Finite Group Databasebeta

← #28487989

Subgroups — #28487989

Order 15104.

642 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
1C2×#16837778C211yesmaximal, characteristic
2#21425023C211yesmaximal, characteristic, complemented
3#21423965C211yesmaximal, characteristic, complemented
4C2×#16837152C211yesmaximal, characteristic, complemented
5C2×#16837146C211yesmaximal, characteristic, complemented
6C2×#16837145C211yesmaximal, characteristic, complemented
7C59×#1982C211yesmaximal, characteristic, complemented
8C22×#11745669—21no—
9C22×#11745669—21no—
10C2×#14273724C2211yescharacteristic
11#16837778—22no—
12#16837778C414yescomplemented
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15C2×#14273647—22no—
16C2×#14273638—21no—
17C2×C4×C59⋊C8C2211yescharacteristic
18#16837157—21no—
19#16837152—22no—
20#16837146—22no—
21#16837145—22no—
22C2×#127×C59C2211yescharacteristic
23C2×C4⋊C8×C59C2211yescharacteristic
24C2×#102×C59—22no—
25C4×C4⋊C4×C59C2211yescharacteristic
26C22×D236—21no—
27C22×D236—21no—
28C22×D236—21no—
29C2×#11745669—24no—
30C2×#11745669—24no—
31C2×#11745669—42no—
32C2×#11745669—42no—
33C2×#11745669—42no—
34C2×#11745669—42no—
35C22×C59⋊C8—21no—
36C22×C59⋊C8—21no—
37C22×C59⋊C8D411yescharacteristic
38C22×C59⋊C8D411yescharacteristic
39#14273724—21no—
40#14273724—21no—
41#14273724—21no—
42#14273724C2×C412yes—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)—21no—
46C4⋊(C59⋊C8)D412yes—
47#14273647—44no—
48#14273638—42no—
49C4×C59⋊C8C2×C412yes—
50C4×C59⋊C8D412yes—
51C22×D4×C59—21no—
52C22×D4×C59—21no—
53C22×D4×C59—21no—
54C22×C8×C59—21no—
55#127×C59—21no—
56#127×C59—21no—
57#127×C59—21no—
58#127×C59C2×C412yes—
59C2×C4⋊C4×C59D412yes—
60C2×C42×C59—21no—
61C2×C42×C59C2311yescharacteristic
62C4⋊C8×C59—21no—
63C4⋊C8×C59—21no—
64#102×C59—44no—
65#95×C59—21no—
66C24×C59—41no—
67C23×D59—41no—
68C2×D4×C59—22no—
69C2×D4×C59—22no—
70C2×D4×C59—22no—
71C2×D4×C59—41no—
72C2×D4×C59—41no—
73C2×D4×C59—41no—
74C2×D4×C59—41no—
75C2×D4×C59—41no—
76C2×D4×C59—42no—
77C2×D4×C59—42no—
78C2×D4×C59—41no—
79C2×D4×C59—41no—
80C2×D4×C59—82no—
81C22×C4×C59—22no—
82C22×C4×C59C2×D411yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59#4111yescharacteristic
85C2×D236—22no—
86C2×D236—22no—
87C2×D236—22no—
88C2×D236—41no—
89C2×D236—41no—
90C2×D236—42no—
91C2×D236—41no—
92C2×D236—41no—
93C2×D236—42no—
94C2×D236—41no—
95C2×D236—41no—
96C2×D236—41no—
97C2×D236—82no—
98C2×C8×C59—41no—
99C2×C8×C59—42no—
100C4⋊C4×C59—22no—
101C4⋊C4×C59—22no—
102C42×C59—41no—
103C42×C59—41no—
104C42×C59C22×C411yescharacteristic
105C42×C59C2×D411yescharacteristic
106C42×C59C2×D411yescharacteristic
107C42×C59#4111yescharacteristic
108#11745669—48no—
109#11745669—48no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—21no—
112C2×C59⋊C8—22no—
113C2×C59⋊C8—41no—
114C2×C59⋊C8—41no—
115C2×C59⋊C8—41no—
116C2×C59⋊C8—42no—
117C2×C59⋊C8C22⋊C412yes—
118C2×C59⋊C8C22⋊C412yes—
119C2×C59⋊C8D812yes—
120C2×C59⋊C8SD1612yes—
121C23×C59—42no—
122C23×C59—81no—
123C23×C59—81no—
124C23×C59—82no—
125C23×C59—82no—
126C23×C59#12111yescharacteristic
127C22×D59—42no—
128C22×D59—81no—
129C22×D59—81no—
130C22×D59—82no—
131C22×D59—82no—
