Finite Group Databasebeta

← #28376359

Subgroups — #28376359

Order 13952.

1556 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D109C212yesmaximal, complemented
2#20295603C214yesmaximal, complemented
3#20295601C212yesmaximal, complemented
4#20295596C212yesmaximal, complemented
5#20295594C212yesmaximal, complemented
6#20295589C212yesmaximal, complemented
7#20295584C214yesmaximal, complemented
8#20295576C211yesmaximal, characteristic, complemented
9#20295572C212yesmaximal, complemented
10#20295535C218yesmaximal, complemented
11C2×#16812656C211yesmaximal, characteristic, complemented
12#532×C109C211yesmaximal, characteristic, complemented
13C2×#13182030C2212yes—
14C2×#13182029C2214yes—
15C2×#13182029C2212yescomplemented
16C2×#13182028C2212yes—
17C2×#13182028C2214yescomplemented
18#41×D109—216no—
19C2×Q8×D109C2212yescomplemented
20C2×D4×D109C2214yes—
21C2×D4×D109C2212yescomplemented
22C22×C4×D109C2211yescharacteristic
23C22×C4×D109C2214yescomplemented
24#16812699C2211yescharacteristic, complemented
25#16812697C2212yescomplemented
26#16812696C2211yescharacteristic, complemented
27#16812695C2212yescomplemented
28#16812688C2212yescomplemented
29#16812687C2212yescomplemented
30#16812686C2212yescomplemented
31#16812684C2218yescomplemented
32#16812681C2218yescomplemented
33#16812677C2218yescomplemented
34#16812676C2218yescomplemented
35#16812673C2214yescomplemented
36#16812672C2218yescomplemented
37#16812671C2218yescomplemented
38#16812671C2218yescomplemented
39#16812669C2218yescomplemented
40#16812667C2218yescomplemented
41C4×D436C2214yescomplemented
42C4×C22⋊D109C2218yescomplemented
43#16812656C2214yescomplemented
44#16812656C2212yescomplemented
45#16812656C2212yescomplemented
46C2×#13182008C2212yescomplemented
47C2×#13182006C2212yes—
48C2×C4×Dic109C2211yescharacteristic, complemented
49C2×#41×C109C2212yescomplemented
50C4⋊Q8×C109C2211yescharacteristic, complemented
51#127×C109C2211yescharacteristic, complemented
52#124×C109C2212yescomplemented
53#121×C109C2214yescomplemented
54C4×D4×C109C2214yescomplemented
55C2×C42×C109C2211yescharacteristic, complemented
56C23×D109C2314yes—
57C23×D109C2311yescharacteristic
58#13182030—28no—
59#13182029—28no—
60#13182029—216no—
61#13182028—216no—
62#13182028—28no—
63C2×Dic218C2314yes—
64C2×Dic218C2312yes—
65Q8×D109—28no—
66C2×D436C2314yes—
67C2×D436C2312yes—
68C2×D436C2314yes—
69C2×C22⋊D109C2314yes—
70C2×C22⋊D109C2314yes—
71C2×C22⋊D109C2314yes—
72C2×C22⋊D109C2318yes—
73D4×D109—216no—
74D4×D109—28no—
75C22×Dic109C2314yes—
76C22×Dic109C2312yes—
77C22×Dic109C2311yescharacteristic
78C2×C4×D109—216no—
79C2×C4×D109D418yescomplemented
80C2×C4×D109C2314yes—
81C2×C4×D109C2314yes—
82C2×C4×D109C2314yes—
83C2×C4×D109C2312yes—
84C2×C4×D109C2312yes—
85C4⋊Dic109C2314yes—
86#13182008C2314yes—
87#13182008C2314yes—
88#13182008C2318yes—
89C22⋊Dic109C2318yes—
90#13182006C2314yes—
91#13182006C2314yes—
92#13182006C2318yes—
93#13182006C2318yes—
94C4×Dic109C2312yes—
95C4×Dic109C2312yes—
96#41×C109—28no—
97C2×Q8×C109C2312yes—
98C2×D4×C109C2314yes—
99C2×D4×C109C2312yes—
100C2×D4×C109C2314yes—
101C22×C4×C109C2314yes—
102C22×C4×C109C2312yes—
103C22×C4×C109C2311yescharacteristic
104C4⋊C4×C109C2314yes—
105C22⋊C4×C109C2318yes—
106C42×C109C2312yes—
107C42×C109C2311yescharacteristic
108C42×C109C2311yescharacteristic
109C23×C109C2414yes—
