Finite Group Databasebeta

← #28485381

Subgroups — #28485381

Order 15104.

490 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435034C211yesmaximal, characteristic
2#21424920C211yesmaximal, characteristic
3#21422765C211yesmaximal, characteristic
4#21422660C211yesmaximal, characteristic, complemented
5#21421956C211yesmaximal, characteristic, complemented
6#21421927C211yesmaximal, characteristic, complemented
7C59×#2961C211yesmaximal, characteristic, complemented
8C2×#14273724C2211yescharacteristic
9#16837776C412yescomplemented
10#16837750C412yescomplemented
11#16837602—22no—
12#16837552—22no—
13#16837455C2211yescharacteristic
14#16837455C2211yescharacteristic
15C2×#14273649—21no—
16C2×#14273649—21no—
17#16837268C2211yescharacteristic
18#16837145—22no—
19#16837139—22no—
20C59×#533C2211yescharacteristic
21C59×#446—22no—
22C59×#446—22no—
23C59×#424—21no—
24C59×#423C2211yescharacteristic
25C59×#405—21no—
26C2×#97×C59C2211yescharacteristic
27C22×D236—21no—
28C22×D236—21no—
29C22×D236—21no—
30C2×#11745671—24no—
31#14273724—21no—
32#14273724—21no—
33#14273724—21no—
34#14273724C2×C412yes—
35C2×C4⋊Dic59—21no—
36C4⋊(C59⋊C8)—21no—
37C4⋊(C59⋊C8)—21no—
38#14273649—42no—
39#14273649—42no—
40#14273647—24no—
41C22⋊(C59⋊C8)—21no—
42C22⋊(C59⋊C8)—21no—
43C4×C59⋊C8—21no—
44C4×C59⋊C8C2×C412yes—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C2×M16×C59—21no—
48C22×C8×C59D411yescharacteristic
49C22×C8×C59D411yescharacteristic
50C4⋊Q8×C59C2×C411yescharacteristic
51C4⋊Q8×C59C2×C411yescharacteristic
52#125×C59—21no—
53C22⋊Q8×C59—21no—
54C22⋊Q8×C59D412yes—
55C4×Q8×C59—21no—
56C4×Q8×C59D412yes—
57#117×C59—21no—
58C2×C42×C59C2311yescharacteristic
59C4⋊C8×C59—24no—
60C4⋊C8×C59—21no—
61C4⋊C8×C59D412yes—
62#97×C59—22no—
63#97×C59—21no—
64#97×C59—21no—
65C4×C8×C59—22no—
66C23×D59—41no—
67C2×Q8×C59—21no—
68C2×Q8×C59C22⋊C412yes—
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59#4111yescharacteristic
72C22×Dic59—41no—
73C2×D236—22no—
74C2×D236—22no—
75C2×D236—22no—
76C2×D236—41no—
77C2×D236—41no—
78C2×D236—41no—
79C2×D236—41no—
80C2×D236—41no—
81C2×D236—41no—
82C2×D236—41no—
83C2×D236—82no—
84C2×D236—82no—
85M16×C59—41no—
86M16×C59—41no—
87C2×C8×C59—21no—
88C2×C8×C59—21no—
89C2×C8×C59—22no—
90C2×C8×C59—21no—
91C2×C8×C59—21no—
92C2×C8×C59—21no—
93C2×C8×C59SD1612yescomplemented
94C4⋊C4×C59—21no—
95C4⋊C4×C59—21no—
96C4⋊C4×C59—21no—
97C4⋊C4×C59—22no—
98C4⋊C4×C59—22no—
99C4⋊C4×C59C22⋊C412yes—
100C22⋊C4×C59—22no—
101C42×C59—22no—
102C42×C59C22×C411yescharacteristic
103C42×C59C2×D411yescharacteristic
104C42×C59C2×D411yescharacteristic
105C42×C59#4111yescharacteristic
106#11745671—84no—
107C4⋊Dic59—41no—
108C4⋊Dic59—41no—
109C2×C59⋊C8—22no—
110C2×C59⋊C8—41no—
111C23×C59#12111yescharacteristic
112C22×D59—42no—
113C22×D59—82no—
114C22×D59—82no—
115C22×D59—81no—
