Finite Group Databasebeta

← #28372551

Subgroups — #28372551

Order 13952.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812731C211yesmaximal, characteristic, complemented
2C2×SD16×D109C211yesmaximal, characteristic, complemented
3#20295594C211yesmaximal, characteristic
4#20295502C211yesmaximal, characteristic, complemented
5#20295251C211yesmaximal, characteristic, complemented
6#20295250C211yesmaximal, characteristic, complemented
7#20295245C211yesmaximal, characteristic, complemented
8#20295227C211yesmaximal, characteristic, complemented
9#20295226C211yesmaximal, characteristic, complemented
10#20295225C211yesmaximal, characteristic, complemented
11#20295224C211yesmaximal, characteristic, complemented
12#20295051C211yesmaximal, characteristic, complemented
13#20295029C211yesmaximal, characteristic, complemented
14#20295021C211yesmaximal, characteristic, complemented
15#464×C109C211yesmaximal, characteristic, complemented
16C2×#13182030C2211yescharacteristic
17C2×Q8×D109C2211yescharacteristic
18C2×D4×D109C2211yescharacteristic
19#16812731—24no—
20C2×#13182019C2211yescharacteristic, complemented
21C2×#13182018C2211yescharacteristic
22C2×#13182018C2211yescharacteristic, complemented
23C2×#13182017C2211yescharacteristic
24C2×#13182016C2211yescharacteristic
25C2×#13182016C2211yescharacteristic, complemented
26SD16×D109—24no—
27C2×#13182012C2211yescharacteristic
28C2×C8×D109C2211yescharacteristic
29#16812696C2211yescharacteristic
30#16812695C2211yescharacteristic
31#16812688C2211yescharacteristic
32#16812672—21no—
33#16812672—21no—
34#16812669—21no—
35#16812669—21no—
36Q8×Dic109C2211yescharacteristic, complemented
37C4×Dic218—21no—
38C4×Dic218C2211yescharacteristic, complemented
39#16812664—21no—
40#16812664C2211yescharacteristic, complemented
41#16812657—21no—
42#16812657C2211yescharacteristic
43#16812656—21no—
44#16812656C2211yescharacteristic
45C4⋊#11724784C2211yescharacteristic, complemented
46#16812628C2211yescharacteristic, complemented
47#16812628C2211yescharacteristic, complemented
48#16812623C2211yescharacteristic, complemented
49#16812621C2211yescharacteristic
50#16812621C2211yescharacteristic, complemented
51#16812620C2211yescharacteristic
52#16812619C2211yescharacteristic
53#16812619C2211yescharacteristic, complemented
54C2×Q16×C109C2211yescharacteristic, complemented
55C2×SD16×C109C2211yescharacteristic
56#124×C109C2211yescharacteristic
57C4×Q8×C109C2211yescharacteristic, complemented
58C4⋊C8×C109C2211yescharacteristic, complemented
59#103×C109C2211yescharacteristic, complemented
60#102×C109C2211yescharacteristic
61C23×D109—21no—
62#13182030—22no—
63#13182030—22no—
64C2×Dic218—21no—
65C2×Dic218C2311yescharacteristic
66Q8×D109—22no—
67Q8×D109D414yescomplemented
68C2×D436—21no—
69C2×D436—21no—
70C2×D436—21no—
71C2×D436C2311yescharacteristic
72C2×D436C2311yescharacteristic
73C2×C22⋊D109—21no—
74C2×C22⋊D109—21no—
75C2×C22⋊D109—21no—
76D4×D109—22no—
77D4×D109—22no—
78C2×C4×D109—21no—
79C2×C4×D109—21no—
80C2×C4×D109C2311yescharacteristic
81#13182019—22no—
82#13182018—22no—
83#13182018—22no—
84#13182017—22no—
85#13182016—22no—
86#13182016—22no—
87#13182012—22no—
88C2×#11724784C2311yescharacteristic
89C2×#11724784C2311yescharacteristic
90C8×D109—22no—
91C4⋊Dic109—21no—
92C4⋊Dic109C2311yescharacteristic
93#13182008—21no—
94#13182008—21no—
95#13182008D411yescharacteristic
96#13182008D411yescharacteristic
97#13182006—21no—
98#13182006—21no—
99#13182006—21no—
100#13182006D411yescharacteristic
101#13182006D411yescharacteristic
102C4×Dic109—21no—
103C4×Dic109—21no—
104C4×Dic109—21no—