110C23×C109C2411yescharacteristic
111C22×D109—28no—
112C22×D109—42no—
113C22×D109—48no—
114C22×D109C2×D414yes—
115C22×D109C2414yes—
116C22×D109C2412yes—
117Q8×C109—24no—
118D4×C109—28no—
119D4×C109—24no—
120D4×C109—44no—
121C2×C4×C109—28no—
122C2×C4×C109—42no—
123C2×C4×C109C2×D414yes—
124C2×C4×C109C2412yes—
125C2×C4×C109C2412yes—
126C2×C4×C109C2414yes—
127C2×C4×C109C2411yescharacteristic
128C2×C4×C109C2411yescharacteristic
129C22⋊D109—28no—
130C22⋊D109—28no—
131C22⋊D109—28no—
132C22⋊D109—48no—
133C2×Dic109—28no—
134C2×Dic109—42no—
135C2×Dic109C2×D414yes—
136C2×Dic109C2414yes—
137C2×Dic109C2414yes—
138C2×Dic109C2412yes—
139D436—28no—
140D436—24no—
141D436—44no—
142C4×D109—28no—
143C4×D109—28no—
144C4×D109—28no—
145C4×D109C2×D418yes—
146C4×D109C2×D418yes—
147Dic218—24no—
148Dic218—28no—
149C22×C109—22no—
150C22×C109—24no—
151C22×C109—28no—
152C22×C109C22×D412yes—
153C22×C109C2511yescharacteristic
154C2×D109—24no—
155C2×D109—24no—
156C2×D109—28no—
157C2×D109—44no—
158C2×D109—416no—
159C2×D109C22×D414yes—
160C4×C109—24no—
161C4×C109—22no—
162C4×C109—22no—
163C4×C109C22×D412yes—
164C4×C109C22×D412yes—
165Dic109—24no—
166Dic109—24no—
167Dic109C22×D414yes—
168C2×C109—22no—
169C2×C109—44no—
170C2×C109C23×D411yescharacteristic
171C2×C109#58512yes—
172D109—24no—
173D109—44no—
174#3449—1091nomaximal, Sylow
175C109#344911yescharacteristic, complemented, Sylow
176C22×#41—1092no—
177#537—1094no—
178#536—1092no—
179#536—1092no—
180#535—1092no—
181#535—1092no—
182#534—1094no—
183#532—1092no—
184#532—1091no—
185#532D10911yescharacteristic, complemented
186#525—1098no—
187C2×#117—1091no—
188C2×#41—1092no—
189C2×#41—1092no—
190C2×#41—1094no—
191C2×#41—1094no—
192C2×#41—1092no—
193C2×#41—21816no—
194C2×#41C2×D10912yescomplemented
195C22×Q8—1092no—
196C22×D4—1094no—
197C22×D4—1092no—
198C23×C4—1091no—
199C23×C4—1094no—
200C4⋊Q8—1092no—
201C4⋊Q8—1091no—
202C4⋊Q8C2×D10911yescharacteristic, complemented
203#127—1092no—
204#127—1091no—
205#127C2×D10911yescharacteristic, complemented
206#124—1092no—
207#124—1092no—
208#124—1092no—
209#124C2×D10912yescomplemented
210#123—1098no—
211#123—1098no—
212C22⋊Q8—1098no—
213C22⋊Q8—1098no—
214#121—1098no—
215#121—1098no—
216#121—1094no—
217#121—1098no—
218#121C2×D10914yescomplemented
219#120—1098no—
220#120—1098no—
221C4×D4—1098no—
222C4×D4—1094no—
223C4×D4C2×D10914yescomplemented
224#117—1094no—
225#117—1092no—
226#117—1092no—
227C2×C4⋊C4—1092no—
228C2×C22⋊C4—1092no—
229C2×C42—1091no—
230C2×C42C2×D10911yescharacteristic, complemented
231C24—1091no—
232C24—1094no—
233#41—28no—
234#41—21816no—
235#41—21816no—
236#41—2188no—
237#41—2188no—
238#41—2188no—
239C2×Q8—1094no—
240C2×Q8—1092no—
241C2×Q8—2188no—
242C2×Q8C22×D10912yes—
243C2×D4—1098no—
244C2×D4—1094no—
245C2×D4—1094no—
246C2×D4—1094no—
247C2×D4—1094no—
248C2×D4—1094no—
249C2×D4—1092no—
250C2×D4—2188no—
251C2×D4—21816no—
252C2×D4C22×D10914yes—
253C2×D4C22×D10912yes—
254C2×D4C22×D10914yes—
255C22×C4—1091no—
256C22×C4—1098no—
257C22×C4—1092no—
258C22×C4—1092no—
259C22×C4—1092no—
260C22×C4—1094no—
261C22×C4—1094no—
262C22×C4—1094no—
263C22×C4—1094no—
264C22×C4—21816no—
265C22×C4C22×D10914yes—
266C22×C4C22×D10912yes—
267C22×C4C22×D10911yescharacteristic
268C4⋊C4—1094no—
269C4⋊C4—1094no—
270C4⋊C4—1098no—
271C4⋊C4—1094no—
272C4⋊C4C22×D10914yes—
273C22⋊C4—1098no—
274C22⋊C4—1094no—
