Finite Group Databasebeta

← #34058376

Subgroups — #34058376

Order 19072.

1328 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750304C214yesmaximal, complemented
2#23750299C214yesmaximal, complemented
3#23750271C214yesmaximal, complemented
4#23750265C214yesmaximal, complemented
5#23750251C211yesmaximal, characteristic, complemented
6#23750208C214yesmaximal, complemented
7#23750184C211yesmaximal, characteristic, complemented
8#23750134C214yesmaximal, complemented
9C2×#18011881C211yesmaximal, characteristic, complemented
10C2×#18011879C211yesmaximal, characteristic, complemented
11#23750098C211yesmaximal, characteristic, complemented
12C2×C22⋊C4×D149C211yesmaximal, characteristic, complemented
13#518×C149C211yesmaximal, characteristic, complemented
14C24×D149C2211yescharacteristic
15C2×#14558736C2214yescomplemented
16C22×D596C2212yescomplemented
17C22×C22⋊D149C2211yescharacteristic
18C22×C22⋊D149C2212yescomplemented
19C22×C4×D149C2212yescomplemented
20#18011906C2214yescomplemented
21#18011905C2214yescomplemented
22#18011899C2214yescomplemented
23#18011897C2214yescomplemented
24#18011896C2214yescomplemented
25#18011896C2214yescomplemented
26#18011894C2212yescomplemented
27#18011894C2214yescomplemented
28#18011893C2214yescomplemented
29#18011892C2212yescomplemented
30#18011890C2214yescomplemented
31#18011889C2214yescomplemented
32#18011888C2214yescomplemented
33#18011888C2214yescomplemented
34#18011885C2214yescomplemented
35#18011884C2214yescomplemented
36#18011884C2214yescomplemented
37#18011883C2214yescomplemented
38#18011881C2214yes—
39#18011881C2212yescomplemented
40#18011881C2212yescomplemented
41#18011880C2212yescomplemented
42#18011879C2214yes—
43#18011879C2212yescomplemented
44#18011879C2212yescomplemented
45#18011879C2212yescomplemented
46C4×D596C2214yescomplemented
47C4×C22⋊D149C2214yescomplemented
48C4×C22⋊D149C2214yescomplemented
49#18011871C2214yescomplemented
50#18011868C2214yescomplemented
51#18011867C2212yescomplemented
52C2×C22⋊Dic149C2211yescharacteristic
53C2×C22⋊Dic149C2211yescharacteristic
54C2×#14558713C2212yescomplemented
55C2×#14558713C2212yescomplemented
56C2×#14558713C2212yescomplemented
57C22⋊C4×D149C2214yescomplemented
58C22⋊C4×D149C2214yescomplemented
59C22⋊C4×D149C2212yescomplemented
60C22×D4×C149C2211yescharacteristic
61C4×D4×C149C2214yescomplemented
62C4×D4×C149C2214yescomplemented
63#117×C149C2212yescomplemented
64C2×C22⋊C4×C149C2211yescharacteristic
65C2×C22⋊C4×C149C2211yescharacteristic
66C2×C22⋊C4×C149C2212yescomplemented
67C23×D149—24no—
68C23×D149—44no—
69C23×D149C2311yescharacteristic
70C23×D149C2312yes—
71C23×D149C2312yes—
72C23×D149C2311yescharacteristic
73C23×D149C2311yescharacteristic
74#14558736—48no—
75C2×Dic298C2314yescomplemented
76C2×D596—44no—
77C2×D596C2314yes—
78C2×D596C2314yes—
79C2×C22⋊D149—24no—
80C2×C22⋊D149—44no—
81C2×C22⋊D149C2312yes—
82C2×C22⋊D149C2312yes—
83C2×C22⋊D149C2314yes—
84C2×C22⋊D149C2314yes—
85C2×C22⋊D149C2314yescomplemented
86C22×Dic149C2311yescharacteristic
87C22×Dic149C2312yes—
88C22×Dic149C2312yes—
89C2×C4×D149—44no—
90C2×C4×D149C2314yes—
91C2×C4×D149C2314yes—
92C2×C4×D149C2314yescomplemented
93C4⋊Dic149C2314yescomplemented
