Finite Group Databasebeta

← #28505872

Subgroups — #28505872

Order 15104.

476 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437230C211yesmaximal, characteristic
2#21425832C211yesmaximal, characteristic, complemented
3C2×#16837211C211yesmaximal, characteristic, complemented
4#21424994C211yesmaximal, characteristic, complemented
5#21422203C211yesmaximal, characteristic, complemented
6#21422182C211yesmaximal, characteristic, complemented
7C59×#2952C211yesmaximal, characteristic, complemented
8C2×#14273749C2211yescharacteristic
9#16837843—21no—
10#16837842C412yescomplemented
11#16837838—21no—
12#16837837C412yescomplemented
13#16837832—22no—
14C2×#14273667C2211yescharacteristic
15C2×#14273667C2211yescharacteristic
16#16837440C2211yescharacteristic
17#16837418—21no—
18#16837375—21no—
19#16837212—22no—
20#16837211—22no—
21C2×#135×C59C2211yescharacteristic
22C59×#520—21no—
23C59×#442—21no—
24C59×#441—21no—
25C4×SD16×C59—22no—
26C2×#106×C59C2211yescharacteristic
27C59×#417—21no—
28C59×#405C2211yescharacteristic
29C2×#11745685—21no—
30#14273749—21no—
31#14273749—21no—
32#14273749—21no—
33#14273749—21no—
34C2×Dic236C2×C411yescharacteristic, complemented
35C2×#11745660—21no—
36C2×D472C2×C411yescharacteristic, complemented
37#14273686—21no—
38#14273673—22no—
39#14273671—22no—
40#14273670—22no—
41#14273668—22no—
42#14273667—21no—
43#14273667—21no—
44#14273667—21no—
45#14273667C2×C412yescomplemented
46C2×C59⋊C16—21no—
47C2×C59⋊C16—21no—
48#14273661—21no—
49#14273659—21no—
50#14273653—42no—
51#14273641—42no—
52C2×#41×C59—21no—
53#135×C59—21no—
54#135×C59—21no—
55#135×C59C2×C412yes—
56#135×C59D412yes—
57C2×Q16×C59D411yescharacteristic
58C2×SD16×C59—21no—
59C2×D8×C59D411yescharacteristic
60C2×M16×C59D411yescharacteristic
61C22×C8×C59C2311yescharacteristic
62C4×Q8×C59—22no—
63C4×D4×C59—22no—
64C4×D4×C59—42no—
65#117×C59—41no—
66#117×C59D411yescharacteristic
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59—41no—
69#106×C59—21no—
70#106×C59D412yes—
71#103×C59—22no—
72#102×C59—22no—
73#97×C59—21no—
74C4×C8×C59—21no—
75#41×C59—22no—
76#41×C59—22no—
77#41×C59—41no—
78#41×C59—41no—
79C2×Q8×C59—21no—
80C2×D4×C59—21no—
81C2×D4×C59—41no—
82C22×C4×C59—42no—
83C22×C4×C59—41no—
84C22×C4×C59—41no—
85C22×C4×C59C2×D411yescharacteristic
86C2×C22⋊D59—41no—
87#11745685—41no—
88#11745685—41no—
89#11745685—41no—
90#11745685—41no—
91C2×D236—21no—
92C2×C4×D59—41no—
93C2×Dic118—21no—
94Q16×C59—21no—
95Q16×C59C22⋊C412yes—
96SD16×C59—22no—
97SD16×C59—22no—
98D8×C59—21no—
99D8×C59C22⋊C412yes—
100M16×C59—21no—
101M16×C59—21no—
102C2×C8×C59—21no—
103C2×C8×C59C22×C411yescharacteristic
104C2×C8×C59C2×D411yescharacteristic
105C2×C8×C59C2×D411yescharacteristic
106C2×C8×C59C2×D411yescharacteristic
107C2×C8×C59#4111yescharacteristic
108C2×C8×C59#4111yescharacteristic
109C4⋊C4×C59—21no—
110C4⋊C4×C59—22no—
111C4⋊C4×C59—42no—
112C4⋊C4×C59—41no—
113C22⋊C4×C59—42no—
114C22⋊C4×C59D812yes—
115C42×C59—21no—
116C42×C59—42no—
117C22⋊Dic59—41no—
118C4⋊Dic59—41no—
