Finite Group Databasebeta

← #28377173

Subgroups — #28377173

Order 13952.

1058 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20295775C212yesmaximal, complemented
2#20295774C212yesmaximal, complemented
3#20295772C214yesmaximal, complemented
4#20295768C211yesmaximal, characteristic, complemented
5#20295767C214yesmaximal, complemented
6#20295766C212yesmaximal, complemented
7#20295577C212yesmaximal, complemented
8#20295575C211yesmaximal, characteristic, complemented
9#20295561C212yesmaximal, complemented
10#20295559C214yesmaximal, complemented
11#20295539C214yesmaximal, complemented
12#20295534C212yesmaximal, complemented
13#563×C109C211yesmaximal, characteristic, complemented
14C2×#13182030C2212yes—
15C2×#13182030C2211yescharacteristic, complemented
16C2×#13182028C2211yescharacteristic
17C22×C4×D109C2212yescomplemented
18#16812697C2211yescharacteristic, complemented
19#16812696C2212yes—
20#16812695C2211yescharacteristic
21#16812694C2214yescomplemented
22#16812693C2214yescomplemented
23#16812689C2214yescomplemented
24#16812688C2212yes—
25#16812688C2212yescomplemented
26#16812687C2212yes—
27#16812687C2211yescharacteristic, complemented
28#16812685C2214yes—
29#16812684C2214yes—
30#16812684C2214yescomplemented
31#16812684C2214yescomplemented
32#16812684C2214yescomplemented
33#16812682C2212yescomplemented
34#16812681C2214yescomplemented
35#16812679C2212yescomplemented
36#16812677C2214yescomplemented
37#16812677C2214yescomplemented
38#16812675C2214yescomplemented
39#16812674C2212yes—
40#16812674C2214yes—
41#16812674C2214yescomplemented
42#16812674C2214yescomplemented
43#16812673C2214yes—
44#16812672C2214yes—
45#16812672C2214yes—
46#16812672C2214yescomplemented
47#16812669C2214yes—
48#16812669C2214yescomplemented
49#16812669C2212yescomplemented
50#16812667C2212yes—
51#16812664C2212yescomplemented
52#16812661C2212yes—
53#16812661C2214yescomplemented
54#16812661C2214yescomplemented
55#16812659C2214yes—
56#16812659C2214yescomplemented
57#16812657C2214yescomplemented
58#16812655C2212yescomplemented
59C42×D109C2212yescomplemented
60C42×D109C2211yescharacteristic, complemented
61C4⋊Q8×C109C2212yescomplemented
62#126×C109C2214yescomplemented
63#125×C109C2212yescomplemented
64#124×C109C2211yescharacteristic
65#123×C109C2212yes—
66C22⋊Q8×C109C2214yescomplemented
67C23×D109C2312yes—
68#13182030—44no—
69#13182030—42no—
70#13182028—42no—
71C2×Dic218C2312yes—
72C2×Dic218C2311yescharacteristic
73C2×D436C2312yes—
74C2×D436C2314yes—
75C2×D436C2314yes—
76C2×D436C2312yes—
77C2×D436C2314yes—
78C2×D436C2312yes—
79C2×D436C2311yescharacteristic
80C2×C22⋊D109C2314yes—
81C2×C22⋊D109C2314yes—
82C2×C22⋊D109C2312yes—
83C22×Dic109C2312yes—
84C2×C4×D109—44no—
85C2×C4×D109C2312yes—
86C2×C4×D109C2311yescharacteristic
87C2×C4×D109C2314yes—
88C2×C4×D109C2314yes—
89C2×C4×D109C2312yes—
90C2×C4×D109C2314yes—
91C4⋊Dic109C2314yes—
92#13182008C2314yes—
93#13182008C2314yes—
94#13182008C2314yes—
95#13182008C2314yes—
96C22⋊Dic109C2312yes—
97#13182006C2314yes—
98#13182006C2314yes—
99#13182006C2314yes—
100#13182006C2314yes—
101#13182006C2314yes—
102#13182006C2314yes—
103#13182006C2314yes—
104#13182006C2314yes—
105#13182006C2314yes—
106C4×Dic109C2314yes—
107C4×Dic109C2312yes—
108C4×Dic109C2312yes—
109C4×Dic109C2311yescharacteristic
110C2×Q8×C109C2312yes—
111C2×Q8×C109C2311yescharacteristic
112C2×D4×C109C2311yescharacteristic
