Finite Group Databasebeta

← #28485493

Subgroups — #28485493

Order 15104.

470 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433568C211yesmaximal, characteristic
2#21424111C211yesmaximal, characteristic, complemented
3#21422657C211yesmaximal, characteristic
4#21422496C211yesmaximal, characteristic
5#21421992C211yesmaximal, characteristic, complemented
6#21421955C211yesmaximal, characteristic, complemented
7C59×#2983C211yesmaximal, characteristic, complemented
8C2×#14273758—21no—
9#16837952C2211yescharacteristic
10#16837624—22no—
11#16837612C412yescomplemented
12#16837602C412yescomplemented
13#16837426C2211yescharacteristic
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15#16837401—21no—
16C2×#14273651—21no—
17#16837268C2211yescharacteristic
18#16837164—21no—
19#16837137—22no—
20#16837131—22no—
21C2×C4×Q8×C59C2211yescharacteristic
22C59×#446—22no—
23C59×#446—22no—
24C59×#432—22no—
25C59×#431C2211yescharacteristic
26C59×#431C2211yescharacteristic
27C2×#11745685—21no—
28#14273758—42no—
29#14273758—42no—
30C2×#11745672—22no—
31C2×#11745671—22no—
32C2×#11745664—21no—
33C22×C59⋊C8—21no—
34#14273727C2×C411yescharacteristic
35#14273724C2×C411yescharacteristic
36#14273715—21no—
37#14273701—21no—
38#14273701—21no—
39C4×D236—21no—
40C4×D236—21no—
41C2×C4⋊Dic59—21no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)C2×C412yes—
46#14273651—42no—
47#14273650—22no—
48#14273647—22no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C22×Q8×C59D411yescharacteristic
52C4×Q8×C59—21no—
53C4×Q8×C59—21no—
54C4×Q8×C59C2×C412yes—
55C4×Q8×C59D412yes—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59D411yescharacteristic
58C2×C42×C59—21no—
59C2×C42×C59C2311yescharacteristic
60C4⋊C8×C59—24no—
61C4⋊C8×C59—21no—
62C4⋊C8×C59—21no—
63C22⋊C8×C59—21no—
64C22⋊C8×C59D412yes—
65#97×C59—24no—
66C4×C8×C59—21no—
67C4×C8×C59D412yes—
68C2×Q8×C59—21no—
69C2×Q8×C59—21no—
70C2×Q8×C59—21no—
71C2×Q8×C59—21no—
72C2×Q8×C59—21no—
73C2×Q8×C59C22⋊C412yes—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59#4111yescharacteristic
79C2×C22⋊D59—41no—
80C22×Dic59—41no—
81#11745685—41no—
82#11745685—41no—
83#11745685—41no—
84#11745685—41no—
85C2×D236—21no—
86C2×D236—21no—
87C2×D236—21no—
88C2×C4×D59—41no—
89C2×Dic118—21no—
90C2×C8×C59—22no—
91C2×C8×C59—22no—
92C4⋊C4×C59—22no—
93C4⋊C4×C59—21no—
94C4⋊C4×C59—22no—
95C4⋊C4×C59C22⋊C412yes—
96C42×C59—22no—
97C42×C59—22no—
98C42×C59C22×C411yescharacteristic
99C42×C59C2×D411yescharacteristic
100C42×C59C2×D411yescharacteristic
101C42×C59#4111yescharacteristic
102#11745672—44no—
103#11745671—44no—
104(C2×D59)⋊C4—41no—
105(C2×D59)⋊C4—41no—
106C4⋊Dic59—21no—
107C4⋊Dic59—21no—
108C4⋊Dic59—41no—
109C4⋊Dic59—41no—
110#11745664—41no—
111#11745664—41no—
112C2×C59⋊C8—22no—
113C2×C59⋊C8—41no—
114C2×C59⋊C8—41no—
115C2×C59⋊C8—41no—
116C23×C59#12111yescharacteristic
117C22×D59—41no—
118Q8×C59—22no—
119Q8×C59—22no—
120Q8×C59—41no—
121Q8×C59—41no—
122C2×C4×C59—22no—
123C2×C4×C59—22no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—81no—
129C2×C4×C59C2×C22⋊C411yescharacteristic
130C2×C4×C59C4×D411yescharacteristic
131C2×C4×C59C4×D411yescharacteristic
132C2×C4×C59#12011yescharacteristic
133C2×C4×C59#12111yescharacteristic
134C2×C4×C59#12311yescharacteristic
135C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C59#13711yescharacteristic
139C22⋊D59—81no—
140C22⋊D59—81no—
141C2×Dic59—41no—
142C2×Dic59—81no—
143C2×Dic59—81no—
144C2×Dic59—81no—
145D236—42no—
146D236—41no—
147D236—41no—
148D236—81no—
149C4×D59—81no—
150C4×D59—81no—
151Dic118—41no—
152Dic118—41no—
153C8×C59—42no—
154C8×C59—42no—
155C59⋊C8—42no—
156C59⋊C8—81no—
157#9310—591nomaximal, Sylow
158C22×C59—21no—
159C22×C59—21no—
160C22×C59#38611yescharacteristic
161C22×C59#46811yescharacteristic
162C22×C59#46911yescharacteristic
163C2×D59—81no—
164C2×D59—81no—
165C2×D59—81no—
166C4×C59—21no—
167C4×C59—21no—
168C4×C59—21no—
169C4×C59—21no—
170C4×C59—42no—
171C4×C59—42no—
172C4×C59#41711yescharacteristic
173C4×C59#41811yescharacteristic
174C4×C59#46811yescharacteristic
175C4×C59#46911yescharacteristic
176Dic59—81no—
177Dic59—161no—
178#3037—591no—
179#2983D5911yescharacteristic, complemented
180#1805—591no—
181#1517—591no—
182#1480—591no—
183#1301—591no—