119C4×Dic59—41no—
120Dic236—21no—
121Dic236—21no—
122D472—21no—
123D472—21no—
124#11745660—41no—
125#11745660—41no—
126C59⋊C16—22no—
127C59⋊C16—41no—
128C23×C59—41no—
129C23×C59C2×D811yescharacteristic
130C22×D59—41no—
131Q8×C59—22no—
132Q8×C59—41no—
133D4×C59—22no—
134D4×C59—42no—
135D4×C59—42no—
136D4×C59—41no—
137C2×C4×C59—22no—
138C2×C4×C59—41no—
139C2×C4×C59—42no—
140C2×C4×C59—42no—
141C2×C4×C59—42no—
142C2×C4×C59—162no—
143C2×C4×C59—161no—
144C2×C4×C59C4×D411yescharacteristic
145C2×C4×C59#12011yescharacteristic
146C2×C4×C59#12111yescharacteristic
147C2×C4×C59#13511yescharacteristic
148C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
150C22⋊D59—81no—
151C22⋊D59—81no—
152C2×Dic59—41no—
153C2×Dic59—81no—
154D236—41no—
155D236—41no—
156C4×D59—81no—
157C4×D59—81no—
158Dic118—41no—
159Dic118—41no—
160C8×C59—41no—
161C8×C59C2×C22⋊C411yescharacteristic
162C8×C59C4×D411yescharacteristic
163C8×C59#12111yescharacteristic
164C8×C59#12311yescharacteristic
165#13030—591nomaximal, Sylow
166C22×C59—41no—
167C22×C59—42no—
168C22×C59—81no—
169C22×C59—81no—
170C22×C59C4×D811yescharacteristic
171C22×C59#44711yescharacteristic
172C22×C59#45911yescharacteristic
173C2×D59—81no—
174C2×D59—161no—
175C4×C59—42no—
176C4×C59—42no—
177C4×C59—82no—
178C4×C59#38611yescharacteristic
179C4×C59#44211yescharacteristic
180C4×C59#44911yescharacteristic
181C4×C59#46311yescharacteristic
182Dic59—81no—
183Dic59—161no—
184#3414—591no—
185#2952D5911yescharacteristic, complemented
186#2145—591no—
187C2×#360—591no—
188#1976—591no—
189#1386—591no—
190#1378—591no—
191C2×C59—21no—
192C2×C59—21no—
193C2×C59—21no—
194C2×C59—82no—
195C2×C59#187911yescharacteristic
196D59—161no—
197C2×#135—591no—
198C2×#135C2×D5911yescharacteristic
199#520—21no—
200#510—592no—
201#510—1181no—
202#509—592no—
203#509—1181no—
204#508—1182no—
205C2×M32—591no—
206C2×M32—591no—
207#442—21no—
208#441—21no—
209C4×SD16—22no—
210#427—591no—
211C2×#106C2×D5911yescharacteristic
212#421—1181no—
213#417—21no—
214#410—1181no—
215#405C2×D5911yescharacteristic
216#360—1182no—
217#360—1182no—
218C59#1303011yescharacteristic, complemented, Sylow
219C2×#41—21no—
220C2×#41—1181no—
221#135—21no—
222#135—21no—
223#135—1181no—
224#135—1181no—
225#135—1181no—
226#135—1181no—
227#135C4×D5912yes—
228#135D23612yes—
229C2×Q16—591no—
230C2×Q16C22⋊D5911yescharacteristic
231C2×SD16—21no—
232C2×SD16—1181no—
233C2×D8—591no—
234C2×D8C22⋊D5911yescharacteristic
235C2×M16C22⋊D5911yescharacteristic
236C22×C8C22×D5911yescharacteristic
237C4×Q8—22no—
238C4×D4—22no—
239C4×D4—42no—
240#117—41no—
241#117—1181no—
242#117C22⋊D5911yescharacteristic
243C2×C4⋊C4—21no—
244C2×C4⋊C4—41no—
245Q32—1182no—
246SD32—1182no—
247SD32—1182no—
248D16—1182no—
249M32—592no—
250M32—1181no—
251M32—1181no—
252M32—1181no—
253C2×C16—1181no—
254C2×C16—1181no—
255#107—1181no—
256#106—21no—
257#106—1181no—
258#106C22⋊D5912yes—
259#104—2362no—
260#103—22no—
261#102—22no—
262#99—2362no—
263#97—21no—
264C4×C8—21no—
265#41—22no—
266#41—22no—
267#41—41no—
