Finite Group Databasebeta

← #28486458

Subgroups — #28486458

Order 15104.

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

normal subgroups only

This group has 578 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#21428413C211yesmaximal, characteristic
2#21424187C212yesmaximal, complemented
3#21424084C211yesmaximal, characteristic
4#21421964C212yesmaximal, complemented
5C59×#2972C211yesmaximal, characteristic
6C22×C4⋊Dic59C2211yescharacteristic
7C2×#14273642—21no—
8#16837341C412yescomplemented
9#16837338C412yescomplemented
10#16837300—21no—
11#16837282—21no—
12#16837274—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—
24C2×C4⋊Dic59—22no—
25C2×C4⋊Dic59—22no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59—22no—
29C2×C4⋊Dic59C2×C412yes—
30C2×Dic59⋊C4—22no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—22no—
33C2×C22⋊Dic59—22no—
34#14273642—42no—
35C59⋊#95—22no—
36C59⋊#95—22no—
37C59⋊#95—22no—
38C59⋊#95C2×C412yes—
39C22×D4×C59D411yescharacteristic
40C23×C4×C59D411yescharacteristic
41C23×C4×C59C2311yescharacteristic
42C22×C8×C59—22no—
43C22×C8×C59D414yes—
44C4×D4×C59—22no—
45C4×D4×C59—22no—
46C2×C4⋊C4×C59C2×C411yescharacteristic
47C2×C22⋊C4×C59C2×C411yescharacteristic
48C2×C22⋊C4×C59D411yescharacteristic
49C2×C42×C59D411yescharacteristic
50C4⋊C8×C59—22no—
51C22⋊C8×C59—22no—
52C22⋊C8×C59—22no—
53C22⋊C8×C59—22no—
54C4×C8×C59—22no—
55C24×C59C22⋊C411yescharacteristic
56C24×C59C2×D411yescharacteristic
57C2×D4×C59—22no—
58C2×D4×C59—22no—
59C2×D4×C59SD1612yes—
60C2×D4×C59Q1612yes—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—42no—
64C22×C4×C59—42no—
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D412yes—
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59#4112yes—
72C22×Dic59—22no—
73C22×Dic59—22no—
74C22×Dic59—42no—
75C22×Dic59—42no—
76C22×Dic59—82no—
77C2×C8×C59—24no—
78C2×C8×C59—44no—
79C2×C8×C59—82no—
80C4⋊C4×C59—22no—
81C22⋊C4×C59—22no—
82C22⋊C4×C59—22no—
83C42×C59—21no—
84C42×C59—21no—
85C22⋊Dic59—42no—
86C22⋊Dic59—42no—
87C22⋊Dic59—42no—
88C22⋊Dic59—82no—
89C4⋊Dic59—24no—
90C4⋊Dic59—42no—
91C4⋊Dic59—41no—
92C4⋊Dic59—41no—
93C4⋊Dic59—44no—
94Dic59⋊C4—82no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—41no—
98C23×C59—41no—
99C23×C59—41no—
100C23×C59—41no—
101C23×C59#10312yes—
102C23×C59C2×C22⋊C411yescharacteristic
103C23×C59#12112yes—
104C23×C59C2×SD1611yescharacteristic
105C23×C59C2×Q1611yescharacteristic
106C23×C59#13512yes—
107D4×C59—24no—
108D4×C59—24no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—24no—
112C2×C4×C59—22no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—41no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—81no—
121C2×C4×C59#10312yes—
122C2×C4×C59C4×D412yes—
123C2×C4×C59#12011yescharacteristic
124C2×C4×C59#12311yescharacteristic
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129C2×Dic59—42no—
130C2×Dic59—42no—
131C2×Dic59—84no—
132C2×Dic59—82no—
133C2×Dic59—82no—
134C2×Dic59—82no—
135C8×C59—44no—
136#9469—591nomaximal, Sylow
137C22×C59—21no—
138C22×C59—22no—
139C22×C59—21no—
140C22×C59—22no—
141C22×C59—22no—
142C22×C59—41no—
143C22×C59—41no—
144C22×C59—42no—
145C22×C59—42no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—81no—
149C22×C59#38611yescharacteristic
150C22×C59C2×#10311yescharacteristic
151C22×C59#41611yescharacteristic
152C22×C59#46512yes—
153C22×C59#46712yes—
154C4×C59—21no—
155C4×C59—22no—
156C4×C59—22no—
157C4×C59—41no—
158C4×C59—41no—
159C4×C59#45812yes—
160Dic59—82no—
161Dic59—82no—
162Dic59—162no—
163#2972D5911yescharacteristic
164#2487—591no—
165#1824—592no—
166#1799—591no—
167#1289—592no—