94#14558715C2314yescomplemented
95#14558715C2314yescomplemented
96#14558715C2314yescomplemented
97#14558715C2314yescomplemented
98C22⋊Dic149C2312yes—
99C22⋊Dic149C2312yes—
100C22⋊Dic149C2312yes—
101C22⋊Dic149C2312yes—
102C22⋊Dic149C2314yescomplemented
103#14558713C2314yes—
104#14558713C2314yes—
105#14558713C2314yes—
106#14558713C2314yes—
107#14558713C2314yes—
108#14558713C2314yes—
109#14558713C2314yescomplemented
110#14558713C2314yescomplemented
111C4×Dic149C2314yescomplemented
112C24×C149C2311yescharacteristic
113C24×C149C2311yescharacteristic
114C2×D4×C149—24no—
115C2×D4×C149C2312yes—
116C2×D4×C149C2312yes—
117C22×C4×C149C2311yescharacteristic
118C22×C4×C149C2312yes—
119C22×C4×C149C2312yes—
120C22×C4×C149C2314yescomplemented
121C4⋊C4×C149C2314yescomplemented
122C22⋊C4×C149C2312yes—
123C22⋊C4×C149C2312yes—
124C22⋊C4×C149C2312yes—
125C22⋊C4×C149C2312yes—
126C22⋊C4×C149C2314yescomplemented
127C42×C149C2314yescomplemented
128C22×D149—22no—
129C22×D149—24no—
130C22×D149—24no—
131C22×D149—24no—
132C22×D149—22no—
133C22×D149—22no—
134C22×D149—24no—
135C22×D149—24no—
136C22×D149—42no—
137C22×D149—42no—
138C22×D149—42no—
139C22×D149—42no—
140C22×D149—416no—
141C22×D149C2412yes—
142C22×D149C2412yes—
143C22×D149C2412yes—
144Dic298—44no—
145D596—28no—
146D596—44no—
147C22⋊D149—24no—
148C22⋊D149—24no—
149C22⋊D149—28no—
150C22⋊D149—44no—
151C22⋊D149—44no—
152C2×Dic149—22no—
153C2×Dic149—42no—
154C2×Dic149—42no—
155C2×Dic149C2414yes—
156C2×Dic149C2414yes—
157C2×Dic149C2412yes—
158C4×D149—28no—
159C4×D149—44no—
160C4×D149—44no—
161C23×C149—22no—
162C23×C149—42no—
163C23×C149—42no—
164C23×C149#4114yescomplemented
165C23×C149C2412yes—
166C23×C149C2411yescharacteristic
167C23×C149C2411yescharacteristic
168C23×C149C2411yescharacteristic
169D4×C149—24no—
170D4×C149—24no—
171C2×C4×C149—22no—
172C2×C4×C149—42no—
173C2×C4×C149—42no—
174C2×C4×C149—44no—
175C2×C4×C149C2414yes—
176C2×C4×C149C2414yes—
177C2×C4×C149C2412yes—
178C22×C149—22no—
179C22×C149—22no—
180C22×C149—22no—
181C22×C149—22no—
182C22×C149—24no—
183C22×C149—44no—
184C22×C149—82no—
185C22×C149C2×#4112yes—
186C22×C149C2×#4112yes—
187C22×C149C2×#4112yes—
188C22×C149C2511yescharacteristic
189C2×D149—28no—
190C2×D149—22no—
191C2×D149—22no—
192C2×D149—22no—
193C2×D149—22no—
194C2×D149—24no—
195C2×D149—22no—
196C2×D149—22no—
197C2×D149—48no—
198C2×D149—48no—
199C2×D149—44no—
200C2×D149—82no—
201C2×D149H(5,𝔽2)14yes—
202C4×C149—22no—
203C4×C149—24no—
204C4×C149—24no—
205Dic149—22no—
206Dic149—24no—
207Dic149—24no—
208C2×C149—22no—
209C2×C149—22no—
210C2×C149—22no—
211C2×C149—42no—
212C2×C149C22×#4111yescharacteristic
213C2×C149C2×H(5,𝔽2)12yes—
214D149—24no—
215D149—42no—
216D149—42no—
217C149#349011yescharacteristic, complemented, Sylow
218#3490—1491nomaximal, Sylow
219#551—1494no—
220#551—1494no—
221#546—1494no—
222#546—1494no—
223#543—1491no—
224#538—1494no—
225#534—1491no—
226#525—1494no—
227C2×#120—1491no—
228C2×#120—1491no—
229#518—1491no—
230#518D14911yescharacteristic, complemented
231C22×C22⋊C4—1491no—
232C25—1491no—
233C2×#41—1494no—
234C22×D4—1491no—
235C22×D4—1492no—
236C22×D4—1492no—
237C22×D4C2×D14911yescharacteristic
238C23×C4—1492no—
239#126—1494no—