113C22×C4×C109C2312yes—
114C4⋊C4×C109C2312yes—
115C4⋊C4×C109C2314yes—
116C4⋊C4×C109C2314yes—
117C4⋊C4×C109C2314yes—
118C4⋊C4×C109C2312yes—
119C22⋊C4×C109C2314yes—
120C22⋊C4×C109C2314yes—
121C22⋊C4×C109C2312yes—
122C42×C109C2311yescharacteristic
123C42×C109C2312yes—
124C23×C109C2412yes—
125C22×D109—42no—
126C22×D109—44no—
127C22×D109C2414yes—
128C22×D109C2412yes—
129C22×D109C2411yescharacteristic
130Q8×C109—41no—
131Q8×C109—42no—
132D4×C109—41no—
133C2×C4×C109—42no—
134C2×C4×C109C2412yes—
135C2×C4×C109C2414yes—
136C2×C4×C109C2412yes—
137C2×C4×C109C2411yescharacteristic
138C2×C4×C109C2414yes—
139C22⋊D109—44no—
140C22⋊D109—44no—
141C22⋊D109—42no—
142C2×Dic109—24no—
143C2×Dic109C2414yes—
144C2×Dic109C2412yes—
145C2×Dic109C2412yes—
146C2×Dic109C2411yescharacteristic
147D436—42no—
148D436—44no—
149D436—44no—
150D436—41no—
151D436—44no—
152D436—42no—
153D436—42no—
154C4×D109—28no—
155C4×D109—22no—
156C4×D109—24no—
157C4×D109—44no—
158C4×D109—44no—
159C4×D109—42no—
160Dic218—42no—
161Dic218—41no—
162C22×C109—22no—
163C22×C109—24no—
164C22×C109C2511yescharacteristic
165C2×D109—24no—
166C2×D109—22no—
167C2×D109—21no—
168C2×D109—22no—
169C2×D109—24no—
170C2×D109—24no—
171C2×D109—24no—
172C2×D109—48no—
173C4×C109—21no—
174C4×C109—24no—
175C4×C109—22no—
176C4×C109—22no—
177C4×C109—24no—
178Dic109—22no—
179Dic109—24no—
180Dic109—22no—
181Dic109H(5,𝔽2)12yes—
182C2×C109—42no—
183C2×C109C2×H(5,𝔽2)11yescharacteristic
184C2×C109#58512yes—
185D109—42no—
186D109—41no—
187D109—44no—
188#3529—1091nomaximal, Sylow
189C109#352911yescharacteristic, complemented, Sylow
190#563—1092no—
191#563D10911yescharacteristic, complemented
192#562—1092no—
193#562—1094no—
194H(3,𝔽4)—1091no—
195#560—1094no—
196#560—1092no—
197#532—1092no—
198#532—1091no—
199#529—1094no—
200#529—1092no—
201#525—1092no—
202#525—1094no—
203C2×#41—1092no—
204C2×#41—1091no—
205C2×#41—1091no—
206C23×C4—1092no—
207C4⋊Q8—1091no—
208C4⋊Q8C2×D10912yescomplemented
209#127—1091no—
210#127—1092no—
211#126—1094no—
212#126—1094no—
213#126C2×D10914yescomplemented
214#125—1094no—
215#125C2×D10912yescomplemented
216#124—1092no—
217#124—1092no—
218#124—1094no—
219#124—1092no—
220#124—1091no—
221#124C2×D10911yescharacteristic
222#123—1092no—
223#123—1094no—
224#123—1094no—
225#123—1094no—
226#123—1094no—
227#123—1094no—
228#123C2×D10912yes—
229C22⋊Q8—1094no—
230C22⋊Q8—1094no—
231C22⋊Q8—1094no—
232C22⋊Q8—1092no—
233C22⋊Q8C2×D10914yescomplemented
234#121—1094no—
235#121—1094no—
236#121—1092no—
237#121—1094no—
238#121—1094no—
239#121—1094no—
240#121—1094no—
241#121—1094no—
242#120—1092no—
243#120—1094no—
244#120—1092no—
245#120—1094no—
246C4×Q8—1092no—
247C4×D4—1092no—
248C4×D4—1094no—
249C4×D4—1094no—
250C4×D4—1094no—
251C4×D4—1094no—
252#117—1094no—
253#117—1092no—
254C2×C42—1092no—
255C2×C42—1091no—
256C24—1092no—
257#41—4362no—
258#41—4364no—
259#41—4362no—
260C2×Q8—1092no—
261C2×Q8—1091no—
262C2×Q8C22×D10912yes—
263C2×Q8C22×D10911yescharacteristic
264C2×D4—1094no—
265C2×D4—1094no—
266C2×D4—1092no—
267C2×D4—1092no—
268C2×D4—1094no—
269C2×D4—1094no—
270C2×D4—1092no—
271C2×D4—1094no—
272C2×D4—1091no—
273C2×D4—1092no—
274C2×D4C22×D10911yescharacteristic
275C22×C4—1092no—
276C22×C4—1094no—
277C22×C4—1092no—
278C22×C4—1094no—
279C22×C4—1091no—
280C22×C4—1092no—
281C22×C4—1094no—
282C22×C4—4364no—
283C22×C4C22×D10912yes—
284C4⋊C4—1094no—
285C4⋊C4—1094no—
286C4⋊C4—1094no—
