Finite Group Databasebeta

← #28499021

Subgroups — #28499021

Order 15104.

456 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 87 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#21427355C211yesmaximal, characteristic
2#21422417C211yesmaximal, characteristic
3#21422119C212yesmaximal, complemented
4#21422095C212yesmaximal, complemented
5C4×#99×C59C211yesmaximal, characteristic
6C2×#14273686C411yescharacteristic, complemented
7#16837455—21no—
8#16837341C411yescharacteristic, complemented
9#16837338C2211yescharacteristic
10#16837332C411yescharacteristic, complemented
11#16837330—21no—
12#16837327C2211yescharacteristic
13#16837316—21no—
14C4×Dic59⋊C4—21no—
15C4×C22⋊Dic59—21no—
16C4×C22⋊Dic59C411yescharacteristic, complemented
17#16837126—21no—
18#16837126C2212yes—
19C2×C4×D4×C59C2211yescharacteristic
20C2×#99×C59—21no—
21C4×C22⋊C4×C59—21no—
22C59×#342C2212yes—
23#14273686—41no—
24#14273686—41no—
25C2×C4⋊Dic59C2×C411yescharacteristic
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×Dic59⋊C4—21no—
28C2×Dic59⋊C4—21no—
29C2×C22⋊Dic59—21no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×C4×Dic59—21no—
34C2×C4×Dic59—21no—
35C4⋊(C59⋊C8)—41no—
36C22⋊(C59⋊C8)—22no—
37C4×C59⋊C8—41no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C22×D4×C59D411yescharacteristic
45C23×C4×C59C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C4×D4×C59—41no—
48C4×D4×C59—41no—
49C2×C4⋊C4×C59C2×C411yescharacteristic
50C2×C22⋊C4×C59—22no—
51C2×C22⋊C4×C59D411yescharacteristic
52C2×C22⋊C4×C59C2311yescharacteristic
53C2×C42×C59—41no—
54C2×C42×C59D411yescharacteristic
55#99×C59—24no—
56#95×C59—41no—
57C24×C59C22⋊C411yescharacteristic
58C24×C59C2×D411yescharacteristic
59C2×D4×C59—22no—
60C2×D4×C59—22no—
61C2×D4×C59—41no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—42no—
67C22×C4×C59—42no—
68C22×C4×C59—42no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—21no—
78C4⋊C4×C59—41no—
79C22⋊C4×C59—24no—
80C22⋊C4×C59—41no—
81C22⋊C4×C59—41no—
82C22⋊C4×C59—42no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59—82no—
86C22⋊Dic59—42no—
87C22⋊Dic59—42no—
88C22⋊Dic59—42no—
89C22⋊Dic59—42no—
90C4⋊Dic59—41no—
91C4⋊Dic59—41no—
92Dic59⋊C4—42no—
93Dic59⋊C4—42no—
94C4×Dic59—42no—
95C4×Dic59—41no—
96C4×Dic59—41no—
97C2×C59⋊C8—42no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59C2×C22⋊C411yescharacteristic
108C23×C59#13512yes—
109D4×C59—44no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—22no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—44no—
123C2×C4×C59—81no—
124C2×C4×C59—82no—
125C2×C4×C59—82no—
126C2×C4×C59#9911yescharacteristic
127C2×C4×C59#9911yescharacteristic
128C2×C4×C59#10412yes—
129C2×Dic59—22no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C59⋊C8—82no—
142#11727—591nomaximal, Sylow
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—22no—
146C22×C59—42no—
147C22×C59—41no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—81no—
153C22×C59—81no—
154C22×C59—81no—
155C22×C59C2×#9911yescharacteristic
156C22×C59#41611yescharacteristic
157C22×C59C2×#10411yescharacteristic
158C4×C59—21no—
159C4×C59—22no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—41no—
163C4×C59—84no—
164C4×C59#35212yes—
165Dic59—42no—
166Dic59—81no—
167Dic59—81no—
168Dic59—81no—
169#2395—591no—
170C4×#99D5911yescharacteristic
171#1460—591no—
172#1352—592no—
173#1344—592no—
174C2×C59—21no—
175C2×C59—21no—
176C2×C59—42no—
177C2×C59—81no—
178C2×C59#150111yescharacteristic