116C22×D59—81no—
117Q8×C59—42no—
118Q8×C59—81no—
119C2×C4×C59—22no—
120C2×C4×C59—22no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59C2×C22⋊C411yescharacteristic
124C2×C4×C59C4×D411yescharacteristic
125C2×C4×C59C4×D411yescharacteristic
126C2×C4×C59#12011yescharacteristic
127C2×C4×C59#12111yescharacteristic
128C2×C4×C59#12311yescharacteristic
129C2×C4×C59C2×SD1611yescharacteristic
130C2×C4×C59#13511yescharacteristic
131C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C59#13711yescharacteristic
133C2×Dic59—81no—
134C2×Dic59—81no—
135C2×Dic59—81no—
136D236—44no—
137D236—42no—
138D236—42no—
139D236—82no—
140D236—82no—
141D236—82no—
142D236—81no—
143D236—81no—
144C8×C59—22no—
145C8×C59—41no—
146C8×C59—42no—
147C59⋊C8—42no—
148C59⋊C8—81no—
149#9294—591nomaximal, Sylow
150C22×C59—21no—
151C22×C59—21no—
152C22×C59#38611yescharacteristic
153C22×C59#46011yescharacteristic
154C22×C59#46411yescharacteristic
155C2×D59—82no—
156C2×D59—82no—
157C2×D59—82no—
158C2×D59—162no—
159C2×D59—162no—
160C4×C59—21no—
161C4×C59—21no—
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—42no—
165C4×C59—42no—
166C4×C59C4×SD1611yescharacteristic
167C4×C59#44011yescharacteristic
168C4×C59#44811yescharacteristic
169C4×C59#44911yescharacteristic
170Dic59—161no—
171#3208—591no—
172#2961D5911yescharacteristic, complemented
173#1961—591no—
174#1545—591no—
175#1518—591no—
176#1286—591no—
177#1275—591no—
178C2×C59—21no—
179C2×C59—21no—
180C2×C59#188011yescharacteristic
181C2×C59#190311yescharacteristic
182C2×C59#191011yescharacteristic
183D59—162no—
184#533C2×D5911yescharacteristic
185C2×#127—591no—
186#492—592no—
187#486—592no—
188#460—1182no—
189#450—1182no—
190#446—22no—
191#446—22no—
192#431—591no—
193#431—591no—
194#424—21no—
195#423C2×D5911yescharacteristic
196C2×#102—1181no—
197C2×#102—1181no—
198#405—21no—
199C2×#97C2×D5911yescharacteristic
200#382—591no—
201#331—1182no—
202#328—1182no—
203C59#929411yescharacteristic, complemented, Sylow
204C22×D4—1181no—
205C22×D4—1181no—
206C22×D4—1181no—
207C2×SD16—1184no—
208C2×M16—21no—
209C22×C8C22⋊D5911yescharacteristic
210C22×C8C22⋊D5911yescharacteristic
211C4⋊Q8C4×D5911yescharacteristic
212C4⋊Q8C4×D5911yescharacteristic
213#127—592no—
214#127—1181no—
215#127—1181no—
216#127—1181no—
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8D23612yes—
220C4×Q8—21no—
221C4×Q8C22⋊D5912yes—
222#117—21no—
223C2×C4⋊C4—1181no—
224C2×C42C22×D5911yescharacteristic
225C4⋊C8—24no—
226C4⋊C8—21no—
227C4⋊C8—1181no—
228C4⋊C8—1181no—
229C4⋊C8C22⋊D5912yes—
230#102—1184no—
231#102—2362no—
232#102—2362no—
233C22⋊C8—1181no—
234C22⋊C8—1181no—
235#97—22no—
236#97—21no—
237#97—21no—
238C4×C8—22no—
239C4×C8—592no—
240C4×C8—1181no—
241#95—1181no—
242#95—1181no—
243C24—2361no—
244C2×Q8—21no—
245C2×Q8(C2×D59)⋊C412yes—
246C2×D4—1182no—
