Finite Group Databasebeta

← #28506919

Subgroups — #28506919

Order 15104.

590 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837777C211yesmaximal, characteristic
2C2×#16837204C211yesmaximal, characteristic, complemented
3C2×#16837203C211yesmaximal, characteristic, complemented
4C2×#16837202C211yesmaximal, characteristic, complemented
5C2×#16837177C211yesmaximal, characteristic, complemented
6#21424259C211yesmaximal, characteristic, complemented
7C2×C59×#357C211yesmaximal, characteristic, complemented
8C22×D472C2211yescharacteristic
9C22×#11745669—21no—
10C2×#14273723—21no—
11C22×C59⋊C16C2211yescharacteristic
12#16837777—22no—
13#16837777C414yescomplemented
14C2×#14273661C2211yescharacteristic
15C2×#14273658—21no—
16C2×C8×Dic59C2211yescharacteristic
17#16837267—21no—
18#16837204—22no—
19#16837203—22no—
20#16837202—22no—
21#16837177—22no—
22C22×D8×C59C2211yescharacteristic
23C22×C16×C59C2211yescharacteristic
24C2×#107×C59C2211yescharacteristic
25C59×#357—22no—
26C22×D236—21no—
27C22×C22⋊D59—42no—
28C2×D472—21no—
29C2×D472—21no—
30C2×D472—21no—
31C2×D472—21no—
32C2×D472—21no—
33C2×D472C2×C412yes—
34C2×#11745669—24no—
35C2×#11745669—42no—
36C2×#11745669—42no—
37C22×C59⋊C8D411yescharacteristic
38#14273723—24no—
39#14273723—42no—
40C2×C4⋊Dic59—21no—
41C2×C4×Dic59D411yescharacteristic
42C2×C59⋊C16—21no—
43C2×C59⋊C16—21no—
44C2×C59⋊C16D412yescomplemented
45#14273661—21no—
46#14273661—21no—
47#14273658—42no—
48C8×Dic59C2×C412yes—
49C8×Dic59D412yes—
50C59⋊#95—42no—
51C22×D4×C59—21no—
52C2×D8×C59—21no—
53C2×D8×C59—21no—
54C2×D8×C59—21no—
55C2×D8×C59—21no—
56C2×D8×C59—21no—
57C2×D8×C59C2×C412yes—
58C22×C8×C59C2311yescharacteristic
59C2×C4⋊C4×C59—21no—
60C2×C16×C59—21no—
61C2×C16×C59—21no—
62C2×C16×C59D412yescomplemented
63#107×C59—21no—
64#107×C59—21no—
65C24×C59—41no—
66C23×D59—41no—
67C2×D4×C59—22no—
68C2×D4×C59—41no—
69C2×D4×C59—41no—
70C2×D4×C59—41no—
71C2×D4×C59—41no—
72C2×D4×C59—41no—
73C22×C4×C59—41no—
74C22×C4×C59C2×D411yescharacteristic
75C2×C22⋊D59—48no—
76C2×C22⋊D59—84no—
77C2×C22⋊D59—84no—
78C22×Dic59—41no—
79C22×Dic59D812yes—
80C2×D236—22no—
81C2×D236—41no—
82C2×D236—41no—
83C2×D236—41no—
84C2×D236—41no—
85C2×D236—41no—
86D8×C59—22no—
87D8×C59—22no—
88D8×C59—41no—
89D8×C59—41no—
90C2×C8×C59C22×C411yescharacteristic
91C2×C8×C59C2×D411yescharacteristic
92C2×C8×C59C2×D411yescharacteristic
93C2×C8×C59C2×D411yescharacteristic
94C2×C8×C59#4111yescharacteristic
95C2×C8×C59#4111yescharacteristic
96C4⋊C4×C59—41no—
97C4⋊C4×C59—41no—
98#11745669—48no—
99C4⋊Dic59—41no—
100C4⋊Dic59—41no—
101C4×Dic59C22⋊C412yes—
102C4×Dic59SD1612yes—
103C2×C59⋊C8C22⋊C412yes—
104C2×C59⋊C8D812yes—
105C2×C59⋊C8SD1612yes—
106D472—22no—
107D472—22no—
108D472—41no—
109D472—41no—
110C16×C59—22no—
111C59⋊C16—22no—
112C23×C59—42no—
113C23×C59—81no—
114C23×C59—81no—
115C23×C59—82no—
116C23×C59—82no—
117C23×C59C2×D811yescharacteristic