168C2×C59—21no—
169C2×C59—22no—
170C2×C59—21no—
171C2×C59—22no—
172C2×C59—42no—
173C2×C59#189011yescharacteristic
174C2×C59#189211yescharacteristic
175C2×C59#191011yescharacteristic
176C2×C4×D4C2×D5911yescharacteristic
177C22×C4⋊C4—591no—
178C8×D4—24no—
179#431—21no—
180C2×#99—1181no—
181C2×C22⋊C8—21no—
182C2×C22⋊C8C2×D5912yes—
183#400—592no—
184#399—592no—
185#389—1181no—
186#386—1181no—
187#385—1181no—
188#380—591no—
189#380—592no—
190#380—1181no—
191#342—1181no—
192#340—1184no—
193C59#946911yescharacteristic, complemented, Sylow
194C22×D4D23611yescharacteristic
195C23×C4—1181no—
196C23×C4C22⋊D5911yescharacteristic
197C23×C4C22×D5911yescharacteristic
198C22×C8—22no—
199C22×C8C22⋊D5914yes—
200C4×D4—22no—
201C4×D4—22no—
202C2×C4⋊C4—592no—
203C2×C4⋊C4—1182no—
204C2×C4⋊C4—1182no—
205C2×C4⋊C4—1182no—
206C2×C4⋊C4—1181no—
207C2×C4⋊C4—1181no—
208C2×C4⋊C4—1182no—
209C2×C4⋊C4C4×D5911yescharacteristic
210C2×C22⋊C4—1182no—
211C2×C22⋊C4—1181no—
212C2×C22⋊C4—1182no—
213C2×C22⋊C4C4×D5911yescharacteristic
214C2×C22⋊C4D23611yescharacteristic
215C2×C42C22⋊D5911yescharacteristic
216C4⋊C8—22no—
217#99—2362no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C4×C8—22no—
222#95—592no—
223#95—1182no—
224#95—1182no—
225#95—1182no—
226C24(C2×D59)⋊C411yescharacteristic
227C24C2×D23611yescharacteristic
228C2×D4—22no—
229C2×D4—22no—
230C2×D4#1174566012yes—
231C2×D4Dic23612yes—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—1182no—
237C22×C4—1182no—
238C22×C4—4722no—
239C22×C4—2362no—
240C22×C4—2362no—
241C22×C4(C2×D59)⋊C411yescharacteristic
242C22×C4(C2×D59)⋊C411yescharacteristic
243C22×C4(C2×D59)⋊C411yescharacteristic
244C22×C4C2×C4×D5911yescharacteristic
245C22×C4#1174568512yes—
246C22×C4C2×C22⋊D5912yes—
247C22×C4C2×C22⋊D5911yescharacteristic
248C2×C8—24no—
249C2×C8—44no—
250C2×C8—82no—
251C4⋊C4—22no—
252C4⋊C4—1184no—
253C4⋊C4—4722no—
254C4⋊C4—2364no—
255C4⋊C4—2362no—
256C4⋊C4—2361no—
257C4⋊C4—2361no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—4722no—
261C22⋊C4—2362no—
262C22⋊C4—2362no—
263C22⋊C4—2362no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23#1427365212yes—
273C23C2×(C2×D59)⋊C411yescharacteristic
274C23#1427370112yes—
275C23C2×#1174566011yescharacteristic
276C23C2×Dic23611yescharacteristic
277C23#1427374912yes—
278D4—24no—
279D4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—4722no—
293C2×C4—4722no—
294C2×C4—4722no—
295C2×C4—4724no—
296C2×C4—2362no—
297C2×C4—2362no—
298C2×C4—2362no—
299C2×C4—2362no—
300C2×C4—2362no—
301C2×C4—2362no—
302C2×C4#1427365212yes—
303C2×C4C4×C22⋊D5912yes—
304C2×C4#1427369611yescharacteristic
305C2×C4#1427371011yescharacteristic
306C8—44no—
307C22—21no—
308C22—22no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—81no—
319C22#1683728111yescharacteristic
320C22C2×#1427365211yescharacteristic
321C22#1683739911yescharacteristic
322C22#1683763112yes—
323C22#1683764312yes—
324C4—21no—
325C4—22no—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—9442no—
330C4—4722no—
331C4—4722no—
332C4#1683759212yes—
333C2—22no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—42no—
338C2#2142450911yescharacteristic
339C2#2142451911yescharacteristic
340C2#2142462311yescharacteristic
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#9469
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. #21424187C2semidirect
4. #21421964C2semidirect
8. #16837341C4semidirect
9. #16837338C4semidirect
193. C59#9469semidirect

Maximal subgroups

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

subgroupindex
#214284132
#214241872
#214240842
#214219642
C59×#29722
#946959