Finite Group Databasebeta

← #34056527

Subgroups — #34056527

Order 19072.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750314C211yesmaximal, characteristic, complemented
2#23750263C211yesmaximal, characteristic
3#23749981C211yesmaximal, characteristic, complemented
4#23749954C211yesmaximal, characteristic, complemented
5#23749949C211yesmaximal, characteristic, complemented
6#23749927C211yesmaximal, characteristic, complemented
7#23749865C211yesmaximal, characteristic, complemented
8#23749854C211yesmaximal, characteristic, complemented
9#23749850C211yesmaximal, characteristic, complemented
10#23749787C211yesmaximal, characteristic, complemented
11#23749777C211yesmaximal, characteristic, complemented
12#23749716C211yesmaximal, characteristic, complemented
13C8×C22⋊D149C211yesmaximal, characteristic, complemented
14#102×D149C211yesmaximal, characteristic, complemented
15#416×C149C211yesmaximal, characteristic, complemented
16C2×D4×D149C2211yescharacteristic
17C2×#14558726C2211yescharacteristic, complemented
18C2×#14558725C2211yescharacteristic
19C2×#14558724C2211yescharacteristic
20C2×C8×D149C2211yescharacteristic
21#18011904—21no—
22#18011903—21no—
23#18011902C2211yescharacteristic, complemented
24#18011900C2211yescharacteristic
25#18011894—21no—
26#18011893—21no—
27#18011891C2211yescharacteristic
28#18011890C2211yescharacteristic, complemented
29#18011887C2211yescharacteristic
30#18011885C2211yescharacteristic
31#18011883—21no—
32#18011879—21no—
33#18011876C2211yescharacteristic, complemented
34D4×Dic149C2211yescharacteristic
35C4×C22⋊D149C2211yescharacteristic
36#18011868—21no—
37C4⋊C4×D149C2211yescharacteristic
38C2×#14558713—21no—
39#18011844C2211yescharacteristic, complemented
40#18011843C2211yescharacteristic, complemented
41#18011842C2211yescharacteristic, complemented
42#18011840C2211yescharacteristic, complemented
43#18011836C2211yescharacteristic, complemented
44#18011835C2211yescharacteristic, complemented
45#18011833C2211yescharacteristic
46#18011832C2211yescharacteristic
47#18011831C2211yescharacteristic
48#18011831C2211yescharacteristic
49C22⋊#11924746C2211yescharacteristic, complemented
50#18011823C2211yescharacteristic
51C8×Dic149C2211yescharacteristic, complemented
52C2×#41×C149C2211yescharacteristic
53C22×C8×C149C2211yescharacteristic, complemented
54#117×C149C2211yescharacteristic, complemented
55#103×C149C2211yescharacteristic, complemented
56#103×C149C2211yescharacteristic, complemented
57#102×C149C2211yescharacteristic
58#102×C149C2211yescharacteristic
59C23×D149—21no—
60C2×Dic298C2311yescharacteristic
61C2×D596C2311yescharacteristic
62C2×C22⋊D149—21no—
63C2×C22⋊D149—21no—
64C2×C22⋊D149D411yescharacteristic
65D4×D149—22no—
66D4×D149—41no—
67C22×Dic149—21no—
68C2×C4×D149—21no—
69C2×C4×D149C2311yescharacteristic
70#14558726—41no—
71#14558725—41no—
72#14558724—22no—
73C2×#11924746C2311yescharacteristic
74C2×#11924746C2311yescharacteristic
75C8×D149—41no—
76C4⋊Dic149—21no—
77C4⋊Dic149—21no—
78C4⋊Dic149C2311yescharacteristic
79C4⋊Dic149C2311yescharacteristic
80#14558715—21no—
81#14558715—21no—
82#14558715—21no—
83#14558715—21no—
84#14558715—21no—
85#14558715D411yescharacteristic
86C22⋊Dic149—21no—
87C22⋊Dic149—21no—
88C22⋊Dic149D411yescharacteristic
89#14558713—21no—
90#14558713—21no—
91#14558713—21no—
92#14558713—21no—
93#14558713—21no—
94#14558713D411yescharacteristic
95C4×Dic149—21no—