179C2×C59C2×#35211yescharacteristic
180C2×C59#212611yescharacteristic
181C2×C4×D4C2×D5911yescharacteristic
182C2×#117—591no—
183#431—1181no—
184C2×#99—21no—
185#400—591no—
186#399—591no—
187#398—591no—
188#397—591no—
189#397—1181no—
190#394—1181no—
191C4×C4⋊C4—1181no—
192C4×C22⋊C4—21no—
193C4×C22⋊C4—591no—
194C4×C22⋊C4—1181no—
195#342C2×D5912yes—
196#323—592no—
197#323—1181no—
198C59#1172711yescharacteristic, complemented, Sylow
199C22×D4C22⋊D5911yescharacteristic
200C23×C4C4×D5911yescharacteristic
201C23×C4C22⋊D5911yescharacteristic
202C4×D4—41no—
203C4×D4—41no—
204#117—2361no—
205#117—2361no—
206C2×C4⋊C4—591no—
207C2×C4⋊C4—591no—
208C2×C4⋊C4—1181no—
209C2×C4⋊C4—1181no—
210C2×C4⋊C4C4×D5911yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—1181no—
213C2×C22⋊C4—1181no—
214C2×C22⋊C4—1181no—
215C2×C22⋊C4—1181no—
216C2×C22⋊C4D23611yescharacteristic
217C2×C22⋊C4C22×D5911yescharacteristic
218C2×C42—41no—
219C2×C42—1181no—
220C2×C42—1181no—
221C2×C42D23611yescharacteristic
222C4⋊C8—2361no—
223#99—24no—
224C22⋊C8—1182no—
225C4×C8—2361no—
226#95—41no—
227#95—591no—
228#95—591no—
229#95—1181no—
230#95—1181no—
231#95—1181no—
232#95—1181no—
233C24(C2×D59)⋊C411yescharacteristic
234C24C2×C22⋊D5911yescharacteristic
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—1181no—
246C22×C4—1181no—
247C22×C4—1181no—
248C22×C4—1181no—
249C22×C4(C2×D59)⋊C411yescharacteristic
250C22×C4(C2×D59)⋊C411yescharacteristic
251C22×C4(C2×D59)⋊C411yescharacteristic
252C22×C4C2×C4×D5911yescharacteristic
253C22×C4C2×D23611yescharacteristic
254C2×C8—2362no—
255C4⋊C4—41no—
256C4⋊C4—2361no—
257C4⋊C4—2361no—
258C4⋊C4—2362no—
259C4⋊C4—2362no—
260C22⋊C4—24no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—42no—
264C22⋊C4—2362no—
265C22⋊C4—2362no—
266C22⋊C4—2362no—
267C22⋊C4—2362no—
268C42—21no—
269C42—21no—
270C42—82no—
271C42—2362no—
272C42—2361no—
273C42—2361no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×(C2×D59)⋊C411yescharacteristic
284C23#1427374712yes—
285D4—44no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—44no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—1182no—
303C2×C4—2361no—
304C2×C4—2361no—
305C2×C4—2361no—
306C2×C4—2361no—
307C2×C4—2361no—
308C2×C4—2361no—
309C2×C4—2361no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4—2361no—
314C2×C4#1427364111yescharacteristic
315C2×C4#1427364111yescharacteristic
316C2×C4#1427365512yes—
317C8—4722no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22C2×#1427364111yescharacteristic
331C22#1683739611yescharacteristic
332C22C2×#1427365511yescharacteristic
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—84no—
339C4—4721no—
340C4—4721no—
341C4—4721no—
342C4—2362no—
343C4#1683719112yes—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—81no—
348C2#2142258511yescharacteristic
349C2C2×#1683719111yescharacteristic
350C2#2142575911yescharacteristic
subgroupG/H
centerC2×C4#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC4×#99×C59C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenterC4×#99D59
hypocenterC59#11727
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #21422119C2semidirect
4. #21422095C2semidirect
6. C2×#14273686C4semidirect
8. #16837341C4semidirect
10. #16837332C4semidirect
16. C4×C22⋊Dic59C4semidirect
198. C59#11727semidirect

Maximal subgroups

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

subgroupindex
#214273552
#214224172
#214221192
#214220952
C4×#99×C592
#1172759