247C2×D4—1182no—
248C2×D4—1182no—
249C2×D4—4722no—
250C2×D4—4722no—
251C2×D4—2361no—
252C2×D4—2361no—
253C2×D4—2361no—
254C2×D4—2361no—
255C2×D4—2361no—
256C2×D4—2361no—
257C2×D4—2361no—
258C22×C4—2361no—
259C22×C4C2×D23611yescharacteristic
260C22×C4#1174568511yescharacteristic
261C22×C4C2×C22⋊D5911yescharacteristic
262SD16—4724no—
263M16—41no—
264M16—41no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—1182no—
272C2×C8—2361no—
273C2×C8#1174567112yescomplemented
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—2361no—
280C4⋊C4—2361no—
281C4⋊C4(C2×D59)⋊C412yes—
282C22⋊C4—22no—
283C42—22no—
284C42C2×C4×D5911yescharacteristic
285C42#1174568511yescharacteristic
286C42C2×C22⋊D5911yescharacteristic
287C42C2×C22⋊D5911yescharacteristic
288C23—4722no—
289C23—4722no—
290C23—4721no—
291C23—4721no—
292C23—2362no—
293C23#1427370111yescharacteristic
294Q8—42no—
295Q8—81no—
296D4—4722no—
297D4—4722no—
298D4—4722no—
299D4—4721no—
300D4—4721no—
301D4—2364no—
302D4—2362no—
303D4—2362no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—4721no—
309C2×C4—4721no—
310C2×C4—4721no—
311C2×C4C2×(C2×D59)⋊C411yescharacteristic
312C2×C4C4×C22⋊D5911yescharacteristic
313C2×C4C4×C22⋊D5911yescharacteristic
314C2×C4#1427369611yescharacteristic
315C2×C4#1427370111yescharacteristic
316C2×C4#1427371011yescharacteristic
317C2×C4C2×#1174567111yescharacteristic
318C2×C4#1427374711yescharacteristic
319C2×C4#1427375211yescharacteristic
320C2×C4#1427375811yescharacteristic
321C8—22no—
322C8—41no—
323C8—42no—
324C8—4721no—
325C8—2362no—
326C22—21no—
327C22—21no—
328C22—9442no—
329C22—9442no—
330C22—4722no—
331C22—4722no—
332C22—4722no—
333C22#1683728111yescharacteristic
334C22#1683760211yescharacteristic
335C22#1683762411yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—9441no—
343C4C4×#1174567111yescharacteristic
344C4#1683750011yescharacteristic
345C4#1683754111yescharacteristic
346C4#1683754711yescharacteristic
347C2—21no—
348C2—21no—
349C2—9442no—
350C2#2142445111yescharacteristic
351C2#2142457611yescharacteristic
352C2#2142462311yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2961C2
socleC22×C59#386
prime radicalC2×#14273724C22
cosocleC2×C42×C59C23
hypercenter#2961D59
hypocenterC59#9294
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2961

Complements

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

NH ≅ G/Ntype
4. #21422660C2semidirect
5. #21421956C2semidirect
6. #21421927C2semidirect
7. C59×#2961C2semidirect
9. #16837776C4semidirect
10. #16837750C4semidirect
93. C2×C8×C59SD16semidirect
172. #2961D59semidirect
203. C59#9294semidirect
273. C2×C8#11745671semidirect

Maximal subgroups

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

subgroupindex
#214350342
#214249202
#214227652
#214226602
#214219562
#214219272
C59×#29612
#929459