Finite Group Databasebeta

← #29469873

Subgroups — #29469873

Order 15104.

1838 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437980C212yesmaximal, complemented
2#21437132C218yesmaximal, complemented
3#21437100C214yesmaximal, complemented
4#21437086C214yesmaximal, complemented
5#21434585C212yesmaximal
6#21433623C214yesmaximal, complemented
7#21433579C212yesmaximal, complemented
8#21433030C211yesmaximal, characteristic, complemented
9#21432964C212yesmaximal, complemented
10C4×#14273749C211yesmaximal, characteristic
11C2×C59×#494C211yesmaximal, characteristic, complemented
12C2×#14273758C2214yes—
13C2×#14273749C2211yescharacteristic
14#16838127C2218yes—
15#16838126C2218yes—
16#16838087—28no—
17#16838013—24no—
18#16837976C2214yes—
19#16837959C2211yescharacteristic
20#16837957—24no—
21#16837880C2214yes—
22C4×#11745685C2212yes—
23#16837795C2218yes—
24#16837787C2214yes—
25#16837780C2214yes—
26#16837626C22116yes—
27#16837610C22116yes—
28#16837609C2218yes—
29#16837604C2218yes—
30#16837532C2214yes—
31C8×Dic118C2214yes—
32#16837509C2218yes—
33C4×Dic236C2212yes—
34Q16×Dic59C2218yes—
35C4×#11745660C2214yes—
36C4×D472C2212yes—
37#16837440C2211yescharacteristic
38#16837428C2212yes—
39#16837426C2211yescharacteristic
40#16837411C2214yes—
41#16837399C2212yes—
42#16837357C2212yes—
43C22×Q16×C59C2212yes—
44C22×Q16×C59C2212yes—
45C2×C4⋊Q8×C59C2212yes—
46C59×#494C2214yes—
47C59×#494C2214yes—
48C2×C4×C8×C59C2211yescharacteristic
49C2×#11745685C2312yes—
50#14273758—48no—
51#14273749—24no—
52C2×Dic236C2311yescharacteristic
53C2×#11745672C2318yes—
54C2×#11745660C2312yes—
55C2×#11745671C2318yes—
56C2×D472C2311yescharacteristic
57C2×#11745664C2312yes—
58#14273727D412yes—
59#14273725C2318yes—
60#14273722—216no—
61#14273719D412yes—
62#14273718—28no—
63#14273717—216no—
64#14273716C2318yes—
65#14273710—28no—
66#14273707—28no—
67#14273706D414yes—
68#14273703—28no—
69Q8×Dic59—28no—
70Q8×Dic59C23116yes—
71C4×Dic118C2314yes—
72C4×Dic118C2314yes—
73#14273692—28no—
74C4×D236C2314yes—
75C4×C22⋊D59—24no—
76#14273686—28no—
77#14273686C2312yes—
78#14273686C2312yes—
79#14273685—28no—
80#14273684—22no—
81C42×D59—22no—
82#14273661C2312yes—
83#14273659C2312yes—
84C4⋊(C59⋊C8)C2314yes—
85Dic59⋊C8C2318yes—
86#14273652C2314yes—
87#14273651C23116yes—
88#14273649C2314yes—
89#14273637C2314yes—
90C8×Dic59C2314yes—
91C22×Q8×C59C2314yes—
92C2×Q16×C59—28no—
93C2×Q16×C59—44no—
94C2×Q16×C59C2314yes—
95C2×Q16×C59C2314yes—
96C2×Q16×C59C2318yes—
97C22×C8×C59C2311yescharacteristic
98C22×C8×C59C2311yescharacteristic
99C4⋊Q8×C59—22no—
100C4⋊Q8×C59—22no—
101C4⋊Q8×C59C2318yes—
102C2×C4⋊C4×C59—24no—
103C2×C42×C59C2311yescharacteristic
104C4×C8×C59C2312yes—
105C4×C8×C59C2312yes—
106C2×Q8×C59—24no—
107C2×Q8×C59—28no—
108C2×Q8×C59—28no—
109C2×Q8×C59C2418yes—
110C22×C4×C59—24no—
111C22×C4×C59C2×D412yes—
112C22×C4×C59C2411yescharacteristic
113C2×C22⋊D59—22no—
114#11745685—24no—
115#11745685—24no—
116C2×D236C2412yes—
117C2×C4×D59—22no—
118C2×C4×D59—24no—
119C2×Dic118C2×D414yes—
120C2×Dic118C2412yes—
121Q16×C59—28no—
122Q16×C59—28no—
123Q16×C59—48no—
124C2×C8×C59—22no—
125C2×C8×C59#4114yes—
