. #16812620C22semidirect
46. #16812620C22semidirect
48. #16812619C22semidirect
49. C22⋊#11724784C22semidirect
50. #16812611C22semidirect
51. C8×Dic109C22semidirect
52. C2×SD16×C109C22semidirect
53. C22×C8×C109C22semidirect
54. C22⋊Q8×C109C22semidirect
56. #106×C109C22semidirect
57. #103×C109C22semidirect
58. #102×C109C22semidirect
138. C22⋊D109SD16semidirect
198. C109#3295semidirect
206. #465D109semidirect
219. C2×SD16C2×D109semidirect
221. C22×C8C2×D109semidirect
228. C22⋊Q8C2×D109semidirect
242. #106C2×D109semidirect
244. #103C2×D109semidirect
253. #102C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202957342
#202956602
#202953712
#202953462
#202953422
#202953152
#202952582
#202952242
#202951762
#202951752
#202951112
C8×C22⋊D1092
C2×#168126212
#102×D1092
#465×C1092
#3295109