118C22×D59—42no—
119C22×D59—81no—
120C22×D59—81no—
121C22×D59—82no—
122C22×D59—82no—
123D4×C59—42no—
124D4×C59—42no—
125D4×C59—81no—
126D4×C59—81no—
127C2×C4×C59—81no—
128C2×C4×C59—82no—
129C2×C4×C59C2×C22⋊C411yescharacteristic
130C2×C4×C59#12411yescharacteristic
131C2×C4×C59#12711yescharacteristic
132C2×C4×C59C2×D811yescharacteristic
133C2×C4×C59C2×SD1611yescharacteristic
134C2×C4×C59C2×SD1611yescharacteristic
135C22⋊D59—816no—
136C2×Dic59—22no—
137C2×Dic59—22no—
138C2×Dic59—81no—
139C2×Dic59—82no—
140C2×Dic59#10214yes—
141D236—42no—
142D236—42no—
143D236—81no—
144D236—81no—
145C8×C59C4×D411yescharacteristic
146C8×C59C4×D411yescharacteristic
147C8×C59#12111yescharacteristic
148C8×C59#12111yescharacteristic
149C59⋊C8#10214yes—
150#13206—591nomaximal, Sylow
151C22×C59—84no—
152C22×C59—82no—
153C22×C59—162no—
154C22×C59—161no—
155C22×C59—161no—
156C22×C59C2×#10211yescharacteristic
157C22×C59#48611yescharacteristic
158C22×C59#49311yescharacteristic
159C22×C59#50912yes—
160C22×C59#50912yes—
161C2×D59—84no—
162C2×D59—82no—
163C2×D59—162no—
164C2×D59—161no—
165C2×D59—161no—
166C4×C59—161no—
167C4×C59#39011yescharacteristic
168C4×C59C2×#10211yescharacteristic
169C4×C59#48611yescharacteristic
170C4×C59#49211yescharacteristic
171Dic59—24no—
172Dic59—161no—
173C2×#493—591no—
174C2×#357—591no—
175C2×#357—591no—
176C2×#357—591no—
177C2×#357D5911yescharacteristic, complemented
178C2×#346—591no—
179#1846—591no—
180C2×C59—162no—
181C2×C59#195711yescharacteristic
182C2×C59#214012yes—
183C2×C59#224412yes—
184C2×C59#225011yescharacteristic
185C2×C59#225011yescharacteristic
186D59—162no—
187C22×D8—591no—
188C22×D8—1181no—
189C22×D8C2×D5911yescharacteristic
190C2×#127—1181no—
191C22×C16—591no—
192C22×C16C2×D5911yescharacteristic
193#493—594no—
194#493—1182no—
195C2×#107—591no—
196C2×#107C2×D5911yescharacteristic
197C2×#106—1181no—
198C2×C4×C8—591no—
199#382—1181no—
200#357—22no—
201#357—1182no—
202#357—1182no—
203#357—1182no—
204#346—1182no—
205C59#1320611yescharacteristic, complemented, Sylow
206C22×D4—21no—
207C22×D4—1181no—
208C22×D4—2362no—
209C2×D8—21no—
210C2×D8—21no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8—21no—
214C2×D8—592no—
215C2×D8—1184no—
216C2×D8—1181no—
217C2×D8—1181no—
218C2×D8—1181no—
219C2×D8—1181no—
220C2×D8—1181no—
221C2×D8—2362no—
222C2×D8—2362no—
223C2×D8C4×D5912yes—
224C22×C8—591no—
225C22×C8C22×D5911yescharacteristic
226#127—1184no—
227#127—2362no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—1181no—
230C2×C42—591no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—592no—
234C2×C16—1181no—
235C2×C16—1181no—
236C2×C16C22⋊D5912yescomplemented
237#107—21no—
238#107—21no—
239#107—1181no—
240#107—1181no—
241#106—2362no—
242C4×C8—592no—
243C4×C8—592no—
244#95—2362no—
245C24—41no—
246C24—2361no—
247C2×D4—22no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—1182no—
