Finite Group Databasebeta

← #28505703

Subgroups — #28505703

Order 15104.

492 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433816C211yesmaximal, characteristic
2#21425831C211yesmaximal, characteristic, complemented
3C2×#16837211C211yesmaximal, characteristic, complemented
4#21425724C211yesmaximal, characteristic, complemented
5#21422212C211yesmaximal, characteristic, complemented
6#21422194C211yesmaximal, characteristic, complemented
7C59×#2912C211yesmaximal, characteristic, complemented
8#16838200—21no—
9C2×#14273749C2211yescharacteristic
10C2×#14273667C2211yescharacteristic
11C2×#14273667C2211yescharacteristic
12#16837677C412yescomplemented
13#16837672C412yescomplemented
14#16837665—22no—
15C2×#14273663C2211yescharacteristic
16#16837384—21no—
17#16837211—22no—
18#16837211—21no—
19#16837211—21no—
20#16837186—22no—
21C2×#131×C59C2211yescharacteristic
22C59×#520—21no—
23C59×#428—21no—
24C59×#428—22no—
25C59×#427—21no—
26C59×#427C2211yescharacteristic
27C2×#106×C59—21no—
28C2×#106×C59C2211yescharacteristic
29C2×#11745685—21no—
30#14273758—41no—
31#14273757—22no—
32#14273753—41no—
33#14273752—22no—
34#14273749—21no—
35#14273749—21no—
36#14273749—21no—
37#14273749—21no—
38#14273747—42no—
39C2×Dic236C2×C411yescharacteristic, complemented
40C2×#11745660—21no—
41C2×D472C2×C411yescharacteristic, complemented
42C2×#11745664—21no—
43#14273667—21no—
44#14273667—21no—
45#14273667—21no—
46#14273667—21no—
47C2×C59⋊C16—21no—
48C2×C59⋊C16—21no—
49#14273663—21no—
50#14273663C2×C412yescomplemented
51#14273645—22no—
52#14273643—22no—
53C2×#41×C59D411yescharacteristic
54#131×C59—21no—
55#131×C59—21no—
56#131×C59C2×C412yes—
57#131×C59D412yes—
58C2×M16×C59—21no—
59C2×M16×C59D411yescharacteristic
60C22×C8×C59—21no—
61C22×C8×C59C2311yescharacteristic
62C4×Q8×C59—41no—
63C4×D4×C59—42no—
64C4×D4×C59—41no—
65#117×C59—22no—
66#117×C59—21no—
67C2×C4⋊C4×C59—22no—
68C2×C4⋊C4×C59—21no—
69#107×C59—22no—
70#107×C59—21no—
71#106×C59—22no—
72#106×C59—21no—
73#106×C59D411yescharacteristic
74#106×C59D412yes—
75#106×C59D411yescharacteristic
76#41×C59—21no—
77#41×C59—21no—
78#41×C59C22⋊C412yes—
79#41×C59SD1612yes—
80C2×Q8×C59D811yescharacteristic
81C2×D4×C59—21no—
82C2×D4×C59D811yescharacteristic
83C22×C4×C59—21no—
84C22×C4×C59—42no—
85C22×C4×C59—41no—
86C22×C4×C59C2×D411yescharacteristic
87C2×C22⋊D59—41no—
88#11745685—41no—
89#11745685—41no—
90#11745685—41no—
91#11745685—41no—
92C2×D236—21no—
93C2×C4×D59—41no—
94C2×Dic118—21no—
95M16×C59—21no—
96M16×C59—22no—
97M16×C59—22no—
98M16×C59C22⋊C412yes—
99C2×C8×C59—21no—
100C2×C8×C59—22no—
101C2×C8×C59—22no—
102C2×C8×C59C22×C411yescharacteristic
103C2×C8×C59C2×D411yescharacteristic
104C2×C8×C59C2×D411yescharacteristic
105C2×C8×C59C2×D411yescharacteristic
106C2×C8×C59#4111yescharacteristic
107C2×C8×C59#4111yescharacteristic
108C4⋊C4×C59—21no—
109C4⋊C4×C59—22no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—42no—
112C4⋊C4×C59—41no—
113C22⋊C4×C59—42no—
114C22⋊C4×C59—41no—
115C42×C59—42no—
116C42×C59—41no—
117#11745672—42no—
118#11745671—42no—
119#11745670—42no—
120#11745669—42no—
121#11745664—22no—