132D4×C59—42no—
133D4×C59—42no—
134D4×C59—44no—
135D4×C59—82no—
136D4×C59—82no—
137D4×C59—81no—
138D4×C59—81no—
139D4×C59—81no—
140D4×C59—81no—
141C2×C4×C59—22no—
142C2×C4×C59—42no—
143C2×C4×C59—42no—
144C2×C4×C59—42no—
145C2×C4×C59C2×C22⋊C411yescharacteristic
146C2×C4×C59C4×D411yescharacteristic
147C2×C4×C59C4×D411yescharacteristic
148C2×C4×C59#12111yescharacteristic
149C2×C4×C59#12411yescharacteristic
150C2×C4×C59#12711yescharacteristic
151C2×C4×C59C2×D811yescharacteristic
152C2×C4×C59C2×SD1611yescharacteristic
153C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C59AGL(1,ℤ/8)12yes—
155C2×C4×C59AGL(1,ℤ/8)12yes—
156C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
157D236—42no—
158D236—42no—
159D236—44no—
160D236—82no—
161D236—82no—
162D236—81no—
163D236—81no—
164D236—81no—
165D236—81no—
166C8×C59—81no—
167C59⋊C8—22no—
168C59⋊C8—42no—
169C59⋊C8—81no—
170C59⋊C8#10214yescomplemented
171#9743—591nomaximal, Sylow
172C22×C59—82no—
173C22×C59—82no—
174C22×C59—82no—
175C22×C59—162no—
176C22×C59—162no—
177C22×C59#39011yescharacteristic
178C22×C59#45911yescharacteristic
179C22×C59#46011yescharacteristic
180C22×C59#46912yes—
181C22×C59#47212yes—
182C2×D59—82no—
183C2×D59—82no—
184C2×D59—82no—
185C2×D59—162no—
186C2×D59—162no—
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—82no—
190C4×C59C2×#10211yescharacteristic
191C4×C59#41811yescharacteristic
192C4×C59#44212yes—
193C4×C59#45911yescharacteristic
194C4×C59#46011yescharacteristic
195C4×C59#49611yescharacteristic
196C4×C59#49611yescharacteristic
197C2×#493—591no—
198#1982—591no—
199#1982D5911yescharacteristic, complemented
200#1770—591no—
201C2×#334—591no—
202C2×#331—591no—
203C2×#331—591no—
204C2×C59—162no—
205C2×C59#168712yes—
206C2×C59#168812yes—
207C2×C59#197211yescharacteristic
208C2×C59#197811yescharacteristic
209C2×C59#198211yescharacteristic
210D59—162no—
211C22×D8—1181no—
212C22×D8—1181no—
213C2×#127—591no—
214C2×#127C2×D5911yescharacteristic
215#493—594no—
216#493—1182no—
217C2×C4⋊C8—591no—
218C2×C4⋊C8—591no—
219C2×C4⋊C8C2×D5911yescharacteristic
220C2×#102—22no—
221C2×#102—1182no—
222C2×#97—1181no—
223C2×C4×C8—591no—
224C4×C4⋊C4C2×D5911yescharacteristic
225#336—1181no—
226#334—1182no—
227#331—1182no—
228#331—1182no—
229C59#974311yescharacteristic, complemented, Sylow
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—1181no—
234C22×D4—1181no—
235C22×D4—1181no—
236C2×D8—1184no—
237C2×D8—1184no—
238C2×D8—2362no—
239C2×D8—2362no—
240C2×D8—2362no—
241C2×D8—2362no—
242C22×C8—21no—
243C22×C8—591no—
244C22×C8—591no—
245C22×C8—1181no—
246C22×C8—1181no—
247#127—21no—
248#127—21no—
249#127—21no—
250#127—592no—
251#127—1181no—
252#127—1181no—
253#127—1181no—
254#127C4×D5912yes—
255C2×C4⋊C4C22⋊D5912yes—
256C2×C42—21no—
257C2×C42C22×D5911yescharacteristic
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—592no—
261C4⋊C8—1181no—
262C4⋊C8—1181no—
263C4⋊C8—1181no—
264#102—44no—
265#102—2364no—
266#97—2362no—
267C4×C8—592no—
268C4×C8—592no—
269#95—21no—
270C24—41no—
271C24—2361no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—82no—
285C2×D4—1182no—
286C2×D4—1182no—
287C2×D4—1182no—
288C2×D4—4722no—