254C2×D4—4724no—
255C2×D4—4724no—
256C2×D4—2368no—
257C2×D4—2361no—
258C2×D4—2361no—
259C2×D4—2361no—
260C2×D4—2361no—
261C2×D4—2361no—
262C22×C4—41no—
263C22×C4—592no—
264C22×C4—2361no—
265C22×C4D4×D5911yescharacteristic
266D8—22no—
267D8—22no—
268D8—41no—
269D8—41no—
270D8—1182no—
271D8—1182no—
272D8—2368no—
273D8—2361no—
274D8—2361no—
275C2×C8—592no—
276C2×C8—592no—
277C2×C8—592no—
278C2×C8C2×C4×D5911yescharacteristic
279C2×C8#1174568511yescharacteristic
280C2×C8D4×D5911yescharacteristic
281C2×C8D4×D5911yescharacteristic
282C2×C8#1174568711yescharacteristic
283C2×C8C2×C22⋊D5911yescharacteristic
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—2361no—
287C4⋊C4—2361no—
288C42—592no—
289C42—592no—
290C16—22no—
291C16—1182no—
292C23—42no—
293C23—81no—
294C23—81no—
295C23—82no—
296C23—82no—
297C23—4721no—
298C23—4721no—
299C23—4722no—
300C23—4722no—
301C23—2362no—
302C23D8×D5911yescharacteristic
303D4—42no—
304D4—42no—
305D4—81no—
306D4—81no—
307D4—47216no—
308D4—4721no—
309D4—4721no—
310D4—2362no—
311D4—2362no—
312C2×C4—81no—
313C2×C4—82no—
314C2×C4—594no—
315C2×C4—1182no—
316C2×C4—1182no—
317C2×C4—4721no—
318C2×C4—4722no—
319C2×C4C22⋊C4×D5911yescharacteristic
320C2×C4#1427371311yescharacteristic
321C2×C4#1427372311yescharacteristic
322C2×C4D8×D5911yescharacteristic
323C2×C4SD16×D5911yescharacteristic
324C2×C4SD16×D5911yescharacteristic
325C8—594no—
326C8(C22⋊D59)⋊C411yescharacteristic
327C8C4×C22⋊D5911yescharacteristic
328C8#1427369911yescharacteristic
329C8#1427370011yescharacteristic
330C22—82no—
331C22—84no—
332C22—162no—
333C22—161no—
334C22—161no—
335C22—9442no—
336C22—9441no—
337C22—9441no—
338C22—4722no—
339C22—4724no—
340C22#102×D5911yescharacteristic
341C22#1683774711yescharacteristic
342C22#1683777711yescharacteristic
343C22#1683783612yes—
344C22#1683783912yes—
345C4—161no—
346C4—1184no—
347C4—9441no—
348C4#1683730211yescharacteristic
349C4#102×D5911yescharacteristic
350C4#1683774711yescharacteristic
351C4#1683777311yescharacteristic
352C2—162no—
353C2—9442no—
354C2#2142490011yescharacteristic
355C2#2142581012yes—
356C2#2142633612yes—
357C2#2142635911yescharacteristic
358C2#2142636111yescharacteristic
subgroupG/H
centerC23D8×D59
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC2×C59×#357C2
socleC23×C59C2×D8
prime radicalC2×#16837777C2
cosocleC22×C8×C59C23
hypercenterC2×#357D59
hypocenterC59#13206
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#16837204C2semidirect
3. C2×#16837203C2semidirect
4. C2×#16837202C2semidirect
5. C2×#16837177C2semidirect
6. #21424259C2semidirect
7. C2×C59×#357C2semidirect
13. #16837777C4semidirect
44. C2×C59⋊C16D4semidirect
62. C2×C16×C59D4semidirect
177. C2×#357D59semidirect
205. C59#13206semidirect
236. C2×C16C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377772
C2×#168372042
C2×#168372032
C2×#168372022
C2×#168371772
#214242592
C2×C59×#3572
#1320659