Finite Group Databasebeta

← #28504704

Subgroups — #28504704

Order 15104.

434 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425737C212yesmaximal
2#21424146C211yesmaximal, characteristic, complemented
3#21424142C211yesmaximal, characteristic, complemented
4#21422225C212yesmaximal, complemented
5C59×#2929C211yesmaximal, characteristic
6C2×#14273663—21no—
7#16837440—21no—
8#16837386C2212yes—
9#16837376C2212yes—
10#16837192—22no—
11#16837189—22no—
12C59⋊#338—21no—
13#16837159—21no—
14C2×#131×C59C2211yescharacteristic
15C23×C8×C59C411yescharacteristic, complemented
16C59×#525C2211yescharacteristic
17C59×#435—21no—
18C59×#435—21no—
19C8×D4×C59—21no—
20C8×D4×C59—21no—
21C59×#424C411yescharacteristic, complemented
22C59×#408C2211yescharacteristic
23C59×#407C411yescharacteristic, complemented
24C59×#405C411yescharacteristic, complemented
25C2×#11745664—22no—
26#14273686—22no—
27#14273663—41no—
28#14273663—41no—
29#14273661—41no—
30#14273659—41no—
31#14273646—22no—
32#14273644—22no—
33#14273642—22no—
34#14273642—22no—
35C2×#41×C59C2311yescharacteristic
36C23×C4×C59C2×C411yescharacteristic
37#131×C59—41no—
38#131×C59—41no—
39C2×M16×C59C2×C411yescharacteristic
40C2×M16×C59C2×C411yescharacteristic
41C2×M16×C59D411yescharacteristic
42C2×M16×C59D411yescharacteristic
43C22×C8×C59—21no—
44C22×C8×C59—21no—
45C22×C8×C59—41no—
46C22×C8×C59—41no—
47C22×C8×C59C2×C411yescharacteristic
48C22×C8×C59C2×C411yescharacteristic
49C22×C8×C59D411yescharacteristic
50C22×C8×C59D411yescharacteristic
51#123×C59D411yescharacteristic
52#123×C59Q811yescharacteristic
53C22⋊Q8×C59—21no—
54#121×C59—21no—
55#120×C59D411yescharacteristic
56#120×C59Q811yescharacteristic
57C4×D4×C59—21no—
58C4×D4×C59—21no—
59#117×C59C2×C411yescharacteristic
60C4⋊C8×C59—21no—
61C4⋊C8×C59—21no—
62C22⋊C8×C59—21no—
63C22⋊C8×C59—21no—
64C22⋊C8×C59—21no—
65C22⋊C8×C59—21no—
66#97×C59—21no—
67C4×C8×C59—21no—
68C24×C59C4⋊C411yescharacteristic
69#41×C59—41no—
70#41×C59—41no—
71C2×Q8×C59#4111yescharacteristic
72C2×D4×C59—21no—
73C2×D4×C59—21no—
74C2×D4×C59C2×D411yescharacteristic
75C2×D4×C59C2×Q811yescharacteristic
76C2×D4×C59#4111yescharacteristic
77C22×C4×C59—21no—
78C22×C4×C59—21no—
79C22×C4×C59—41no—
80C22×C4×C59—41no—
81C22×C4×C59C4⋊C411yescharacteristic
82C22×C4×C59C22×C411yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59C2×D411yescharacteristic
85M16×C59—41no—
86M16×C59—41no—
87M16×C59—41no—
88M16×C59—41no—
89C2×C8×C59—21no—
90C2×C8×C59—21no—
91C2×C8×C59—21no—
92C2×C8×C59—21no—
93C2×C8×C59—21no—
94C2×C8×C59—21no—
95C2×C8×C59—21no—
96C2×C8×C59—21no—
97C2×C8×C59—42no—
98C2×C8×C59—41no—
99C2×C8×C59—41no—
100C2×C8×C59—41no—
101C2×C8×C59C22⋊C411yescharacteristic
102C2×C8×C59C22⋊C411yescharacteristic
103C2×C8×C59C22⋊C411yescharacteristic
104C2×C8×C59C22⋊C411yescharacteristic
105C4⋊C4×C59—21no—
106C4⋊C4×C59—21no—
107C4⋊C4×C59—21no—
108C22⋊C4×C59—21no—
109C22⋊C4×C59—21no—
110C22⋊C4×C59—21no—
111C22⋊C4×C59—21no—
112C22⋊C4×C59C4⋊C411yescharacteristic
113C22⋊C4×C59C4⋊C411yescharacteristic
114C42×C59—21no—
115C22⋊Dic59—42no—
116C4⋊Dic59—42no—
117C4×Dic59—42no—
118#11745664—42no—
119#11745664—42no—
120C2×C59⋊C8—42no—
121C23×C59—21no—
122C23×C59—41no—
123C23×C59—42no—
124C23×C59C2×C4⋊C411yescharacteristic
125C23×C59#12111yescharacteristic
126C23×C59C22⋊Q811yescharacteristic
127Q8×C59—41no—
128D4×C59—41no—
129D4×C59—41no—
130D4×C59—41no—
131D4×C59—42no—
132D4×C59—42no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—42no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59C2×C22⋊C411yescharacteristic
147C2×C4×C59#11711yescharacteristic
148C2×C4×C59#12111yescharacteristic
149C2×C4×C59C22⋊Q811yescharacteristic
150C2×Dic59—82no—
151C8×C59—21no—
152C8×C59—21no—
153C8×C59—21no—
154C8×C59—41no—
155C8×C59—41no—
156C8×C59#9911yescharacteristic
157C8×C59#9911yescharacteristic
158C59⋊C8—82no—
159#12805—591nomaximal, Sylow
160C22×C59—22no—
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—41no—
168C22×C59—42no—
169C22×C59—42no—
170C22×C59—41no—
171C22×C59#38011yescharacteristic
172C4×C59—21no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—41no—
176C4×C59—41no—
177C4×C59—41no—
178C4×C59C2×#9911yescharacteristic
179C4×C59#41011yescharacteristic
180Dic59—162no—
181#2929D5911yescharacteristic
182#2120—592no—
183#1813—591no—
184#1812—591no—
185#1395—592no—
