Finite Group Databasebeta

← #34056913

Subgroups — #34056913

Order 19072.

402 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#23750309C211yesmaximal, characteristic, complemented
2#23750266C211yesmaximal, characteristic
3#23749981C211yesmaximal, characteristic, complemented
4#23749963C211yesmaximal, characteristic, complemented
5#23749945C211yesmaximal, characteristic, complemented
6#23749924C211yesmaximal, characteristic, complemented
7#23749874C211yesmaximal, characteristic, complemented
8#23749849C211yesmaximal, characteristic, complemented
9#23749786C211yesmaximal, characteristic, complemented
10#23749775C211yesmaximal, characteristic, complemented
11#23749710C211yesmaximal, characteristic, complemented
12#23749701C211yesmaximal, characteristic, complemented
13#23749631C211yesmaximal, characteristic, complemented
14#23749625C211yesmaximal, characteristic, complemented
15#470×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17C22×D596C2211yescharacteristic
18C2×#14558723C2211yescharacteristic
19C2×#14558723C2211yescharacteristic
20C2×#14558719C2211yescharacteristic, complemented
21#18011904—21no—
22#18011902C2211yescharacteristic, complemented
23#18011901—21no—
24#18011900C2211yescharacteristic
25#18011896—21no—
26#18011893—21no—
27#18011887C2211yescharacteristic, complemented
28#18011886—21no—
29#18011886C2211yescharacteristic
30#18011881—21no—
31Q8×Dic149C2211yescharacteristic, complemented
32#18011873C2211yescharacteristic
33C4×C22⋊D149C2211yescharacteristic
34#18011870C2211yescharacteristic
35#18011869—21no—
36C4⋊C4×D149C2211yescharacteristic, complemented
37C22⋊C4×D149—21no—
38#18011844C2211yescharacteristic, complemented
39#18011842C2211yescharacteristic, complemented
40#18011840C2211yescharacteristic, complemented
41#18011835C2211yescharacteristic, complemented
42#18011835C2211yescharacteristic, complemented
43#18011833C2211yescharacteristic
44#18011833C2211yescharacteristic
45#18011833C2211yescharacteristic
46#18011833C2211yescharacteristic
47#18011831C2211yescharacteristic
48#18011831C2211yescharacteristic
49C22⋊#11924746C2211yescharacteristic
50#18011823C2211yescharacteristic, complemented
51C8×Dic149C2211yescharacteristic, complemented
52C2×Q16×C149C2211yescharacteristic, complemented
53C2×M16×C149C2211yescharacteristic
54C22⋊Q8×C149C2211yescharacteristic
55C22⋊Q8×C149C2211yescharacteristic
56#106×C149C2211yescharacteristic, complemented
57#103×C149C2211yescharacteristic, complemented
58#103×C149C2211yescharacteristic, complemented
59C23×D149—21no—
60#14558737—22no—
61#14558737—41no—
62C2×D596—22no—
63C2×D596—21no—
64C2×D596—21no—
65C2×D596—21no—
66C2×D596—41no—
67C2×D596C2311yescharacteristic
68C2×D596C2311yescharacteristic
69C2×C22⋊D149D411yescharacteristic
70C2×C4×D149—21no—
71C2×C4×D149—21no—
72C2×C4×D149—21no—
73C2×C4×D149C2311yescharacteristic
74#14558723—22no—
75#14558723—41no—
76#14558719—41no—
77C2×#11924746C2311yescharacteristic
78C2×#11924746C2311yescharacteristic
79C4⋊Dic149—21no—
80C4⋊Dic149—21no—
81C4⋊Dic149C2311yescharacteristic
82C4⋊Dic149C2311yescharacteristic
83#14558715—21no—