289C2×D4—2361no—
290C2×D4—2361no—
291C2×D4—2361no—
292C2×D4—2361no—
293C2×D4—2361no—
294C2×D4—2361no—
295C2×D4—2362no—
296C2×D4—2362no—
297C2×D4—2361no—
298C22×C4—22no—
299C22×C4D4×D5911yescharacteristic
300C22×C4#1174568711yescharacteristic
301C22×C4C2×C22⋊D5911yescharacteristic
302D8—2368no—
303D8—2368no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—592no—
307C2×C8—592no—
308C2×C8—592no—
309C2×C8—592no—
310C2×C8—1181no—
311C2×C8—1181no—
312C2×C8—1182no—
313C2×C8—2361no—
314C2×C8—2362no—
315C2×C8—2361no—
316C2×C8—2361no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C42—41no—
320C42—41no—
321C42C2×C4×D5911yescharacteristic
322C42#1174568511yescharacteristic
323C42D4×D5911yescharacteristic
324C42D4×D5911yescharacteristic
325C23—42no—
326C23—82no—
327C23—82no—
328C23—81no—
329C23—81no—
330C23—4722no—
331C23—4722no—
332C23—4721no—
333C23—4721no—
334C23—2362no—
335C23#1427369911yescharacteristic
336D4—42no—
337D4—42no—
338D4—44no—
339D4—82no—
340D4—82no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345D4—4722no—
346D4—4722no—
347D4—4721no—
348D4—4721no—
349D4—4721no—
350D4—4721no—
351D4—2362no—
352D4—2362no—
353D4—2364no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4C22⋊C4×D5911yescharacteristic
359C2×C4(C22⋊D59)⋊C411yescharacteristic
360C2×C4C4×C22⋊D5911yescharacteristic
361C2×C4#1427370011yescharacteristic
362C2×C4#1427371311yescharacteristic
363C2×C4#1427372311yescharacteristic
364C2×C4D8×D5911yescharacteristic
365C2×C4SD16×D5911yescharacteristic
366C2×C4#1427375111yescharacteristic
367C2×C4#1427375212yes—
368C2×C4#1427375312yes—
369C2×C4#1427375411yescharacteristic
370C8—81no—
371C8—594no—
372C8—1182no—
373C8—4721no—
374C8—2362no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22—162no—
379C22—162no—
380C22—9442no—
381C22—9442no—
382C22—4722no—
383C22—4722no—
384C22—4722no—
385C22#1683730211yescharacteristic
386C22#1683759511yescharacteristic
387C22#1683760011yescharacteristic
388C22#1683765312yes—
389C22#1683767012yes—
390C4—21no—
391C4—21no—
392C4—82no—
393C4#102×D5911yescharacteristic
394C4#1683740511yescharacteristic
395C4#1683751212yes—
396C4#1683759611yescharacteristic
397C4#1683760111yescharacteristic
398C4#1683778911yescharacteristic
399C4#1683779211yescharacteristic
400C2—162no—
401C2—9442no—
402C2#2142357312yes—
403C2#2142357812yes—
404C2#2142497111yescharacteristic
405C2#2142500511yescharacteristic
406C2#2142502111yescharacteristic
subgroupG/H
centerC23#14273699
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1982C2
socleC23×C59#121
prime radicalC2×#16837778C2
cosocleC2×C42×C59C23
hypercenter#1982D59
hypocenterC59#9743
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
2. #21425023C2semidirect
3. #21423965C2semidirect
4. C2×#16837152C2semidirect
5. C2×#16837146C2semidirect
6. C2×#16837145C2semidirect
7. C59×#1982C2semidirect
12. #16837778C4semidirect
170. C59⋊C8#102semidirect
199. #1982D59semidirect
229. C59#9743semidirect

Maximal subgroups

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

subgroupindex
C2×#168377782
#214250232
#214239652
C2×#168371522
C2×#168371462
C2×#168371452
C59×#19822
#974359