240#126—1494no—
241#124—1494no—
242#124—1494no—
243#123—1492no—
244#123—1492no—
245#123—1494no—
246#123—1494no—
247#123—1494no—
248#123—1494no—
249C22⋊Q8—1494no—
250C22⋊Q8—1494no—
251C22⋊Q8—1494no—
252C22⋊Q8—1494no—
253#121—1494no—
254#121—1494no—
255#121—1494no—
256#121—1494no—
257#120—1492no—
258#120—1492no—
259#120—1492no—
260#120—1494no—
261#120—1492no—
262#120—1492no—
263#120—1492no—
264#120—1494no—
265C4×D4—1494no—
266C4×D4—1494no—
267C4×D4—1494no—
268C4×D4—1494no—
269C4×D4C2×D14914yescomplemented
270C4×D4C2×D14914yescomplemented
271#117—1492no—
272#117—1494no—
273#117C2×D14912yescomplemented
274C2×C22⋊C4—1491no—
275C2×C22⋊C4—1491no—
276C2×C22⋊C4—1492no—
277C2×C22⋊C4—1492no—
278C2×C22⋊C4—1492no—
279C2×C22⋊C4—1494no—
280C2×C22⋊C4—1494no—
281C2×C22⋊C4—1492no—
282C2×C22⋊C4C2×D14911yescharacteristic
283C2×C22⋊C4C2×D14911yescharacteristic
284C2×C22⋊C4C2×D14912yescomplemented
285C24—2984no—
286C24—5964no—
287C24—1491no—
288C24—1492no—
289C24—1492no—
290C24—1491no—
291C24—1491no—
292C24C22×D14911yescharacteristic
293C24C22×D14911yescharacteristic
294#41—5968no—
295C2×Q8—1494no—
296C2×D4—24no—
297C2×D4—2984no—
298C2×D4—5964no—
299C2×D4—5964no—
300C2×D4—1492no—
301C2×D4—1492no—
302C2×D4—1494no—
303C2×D4—1494no—
304C2×D4—1494no—
305C2×D4—1494no—
306C2×D4—1494no—
307C2×D4C22×D14912yes—
308C2×D4C22×D14912yes—
309C22×C4—5964no—
310C22×C4—1491no—
311C22×C4—1494no—
312C22×C4—1492no—
313C22×C4—1492no—
314C22×C4—1494no—
315C22×C4—1494no—
316C22×C4C22×D14911yescharacteristic
317C22×C4C22×D14912yes—
318C22×C4C22×D14912yes—
319C22×C4C22×D14914yescomplemented
320C4⋊C4—1494no—
321C4⋊C4—1494no—
322C4⋊C4—1494no—
323C4⋊C4—1494no—
324C4⋊C4—1494no—
325C4⋊C4C22×D14914yescomplemented
326C22⋊C4—1492no—
327C22⋊C4—1492no—
328C22⋊C4—1492no—
329C22⋊C4—1492no—
330C22⋊C4—1494no—
331C22⋊C4—1494no—
332C22⋊C4—1494no—
333C22⋊C4—1494no—
334C22⋊C4—1494no—
335C22⋊C4—1494no—
336C22⋊C4—1494no—
337C22⋊C4—1494no—
338C22⋊C4—1494no—
339C22⋊C4C22×D14912yes—
340C22⋊C4C22×D14912yes—
341C22⋊C4C22×D14912yes—
342C22⋊C4C22×D14912yes—
343C22⋊C4C22×D14914yescomplemented
344C42—1494no—
345C42C22×D14914yescomplemented
346C23—22no—
347C23—42no—
348C23—42no—
349C23—2984no—
350C23—2984no—
351C23—2984no—
352C23—2982no—
353C23—2984no—
354C23—2984no—
355C23—2982no—
356C23—2982no—
357C23—5962no—
358C23—5962no—
359C23—5962no—
360C23—5962no—
361C23—59616no—
362C23—1492no—
363C23—1492no—
364C23—1492no—
365C23#1455873614yescomplemented
366C23C23×D14912yes—
367C23C23×D14911yescharacteristic
368C23C23×D14911yescharacteristic
369C23C23×D14911yescharacteristic
370Q8—5964no—
371D4—24no—
372D4—24no—
373D4—2988no—
374D4—2988no—
375D4—2984no—
376D4—2984no—
377D4—5964no—
378D4—5964no—
379D4—5964no—
380C2×C4—22no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—2988no—
385C2×C4—2982no—
386C2×C4—5964no—
387C2×C4—5964no—
388C2×C4—5962no—
389C2×C4—5962no—
390C2×C4—1492no—
391C2×C4—1494no—
392C2×C4—1494no—
393C2×C4C23×D14914yes—
394C2×C4C23×D14912yes—
395C2×C4C23×D14914yes—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—24no—
401C22—44no—
402C22—82no—
403C22—2988no—
404C22—2982no—
405C22—2982no—
406C22—2982no—
407C22—2982no—
408C22—2984no—
409C22—2982no—
410C22—2982no—
