Finite Group Databasebeta

← #28507337

Subgroups — #28507337

Order 15104.

612 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427461C211yesmaximal, characteristic
2#21424084C211yesmaximal, characteristic
3#21422215C214yesmaximal
4C4×#100×C59C211yesmaximal, characteristic
5C23×Dic118C411yescharacteristic, complemented
6C2×#14273706—21no—
7C2×#14273642C2212yes—
8#16837322—21no—
9#16837314—21no—
10#16837308—21no—
11#16837282C2212yes—
12#16837274C412yescomplemented
13#16837262—21no—
14#16837262C411yescharacteristic, complemented
15#16837169—21no—
16C2×C4×D4×C59C2211yescharacteristic
17C2×#100×C59C2212yes—
18C4×M16×C59—21no—
19C59×#343—21no—
20C22×Dic118—21no—
21C22×Dic118—21no—
22C22×Dic118—44no—
23C22×Dic118—41no—
24C22×Dic118—41no—
25C23×Dic59C2×C412yes—
26#14273706—44no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59—41no—
29C2×Dic59⋊C4—22no—
30C2×C22⋊Dic59—22no—
31C2×C22⋊Dic59—22no—
32C2×C22⋊Dic59C2×C412yes—
33#14273642—24no—
34C59⋊#95—21no—
35C59⋊#95—22no—
36C59⋊#95—22no—
37C59⋊#95—41no—
38C22×D4×C59C2311yescharacteristic
39C23×C4×C59C2×C411yescharacteristic
40C23×C4×C59D411yescharacteristic
41C2×M16×C59—22no—
42C4×D4×C59—41no—
43C4×D4×C59—41no—
44C2×C4⋊C4×C59D411yescharacteristic
45C2×C22⋊C4×C59C2×C411yescharacteristic
46C2×C22⋊C4×C59D411yescharacteristic
47C2×C42×C59D411yescharacteristic
48#100×C59—22no—
49#100×C59Q814yescomplemented
50#97×C59—41no—
51C4×C8×C59—41no—
52C24×C59C22×C411yescharacteristic
53C24×C59C2×D411yescharacteristic
54C2×D4×C59—41no—
55C2×D4×C59—41no—
56C2×D4×C59C2×Q812yes—
57C2×D4×C59#4112yes—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—42no—
63C22×C4×C59—42no—
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×Dic59—22no—
70C22×Dic59—22no—
71C22×Dic59—44no—
72C22×Dic59—42no—
73C22×Dic59—42no—
74C2×Dic118—24no—
75C2×Dic118—44no—
76C2×Dic118—42no—
77C2×Dic118—42no—
78C2×Dic118—48no—
79C2×Dic118—44no—
80C2×Dic118—41no—
81C2×Dic118—41no—
82C2×Dic118—44no—
83M16×C59—24no—
84M16×C59—42no—
85C2×C8×C59—42no—
86C4⋊C4×C59—41no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—41no—
89C42×C59—21no—
90C42×C59—21no—
91C22⋊Dic59—44no—
92C22⋊Dic59—42no—
93C22⋊Dic59—42no—
94C22⋊Dic59—42no—
95C4⋊Dic59—42no—
96C4⋊Dic59—82no—
97Dic59⋊C4—44no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59C2×C22⋊C411yescharacteristic
107C23×C59C4×Q812yes—
108C23×C59#12411yescharacteristic
109C23×C59C4⋊Q811yescharacteristic
110D4×C59—42no—
111D4×C59—42no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—22no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—42no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—81no—
125C2×C4×C59#9911yescharacteristic
126C2×C4×C59#9911yescharacteristic
127C2×C4×C59C22⋊Q812yes—
128C2×Dic59—24no—
129C2×Dic59—42no—
130C2×Dic59—44no—
131C2×Dic59—42no—
132C2×Dic59—44no—
133C2×Dic59—42no—
134C2×Dic59—42no—
135C2×Dic59—42no—
136C2×Dic59—82no—
137C2×Dic59—82no—
138C2×Dic59—82no—
139Dic118—44no—
140Dic118—48no—
141Dic118—44no—
142C8×C59—44no—
143#13293—591nomaximal, Sylow
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—21no—
147C22×C59—22no—
148C22×C59—21no—
149C22×C59—42no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—42no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—81no—
158C22×C59#39111yescharacteristic
159C22×C59C2×#9911yescharacteristic
160C22×C59#41011yescharacteristic
161C4×C59—21no—
162C4×C59—22no—
163C4×C59—41no—
164C4×C59—41no—
165C4×C59—41no—
166C4×C59AΣL(1,𝔽16)12yes—
167Dic59—44no—
168Dic59—82no—
169Dic59—162no—
170#2409—591no—
171#1799—591no—
172C4×#100D5911yescharacteristic
173#1390—594no—
174C2×C59—21no—
175C2×C59—21no—
176C2×C59—41no—
177C2×C59—42no—
