Finite Group Databasebeta

← #34058018

Subgroups — #34058018

Order 19072.

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×D149C212yesmaximal, complemented
2#23750202C214yesmaximal, complemented
3#23750200C212yesmaximal, complemented
4#23750195C212yesmaximal, complemented
5#23750193C212yesmaximal, complemented
6#23750188C212yesmaximal, complemented
7#23750183C214yesmaximal, complemented
8#23750175C211yesmaximal, characteristic, complemented
9#23750171C212yesmaximal, complemented
10#23750134C218yesmaximal, complemented
11C2×#18011868C211yesmaximal, characteristic, complemented
12#532×C149C211yesmaximal, characteristic, complemented
13C2×#14558737C2212yes—
14C2×#14558736C2214yes—
15C2×#14558736C2212yescomplemented
16C2×#14558735C2212yes—
17C2×#14558735C2214yescomplemented
18#41×D149—216no—
19C2×Q8×D149C2212yescomplemented
20C2×D4×D149C2214yes—
21C2×D4×D149C2212yescomplemented
22C22×C4×D149C2211yescharacteristic
23C22×C4×D149C2214yescomplemented
24#18011911C2211yescharacteristic, complemented
25#18011909C2212yescomplemented
26#18011908C2211yescharacteristic, complemented
27#18011907C2212yescomplemented
28#18011900C2212yescomplemented
29#18011899C2212yescomplemented
30#18011898C2212yescomplemented
31#18011896C2218yescomplemented
32#18011893C2218yescomplemented
33#18011889C2218yescomplemented
34#18011888C2218yescomplemented
35#18011885C2214yescomplemented
36#18011884C2218yescomplemented
37#18011883C2218yescomplemented
38#18011883C2218yescomplemented
39#18011881C2218yescomplemented
40#18011879C2218yescomplemented
41C4×D596C2214yescomplemented
42C4×C22⋊D149C2218yescomplemented
43#18011868C2212yescomplemented
44#18011868C2212yescomplemented
45#18011868C2214yescomplemented
46C2×#14558715C2212yescomplemented
47C2×#14558713C2212yes—
48C2×C4×Dic149C2211yescharacteristic, complemented
49C2×#41×C149C2212yescomplemented
50C4⋊Q8×C149C2211yescharacteristic, complemented
51#127×C149C2211yescharacteristic, complemented
52#124×C149C2212yescomplemented
53#121×C149C2214yescomplemented
54C4×D4×C149C2214yescomplemented
55C2×C42×C149C2211yescharacteristic, complemented
56C23×D149C2314yes—
57C23×D149C2311yescharacteristic
58#14558737—28no—
59#14558736—216no—
60#14558736—28no—
61#14558735—216no—
62#14558735—28no—
63C2×Dic298C2312yes—
64C2×Dic298C2314yes—
65Q8×D149—28no—
66C2×D596C2312yes—
67C2×D596C2314yes—
68C2×D596C2314yes—
69C2×C22⋊D149C2314yes—
70C2×C22⋊D149C2314yes—
71C2×C22⋊D149C2314yes—
72C2×C22⋊D149C2318yes—
73D4×D149—216no—
74D4×D149—28no—
75C22×Dic149C2314yes—
76C22×Dic149C2312yes—
77C22×Dic149C2311yescharacteristic
78C2×C4×D149—216no—
79C2×C4×D149D418yescomplemented
80C2×C4×D149C2314yes—
81C2×C4×D149C2314yes—
82C2×C4×D149C2314yes—
83C2×C4×D149C2312yes—
84C2×C4×D149C2312yes—
85C4⋊Dic149C2314yes—
86#14558715C2318yes—
87#14558715C2314yes—
88#14558715C2314yes—
89C22⋊Dic149C2318yes—
90#14558713C2314yes—
91#14558713C2314yes—
92#14558713C2318yes—
93#14558713C2318yes—
94C4×Dic149C2312yes—
95C4×Dic149C2312yes—
96#41×C149—28no—
97C2×Q8×C149C2312yes—
98C2×D4×C149C2312yes—
99C2×D4×C149C2314yes—
100C2×D4×C149C2314yes—
101C22×C4×C149C2314yes—
102C22×C4×C149C2312yes—
103C22×C4×C149C2311yescharacteristic
104C4⋊C4×C149C2314yes—
105C22⋊C4×C149C2318yes—
106C42×C149C2311yescharacteristic
107C42×C149C2312yes—
108C42×C149C2311yescharacteristic
109C22×D149—28no—
110C22×D149—42no—