122#11745664—41no—
123C2×C59⋊C8—41no—
124Dic236—21no—
125Dic236—21no—
126D472—21no—
127D472—21no—
128#11745660—41no—
129#11745660—41no—
130C59⋊C16—41no—
131C59⋊C16—41no—
132C23×C59—21no—
133C23×C59AGL(1,ℤ/8)11yescharacteristic
134C22×D59—41no—
135Q8×C59—21no—
136Q8×C59#10212yes—
137D4×C59—22no—
138D4×C59—22no—
139D4×C59—21no—
140D4×C59#10212yes—
141C2×C4×C59—21no—
142C2×C4×C59—22no—
143C2×C4×C59—22no—
144C2×C4×C59—41no—
145C2×C4×C59—42no—
146C2×C4×C59—41no—
147C2×C4×C59—161no—
148C2×C4×C59—162no—
149C2×C4×C59C2×C22⋊C411yescharacteristic
150C2×C4×C59#12011yescharacteristic
151C2×C4×C59#12311yescharacteristic
152C2×C4×C59C2×D811yescharacteristic
153C2×C4×C59C2×SD1611yescharacteristic
154C2×C4×C59#13711yescharacteristic
155C22⋊D59—81no—
156C22⋊D59—81no—
157C2×Dic59—41no—
158D236—41no—
159D236—41no—
160C4×D59—81no—
161C4×D59—81no—
162Dic118—41no—
163Dic118—41no—
164C8×C59—22no—
165C8×C59C4×D411yescharacteristic
166C8×C59C4×D411yescharacteristic
167C8×C59#12111yescharacteristic
168C8×C59#12111yescharacteristic
169C59⋊C8—42no—
170#13005—591nomaximal, Sylow
171C22×C59—22no—
172C22×C59—41no—
173C22×C59—42no—
174C22×C59#41711yescharacteristic
175C22×C59#44711yescharacteristic
176C22×C59#48111yescharacteristic
177C2×D59—81no—
178C2×D59—161no—
179C4×C59—22no—
180C4×C59—81no—
181C4×C59—82no—
182C4×C59—81no—
183C4×C59#38611yescharacteristic
184C4×C59C2×#10211yescharacteristic
185C4×C59#44811yescharacteristic
186C4×C59#48011yescharacteristic
187Dic59—81no—
188#3067—591no—
189#2912D5911yescharacteristic, complemented
190#2145—591no—
191C2×#360—591no—
192#2117—591no—
193#1389—591no—
194#1383—591no—
195C2×C59—21no—
196C2×C59—21no—
197C2×C59—21no—
198C2×C59—42no—
199C2×C59#187411yescharacteristic
200D59—161no—
201#575—1181no—
202C2×#135—591no—
203C2×#131C2×D5911yescharacteristic
204#520—21no—
205C2×M32—591no—
206C2×M32—591no—
207#473—592no—
208#472—592no—
209#471—1182no—
210C2×#108—591no—
211#428—21no—
212#428—22no—
213#427—21no—
214#427C2×D5911yescharacteristic
215C2×#106—21no—
216C2×#106C2×D5911yescharacteristic
217#413—1181no—
218#360—1182no—
219#360—1181no—
220#360—1181no—
221#350—1182no—
222C59#1300511yescharacteristic, complemented, Sylow
223C2×#41—1181no—
224C2×#41C22⋊D5911yescharacteristic
225#137—1182no—
226#137—2361no—
227AGL(1,ℤ/8)—1182no—
228AGL(1,ℤ/8)—2361no—
229#135—1181no—
230#135—1181no—
231#135—1181no—
232#135—1181no—
233#135—2362no—
234C2×Q16—591no—
235C2×SD16—1181no—
236C2×D8—591no—
237#131—21no—
238#131—21no—
239#131C4×D5912yes—
240#131D23612yes—
241C2×M16—21no—
242C2×M16—1181no—
243C2×M16C22⋊D5911yescharacteristic
244C22×C8—21no—
245C22×C8C22×D5911yescharacteristic
246C4×Q8—41no—
247C4×D4—42no—
248C4×D4—41no—
249#117—22no—
250#117—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253M32—1181no—
254M32—1181no—
255M32—1181no—
256M32—1181no—
257C2×C16—1181no—
258C2×C16—1181no—
259#108—592no—
260#108—1181no—
261#107—22no—
262#107—21no—
263#106—22no—
264#106—21no—
265#106C22⋊D5911yescharacteristic
266#106C22⋊D5912yes—
267#106C22⋊D5911yescharacteristic
268#101—1182no—
269#100—1182no—
270#41—21no—
271#41—21no—
272#41—2361no—
273#41—2361no—