105C4×Dic109—21no—
106C4×Dic109C2311yescharacteristic
107C2×Q8×C109C2311yescharacteristic
108C2×Q8×C109C2311yescharacteristic
109C2×D4×C109C2311yescharacteristic
110Q16×C109—22no—
111SD16×C109—22no—
112C2×C8×C109C2311yescharacteristic
113C2×C8×C109C2311yescharacteristic
114C4⋊C4×C109—21no—
115C4⋊C4×C109C2311yescharacteristic
116C22⋊C4×C109—21no—
117C22⋊C4×C109—21no—
118C42×C109—21no—
119C42×C109C2311yescharacteristic
120C23×C109—21no—
121C22×D109—21no—
122C22×D109—21no—
123C22×D109—42no—
124C22×D109—81no—
125C22×D109—81no—
126C22×D109C2×D411yescharacteristic
127Q8×C109—21no—
128Q8×C109—21no—
129Q8×C109—21no—
130Q8×C109C2×D412yes—
131D4×C109—21no—
132D4×C109—21no—
133C2×C4×C109—21no—
134C2×C4×C109—21no—
135C2×C4×C109—21no—
136C2×C4×C109C2×D411yescharacteristic
137C2×C4×C109C2×D411yescharacteristic
138C2×C4×C109C2411yescharacteristic
139C22⋊D109—42no—
140C22⋊D109—81no—
141C22⋊D109—81no—
142C2×Dic109—21no—
143C2×Dic109—21no—
144C2×Dic109C2×D411yescharacteristic
145C2×Dic109C2×D411yescharacteristic
146C2×Dic109C2×D411yescharacteristic
147D436—21no—
148D436—21no—
149D436—21no—
150D436—21no—
151D436—42no—
152D436—81no—
153D436—81no—
154C4×D109—24no—
155C4×D109—42no—
156C4×D109C2×D412yes—
157C4×D109#4112yes—
158Dic218—21no—
159Dic218—24no—
160Dic218C2×D412yes—
161C8×C109—21no—
162C8×C109—21no—
163#11724784—21no—
164#11724784—21no—
165C22×C109—41no—
166C22×C109—41no—
167C22×C109—41no—
168C22×C109C22×D411yescharacteristic
169C2×D109—21no—
170C2×D109—21no—
171C2×D109—41no—
172C2×D109—41no—
173C2×D109—41no—
174C2×D109—41no—
175C2×D109—41no—
176C2×D109—41no—
177C2×D109—84no—
178C2×D109#12112yes—
179C4×C109—22no—
180C4×C109—21no—
181C4×C109—21no—
182C4×C109—41no—
183C4×C109—41no—
184C4×C109C22×D411yescharacteristic
185C4×C109C2×#4111yescharacteristic
186Dic109—22no—
187Dic109—21no—
188Dic109—21no—
189Dic109—41no—
190Dic109#12112yes—
191C2×C109—81no—
192C2×C109C2×#12111yescharacteristic
193C2×C109#57711yescharacteristic
194C2×C109#57711yescharacteristic
195D109—22no—
196D109—81no—
197D109—81no—
198#3043—1091nomaximal, Sylow
199C109#304311yescharacteristic, complemented, Sylow
200C2×#137—1091no—
201C22×SD16—1091no—
202#535—1091no—
203#519—1091no—
204#464—1091no—
205#464—1091no—
206#464—1091no—
207#464D10911yescharacteristic, complemented
208#460—1091no—
209#460—1091no—
210#460—1091no—
211#460—1091no—
212#424—1091no—
213#418—1091no—
214#416—1091no—
215C2×#41—1091no—
216C22×Q8—1091no—
217C22×D4—1091no—
218#137—2184no—
219C2×Q16—1091no—
220C2×Q16C2×D10911yescharacteristic, complemented
221C2×SD16—1091no—
222C2×SD16—1091no—
223C2×SD16—1091no—
224C2×SD16—1091no—
225C2×SD16—1091no—
226C2×SD16—2184no—
227C2×SD16C2×D10911yescharacteristic
228C2×M16—1091no—
229C22×C8—1091no—
230#127—1091no—
231#127—1091no—
232#124—1091no—
233#124C2×D10911yescharacteristic
234#121—2181no—
235#121—2181no—
236#120—2181no—
237#120—2181no—
238C4×Q8—1091no—
239C4×Q8—1091no—
240C4×Q8—1091no—
241C4×Q8—2181no—
242C4×Q8—2181no—
243C4×Q8C2×D10911yescharacteristic, complemented
244#117—1091no—
245#117—1091no—
246#117—2181no—
247#117—2181no—
248C4⋊C8—1091no—
249C4⋊C8—1091no—
250C4⋊C8—1091no—
251C4⋊C8C2×D10911yescharacteristic, complemented
252#103—1091no—
253#103C2×D10911yescharacteristic, complemented
254#102—1091no—
255#102—1091no—
256#102—1091no—
257#102—1091no—
258#102—1091no—
259#102C2×D10911yescharacteristic
260C24—2181no—
261#41—2182no—
262#41—2182no—
263C2×Q8—1094no—
264C2×Q8—1091no—
265C2×Q8—2181no—
266C2×Q8—2182no—
267C2×Q8C22×D10911yescharacteristic
268C2×Q8C22×D10911yescharacteristic
269C2×D4—1091no—
270C2×D4—1091no—