111C22×D149—48no—
112C22×D149C2×D414yes—
113C22×D149C2414yes—
114C22×D149C2412yes—
115Dic298—24no—
116Dic298—28no—
117D596—28no—
118D596—24no—
119D596—44no—
120C22⋊D149—28no—
121C22⋊D149—28no—
122C22⋊D149—28no—
123C22⋊D149—48no—
124C2×Dic149—28no—
125C2×Dic149—42no—
126C2×Dic149C2×D414yes—
127C2×Dic149C2414yes—
128C2×Dic149C2414yes—
129C2×Dic149C2412yes—
130C4×D149—28no—
131C4×D149—28no—
132C4×D149—28no—
133C4×D149C2×D418yes—
134C4×D149C2×D418yes—
135C23×C149C2414yes—
136C23×C149C2411yescharacteristic
137Q8×C149—24no—
138D4×C149—28no—
139D4×C149—24no—
140D4×C149—44no—
141C2×C4×C149—28no—
142C2×C4×C149—42no—
143C2×C4×C149C2×D414yes—
144C2×C4×C149C2414yes—
145C2×C4×C149C2412yes—
146C2×C4×C149C2412yes—
147C2×C4×C149C2411yescharacteristic
148C2×C4×C149C2411yescharacteristic
149C22×C149—22no—
150C22×C149—24no—
151C22×C149—28no—
152C22×C149C22×D412yes—
153C22×C149C2511yescharacteristic
154C2×D149—24no—
155C2×D149—24no—
156C2×D149—28no—
157C2×D149—44no—
158C2×D149—416no—
159C2×D149C22×D414yes—
160C4×C149—24no—
161C4×C149—22no—
162C4×C149—22no—
163C4×C149C22×D412yes—
164C4×C149C22×D412yes—
165Dic149—24no—
166Dic149—24no—
167Dic149C22×D414yes—
168C2×C149—22no—
169C2×C149—44no—
170C2×C149C23×D411yescharacteristic
171C2×C149#58512yes—
172D149—24no—
173D149—44no—
174C149#344911yescharacteristic, complemented, Sylow
175#3449—1491nomaximal, Sylow
176C22×#41—1492no—
177#537—1494no—
178#536—1492no—
179#536—1492no—
180#535—1492no—
181#535—1492no—
182#534—1494no—
183#532—1492no—
184#532—1491no—
185#532D14911yescharacteristic, complemented
186#525—1498no—
187C2×#117—1491no—
188C2×#41—29816no—
189C2×#41—1492no—
190C2×#41—1492no—
191C2×#41—1494no—
192C2×#41—1494no—
193C2×#41—1492no—
194C2×#41C2×D14912yescomplemented
195C22×Q8—1492no—
196C22×D4—1494no—
197C22×D4—1492no—
198C23×C4—1494no—
199C23×C4—1491no—
200C4⋊Q8—1492no—
201C4⋊Q8—1491no—
202C4⋊Q8C2×D14911yescharacteristic, complemented
203#127—1492no—
204#127—1491no—
205#127C2×D14911yescharacteristic, complemented
206#124—1492no—
207#124—1492no—
208#124—1492no—
209#124C2×D14912yescomplemented
210#123—1498no—
211#123—1498no—
212C22⋊Q8—1498no—
213C22⋊Q8—1498no—
214#121—1498no—
215#121—1498no—
216#121—1494no—
217#121—1498no—
218#121C2×D14914yescomplemented
219#120—1498no—
220#120—1498no—
221C4×D4—1498no—
222C4×D4—1494no—
223C4×D4C2×D14914yescomplemented
224#117—1494no—
225#117—1492no—
226#117—1492no—
227C2×C4⋊C4—1492no—
228C2×C22⋊C4—1492no—
229C2×C42—1491no—
230C2×C42C2×D14911yescharacteristic, complemented
231C24—1494no—
232C24—1491no—
233#41—28no—
234#41—29816no—
235#41—29816no—
236#41—2988no—
237#41—2988no—
238#41—2988no—
239C2×Q8—2988no—
240C2×Q8—1494no—
241C2×Q8—1492no—
242C2×Q8C22×D14912yes—
243C2×D4—29816no—
244C2×D4—2988no—
245C2×D4—1494no—
246C2×D4—1494no—
247C2×D4—1494no—
248C2×D4—1498no—
249C2×D4—1494no—
250C2×D4—1492no—
251C2×D4—1494no—
252C2×D4C22×D14914yes—
253C2×D4C22×D14912yes—
254C2×D4C22×D14914yes—
255C22×C4—29816no—
256C22×C4—1494no—
257C22×C4—1494no—
258C22×C4—1494no—
259C22×C4—1494no—
260C22×C4—1492no—
261C22×C4—1491no—
262C22×C4—1498no—
263C22×C4—1492no—
264C22×C4—1492no—
265C22×C4C22×D14914yes—
266C22×C4C22×D14912yes—
267C22×C4C22×D14911yescharacteristic
268C4⋊C4—1494no—
269C4⋊C4—1494no—
270C4⋊C4—1498no—
271C4⋊C4—1494no—
272C4⋊C4C22×D14914yes—
273C22⋊C4—1498no—
274C22⋊C4—1494no—
