Finite Group Databasebeta

← #28592116

Subgroups — #28592116

Order 15104.

890 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837449C211yesmaximal, characteristic
2#21432684C211yesmaximal, characteristic
3#424×D59C212yesmaximal
4C2×#16837359C211yesmaximal, characteristic
5#21424849C211yesmaximal, characteristic, complemented
6#21424846C212yesmaximal, complemented
7#21424845C211yesmaximal, characteristic, complemented
8#21424843C212yesmaximal, complemented
9#21424842C211yesmaximal, characteristic, complemented
10#21423804C212yesmaximal
11C59×#1946C211yesmaximal, characteristic, complemented
12C22×#11745659C2211yescharacteristic
13C2×M16×D59C2212yes—
14C22×C8×D59C2211yescharacteristic
15C2×#14273684C2211yescharacteristic
16#117×D59C2212yes—
17#16837449—24no—
18C2×#14273663C2211yescharacteristic
19C2×#14273662C2212yes—
20#16837443C414yescomplemented
21#16837441C414yescomplemented
22#16837440C2211yescharacteristic
23#16837438C2212yes—
24#16837430C2212yes—
25#16837429C2212yes—
26#16837428C2212yes—
27#16837427—24no—
28C4⋊C8×D59—24no—
29#16837359—24no—
30C2×#14273637C2211yescharacteristic
31C2×C4×C59⋊C8C2211yescharacteristic
32C2×C8×Dic59C2211yescharacteristic
33#16837163C2212yes—
34#16837162C2212yes—
35#16837161C2212yes—
36C2×#108×C59C2211yescharacteristic
37C59×#427C2211yescharacteristic
38C59×#424C2212yes—
39C2×#97×C59C2211yescharacteristic
40C59×#339C2212yes—
41C22×C4×D59C2311yescharacteristic
42C2×#11745664C2312yes—
43C2×#11745659—22no—
44C2×#11745659C2×C412yes—
45C2×#11745659C2×C412yes—
46C2×#11745659D414yes—
47C2×#11745659Q814yes—
48M16×D59—24no—
49M16×D59—24no—
50C22×C59⋊C8C2311yescharacteristic
51C22×C59⋊C8C2311yescharacteristic
52C2×C8×D59—22no—
53C2×C8×D59—22no—
54C2×C8×D59—22no—
55C2×C8×D59D414yes—
56#14273686C2312yes—
57#14273685C2×C414yes—
58#14273684—22no—
59#14273684—22no—
60#14273684Q814yes—
61#14273683—22no—
62C2×Dic59⋊C4—21no—
63C4⋊C4×D59C2×C414yes—
64C2×(C2×D59)⋊C4—21no—
65C22⋊C4×D59—22no—
66C2×C4×Dic59C2311yescharacteristic
67C42×D59—22no—
68#14273663—22no—
69#14273662—24no—
70#14273661C2×C412yes—
71#14273660C2×C414yes—
72#14273659C2×C412yes—
73#14273658C2×C414yes—
74C4⋊(C59⋊C8)—24no—
75Dic59⋊C8—24no—
76Dic59⋊C8—24no—
77#14273637—22no—
78C4×C59⋊C8—22no—
79C8×Dic59—22no—
80C2×M16×C59C2312yes—
81C22×C8×C59C2311yescharacteristic
82C22×C8×C59C2311yescharacteristic
83#117×C59C2312yes—
84C2×C42×C59C2311yescharacteristic
85#108×C59—22no—
86#107×C59C2×C412yes—
87#106×C59C2×C412yes—
88C4⋊C8×C59—24no—
89#97×C59—22no—
90C23×D59#4111yescharacteristic
91C22×C4×C59—21no—
92C22×C4×C59C2411yescharacteristic
93C22×Dic59—21no—
94C22×Dic59#4111yescharacteristic
95C2×C4×D59—24no—
96C2×C4×D59C22×C412yes—
97C2×C4×D59C2×D412yes—
98C2×C4×D59C2×D412yes—
99C2×C4×D59C2×Q812yes—
100C2×C4×D59C2×Q812yes—
101C2×C4×D59#4112yes—
102M16×C59—22no—
103M16×C59—22no—
104C2×C8×C59—21no—
105C2×C8×C59—21no—
106C2×C8×C59—22no—
107C2×C8×C59C22×C411yescharacteristic
108C2×C8×C59C22×C411yescharacteristic
109C2×C8×C59C2×D412yes—
110C2×C8×C59C2×D412yes—
111C2×C8×C59C2×Q812yes—
112C4⋊C4×C59C22×C414yes—
113C22⋊C4×C59—22no—
114C42×C59—21no—
115C42×C59—22no—
116C42×C59C2×Q812yes—
117C22⋊Dic59—22no—
118(C2×D59)⋊C4—24no—