96C4×Dic149C2311yescharacteristic
97#41×C149—22no—
98#41×C149—41no—
99C2×Q8×C149C2311yescharacteristic
100C2×D4×C149—21no—
101C2×D4×C149C2311yescharacteristic
102C22×C4×C149—21no—
103C22×C4×C149C2311yescharacteristic
104C2×C8×C149—41no—
105C2×C8×C149C2311yescharacteristic
106C2×C8×C149C2311yescharacteristic
107C4⋊C4×C149C2311yescharacteristic
108C4⋊C4×C149C2311yescharacteristic
109C22⋊C4×C149—21no—
110C42×C149—21no—
111C22×D149—21no—
112C22×D149—42no—
113C22×D149—42no—
114C22×D149—81no—
115C22×D149C2×D411yescharacteristic
116Dic298—21no—
117Dic298—21no—
118D596—21no—
119D596—21no—
120C22⋊D149—22no—
121C22⋊D149—42no—
122C22⋊D149—81no—
123C2×Dic149—21no—
124C2×Dic149—21no—
125C2×Dic149—21no—
126C2×Dic149—81no—
127C2×Dic149C2×D411yescharacteristic
128C2×Dic149C2×D411yescharacteristic
129C2×Dic149C2×D411yescharacteristic
130C4×D149—21no—
131C4×D149—21no—
132C4×D149—42no—
133#11924746—21no—
134#11924746—21no—
135C23×C149—21no—
136C23×C149C2×D411yescharacteristic
137Q8×C149—21no—
138Q8×C149—21no—
139D4×C149—21no—
140D4×C149—42no—
141D4×C149#4112yes—
142C2×C4×C149—21no—
143C2×C4×C149—21no—
144C2×C4×C149—21no—
145C2×C4×C149—21no—
146C2×C4×C149—21no—
147C2×C4×C149—42no—
148C2×C4×C149C2×D411yescharacteristic
149C2×C4×C149C2411yescharacteristic
150C8×C149—21no—
151C8×C149—21no—
152C22×C149—21no—
153C22×C149—22no—
154C22×C149—21no—
155C22×C149—21no—
156C22×C149—41no—
157C22×C149—41no—
158C22×C149C22×D411yescharacteristic
159C2×D149—21no—
160C2×D149—21no—
161C2×D149—41no—
162C2×D149—41no—
163C2×D149—41no—
164C2×D149—84no—
165C2×D149#13512yes—
166C4×C149—21no—
167C4×C149—41no—
168C4×C149—41no—
169C4×C149—41no—
170C4×C149C2×#4111yescharacteristic
171C4×C149H(5,𝔽2)11yescharacteristic
172Dic149—21no—
173Dic149—21no—
174Dic149—21no—
175Dic149—41no—
176Dic149—41no—
177Dic149—41no—
178C2×C149—42no—
179C2×C149—41no—
180C2×C149#54611yescharacteristic
181C2×C149C2×#13511yescharacteristic
182C2×C149#57711yescharacteristic
183D149—22no—
184D149—81no—
185C149#329611yescharacteristic, complemented, Sylow
186#3296—1491nomaximal, Sylow
187#552—1491no—
188#546—1491no—
189#488—1491no—
190#483—1491no—
191#482—1491no—
192#478—1491no—
193#467—1491no—
194#465—1491no—
195#464—1491no—
196#452—1491no—
197#450—1491no—
198C4×SD16—1491no—
199C8×D4—1491no—
200#416D14911yescharacteristic, complemented
201C2×#102—1491no—
202C2×#41C2×D14911yescharacteristic
203C22×D4—1491no—
204C2×Q16—1491no—
205C2×SD16—1491no—
206C2×SD16—1491no—
207C22×C8—1491no—
208C22×C8C2×D14911yescharacteristic, complemented
209#126—2981no—
210#125—2981no—
211#125—1491no—
212#124—1491no—
213#123—2981no—
214#123—2981no—
215C22⋊Q8—1491no—
216C22⋊Q8—1491no—
217C22⋊Q8—1491no—
218#121—2981no—
219#121—1491no—
220#120—2981no—
221C4×Q8—1491no—
222C4×D4—1491no—
223C4×D4—1491no—
224#117—2981no—
225#117C2×D14911yescharacteristic, complemented
226C2×C4⋊C4—1491no—
227C2×C22⋊C4—2981no—
228#107—1491no—
229#106—1491no—
230#106—1491no—
231C4⋊C8—1491no—
232#103—1491no—
233#103—1491no—
234#103C2×D14911yescharacteristic, complemented
235#103C2×D14911yescharacteristic, complemented
236#102—1491no—
237#102—1491no—
238#102—1491no—
239#102—1491no—
240#102C2×D14911yescharacteristic
241#102C2×D14911yescharacteristic
242C22⋊C8—1491no—
243C22⋊C8—1491no—
244C4×C8—1491no—
245C24—2981no—
246#41—22no—
247#41—41no—
248C2×Q8—1491no—
249C2×Q8C22×D14911yescharacteristic
250C2×D4—21no—
251C2×D4—2982no—
252C2×D4—2981no—