411C22—5968no—
412C22—5968no—
413C22—5964no—
414C22—1494no—
415C22—11922no—
416C22C2×#1455873612yes—
417C22C2×#1455873612yes—
418C22C2×#1455873612yes—
419C22C24×D14911yescharacteristic
420C4—22no—
421C4—24no—
422C4—24no—
423C4—2982no—
424C4—2984no—
425C4—2984no—
426C2—22no—
427C2—22no—
428C2—22no—
429C2—42no—
430C2—2984no—
431C2—5962no—
432C2—5962no—
433C2C22×#1455873611yescharacteristic
434C2H(5,𝔽2)×D14912yes—
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#518×C149C2
hypercenter#518D149
hypocenterC149#3490
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #23750304C2semidirect
2. #23750299C2semidirect
3. #23750271C2semidirect
4. #23750265C2semidirect
5. #23750251C2semidirect
6. #23750208C2semidirect
7. #23750184C2semidirect
8. #23750134C2semidirect
9. C2×#18011881C2semidirect
10. C2×#18011879C2semidirect
11. #23750098C2semidirect
12. C2×C22⋊C4×D149C2semidirect
13. #518×C149C2semidirect
15. C2×#14558736C22semidirect
16. C22×D596C22semidirect
18. C22×C22⋊D149C22semidirect
19. C22×C4×D149C22semidirect
20. #18011906C22semidirect
21. #18011905C22semidirect
22. #18011899C22semidirect
23. #18011897C22semidirect
24. #18011896C22semidirect
25. #18011896C22semidirect
26. #18011894C22semidirect
27. #18011894C22semidirect
28. #18011893C22semidirect
29. #18011892C22semidirect
30. #18011890C22semidirect
31. #18011889C22semidirect
32. #18011888C22semidirect
33. #18011888C22semidirect
34. #18011885C22semidirect
35. #18011884C22semidirect
36. #18011884C22semidirect
37. #18011883C22semidirect
39. #18011881C22semidirect
40. #18011881C22semidirect
41. #18011880C22semidirect
43. #18011879C22semidirect
44. #18011879C22semidirect
45. #18011879C22semidirect
46. C4×D596C22semidirect
47. C4×C22⋊D149C22semidirect
48. C4×C22⋊D149C22semidirect
49. #18011871C22semidirect
50. #18011868C22semidirect
51. #18011867C22semidirect
54. C2×#14558713C22semidirect
55. C2×#14558713C22semidirect
56. C2×#14558713C22semidirect
57. C22⋊C4×D149C22semidirect
58. C22⋊C4×D149C22semidirect
59. C22⋊C4×D149C22semidirect
61. C4×D4×C149C22semidirect
62. C4×D4×C149C22semidirect
63. #117×C149C22semidirect
66. C2×C22⋊C4×C149C22semidirect
75. C2×Dic298C23semidirect
85. C2×C22⋊D149C23semidirect
92. C2×C4×D149C23semidirect
93. C4⋊Dic149C23semidirect
94. #14558715C23semidirect
95. #14558715C23semidirect
96. #14558715C23semidirect
97. #14558715C23semidirect
102. C22⋊Dic149C23semidirect
109. #14558713C23semidirect
110. #14558713C23semidirect
111. C4×Dic149C23semidirect
120. C22×C4×C149C23semidirect
121. C4⋊C4×C149C23semidirect
126. C22⋊C4×C149C23semidirect
127. C42×C149C23semidirect
164. C23×C149#41semidirect
217. C149#3490semidirect
230. #518D149semidirect
269. C4×D4C2×D149semidirect
270. C4×D4C2×D149semidirect
273. #117C2×D149semidirect
284. C2×C22⋊C4C2×D149semidirect
319. C22×C4C22×D149semidirect
325. C4⋊C4C22×D149semidirect
343. C22⋊C4C22×D149semidirect
345. C42C22×D149semidirect
365. C23#14558736semidirect

Maximal subgroups

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

subgroupindex
#237503042
#237502992
#237502712
#237502652
#237502512
#237502082
#237501842
#237501342
C2×#180118812
C2×#180118792
#237500982
C2×C22⋊C4×D1492
#518×C1492
#3490149