119(C2×D59)⋊C4—24no—
120C4⋊Dic59C22×C414yes—
121Dic59⋊C4—24no—
122Dic59⋊C4—24no—
123C4×Dic59—21no—
124C4×Dic59—22no—
125C4×Dic59—24no—
126C4×Dic59C2×Q812yes—
127#11745664—22no—
128#11745664—22no—
129C2×C59⋊C8—21no—
130C2×C59⋊C8—21no—
131C2×C59⋊C8—22no—
132C2×C59⋊C8C22×C412yes—
133C2×C59⋊C8C2×D412yes—
134C2×C59⋊C8C2×D412yes—
135C2×C59⋊C8C2×Q812yes—
136#11745659—44no—
137#11745659C4⋊C418yes—
138#11745659C4⋊C418yes—
139C8×D59—28no—
140C8×D59—44no—
141C23×C59C2×#4111yescharacteristic
142C22×D59—42no—
143C22×D59C4×D412yes—
144C22×D59C22⋊Q812yes—
145C22×D59#12512yes—
146C2×C4×C59—24no—
147C2×C4×C59—22no—
148C2×C4×C59—42no—
149C2×C4×C59C23×C411yescharacteristic
150C2×C4×C59C22×D411yescharacteristic
151C2×C4×C59C22×D411yescharacteristic
152C2×C4×C59C22×Q811yescharacteristic
153C2×C4×C59C22×Q811yescharacteristic
154C2×C4×C59C2×#4111yescharacteristic
155C2×Dic59—24no—
156C2×Dic59—22no—
157C2×Dic59—42no—
158C2×Dic59C4×D412yes—
159C2×Dic59C22⋊Q812yes—
160C2×Dic59#12512yes—
161C4×D59—48no—
162C4×D59C2×C4⋊C414yes—
163C4×D59C4×Q814yes—
164C4×D59#12114yes—
165C4×D59C4⋊Q814yes—
166C8×C59—22no—
167C8×C59—42no—
168C8×C59C2×C4⋊C412yes—
169C8×C59C2×C4⋊C412yes—
170C59⋊C8—22no—
171C59⋊C8—42no—
172C59⋊C8C2×C4⋊C414yes—
173#21164—591nomaximal, Sylow
174C22×C59—22no—
175C22×C59C2×C4×D411yescharacteristic
176C22×C59C2×C22⋊Q811yescharacteristic
177C22×C59C2×#12511yescharacteristic
178C2×D59—24no—
179C2×D59—42no—
180C2×D59—42no—
181C2×D59#38914yes—
182C4×C59—42no—
183C4×C59—44no—
184C4×C59C22×C4⋊C411yescharacteristic
185C4×C59C2×C4×Q811yescharacteristic
186C4×C59C2×#12111yescharacteristic
187C4×C59C2×C4⋊Q811yescharacteristic
188Dic59—42no—
189Dic59—44no—
190Dic59#38914yes—
191C2×#430—591no—
192#2911—591no—
193C2×#424—592no—
194C2×#405—591no—
195#1946—592no—
196#1946—591no—
197#1946D5911yescharacteristic, complemented
198#1945—591no—
199#1945—591no—
200#1945—592no—
201#1733—592no—
202C2×C59—21no—
203C2×C59—21no—
204C2×C59C2×#38911yescharacteristic
205C2×C59#295512yes—
206D59—24no—
207C22×M16—591no—
208C22×M16—592no—
209C23×C8—591no—
210C2×#117—591no—
211C2×#117—592no—
212#430—1184no—
213C2×#108—592no—
214C2×#108—591no—
215C2×#108C2×D5911yescharacteristic
216#428—594no—
217#428—594no—
218#427—591no—
219#427—592no—
220#427C2×D5911yescharacteristic
221#424—592no—
222#424—592no—
223#424—592no—
224#424—1184no—
225#424C2×D5912yes—
226C2×C4⋊C8—1184no—
227#405—1184no—
228C2×#97—591no—
229C2×#97C2×D5911yescharacteristic
230C2×C4×C8—591no—
231C2×C4×C8—591no—
232#339—592no—
233#339—592no—
234#339—592no—
235#339C2×D5912yes—
236C59#2116411yescharacteristic, complemented, Sylow
237C23×C4—591no—
238C2×M16—592no—
239C2×M16—592no—
240C2×M16—594no—
241C2×M16—594no—
242C2×M16—592no—
243C2×M16—1182no—
244C2×M16—1184no—
245C2×M16—1184no—
246C2×M16C22×D5912yes—
247C22×C8—591no—
248C22×C8—591no—
249C22×C8—594no—
250C22×C8—1182no—
251C22×C8—1182no—
252C22×C8—1182no—
253C22×C8C22×D5911yescharacteristic
254C22×C8C22×D5911yescharacteristic
255#117—594no—
256#117—592no—
257#117—594no—
258#117—1182no—
259#117—1182no—
260#117—1182no—
261#117C22×D5912yes—
262C2×C4⋊C4—594no—
263C2×C4⋊C4—1181no—
264C2×C22⋊C4—1182no—
265C2×C22⋊C4—1181no—
266C2×C42—591no—