126C2×C8×C59C2412yes—
127C2×C8×C59C2411yescharacteristic
128C2×C8×C59C2411yescharacteristic
129C4⋊C4×C59—28no—
130C4⋊C4×C59—28no—
131C42×C59C2×D411yescharacteristic
132C42×C59C2×D411yescharacteristic
133C42×C59C2412yes—
134C22⋊Dic59—22no—
135C22⋊Dic59C2×D414yes—
136#11745672—48no—
137#11745671—48no—
138(C2×D59)⋊C4—24no—
139(C2×D59)⋊C4—216no—
140C4⋊Dic59—216no—
141C4⋊Dic59C2×D414yes—
142C4⋊Dic59C2×D414yes—
143C4⋊Dic59C2414yes—
144Dic59⋊C4—24no—
145Dic59⋊C4—216no—
146Dic59⋊C4C2×D418yes—
147C4×Dic59—22no—
148C4×Dic59—24no—
149C4×Dic59—216no—
150C4×Dic59C2414yes—
151#11745664—24no—
152C2×C59⋊C8C2414yes—
153Dic236—22no—
154D472—22no—
155#11745660—24no—
156C23×C59C22×D411yescharacteristic
157C22×D59—22no—
158Q8×C59—28no—
159Q8×C59—28no—
160Q8×C59—48no—
161C2×C4×C59—24no—
162C2×C4×C59—28no—
163C2×C4×C59—48no—
164C2×C4×C59C22×D412yes—
165C2×C4×C59C22×D411yescharacteristic
166C2×C4×C59C22×D412yes—
167C2×C4×C59C2×#4112yes—
168C2×C4×C59H(5,𝔽2)12yes—
169C2×C4×C59C2511yescharacteristic
170C22⋊D59—44no—
171C2×Dic59—24no—
172C2×Dic59—22no—
173C2×Dic59C22×D418yes—
174D236—22no—
175D236—22no—
176C4×D59—44no—
177C4×D59—48no—
178Dic118—28no—
179Dic118—22no—
180Dic118—22no—
181C8×C59—22no—
182C8×C59C2×#4112yes—
183C8×C59C2×#4112yes—
184C59⋊C8—24no—
185#61411—591nomaximal, Sylow
186C22×C59—22no—
187C22×C59#55012yes—
188C22×C59C23×D411yescharacteristic
189C2×D59—42no—
190C2×D59—44no—
191C4×C59—22no—
192C4×C59—48no—
193C4×C59#55012yes—
194C4×C59#57814yes—
195C4×C59C22×#4111yescharacteristic
196C4×C59C2×H(5,𝔽2)11yescharacteristic
197Dic59—28no—
198Dic59—42no—
199Dic59—44no—
200#3509—592no—
201#3398—598no—
202#3395—594no—
203#3393—594no—
204#3156—592no—
205C2×#494D5911yescharacteristic, complemented
206#3044—594no—
207#3039—592no—
208#2960—591no—
209#2950—592no—
210C4×#135—591no—
211C2×C59—22no—
212C2×C59C2×#55011yescharacteristic
213C2×C59C2×#57812yes—
214D59—82no—
215C2×#137—594no—
216C2×#135—591no—
217C22×Q16C2×D5912yes—
218C22×Q16C2×D5912yes—
219#557—598no—
220#556—598no—
221#552—1188no—
222#541—1184no—
223#536—594no—
224#533—591no—
225#533—1184no—
226C2×C4⋊Q8C2×D5912yes—
227#519—594no—
228C4×#41—592no—
229#497—598no—
230#495—594no—
231#494—594no—
232#494C2×D5914yes—
233#494C2×D5914yes—
234#464—5916no—
235#462—5916no—
236#462—598no—
237#461—598no—
238#446—594no—
239C8×Q8—594no—
240#441—598no—
241C4×Q16—598no—
242C4×Q16—592no—
243C4×SD16—594no—
244C4×D8—592no—
245#427—591no—
246#424—592no—
247#423—591no—
248#419—594no—
249#416—592no—
250#405—592no—
251C2×C4×C8C2×D5911yescharacteristic
252C59#6141111yescharacteristic, complemented, Sylow
253C2×#41—592no—
254C22×Q8C22×D5914yes—
255#137—2368no—
256#135—1184no—
257C2×Q16—28no—
258C2×Q16—44no—
259C2×Q16—598no—
260C2×Q16—591no—
261C2×Q16C22×D5914yes—
262C2×Q16C22×D5914yes—
263C2×Q16C22×D5918yes—
264C2×SD16—598no—
265C2×SD16—592no—
266C2×D8—591no—
267C2×M16—592no—
268C22×C8C22×D5911yescharacteristic
269C22×C8C22×D5911yescharacteristic
270C4⋊Q8—22no—
271C4⋊Q8—22no—
272C4⋊Q8—598no—
273C4⋊Q8—592no—
274C4⋊Q8C22×D5918yes—
275#126—11816no—
276#125—592no—
277#125—1188no—
278#125—11816no—
279#124—598no—
280#123—1188no—