253C2×D4—2981no—
254C2×D4—5961no—
255C2×D4—1491no—
256C2×D4—1491no—
257C2×D4C22×D14911yescharacteristic
258C22×C4—21no—
259C22×C4—2981no—
260C22×C4—2981no—
261C22×C4—1491no—
262C22×C4C22×D14911yescharacteristic
263Q16—5961no—
264SD16—2982no—
265SD16—5961no—
266C2×C8—41no—
267C2×C8—5961no—
268C2×C8—1491no—
269C2×C8—1491no—
270C2×C8C22×D14911yescharacteristic
271C2×C8C22×D14911yescharacteristic
272C4⋊C4—2981no—
273C4⋊C4—2981no—
274C4⋊C4—2981no—
275C4⋊C4—2981no—
276C4⋊C4—2981no—
277C4⋊C4—2981no—
278C4⋊C4—2981no—
279C4⋊C4—1491no—
280C4⋊C4—1491no—
281C4⋊C4—1491no—
282C4⋊C4C22×D14911yescharacteristic
283C4⋊C4C22×D14911yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—2981no—
286C22⋊C4—2981no—
287C22⋊C4—2981no—
288C22⋊C4—2981no—
289C22⋊C4—2981no—
290C22⋊C4—2981no—
291C22⋊C4—2981no—
292C22⋊C4—1491no—
293C22⋊C4—1491no—
294C42—21no—
295C42—2981no—
296C42—1491no—
297C23—21no—
298C23—2981no—
299C23—5962no—
300C23—5962no—
301C23—1491no—
302C23—11921no—
303C23D4×D14911yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2981no—
307Q8—2981no—
308D4—21no—
309D4—42no—
310D4—2982no—
311D4—2981no—
312D4—2981no—
313D4—5962no—
314D4—11921no—
315D4#1455873612yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—2981no—
323C2×C4—2981no—
324C2×C4—2981no—
325C2×C4—2981no—
326C2×C4—2981no—
327C2×C4—5962no—
328C2×C4—1491no—
329C2×C4—1491no—
330C2×C4—1491no—
331C2×C4—11921no—
332C2×C4D4×D14911yescharacteristic
333C2×C4C23×D14911yescharacteristic
334C8—21no—
335C8—21no—
336C8—2981no—
337C8—2981no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—2981no—
345C22—2981no—
346C22—5961no—
347C22—5961no—
348C22—5961no—
349C22—1492no—
350C22—11924no—
351C22C2×D4×D14911yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—2981no—
357C4—2981no—
358C4—2981no—
359C4—5961no—
360C4—5961no—
361C4—5961no—
362C4C2×#1455873611yescharacteristic
363C4#1801195611yescharacteristic
364C2—41no—
365C2—42no—
366C2—2982no—
367C2—11921no—
368C2#2375026511yescharacteristic
369C2#135×D14911yescharacteristic
370C2#2375043211yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#416×C149C2
socleC22×C149C22×D4
prime radical#23750263C2
hypercenter#416D149
hypocenterC149#3296
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #23750314C2semidirect
3. #23749981C2semidirect
4. #23749954C2semidirect
5. #23749949C2semidirect
6. #23749927C2semidirect
7. #23749865C2semidirect
8. #23749854C2semidirect
9. #23749850C2semidirect
10. #23749787C2semidirect
11. #23749777C2semidirect
12. #23749716C2semidirect
13. C8×C22⋊D149C2semidirect
14. #102×D149C2semidirect
15. #416×C149C2semidirect
17. C2×#14558726C22semidirect
23. #18011902C22semidirect
28. #18011890C22semidirect
33. #18011876C22semidirect
39. #18011844C22semidirect
40. #18011843C22semidirect
41. #18011842C22semidirect
42. #18011840C22semidirect
43. #18011836C22semidirect
44. #18011835C22semidirect
49. C22⋊#11924746C22semidirect
51. C8×Dic149C22semidirect
53. C22×C8×C149C22semidirect
54. #117×C149C22semidirect
55. #103×C149C22semidirect
56. #103×C149C22semidirect
185. C149#3296semidirect
200. #416D149semidirect
208. C22×C8C2×D149semidirect
225. #117C2×D149semidirect
234. #103C2×D149semidirect
235. #103C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237503142
#237502632
#237499812
#237499542
#237499492
#237499272
#237498652
#237498542
#237498502
#237497872
#237497772
#237497162
C8×C22⋊D1492
#102×D1492
#416×C1492
#3296149