271C2×D4—2181no—
272C2×D4—2182no—
273C2×D4—2182no—
274C2×D4—2181no—
275C2×D4—2181no—
276C2×D4—2181no—
277C2×D4—2181no—
278C2×D4—2181no—
279C2×D4C22×D10911yescharacteristic
280C22×C4—1091no—
281C22×C4—2181no—
282C22×C4—2181no—
283Q16—22no—
284Q16—2182no—
285SD16—22no—
286SD16—2182no—
287SD16—2182no—
288SD16—2182no—
289SD16—2182no—
290SD16—2182no—
291M16—2182no—
292C2×C8—1091no—
293C2×C8—1091no—
294C2×C8—2182no—
295C2×C8C22×D10911yescharacteristic
296C2×C8C22×D10911yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—1091no—
299C4⋊C4—1091no—
300C4⋊C4—1091no—
301C4⋊C4—2181no—
302C4⋊C4—2181no—
303C4⋊C4—2181no—
304C4⋊C4C22×D10911yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—1091no—
308C22⋊C4—1091no—
309C22⋊C4—2181no—
310C22⋊C4—2181no—
311C22⋊C4—2181no—
312C42—21no—
313C42—1091no—
314C42—2181no—
315C42—2181no—
316C42—2181no—
317C42—2181no—
318C42C22×D10911yescharacteristic
319C23—21no—
320C23—8721no—
321C23—8721no—
322C23—1091no—
323C23—4362no—
324C23—2181no—
325C23—2181no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—1092no—
330Q8—2184no—
331Q8—2181no—
332Q8D4×D10912yes—
333D4—21no—
334D4—21no—
335D4—8721no—
336D4—8721no—
337D4—8721no—
338D4—8721no—
339D4—4362no—
340D4—4362no—
341D4—2181no—
342D4—2181no—
343D4—2181no—
344D4—2181no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—1091no—
349C2×C4—1092no—
350C2×C4—1092no—
351C2×C4—1091no—
352C2×C4—1091no—
353C2×C4—4362no—
354C2×C4—2184no—
355C2×C4—2181no—
356C2×C4—2181no—
357C2×C4D4×D10911yescharacteristic
358C2×C4D4×D10911yescharacteristic
359C2×C4C23×D10911yescharacteristic
360C8—21no—
361C8—21no—
362C8—2181no—
363C8—2181no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—8724no—
368C22—1092no—
369C22—4361no—
370C22—4361no—
371C22—4361no—
372C22—4361no—
373C22—4361no—
374C22—4361no—
375C22—2181no—
376C22—2181no—
377C22C2×D4×D10911yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—1092no—
384C4—4361no—
385C4—2182no—
386C4—2181no—
387C4—2181no—
388C4C2×D4×D10911yescharacteristic
389C4#41×D10911yescharacteristic
390C2—81no—
391C2—8721no—
392C2—8721no—
393C2—2182no—
394C2#121×D10911yescharacteristic
395C2#2029583311yescharacteristic
396C2#2029583511yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#464×C109C2
socleC22×C109C22×D4
prime radical#20295594C2
hypercenter#464D109
hypocenterC109#3043
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#16812731C2semidirect
2. C2×SD16×D109C2semidirect
4. #20295502C2semidirect
5. #20295251C2semidirect
6. #20295250C2semidirect
7. #20295245C2semidirect
8. #20295227C2semidirect
9. #20295226C2semidirect
10. #20295225C2semidirect
11. #20295224C2semidirect
12. #20295051C2semidirect
13. #20295029C2semidirect
14. #20295021C2semidirect
15. #464×C109C2semidirect
20. C2×#13182019C22semidirect
22. C2×#13182018C22semidirect
25. C2×#13182016C22semidirect
36. Q8×Dic109C22semidirect
38. C4×Dic218C22semidirect
40. #16812664C22semidirect
45. C4⋊#11724784C22semidirect
46. #16812628C22semidirect
47. #16812628C22semidirect
48. #16812623C22semidirect
50. #16812621C22semidirect
53. #16812619C22semidirect
54. C2×Q16×C109C22semidirect
57. C4×Q8×C109C22semidirect
58. C4⋊C8×C109C22semidirect
59. #103×C109C22semidirect
67. Q8×D109D4semidirect
199. C109#3043semidirect
207. #464D109semidirect
220. C2×Q16C2×D109semidirect
243. C4×Q8C2×D109semidirect
251. C4⋊C8C2×D109semidirect
253. #103C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127312
C2×SD16×D1092
#202955942
#202955022
#202952512
#202952502
#202952452
#202952272
#202952262
#202952252
#202952242
#202950512
#202950292
#202950212
#464×C1092
#3043109