268#41—41no—
269#41—2361no—
270#41—2361no—
271#41—2361no—
272#41—2361no—
273C2×Q8—21no—
274C2×Q8—1181no—
275C2×D4—21no—
276C2×D4—41no—
277C2×D4—1181no—
278C2×D4—2361no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—2361no—
283C22×C4C2×C22⋊D5911yescharacteristic
284Q16—21no—
285Q16—1181no—
286Q16—1181no—
287Q16(C2×D59)⋊C412yes—
288SD16—22no—
289SD16—22no—
290SD16—2361no—
291SD16—2361no—
292D8—21no—
293D8—1181no—
294D8—1181no—
295D8(C2×D59)⋊C412yes—
296M16—21no—
297M16—21no—
298C2×C8—21no—
299C2×C8C2×C4×D5911yescharacteristic
300C2×C8C2×D23611yescharacteristic
301C2×C8#1174568511yescharacteristic
302C2×C8#1174568511yescharacteristic
303C2×C8C2×C22⋊D5911yescharacteristic
304C2×C8C2×C22⋊D5911yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—2361no—
310C22⋊C4—42no—
311C22⋊C4—2361no—
312C22⋊C4#1174566912yes—
313C42—21no—
314C42—42no—
315C42—2361no—
316C16—1182no—
317C16—2361no—
318C23—41no—
319C23—2361no—
320C23C2×#1174566911yescharacteristic
321Q8—22no—
322Q8—41no—
323Q8—2361no—
324Q8—2361no—
325D4—22no—
326D4—42no—
327D4—42no—
328D4—41no—
329D4—4721no—
330D4—4721no—
331D4—2361no—
332D4—2361no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—162no—
339C2×C4—161no—
340C2×C4—4721no—
341C2×C4—4721no—
342C2×C4—4721no—
343C2×C4—2361no—
344C2×C4C4×C22⋊D5911yescharacteristic
345C2×C4#1427369611yescharacteristic
346C2×C4#1427370111yescharacteristic
347C2×C4#1427374711yescharacteristic
348C2×C4#1427375211yescharacteristic
349C2×C4#1427375311yescharacteristic
350C8—41no—
351C8C2×(C2×D59)⋊C411yescharacteristic
352C8C4×C22⋊D5911yescharacteristic
353C8#1427370111yescharacteristic
354C8#1427371011yescharacteristic
355C22—41no—
356C22—42no—
357C22—81no—
358C22—81no—
359C22—9441no—
360C22—4721no—
361C22C4×#1174566911yescharacteristic
362C22#1683753611yescharacteristic
363C22#1683759711yescharacteristic
364C4—42no—
365C4—42no—
366C4—82no—
367C4—9441no—
368C4—4721no—
369C4#1683728111yescharacteristic
370C4#1683751211yescharacteristic
371C4#1683754711yescharacteristic
372C4#1683761711yescharacteristic
373C2—21no—
374C2—21no—
375C2—21no—
376C2—82no—
377C2—9441no—
378C2#2142444411yescharacteristic
subgroupG/H
centerC2#21424444
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#2952C2
socleC2×C59#1879
prime radical#21437230C2
cosocleC22×C8×C59C23
hypercenter#2952D59
hypocenterC59#13030
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2952

Complements

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

NH ≅ G/Ntype
2. #21425832C2semidirect
3. C2×#16837211C2semidirect
4. #21424994C2semidirect
5. #21422203C2semidirect
6. #21422182C2semidirect
7. C59×#2952C2semidirect
10. #16837842C4semidirect
12. #16837837C4semidirect
34. C2×Dic236C2×C4semidirect
36. C2×D472C2×C4semidirect
45. #14273667C2×C4semidirect
185. #2952D59semidirect
218. C59#13030semidirect

Maximal subgroups

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

subgroupindex
#214372302
#214258322
C2×#168372112
#214249942
#214222032
#214221822
C59×#29522
#1303059