281C22⋊Q8—594no—
282C22⋊Q8—1188no—
283C22⋊Q8—1188no—
284C4×Q8—594no—
285C4×Q8—5916no—
286C4×Q8—594no—
287C4×Q8—1188no—
288C4×Q8—1188no—
289C4×D4—594no—
290C4×D4—1184no—
291#117—592no—
292#117—592no—
293#117—1188no—
294#117—1188no—
295#117—1182no—
296C2×C4⋊C4—24no—
297C2×C42—1182no—
298C2×C42C22×D5911yescharacteristic
299#107—592no—
300#106—592no—
301C4⋊C8—598no—
302C4⋊C8—594no—
303#103—5916no—
304#103—594no—
305#102—594no—
306#97—594no—
307C4×C8—594no—
308C4×C8C22×D5912yes—
309C4×C8C22×D5912yes—
310#41—1184no—
311#41—1184no—
312C2×Q8—24no—
313C2×Q8—28no—
314C2×Q8—28no—
315C2×Q8—594no—
316C2×Q8—592no—
317C2×Q8C23×D5918yes—
318C2×D4—592no—
319C2×D4—1182no—
320C22×C4—24no—
321C22×C4—1184no—
322C22×C4—1182no—
323C22×C4D4×D5912yes—
324C22×C4C23×D5911yescharacteristic
325Q16—28no—
326Q16—28no—
327Q16—48no—
328Q16—1182no—
329Q16—2368no—
330SD16—1184no—
331SD16—2368no—
332D8—1182no—
333M16—1184no—
334C2×C8—22no—
335C2×C8—594no—
336C2×C8#1174568714yes—
337C2×C8C23×D5912yes—
338C2×C8C23×D5911yescharacteristic
339C2×C8C23×D5911yescharacteristic
340C4⋊C4—28no—
341C4⋊C4—28no—
342C4⋊C4—594no—
343C4⋊C4—594no—
344C4⋊C4—598no—
345C4⋊C4—594no—
346C4⋊C4—11816no—
347C4⋊C4—11816no—
348C4⋊C4—1184no—
349C22⋊C4—594no—
350C22⋊C4—11816no—
351C22⋊C4—1182no—
352C22⋊C4—1184no—
353C42—594no—
354C42—11816no—
355C42—1182no—
356C42—1184no—
357C42D4×D5911yescharacteristic
358C42D4×D5911yescharacteristic
359C42C23×D5912yes—
360C23—1182no—
361C23C2×D4×D5911yescharacteristic
362Q8—28no—
363Q8—28no—
364Q8—48no—
365Q8—1188no—
366Q8—1182no—
367Q8—1182no—
368D4—1182no—
369D4—1182no—
370D4—2364no—
371C2×C4—24no—
372C2×C4—28no—
373C2×C4—48no—
374C2×C4—598no—
375C2×C4—1184no—
376C2×C4—1182no—
377C2×C4—2368no—
378C2×C4—2364no—
379C2×C4C2×D4×D5912yes—
380C2×C4C2×D4×D5912yes—
381C2×C4C2×D4×D5911yescharacteristic
382C2×C4C2×#1174568712yes—
383C2×C4#1427377212yes—
384C2×C4C24×D5911yescharacteristic
385C8—22no—
386C8—1184no—
387C8C2×#1174568712yes—
388C8C2×#1174568712yes—
389C22—22no—
390C22—2362no—
391C22—2364no—
392C22#1683807212yes—
393C22C22×D4×D5911yescharacteristic
394C4—22no—
395C4—48no—
396C4—1188no—
397C4—2362no—
398C4—2364no—
399C4#1683807212yes—
400C4#1683821514yes—
401C4C22×#1174568711yescharacteristic
402C4C2×#1427377211yescharacteristic
403C2—22no—
404C2—4722no—
405C2C2×#1683807211yescharacteristic
406C2C2×#1683821512yes—
subgroupG/H
centerC22C22×D4×D59
commutator, cosocleC2×C4×C59C25
FrattiniC2×C4C24×D59
FittingC2×C59×#494C2
socleC22×C59C23×D4
prime radicalC2×#14273749C22
hypercenterC2×#494D59
hypocenterC59#61411
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
1. #21437980C2semidirect
2. #21437132C2semidirect
3. #21437100C2semidirect
4. #21437086C2semidirect
6. #21433623C2semidirect
7. #21433579C2semidirect
8. #21433030C2semidirect
9. #21432964C2semidirect
11. C2×C59×#494C2semidirect
205. C2×#494D59semidirect
252. C59#61411semidirect

Maximal subgroups

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

subgroupindex
#214379802
#214371322
#214371002
#214370862
#214345852
#214336232
#214335792
#214330302
#214329642
C4×#142737492
C2×C59×#4942
#6141159