178C2×C59—41no—
179C2×C59#199611yescharacteristic
180C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
181C2×C59#212311yescharacteristic
182C23×Q8—591no—
183C2×C22⋊Q8—1181no—
184C2×C4×D4C2×D5911yescharacteristic
185C2×#100C2×D5912yes—
186C2×#99—592no—
187C4×M16—21no—
188#395—1181no—
189#393—1181no—
190#391—1181no—
191#386—592no—
192#385—592no—
193#380—591no—
194#380—1181no—
195#343—21no—
196#342—1181no—
197C59#1329311yescharacteristic, complemented, Sylow
198C22×Q8—1181no—
199C22×Q8—1181no—
200C22×Q8—2361no—
201C22×Q8—2361no—
202C22×Q8—2364no—
203C22×D4C22×D5911yescharacteristic
204C23×C4—592no—
205C23×C4C4×D5911yescharacteristic
206C23×C4D23611yescharacteristic
207C2×M16—22no—
208C22⋊Q8—2364no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—1182no—
212C2×C4⋊C4—1181no—
213C2×C4⋊C4—2361no—
214C2×C4⋊C4C22⋊D5911yescharacteristic
215C2×C22⋊C4—592no—
216C2×C22⋊C4—1182no—
217C2×C22⋊C4—1182no—
218C2×C22⋊C4C4×D5911yescharacteristic
219C2×C22⋊C4D23611yescharacteristic
220C2×C42C22⋊D5911yescharacteristic
221#100—22no—
222#100Dic11814yescomplemented
223#99—1184no—
224#97—41no—
225C4×C8—41no—
226#95—1181no—
227#95—1182no—
228#95—1182no—
229#95—2361no—
230C24C2×C4×D5911yescharacteristic
231C24C2×D23611yescharacteristic
232C2×Q8—1184no—
233C2×Q8—2362no—
234C2×Q8—2362no—
235C2×Q8—2364no—
236C2×Q8—2364no—
237C2×Q8—2368no—
238C2×Q8—2364no—
239C2×Q8—2361no—
240C2×Q8—2361no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4C2×Dic11812yes—
244C2×D4#1174568512yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—1182no—
252C22×C4—1182no—
253C22×C4—2364no—
254C22×C4—2362no—
255C22×C4—2362no—
256C22×C4(C2×D59)⋊C411yescharacteristic
257C22×C4(C2×D59)⋊C411yescharacteristic
258C22×C4(C2×D59)⋊C411yescharacteristic
259C22×C4(C2×D59)⋊C411yescharacteristic
260C22×C4C2×C22⋊D5911yescharacteristic
261M16—24no—
262M16—42no—
263C2×C8—42no—
264C4⋊C4—41no—
265C4⋊C4—4722no—
266C4⋊C4—2362no—
267C4⋊C4—2364no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—2364no—
271C22⋊C4—2362no—
272C22⋊C4—2362no—
273C22⋊C4—2362no—
274C42—21no—
275C42—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23C2×(C2×D59)⋊C411yescharacteristic
285C23C4×Dic11812yes—
286C23#1427371511yescharacteristic
287C23#1427372711yescharacteristic
288Q8—2364no—
289Q8—2368no—
290Q8—2364no—
291D4—42no—
292D4—42no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—81no—
306C2×C4—1184no—
307C2×C4—4722no—
308C2×C4—4722no—
309C2×C4—4722no—
310C2×C4—2362no—
311C2×C4—2362no—
312C2×C4—2362no—
313C2×C4—2362no—
314C2×C4—2364no—
315C2×C4—2362no—
316C2×C4—2364no—
317C2×C4#1427364111yescharacteristic
318C2×C4#1427364111yescharacteristic
319C2×C4#1427370612yes—
320C8—44no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—21no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22#1683730811yescharacteristic
336C22C2×#1427364111yescharacteristic
337C22#1683737511yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—9442no—
344C4—4722no—
345C4—2364no—
346C4#1683719012yes—
347C2—21no—
348C2—21no—
349C2—41no—
350C2—42no—
351C2—41no—
352C2#2142508111yescharacteristic
353C2C2×#1683719011yescharacteristic
354C2#2142575111yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
FittingC4×#100×C59C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenterC4×#100D59
hypocenterC59#13293
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
5. C23×Dic118C4semidirect
12. #16837274C4semidirect
14. #16837262C4semidirect
49. #100×C59Q8semidirect
197. C59#13293semidirect
222. #100Dic118semidirect

Maximal subgroups

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

subgroupindex
#214274612
#214240842
#214222152
C4×#100×C592
#1329359