274#41—2361no—
275#41—2361no—
276#41(C2×D59)⋊C412yes—
277#41#1174567112yes—
278C2×Q8—1181no—
279C2×Q8#1174566911yescharacteristic
280C2×D4—21no—
281C2×D4—1181no—
282C2×D4—2361no—
283C2×D4#1174566911yescharacteristic
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—2361no—
288C22×C4C2×C22⋊D5911yescharacteristic
289Q16—1181no—
290Q16—1181no—
291Q16—2362no—
292SD16—2362no—
293SD16—2362no—
294SD16—2361no—
295SD16—2361no—
296D8—1181no—
297D8—1181no—
298D8—2362no—
299M16—21no—
300M16—22no—
301M16—22no—
302M16—1182no—
303M16—2361no—
304M16(C2×D59)⋊C412yes—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—2361no—
309C2×C8C2×C4×D5911yescharacteristic
310C2×C8C2×D23611yescharacteristic
311C2×C8#1174568511yescharacteristic
312C2×C8#1174568511yescharacteristic
313C2×C8C2×C22⋊D5911yescharacteristic
314C2×C8C2×C22⋊D5911yescharacteristic
315C4⋊C4—21no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C22⋊C4—42no—
321C22⋊C4—41no—
322C42—42no—
323C42—41no—
324C16—2361no—
325C16—2361no—
326C23—21no—
327C23—2361no—
328C23#1427375311yescharacteristic
329Q8—21no—
330Q8—2361no—
331Q8—2361no—
332Q8#1427364712yes—
333D4—22no—
334D4—22no—
335D4—21no—
336D4—4721no—
337D4—4721no—
338D4—2361no—
339D4—2361no—
340D4#1427364712yes—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—161no—
348C2×C4—162no—
349C2×C4—4721no—
350C2×C4—4721no—
351C2×C4—2361no—
352C2×C4C2×(C2×D59)⋊C411yescharacteristic
353C2×C4#1427369611yescharacteristic
354C2×C4#1427371011yescharacteristic
355C2×C4C2×#1174566911yescharacteristic
356C2×C4C2×#1174567111yescharacteristic
357C2×C4#1427375811yescharacteristic
358C8—22no—
359C8—2362no—
360C8C4×C22⋊D5911yescharacteristic
361C8C4×C22⋊D5911yescharacteristic
362C8#1427370111yescharacteristic
363C8#1427370111yescharacteristic
364C22—22no—
365C22—41no—
366C22—42no—
367C22—9441no—
368C22—4721no—
369C22#1683740111yescharacteristic
370C22#1683753611yescharacteristic
371C22#1683772011yescharacteristic
372C4—22no—
373C4—82no—
374C4—81no—
375C4—81no—
376C4—4721no—
377C4#1683728111yescharacteristic
378C4C2×#1427364711yescharacteristic
379C4#1683754111yescharacteristic
380C4#1683771511yescharacteristic
381C2—21no—
382C2—21no—
383C2—21no—
384C2—42no—
385C2—9441no—
386C2#2142441311yescharacteristic
subgroupG/H
centerC2#21424413
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#2912C2
socleC2×C59#1874
prime radical#21433816C2
cosocleC22×C8×C59C23
hypercenter#2912D59
hypocenterC59#13005
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #2912

Complements

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

NH ≅ G/Ntype
2. #21425831C2semidirect
3. C2×#16837211C2semidirect
4. #21425724C2semidirect
5. #21422212C2semidirect
6. #21422194C2semidirect
7. C59×#2912C2semidirect
12. #16837677C4semidirect
13. #16837672C4semidirect
39. C2×Dic236C2×C4semidirect
41. C2×D472C2×C4semidirect
50. #14273663C2×C4semidirect
189. #2912D59semidirect
222. C59#13005semidirect

Maximal subgroups

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

subgroupindex
#214338162
#214258312
C2×#168372112
#214257242
#214222122
#214221942
C59×#29122
#1300559