Finite Group Databasebeta

← #28486463

Subgroups — #28486463

Order 15104.

598 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 135 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#21428404C211yesmaximal, characteristic
2#21424183C212yesmaximal, complemented
3#21424084C211yesmaximal, characteristic
4#21421964C212yesmaximal, complemented
5C59×#2972C211yesmaximal, characteristic
6C2×#14273708—21no—
7C22×C4⋊Dic59C2211yescharacteristic
8C2×#14273642—21no—
9#16837332C412yescomplemented
10#16837327C412yescomplemented
11#16837300—21no—
12#16837263—21no—
13#16837262—21no—
14#16837262C2212yes—
15#16837262C2211yescharacteristic
16#16837169—21no—
17#16837165—24no—
18C2×C4×D4×C59C2211yescharacteristic
19C8×D4×C59—24no—
20C59×#431—21no—
21C2×C22⋊C8×C59—21no—
22C2×C22⋊C8×C59C2212yes—
23C23×Dic59—21no—
24#14273708—42no—
25#14273708—42no—
26C2×C4⋊Dic59—22no—
27C2×C4⋊Dic59—22no—
28C2×C4⋊Dic59—21no—
29C2×C4⋊Dic59—21no—
30C2×C4⋊Dic59—22no—
31C2×C4⋊Dic59C2×C412yes—
32C2×Dic59⋊C4—22no—
33C2×C22⋊Dic59—22no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—22no—
36#14273642—42no—
37C59⋊#95—22no—
38C59⋊#95—22no—
39C59⋊#95—22no—
40C59⋊#95C2×C412yes—
41C22×D4×C59D411yescharacteristic
42C23×C4×C59D411yescharacteristic
43C23×C4×C59C2311yescharacteristic
44C22×C8×C59—22no—
45C22×C8×C59D414yes—
46C4×D4×C59—22no—
47C4×D4×C59—22no—
48C2×C4⋊C4×C59C2×C411yescharacteristic
49C2×C22⋊C4×C59C2×C411yescharacteristic
50C2×C22⋊C4×C59D411yescharacteristic
51C2×C42×C59D411yescharacteristic
52C4⋊C8×C59—22no—
53C22⋊C8×C59—22no—
54C22⋊C8×C59—22no—
55C22⋊C8×C59—22no—
56C4×C8×C59—22no—
57C24×C59C22⋊C411yescharacteristic
58C24×C59C2×D411yescharacteristic
59C2×D4×C59—22no—
60C2×D4×C59—22no—
61C2×D4×C59D812yes—
62C2×D4×C59SD1612yes—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—42no—
66C22×C4×C59—42no—
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D412yes—
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59#4112yes—
74C22×Dic59—22no—
75C22×Dic59—22no—
76C22×Dic59—42no—
77C22×Dic59—41no—
78C22×Dic59—41no—
79C22×Dic59—42no—
80C22×Dic59—42no—
81C2×C8×C59—24no—
82C2×C8×C59—44no—
83C2×C8×C59—82no—
84C4⋊C4×C59—22no—
85C22⋊C4×C59—22no—
86C22⋊C4×C59—22no—
87C42×C59—21no—
88C42×C59—21no—
89C22⋊Dic59—44no—
90C22⋊Dic59—41no—
91C22⋊Dic59—41no—
92C22⋊Dic59—42no—
93C22⋊Dic59—42no—
94C4⋊Dic59—24no—
95C4⋊Dic59—42no—
96C4⋊Dic59—41no—
97C4⋊Dic59—41no—
98C4⋊Dic59—44no—
99Dic59⋊C4—44no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59#10212yes—
107C23×C59C2×C22⋊C411yescharacteristic
108C23×C59#12112yes—
109C23×C59C2×D811yescharacteristic
110C23×C59C2×SD1611yescharacteristic
111C23×C59#13512yes—
112D4×C59—24no—
113D4×C59—24no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—24no—
117C2×C4×C59—22no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—41no—
121C2×C4×C59—42no—
122C2×C4×C59—42no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—81no—
126C2×C4×C59#10212yes—
127C2×C4×C59C4×D412yes—
128C2×C4×C59#12011yescharacteristic
129C2×C4×C59#12311yescharacteristic
130C2×Dic59—42no—
131C2×Dic59—42no—
132C2×Dic59—42no—
133C2×Dic59—42no—
134C2×Dic59—42no—
135C2×Dic59—42no—
136C2×Dic59—82no—
137C2×Dic59—82no—
138C2×Dic59—82no—
139C2×Dic59—82no—
140C2×Dic59—82no—
141C8×C59—44no—
142#9470—591nomaximal, Sylow
143C22×C59—21no—
144C22×C59—22no—
145C22×C59—21no—
146C22×C59—22no—
147C22×C59—22no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—42no—
151C22×C59—42no—
152C22×C59—41no—
153C22×C59—41no—
154C22×C59—81no—
155C22×C59#38611yescharacteristic
156C22×C59C2×#10211yescharacteristic
157C22×C59#41611yescharacteristic
158C22×C59#46512yes—
159C22×C59#46612yes—
160C4×C59—21no—
161C4×C59—22no—
162C4×C59—22no—
163C4×C59—41no—
164C4×C59—41no—
165C4×C59#45812yes—
166Dic59—82no—
167Dic59—82no—
168Dic59—162no—
169#2972D5911yescharacteristic
170#2486—591no—