84#14558715—21no—
85#14558715—21no—
86#14558715D411yescharacteristic
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×Dic149—21no—
97C4×Dic149C2311yescharacteristic
98C2×Q8×C149C2311yescharacteristic
99C2×Q8×C149C2311yescharacteristic
100C22×C4×C149C2311yescharacteristic
101Q16×C149—41no—
102M16×C149—41no—
103C2×C8×C149C2311yescharacteristic
104C2×C8×C149C2311yescharacteristic
105C4⋊C4×C149—21no—
106C4⋊C4×C149—21no—
107C4⋊C4×C149C2311yescharacteristic
108C4⋊C4×C149C2311yescharacteristic
109C22⋊C4×C149—21no—
110C22⋊C4×C149—21no—
111C22×D149—21no—
112C22×D149—21no—
113C22×D149—42no—
114C22×D149—42no—
115C22×D149—81no—
116C22×D149C2×D411yescharacteristic
117D596—21no—
118D596—21no—
119D596—21no—
120D596—42no—
121D596—42no—
122D596—81no—
123D596#4112yes—
124C22⋊D149—41no—
125C2×Dic149—21no—
126C2×Dic149—21no—
127C2×Dic149C2×D411yescharacteristic
128C2×Dic149C2×D411yescharacteristic
129C2×Dic149C2×D411yescharacteristic
130C4×D149—21no—
131C4×D149—21no—
132C4×D149—42no—
133C4×D149—42no—
134C4×D149—81no—
135#11924746—21no—
136#11924746—21no—
137C23×C149C2×D411yescharacteristic
138Q8×C149—21no—
139Q8×C149—21no—
140Q8×C149—21no—
141Q8×C149—21no—
142C2×C4×C149—21no—
143C2×C4×C149—21no—
144C2×C4×C149—21no—
145C2×C4×C149—21no—
146C2×C4×C149—21no—
147C2×C4×C149—21no—
148C2×C4×C149C2×D411yescharacteristic
149C2×C4×C149C2411yescharacteristic
150C8×C149—21no—
151C8×C149—21no—
152C22×C149—21no—
153C22×C149—21no—
154C22×C149—21no—
155C22×C149C22×D411yescharacteristic
156C2×D149—21no—
157C2×D149—22no—
158C2×D149—21no—
159C2×D149—21no—
160C2×D149—41no—
161C2×D149—41no—
162C2×D149—41no—
163C2×D149—41no—
164C2×D149—41no—
165C2×D149—84no—
166C4×C149—21no—
167C4×C149—41no—
168C4×C149—41no—
169C4×C149—41no—
170C4×C149—41no—
171C4×C149C2×#4111yescharacteristic
172C4×C149H(5,𝔽2)11yescharacteristic
173Dic149—21no—
174Dic149—21no—
175Dic149—21no—
176Dic149—41no—
177Dic149—41no—
178C2×C149—41no—
179C2×C149#54611yescharacteristic
180C2×C149#57511yescharacteristic
181C2×C149#57711yescharacteristic
182D149—41no—
183D149—42no—
184D149—81no—
185C149#332311yescharacteristic, complemented, Sylow
186#3323—1491nomaximal, Sylow
187#552—1491no—
188#546—1491no—
189#488—1491no—
190#485—1491no—
191#481—1491no—
192#478—1491no—
193#470D14911yescharacteristic, complemented
194#468—1491no—
195#464—1491no—
196#452—1491no—
197#450—1491no—
198C4×SD16—1491no—
199#436—1491no—
200#418—1491no—
201#417—1491no—
202C2×#41—1491no—
203C22×D4—1491no—
204C2×Q16C2×D14911yescharacteristic, complemented
205C2×SD16—1491no—
206C2×SD16—1491no—
207C2×M16—1491no—
208C2×M16C2×D14911yescharacteristic
209#126—2981no—
210#125—2981no—
211#125—1491no—
212#124—1491no—
213#123—2981no—
214#123—2981no—
215C22⋊Q8—1491no—
216C22⋊Q8C2×D14911yescharacteristic
217C22⋊Q8C2×D14911yescharacteristic
218#121—2981no—
219#121—1491no—
220#120—2981no—
221C4×Q8—1491no—
222C4×D4—1491no—
223C4×D4—1491no—
224#117—2981no—
225#117—1491no—
226C2×C4⋊C4—1491no—
227C2×C22⋊C4—2981no—
228#107—1491no—
229#106—1491no—
230#106C2×D14911yescharacteristic, complemented
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#102—1491no—