275C22⋊C4—1494no—
276C22⋊C4—1498no—
277C22⋊C4—1498no—
278C22⋊C4C22×D14918yes—
279C42—1492no—
280C42—1492no—
281C42C22×D14911yescharacteristic
282C42C22×D14911yescharacteristic
283C42C22×D14912yes—
284C23—2988no—
285C23—5962no—
286C23—5968no—
287C23—1494no—
288C23—1492no—
289C23—1494no—
290C23C23×D14914yes—
291C23C23×D14911yescharacteristic
292Q8—24no—
293Q8—2988no—
294Q8—2984no—
295D4—28no—
296D4—24no—
297D4—44no—
298D4—2988no—
299D4—2988no—
300D4—2988no—
301D4—2988no—
302D4—2984no—
303D4—5964no—
304D4—5968no—
305C2×C4—28no—
306C2×C4—42no—
307C2×C4—2988no—
308C2×C4—2988no—
309C2×C4—2988no—
310C2×C4—2988no—
311C2×C4—5962no—
312C2×C4—1494no—
313C2×C4—1494no—
314C2×C4—1494no—
315C2×C4—1492no—
316C2×C4—1498no—
317C2×C4—1498no—
318C2×C4D4×D14914yes—
319C2×C4C23×D14912yes—
320C2×C4C23×D14912yes—
321C2×C4C23×D14914yes—
322C2×C4C23×D14911yescharacteristic
323C2×C4C23×D14911yescharacteristic
324C22—22no—
325C22—24no—
326C22—28no—
327C22—2984no—
328C22—2984no—
329C22—2988no—
330C22—5964no—
331C22—59616no—
332C22—1494no—
333C22C2×D4×D14912yes—
334C22C24×D14911yescharacteristic
335C4—24no—
336C4—22no—
337C4—22no—
338C4—2984no—
339C4—2984no—
340C4—1494no—
341C4C2×D4×D14912yes—
342C4C2×D4×D14912yes—
343C2—22no—
344C2—44no—
345C2—2984no—
346C2—5964no—
347C2C22×D4×D14911yescharacteristic
348C2#2375046112yes—
subgroupG/H
centerC2×C4C23×D149
commutator, socle, cosocleC22×C149C25
FrattiniC22C24×D149
Fitting#532×C149C2
hypercenter#532D149
hypocenterC149#3449
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C2×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#41×D149C2semidirect
2. #23750202C2semidirect
3. #23750200C2semidirect
4. #23750195C2semidirect
5. #23750193C2semidirect
6. #23750188C2semidirect
7. #23750183C2semidirect
8. #23750175C2semidirect
9. #23750171C2semidirect
10. #23750134C2semidirect
11. C2×#18011868C2semidirect
12. #532×C149C2semidirect
15. C2×#14558736C22semidirect
17. C2×#14558735C22semidirect
19. C2×Q8×D149C22semidirect
21. C2×D4×D149C22semidirect
23. C22×C4×D149C22semidirect
24. #18011911C22semidirect
25. #18011909C22semidirect
26. #18011908C22semidirect
27. #18011907C22semidirect
28. #18011900C22semidirect
29. #18011899C22semidirect
30. #18011898C22semidirect
31. #18011896C22semidirect
32. #18011893C22semidirect
33. #18011889C22semidirect
34. #18011888C22semidirect
35. #18011885C22semidirect
36. #18011884C22semidirect
37. #18011883C22semidirect
38. #18011883C22semidirect
39. #18011881C22semidirect
40. #18011879C22semidirect
41. C4×D596C22semidirect
42. C4×C22⋊D149C22semidirect
43. #18011868C22semidirect
44. #18011868C22semidirect
45. #18011868C22semidirect
46. C2×#14558715C22semidirect
48. C2×C4×Dic149C22semidirect
49. C2×#41×C149C22semidirect
50. C4⋊Q8×C149C22semidirect
51. #127×C149C22semidirect
52. #124×C149C22semidirect
53. #121×C149C22semidirect
54. C4×D4×C149C22semidirect
55. C2×C42×C149C22semidirect
79. C2×C4×D149D4semidirect
174. C149#3449semidirect
185. #532D149semidirect
194. C2×#41C2×D149semidirect
202. C4⋊Q8C2×D149semidirect
205. #127C2×D149semidirect
209. #124C2×D149semidirect
218. #121C2×D149semidirect
223. C4×D4C2×D149semidirect
230. C2×C42C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1492
#237502022
#237502002
#237501952
#237501932
#237501882
#237501832
#237501752
#237501712
#237501342
C2×#180118682
#532×C1492
#3449149