275C22⋊C4—1094no—
276C22⋊C4—1098no—
277C22⋊C4—1098no—
278C22⋊C4C22×D10918yes—
279C42—1092no—
280C42—1092no—
281C42C22×D10911yescharacteristic
282C42C22×D10911yescharacteristic
283C42C22×D10912yes—
284C23—1094no—
285C23—1092no—
286C23—1094no—
287C23—4362no—
288C23—4368no—
289C23—2188no—
290C23C23×D10914yes—
291C23C23×D10911yescharacteristic
292Q8—24no—
293Q8—2188no—
294Q8—2184no—
295D4—28no—
296D4—24no—
297D4—44no—
298D4—4364no—
299D4—4368no—
300D4—2188no—
301D4—2188no—
302D4—2188no—
303D4—2188no—
304D4—2184no—
305C2×C4—28no—
306C2×C4—42no—
307C2×C4—1094no—
308C2×C4—1092no—
309C2×C4—1098no—
310C2×C4—1098no—
311C2×C4—1094no—
312C2×C4—1094no—
313C2×C4—4362no—
314C2×C4—2188no—
315C2×C4—2188no—
316C2×C4—2188no—
317C2×C4—2188no—
318C2×C4D4×D10914yes—
319C2×C4C23×D10912yes—
320C2×C4C23×D10912yes—
321C2×C4C23×D10914yes—
322C2×C4C23×D10911yescharacteristic
323C2×C4C23×D10911yescharacteristic
324C22—22no—
325C22—24no—
326C22—28no—
327C22—1094no—
328C22—4364no—
329C22—43616no—
330C22—2184no—
331C22—2188no—
332C22—2184no—
333C22C2×D4×D10912yes—
334C22C24×D10911yescharacteristic
335C4—24no—
336C4—22no—
337C4—22no—
338C4—1094no—
339C4—2184no—
340C4—2184no—
341C4C2×D4×D10912yes—
342C4C2×D4×D10912yes—
343C2—22no—
344C2—44no—
345C2—4364no—
346C2—2184no—
347C2C22×D4×D10911yescharacteristic
348C2#2029586212yes—
subgroupG/H
centerC2×C4C23×D109
commutator, socle, cosocleC22×C109C25
FrattiniC22C24×D109
Fitting#532×C109C2
hypercenter#532D109
hypocenterC109#3449
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C2×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#41×D109C2semidirect
2. #20295603C2semidirect
3. #20295601C2semidirect
4. #20295596C2semidirect
5. #20295594C2semidirect
6. #20295589C2semidirect
7. #20295584C2semidirect
8. #20295576C2semidirect
9. #20295572C2semidirect
10. #20295535C2semidirect
11. C2×#16812656C2semidirect
12. #532×C109C2semidirect
15. C2×#13182029C22semidirect
17. C2×#13182028C22semidirect
19. C2×Q8×D109C22semidirect
21. C2×D4×D109C22semidirect
23. C22×C4×D109C22semidirect
24. #16812699C22semidirect
25. #16812697C22semidirect
26. #16812696C22semidirect
27. #16812695C22semidirect
28. #16812688C22semidirect
29. #16812687C22semidirect
30. #16812686C22semidirect
31. #16812684C22semidirect
32. #16812681C22semidirect
33. #16812677C22semidirect
34. #16812676C22semidirect
35. #16812673C22semidirect
36. #16812672C22semidirect
37. #16812671C22semidirect
38. #16812671C22semidirect
39. #16812669C22semidirect
40. #16812667C22semidirect
41. C4×D436C22semidirect
42. C4×C22⋊D109C22semidirect
43. #16812656C22semidirect
44. #16812656C22semidirect
45. #16812656C22semidirect
46. C2×#13182008C22semidirect
48. C2×C4×Dic109C22semidirect
49. C2×#41×C109C22semidirect
50. C4⋊Q8×C109C22semidirect
51. #127×C109C22semidirect
52. #124×C109C22semidirect
53. #121×C109C22semidirect
54. C4×D4×C109C22semidirect
55. C2×C42×C109C22semidirect
79. C2×C4×D109D4semidirect
175. C109#3449semidirect
185. #532D109semidirect
194. C2×#41C2×D109semidirect
202. C4⋊Q8C2×D109semidirect
205. #127C2×D109semidirect
209. #124C2×D109semidirect
218. #121C2×D109semidirect
223. C4×D4C2×D109semidirect
230. C2×C42C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1092
#202956032
#202956012
#202955962
#202955942
#202955892
#202955842
#202955762
#202955722
#202955352
C2×#168126562
#532×C1092
#3449109