184#1286—591no—
185C2×C59—21no—
186C2×C59—21no—
187C2×C59#188111yescharacteristic
188C2×C59#189511yescharacteristic
189C2×C59#190511yescharacteristic
190D59—161no—
191C2×#137—1181no—
192#532—591no—
193C2×C4×Q8C2×D5911yescharacteristic
194#464—1182no—
195#462—592no—
196#460—592no—
197#446—22no—
198#446—22no—
199#432—22no—
200#431C2×D5911yescharacteristic
201#431C2×D5911yescharacteristic
202#423—591no—
203C2×C4⋊C8—591no—
204#417—1181no—
205C2×#103—1181no—
206#382—591no—
207#340—1181no—
208#327—1182no—
209D4⋊C8—1182no—
210C59#931011yescharacteristic, complemented, Sylow
211C2×#41—1181no—
212C22×Q8D23611yescharacteristic
213#137—2362no—
214#137—2362no—
215C2×Q16—1182no—
216C2×SD16—1182no—
217C2×M16—1181no—
218C22×C8—1181no—
219C4⋊Q8—591no—
220#127—591no—
221#124—1181no—
222#121—1181no—
223#121—1181no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8C4×D5912yes—
227C4×Q8C22⋊D5912yes—
228C4×D4—1181no—
229C4×D4—1181no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—1181no—
232C2×C4⋊C4D23611yescharacteristic
233C2×C42—21no—
234C2×C42C22×D5911yescharacteristic
235C4⋊C8—24no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—592no—
239C4⋊C8—1181no—
240C4⋊C8—1181no—
241C4⋊C8—1181no—
242#103—1182no—
243#103—2362no—
244#102—1182no—
245C22⋊C8—21no—
246C22⋊C8C22⋊D5912yes—
247#97—24no—
248C4×C8—21no—
249C4×C8C22⋊D5912yes—
250#95—1181no—
251#95—1181no—
252#41—2361no—
253#41—2361no—
254#41—2361no—
255#41—2361no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—1181no—
262C2×Q8(C2×D59)⋊C412yes—
263C2×D4—1181no—
264C2×D4—1181no—
265C2×D4—1181no—
266C2×D4—2361no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—2361no—
270C22×C4—2361no—
271C22×C4C2×D23611yescharacteristic
272C22×C4#1174568511yescharacteristic
273C22×C4C2×C22⋊D5911yescharacteristic
274Q16—2364no—
275SD16—2364no—
276M16—2361no—
277M16—2361no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—1182no—
281C2×C8—2361no—
282C2×C8—2361no—
283C2×C8—2361no—
284C4⋊C4—22no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—1181no—
288C4⋊C4—1181no—
289C4⋊C4—2361no—
290C4⋊C4—2361no—
291C4⋊C4(C2×D59)⋊C412yes—
292C22⋊C4—2361no—
293C22⋊C4—2361no—
294C42—22no—
295C42—22no—
296C42C2×C4×D5911yescharacteristic
297C42#1174568511yescharacteristic
298C42C2×C22⋊D5911yescharacteristic
299C42C2×C22⋊D5911yescharacteristic
300C23—2361no—
301C23#1427370111yescharacteristic
302Q8—22no—
303Q8—22no—
304Q8—41no—
305Q8—41no—
306Q8—2361no—
307Q8—2361no—
308D4—4721no—
309D4—4721no—
310D4—4721no—
311D4—2362no—
312D4—2361no—
313D4—2361no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—4721no—
322C2×C4—4721no—
323C2×C4—4721no—
324C2×C4—4721no—
325C2×C4—4721no—
326C2×C4—2361no—
327C2×C4C2×(C2×D59)⋊C411yescharacteristic
328C2×C4C4×C22⋊D5911yescharacteristic
329C2×C4C4×C22⋊D5911yescharacteristic
330C2×C4#1427369611yescharacteristic
331C2×C4#1427370111yescharacteristic
332C2×C4#1427371011yescharacteristic
333C2×C4#1427375511yescharacteristic
334C2×C4#1427375511yescharacteristic
335C2×C4#1427375511yescharacteristic
336C2×C4#1427376011yescharacteristic
337C8—42no—
338C8—42no—
339C8—4721no—
340C8—2362no—
341C22—21no—
342C22—21no—
343C22—4721no—
344C22—4721no—
345C22—4721no—
346C22#1683728111yescharacteristic
347C22#1683764811yescharacteristic
348C22#1683765511yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—42no—
354C4—42no—
355C4—9441no—
356C4—4721no—
357C4#1683740411yescharacteristic
358C4#1683740811yescharacteristic
359C4#1683764811yescharacteristic
360C4#1683765511yescharacteristic
361C2—21no—
362C2—21no—
363C2—9441no—
364C2#2142445811yescharacteristic
365C2#2142453411yescharacteristic
366C2#2142459011yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2983C2
socleC22×C59#386
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#2983D59
hypocenterC59#9310
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
2. #21424111C2semidirect
5. #21421992C2semidirect
6. #21421955C2semidirect
7. C59×#2983C2semidirect
11. #16837612C4semidirect
12. #16837602C4semidirect
179. #2983D59semidirect
210. C59#9310semidirect

Maximal subgroups

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

subgroupindex
#214335682
#214241112
#214226572
#214224962
#214219922
#214219552
C59×#29832
#931059