186C2×C59—21no—
187C2×C59—41no—
188C2×C59—41no—
189C2×C59—42no—
190C2×C59—41no—
191C2×C59#179911yescharacteristic
192C2×#131C2×D5911yescharacteristic
193C23×C8Dic5911yescharacteristic, complemented
194#525C2×D5911yescharacteristic
195#435—21no—
196#435—21no—
197C8×D4—21no—
198C8×D4—21no—
199C2×#108—1181no—
200#427—1181no—
201#424Dic5911yescharacteristic, complemented
202#413—592no—
203#410—592no—
204#408C2×D5911yescharacteristic
205#407Dic5911yescharacteristic, complemented
206#405Dic5911yescharacteristic, complemented
207#352—1182no—
208AΣL(1,𝔽16)—1182no—
209#338—1181no—
210H(3,ℤ/4)—1181no—
211C59#1280511yescharacteristic, complemented, Sylow
212C2×#41C22×D5911yescharacteristic
213C23×C4C2×Dic5911yescharacteristic
214#131—41no—
215#131—41no—
216C2×M16—1182no—
217C2×M16C2×Dic5911yescharacteristic
218C2×M16C2×Dic5911yescharacteristic
219C2×M16C22⋊D5911yescharacteristic
220C2×M16C22⋊D5911yescharacteristic
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—41no—
224C22×C8—41no—
225C22×C8C2×Dic5911yescharacteristic
226C22×C8C2×Dic5911yescharacteristic
227C22×C8C22⋊D5911yescharacteristic
228C22×C8C22⋊D5911yescharacteristic
229#123Dic11811yescharacteristic
230#123D23611yescharacteristic
231C22⋊Q8—21no—
232#121—21no—
233#120Dic11811yescharacteristic
234#120D23611yescharacteristic
235C4×D4—21no—
236C4×D4—21no—
237#117—1182no—
238#117C2×Dic5911yescharacteristic
239#108—2361no—
240#108—2361no—
241#107—2361no—
242#106—2361no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245#101—1182no—
246#100—1182no—
247#99—1182no—
248#99—1182no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253#97—21no—
254C4×C8—21no—
255C24C4⋊Dic5911yescharacteristic
256#41—41no—
257#41—41no—
258C2×Q8#1174568511yescharacteristic
259C2×D4—21no—
260C2×D4—21no—
261C2×D4C2×Dic11811yescharacteristic
262C2×D4C2×D23611yescharacteristic
263C2×D4#1174568511yescharacteristic
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4C4⋊Dic5911yescharacteristic
269C22×C4C22×Dic5911yescharacteristic
270C22×C4C2×C22⋊D5911yescharacteristic
271C22×C4C2×C22⋊D5911yescharacteristic
272M16—41no—
273M16—41no—
274M16—41no—
275M16—41no—
276M16—2362no—
277M16—2362no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—2362no—
291C2×C8C22⋊Dic5911yescharacteristic
292C2×C8C22⋊Dic5911yescharacteristic
293C2×C8C22⋊Dic5911yescharacteristic
294C2×C8C22⋊Dic5911yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—2362no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—2362no—
304C22⋊C4C4⋊Dic5911yescharacteristic
305C22⋊C4C4⋊Dic5911yescharacteristic
306C42—21no—
307C42—2362no—
308C23—21no—
309C23—41no—
310C23—42no—
311C23C2×C4⋊Dic5911yescharacteristic
312C23#1427370111yescharacteristic
313C23#1427370611yescharacteristic
314Q8—41no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—42no—
319D4—42no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—4722no—
334C2×C4C2×C22⋊Dic5911yescharacteristic
335C2×C4#1427368611yescharacteristic
336C2×C4#1427370111yescharacteristic
337C2×C4#1427370611yescharacteristic
338C8—21no—
339C8—21no—
340C8—21no—
341C8—41no—
342C8—41no—
343C8—4722no—
344C8#1427364211yescharacteristic
345C8#1427364211yescharacteristic
346C22—22no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—42no—
355C22—42no—
356C22—41no—
357C22#1683726211yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—9442no—
365C4C2×#1427364211yescharacteristic
366C4#1683737611yescharacteristic
367C2—21no—
368C2—41no—
369C2—42no—
370C2—41no—
371C2—41no—
372C2#2142408411yescharacteristic
subgroupG/H
centerC4#16837376
commutatorC22×C4×C59C22×C4
FrattiniC2×#41C22×D59
FittingC59×#2929C2
socleC2×C59#1799
prime radicalC59×#525C22
cosocleC2×#41×C59C23
hypercenter#2929D59
hypocenterC59#12805
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ #2929

Complements

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

NH ≅ G/Ntype
2. #21424146C2semidirect
3. #21424142C2semidirect
4. #21422225C2semidirect
15. C23×C8×C59C4semidirect
21. C59×#424C4semidirect
23. C59×#407C4semidirect
24. C59×#405C4semidirect
193. C23×C8Dic59semidirect
201. #424Dic59semidirect
205. #407Dic59semidirect
206. #405Dic59semidirect
211. C59#12805semidirect

Maximal subgroups

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

subgroupindex
#214257372
#214241462
#214241422
#214222252
C59×#29292
#1280559