287C4⋊C4—1094no—
288C4⋊C4—1094no—
289C4⋊C4C22×D10914yes—
290C4⋊C4C22×D10914yes—
291C4⋊C4C22×D10914yes—
292C4⋊C4C22×D10912yes—
293C4⋊C4C22×D10912yes—
294C22⋊C4—1092no—
295C22⋊C4—1094no—
296C22⋊C4—1094no—
297C22⋊C4—1094no—
298C22⋊C4—1094no—
299C22⋊C4—1094no—
300C22⋊C4—1094no—
301C22⋊C4—1094no—
302C22⋊C4—1094no—
303C22⋊C4—1094no—
304C22⋊C4C22×D10912yes—
305C22⋊C4C22×D10914yes—
306C22⋊C4C22×D10914yes—
307C42—1094no—
308C42—1092no—
309C42—1092no—
310C42—1091no—
311C42C22×D10911yescharacteristic
312C42C22×D10912yes—
313C23—1094no—
314C23—1092no—
315C23—1091no—
316C23—4362no—
317C23—4364no—
318C23C23×D10912yes—
319Q8—42no—
320Q8—41no—
321Q8—4361no—
322Q8—4362no—
323D4—41no—
324D4—4364no—
325D4—4361no—
326D4—4362no—
327D4—4362no—
328D4—4364no—
329D4—4364no—
330D4—4362no—
331D4—4362no—
332D4—4364no—
333D4—4364no—
334C2×C4—42no—
335C2×C4—1091no—
336C2×C4—1094no—
337C2×C4—1092no—
338C2×C4—1092no—
339C2×C4—4364no—
340C2×C4—4364no—
341C2×C4—4362no—
342C2×C4—2188no—
343C2×C4—2182no—
344C2×C4—2184no—
345C2×C4—2184no—
346C2×C4C23×D10912yes—
347C2×C4C23×D10911yescharacteristic
348C2×C4C23×D10914yes—
349C2×C4C23×D10912yes—
350C2×C4C23×D10914yes—
351C22—22no—
352C22—24no—
353C22—4368no—
354C22—2184no—
355C22—2182no—
356C22—2181no—
357C22—2182no—
358C22—2184no—
359C22—2184no—
360C22—2184no—
361C22C24×D10911yescharacteristic
362C4—24no—
363C4—21no—
364C4—22no—
365C4—24no—
366C4—22no—
367C4—1092no—
368C4—2182no—
369C4—2182no—
370C4—2184no—
371C2—42no—
372C2—4364no—
373C2—4362no—
374C2—4361no—
375C2H(5,𝔽2)×D10911yescharacteristic
376C2#2029586312yes—
subgroupG/H
center, FrattiniC22C24×D109
commutator, socle, cosocleC22×C109C25
Fitting#563×C109C2
hypercenter#563D109
hypocenterC109#3529
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #20295775C2semidirect
2. #20295774C2semidirect
3. #20295772C2semidirect
4. #20295768C2semidirect
5. #20295767C2semidirect
6. #20295766C2semidirect
7. #20295577C2semidirect
8. #20295575C2semidirect
9. #20295561C2semidirect
10. #20295559C2semidirect
11. #20295539C2semidirect
12. #20295534C2semidirect
13. #563×C109C2semidirect
15. C2×#13182030C22semidirect
17. C22×C4×D109C22semidirect
18. #16812697C22semidirect
21. #16812694C22semidirect
22. #16812693C22semidirect
23. #16812689C22semidirect
25. #16812688C22semidirect
27. #16812687C22semidirect
30. #16812684C22semidirect
31. #16812684C22semidirect
32. #16812684C22semidirect
33. #16812682C22semidirect
34. #16812681C22semidirect
35. #16812679C22semidirect
36. #16812677C22semidirect
37. #16812677C22semidirect
38. #16812675C22semidirect
41. #16812674C22semidirect
42. #16812674C22semidirect
46. #16812672C22semidirect
48. #16812669C22semidirect
49. #16812669C22semidirect
51. #16812664C22semidirect
53. #16812661C22semidirect
54. #16812661C22semidirect
56. #16812659C22semidirect
57. #16812657C22semidirect
58. #16812655C22semidirect
59. C42×D109C22semidirect
60. C42×D109C22semidirect
61. C4⋊Q8×C109C22semidirect
62. #126×C109C22semidirect
63. #125×C109C22semidirect
66. C22⋊Q8×C109C22semidirect
189. C109#3529semidirect
191. #563D109semidirect
208. C4⋊Q8C2×D109semidirect
213. #126C2×D109semidirect
215. #125C2×D109semidirect
233. C22⋊Q8C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202957752
#202957742
#202957722
#202957682
#202957672
#202957662
#202955772
#202955752
#202955612
#202955592
#202955392
#202955342
#563×C1092
#3529109