171#1823—592no—
172#1799—591no—
173#1289—592no—
174C2×C59—21no—
175C2×C59—22no—
176C2×C59—21no—
177C2×C59—22no—
178C2×C59—42no—
179C2×C59#189011yescharacteristic
180C2×C59#189311yescharacteristic
181C2×C59#190911yescharacteristic
182C2×#123—1181no—
183C2×C4×D4C2×D5911yescharacteristic
184C22×C4⋊C4—591no—
185C8×D4—24no—
186#431—21no—
187C2×#99—1181no—
188C2×C22⋊C8—21no—
189C2×C22⋊C8C2×D5912yes—
190#398—592no—
191#397—592no—
192#389—1181no—
193#381—1181no—
194#380—591no—
195#380—592no—
196#380—1181no—
197#342—1181no—
198#340—1184no—
199C59#947011yescharacteristic, complemented, Sylow
200C22×D4D23611yescharacteristic
201C23×C4—1181no—
202C23×C4C22⋊D5911yescharacteristic
203C23×C4C22×D5911yescharacteristic
204C22×C8—22no—
205C22×C8C22⋊D5914yes—
206#123—2362no—
207#123—2362no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—592no—
211C2×C4⋊C4—1182no—
212C2×C4⋊C4—1182no—
213C2×C4⋊C4—1182no—
214C2×C4⋊C4—1181no—
215C2×C4⋊C4—1181no—
216C2×C4⋊C4—1182no—
217C2×C4⋊C4C4×D5911yescharacteristic
218C2×C22⋊C4—1182no—
219C2×C22⋊C4—1181no—
220C2×C22⋊C4—1182no—
221C2×C22⋊C4C4×D5911yescharacteristic
222C2×C22⋊C4D23611yescharacteristic
223C2×C42C22⋊D5911yescharacteristic
224C4⋊C8—22no—
225#99—2362no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C4×C8—22no—
230#95—592no—
231#95—1182no—
232#95—1182no—
233#95—1182no—
234C24(C2×D59)⋊C411yescharacteristic
235C24C2×D23611yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4#1174566012yes—
239C2×D4D47212yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—1182no—
245C22×C4—1182no—
246C22×C4—2362no—
247C22×C4—2362no—
248C22×C4—2361no—
249C22×C4—2361no—
250C22×C4—2362no—
251C22×C4(C2×D59)⋊C411yescharacteristic
252C22×C4(C2×D59)⋊C411yescharacteristic
253C22×C4(C2×D59)⋊C411yescharacteristic
254C22×C4C2×C4×D5911yescharacteristic
255C22×C4#1174568512yes—
256C22×C4C2×C22⋊D5912yes—
257C22×C4C2×C22⋊D5911yescharacteristic
258C2×C8—24no—
259C2×C8—44no—
260C2×C8—82no—
261C4⋊C4—22no—
262C4⋊C4—1184no—
263C4⋊C4—2364no—
264C4⋊C4—2362no—
265C4⋊C4—2364no—
266C4⋊C4—2361no—
267C4⋊C4—2361no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—2362no—
271C22⋊C4—2362no—
272C22⋊C4—2364no—
273C22⋊C4—2361no—
274C22⋊C4—2361no—
275C42—21no—
276C42—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23#1427364912yes—
284C23C2×(C2×D59)⋊C411yescharacteristic
285C23#1427370112yes—
286C23C2×D47211yescharacteristic
287C23C2×#1174566011yescharacteristic
288C23#1427374912yes—
289D4—24no—
290D4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4—4722no—
304C2×C4—4722no—
305C2×C4—4722no—
306C2×C4—4722no—
307C2×C4—4722no—
308C2×C4—2362no—
309C2×C4—2362no—
310C2×C4—2362no—
311C2×C4—2362no—
312C2×C4—2362no—
313C2×C4—2362no—
314C2×C4#1427364912yes—
315C2×C4C4×C22⋊D5912yes—
316C2×C4#1427369611yescharacteristic
317C2×C4#1427371011yescharacteristic
318C8—44no—
319C22—21no—
320C22—22no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1683728111yescharacteristic
332C22C2×#1427364911yescharacteristic
333C22#1683739911yescharacteristic
334C22#1683763112yes—
335C22#1683763812yes—
336C4—21no—
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—9442no—
342C4—4722no—
343C4—4722no—
344C4#1683759212yes—
345C2—22no—
346C2—21no—
347C2—21no—
348C2—22no—
349C2—42no—
350C2#2142450911yescharacteristic
351C2#2142452411yescharacteristic
352C2#2142461811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2972C2
socleC22×C59#386
prime radicalC22×D4×C59D4
cosocleC23×C4×C59C23
hypercenter#2972D59
hypocenterC59#9470
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #21424183C2semidirect
4. #21421964C2semidirect
9. #16837332C4semidirect
10. #16837327C4semidirect
199. C59#9470semidirect

Maximal subgroups

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

subgroupindex
#214284042
#214241832
#214240842
#214219642
C59×#29722
#947059