241#102—1491no—
242C22⋊C8—1491no—
243C22⋊C8—1491no—
244C4×C8—1491no—
245C24—2981no—
246#41—2982no—
247#41—5961no—
248C2×Q8C22×D14911yescharacteristic
249C2×Q8C22×D14911yescharacteristic
250C2×D4—2982no—
251C2×D4—2981no—
252C2×D4—2981no—
253C2×D4—2981no—
254C2×D4—5961no—
255C2×D4—1491no—
256C2×D4—1491no—
257C2×D4—1491no—
258C22×C4—2981no—
259C22×C4—2981no—
260C22×C4—2981no—
261C22×C4—1491no—
262C22×C4C22×D14911yescharacteristic
263Q16—41no—
264SD16—2982no—
265SD16—5961no—
266M16—41no—
267M16—5961no—
268C2×C8—1491no—
269C2×C8—1491no—
270C2×C8C22×D14911yescharacteristic
271C2×C8C22×D14911yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
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—21no—
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—2981no—
295C42—2981no—
296C42—1491no—
297C23—2981no—
298C23—2981no—
299C23—5962no—
300C23—5962no—
301C23—1491no—
302C23—11921no—
303C23D4×D14911yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—2981no—
309D4—2981no—
310D4—2981no—
311D4—5962no—
312D4—5962no—
313D4—5961no—
314D4—1492no—
315D4—11921no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—2981no—
323C2×C4—2981no—
324C2×C4—2981no—
325C2×C4—2981no—
326C2×C4—5962no—
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—2981no—
342C22—2981no—
343C22—2981no—
344C22—2982no—
345C22—5961no—
346C22—5961no—
347C22—5961no—
348C22—5961no—
349C22—5961no—
350C22—11924no—
351C22C2×D4×D14911yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—2981no—
358C4—2981no—
359C4—2981no—
360C4—5961no—
361C4—5961no—
362C4#41×D14911yescharacteristic
363C4#1801195611yescharacteristic
364C2—41no—
365C2—5961no—
366C2—5962no—
367C2—11921no—
368C2#2375026411yescharacteristic
369C2#2375042411yescharacteristic
370C2#2375043411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#470×C149C2
socleC22×C149C22×D4
prime radical#23750266C2
hypercenter#470D149
hypocenterC149#3323
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #23750309C2semidirect
3. #23749981C2semidirect
4. #23749963C2semidirect
5. #23749945C2semidirect
6. #23749924C2semidirect
7. #23749874C2semidirect
8. #23749849C2semidirect
9. #23749786C2semidirect
10. #23749775C2semidirect
11. #23749710C2semidirect
12. #23749701C2semidirect
13. #23749631C2semidirect
14. #23749625C2semidirect
15. #470×C149C2semidirect
20. C2×#14558719C22semidirect
22. #18011902C22semidirect
27. #18011887C22semidirect
31. Q8×Dic149C22semidirect
36. C4⋊C4×D149C22semidirect
38. #18011844C22semidirect
39. #18011842C22semidirect
40. #18011840C22semidirect
41. #18011835C22semidirect
42. #18011835C22semidirect
50. #18011823C22semidirect
51. C8×Dic149C22semidirect
52. C2×Q16×C149C22semidirect
56. #106×C149C22semidirect
57. #103×C149C22semidirect
58. #103×C149C22semidirect
185. C149#3323semidirect
193. #470D149semidirect
204. C2×Q16C2×D149semidirect
230. #106C2×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
#237503092
#237502662
#237499812
#237499632
#237499452
#237499242
#237498742
#237498492
#237497862
#237497752
#237497102
#237497012
#237496312
#237496252
#470×C1492
#3323149