267C2×C42—1182no—
268C2×C42C22×D5911yescharacteristic
269#108—22no—
270#108—1184no—
271#108—1182no—
272#107—592no—
273#107—594no—
274#107C4×D5912yes—
275#106—592no—
276#106—594no—
277#106C4×D5912yes—
278C4⋊C8—24no—
279C4⋊C8—1184no—
280C4⋊C8—1184no—
281C4⋊C8—1184no—
282#97—22no—
283#97—1182no—
284C4×C8—1182no—
285C4×C8—1182no—
286C24—591no—
287C22×C4—21no—
288C22×C4—591no—
289C22×C4—592no—
290C22×C4—592no—
291C22×C4—592no—
292C22×C4—592no—
293C22×C4—592no—
294C22×C4—592no—
295C22×C4—1184no—
296C22×C4—1181no—
297C22×C4C23×D5911yescharacteristic
298M16—22no—
299M16—22no—
300M16—598no—
301M16—598no—
302M16—1182no—
303M16—1182no—
304M16—2364no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—592no—
309C2×C8—592no—
310C2×C8—592no—
311C2×C8—592no—
312C2×C8—1188no—
313C2×C8—1181no—
314C2×C8—1181no—
315C2×C8—1182no—
316C2×C8—2364no—
317C2×C8C2×C4×D5911yescharacteristic
318C2×C8C2×C4×D5911yescharacteristic
319C2×C8D4×D5912yes—
320C2×C8D4×D5912yes—
321C2×C8Q8×D5912yes—
322C4⋊C4—594no—
323C4⋊C4—1184no—
324C4⋊C4—1184no—
325C4⋊C4C2×C4×D5914yes—
326C22⋊C4—22no—
327C22⋊C4—1184no—
328C22⋊C4—1184no—
329C22⋊C4—1182no—
330C42—21no—
331C42—22no—
332C42—592no—
333C42—1184no—
334C42—1181no—
335C42—1182no—
336C42Q8×D5912yes—
337C23—592no—
338C23—592no—
339C23—592no—
340C23—2362no—
341C23#41×D5911yescharacteristic
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—42no—
345C2×C4—592no—
346C2×C4—592no—
347C2×C4—592no—
348C2×C4—594no—
349C2×C4—594no—
350C2×C4—594no—
351C2×C4—594no—
352C2×C4—1182no—
353C2×C4—1184no—
354C2×C4—2368no—
355C2×C4—2362no—
356C2×C4C22×C4×D5911yescharacteristic
357C2×C4C2×D4×D5911yescharacteristic
358C2×C4C2×D4×D5911yescharacteristic
359C2×C4C2×Q8×D5911yescharacteristic
360C2×C4C2×Q8×D5911yescharacteristic
361C2×C4#41×D5911yescharacteristic
362C8—22no—
363C8—42no—
364C8—594no—
365C8—1182no—
366C8—2362no—
367C8C4⋊C4×D5912yes—
368C8C4⋊C4×D5912yes—
369C22—22no—
370C22—594no—
371C22—1184no—
372C22—2362no—
373C22—2362no—
374C22C4×D4×D5911yescharacteristic
375C22C22⋊Q8×D5911yescharacteristic
376C22#125×D5911yescharacteristic
377C4—42no—
378C4—44no—
379C4—594no—
380C4—2362no—
381C4—2364no—
382C4C2×C4⋊C4×D5911yescharacteristic
383C4C4×Q8×D5911yescharacteristic
384C4#121×D5911yescharacteristic
385C4C4⋊Q8×D5911yescharacteristic
386C2—21no—
387C2—21no—
388C2—1184no—
389C2#389×D5911yescharacteristic
390C2#2143299612yes—
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1946C2
socleC22×C59C2×C4×D4
prime radicalC23×D59#41
cosocleC22×C4×C59C24
hypercenter#1946D59
hypocenterC59#21164
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1946

Complements

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

NH ≅ G/Ntype
5. #21424849C2semidirect
6. #21424846C2semidirect
7. #21424845C2semidirect
8. #21424843C2semidirect
9. #21424842C2semidirect
11. C59×#1946C2semidirect
20. #16837443C4semidirect
21. #16837441C4semidirect
197. #1946D59semidirect
236. C59#21164semidirect

Maximal subgroups

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

subgroupindex
C2×#168374492
#214326842
#424×D592
C2×#168373592
#214248492
#214248462
#214248452
#214248432
#214248422
#214238042
C59×#19462
#2116459