Finite Group Databasebeta

← #28498901

Subgroups — #28498901

Order 15104.

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

normal subgroups only

This group has 552 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#21429846C211yesmaximal, characteristic
2#21423835C211yesmaximal, characteristic, complemented
3#21422100C214yesmaximal, complemented
4C59×#1458C211yesmaximal, characteristic, complemented
5C22×#11745664—21no—
6C2×#14273724C411yescharacteristic
7C2×#14273701—21no—
8#16837319—21no—
9#16837319C2212yes—
10#16837311—21no—
11#16837311C412yes—
12#16837304—21no—
13#16837268C411yescharacteristic
14#16837171—21no—
15#16837171C2212yes—
16C2×#117×C59C2211yescharacteristic
17C59×#431—21no—
18C59×#324C2212yes—
19C59×#323—21no—
20C22×D236—21no—
21C22×D236—21no—
22C22×D236C2×C412yes—
23C22×C22⋊D59—22no—
24C2×#11745664—24no—
25C2×#11745664—22no—
26C2×#11745664—41no—
27C2×#11745664—42no—
28C22×C59⋊C8—41no—
29#14273724—41no—
30#14273724—41no—
31#14273701—42no—
32#14273701—42no—
33C2×C4⋊Dic59—21no—
34C2×(C2×D59)⋊C4—22no—
35C2×(C2×D59)⋊C4—22no—
36C2×(C2×D59)⋊C4—22no—
37C59⋊#95—21no—
38C59⋊#95C2×C412yes—
39C23×C4×C59D411yescharacteristic
40#117×C59—41no—
41#117×C59—41no—
42C2×C4⋊C4×C59D411yescharacteristic
43C2×C4⋊C4×C59C2311yescharacteristic
44C2×C22⋊C4×C59C2×C411yescharacteristic
45C2×C22⋊C4×C59D411yescharacteristic
46C2×C42×C59C2×C411yescharacteristic
47C2×C42×C59D411yescharacteristic
48C4⋊C8×C59—41no—
49C22⋊C8×C59—22no—
50C4×C8×C59—41no—
51C24×C59C22⋊C411yescharacteristic
52C23×D59—22no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—22no—
56C22×C4×C59C22⋊C411yescharacteristic
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59D812yes—
60C22×C4×C59SD1612yes—
61C22×C4×C59C22×C411yescharacteristic
62C22×C4×C59C2×D411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C2×C22⋊D59—44no—
65C2×C22⋊D59—82no—
66C2×C22⋊D59—82no—
67C22×Dic59—22no—
68C2×D236—42no—
69C2×D236—42no—
70C2×D236—41no—
71C2×D236—41no—
72C2×D236—42no—
73C2×D236—42no—
74C2×D236—82no—
75C2×D236—82no—
76C2×C8×C59—42no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—41no—
79C22⋊C4×C59—41no—
80C22⋊C4×C59—41no—
81C42×C59—21no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59—21no—
85(C2×D59)⋊C4—42no—
86(C2×D59)⋊C4—42no—
87(C2×D59)⋊C4—82no—
88(C2×D59)⋊C4—82no—
89C4⋊Dic59—42no—
90#11745664—24no—
91#11745664—44no—
92#11745664—42no—
93C2×C59⋊C8—42no—
94C2×C59⋊C8—44no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—41no—
98C23×C59—41no—
99C23×C59#10212yes—
100C23×C59C2×C22⋊C411yescharacteristic
101C22×D59—42no—
102C22×D59—42no—
103C22×D59—42no—
104C22×D59—42no—
105C22×D59—42no—
106C22×D59—42no—
107C22×D59—42no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—22no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59#9911yescharacteristic
123C2×C4×C59#9911yescharacteristic
124C2×C4×C59#10212yes—
125C2×C4×C59#10412yes—
126C2×C4×C59C2×D811yescharacteristic
127C2×C4×C59C2×SD1611yescharacteristic
128C22⋊D59—88no—
129C2×Dic59—42no—
130C2×Dic59—42no—
131C2×Dic59—42no—
132D236—84no—
133D236—84no—
134D236—82no—
135D236—82no—
136C8×C59—82no—
137C59⋊C8—44no—
138#11699—591nomaximal, Sylow
139C22×C59—21no—
140C22×C59—21no—
141C22×C59—22no—
142C22×C59—42no—
143C22×C59—41no—
144C22×C59—41no—
145C22×C59—81no—
146C22×C59C2×#9911yescharacteristic
147C22×C59C2×#10211yescharacteristic
148C22×C59C2×#10411yescharacteristic
149C2×D59—82no—
150C2×D59—82no—
151C2×D59—82no—
152C2×D59—82no—
153C2×D59—82no—
154C2×D59—82no—
155C2×D59—82no—
156C4×C59—21no—
157C4×C59—21no—
158C4×C59—41no—
159C4×C59—41no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59#32712yes—
163C4×C59#32712yes—
164Dic59—82no—
165#2619—591no—
166#1742—591no—
167#1458D5911yescharacteristic, complemented
168#1345—594no—
169C2×C59—21no—
170C2×C59—21no—
171C2×C59—42no—
172C2×C59C2×#32711yescharacteristic
173C2×C59#212211yescharacteristic
174C2×C59#213411yescharacteristic
175D59—162no—
176C22×M16—1181no—
177C2×#127—591no—
178C2×#121—1181no—
179C2×#117C2×D5911yescharacteristic
180#431—21no—
181#394—592no—
182#394—1181no—
183#392—592no—
184#392—1181no—
185#390—1181no—
186#382—591no—
187#343—592no—
188#343—1181no—
189#324C2×D5912yes—
190#323—21no—
191C59#1169911yescharacteristic, complemented, Sylow
192C22×D4—592no—
193C22×D4—1182no—
194C22×D4—1181no—
195C22×D4—1181no—
196C23×C4D23611yescharacteristic
197C2×M16—1184no—
198C2×M16—1182no—
199C2×M16—2361no—
200C2×M16—2362no—
201C22×C8—2361no—
202#127—2361no—
203#127—2361no—
204#121—2362no—
205#121—2362no—
206#117—41no—
207#117—41no—
208C2×C4⋊C4—1181no—
209C2×C4⋊C4D23611yescharacteristic
210C2×C4⋊C4C22×D5911yescharacteristic
211C2×C22⋊C4—1182no—
212C2×C22⋊C4—1182no—
213C2×C22⋊C4—1182no—
214C2×C22⋊C4C4×D5911yescharacteristic
215C2×C22⋊C4C22⋊D5911yescharacteristic
216C2×C42C4×D5911yescharacteristic
217C2×C42C22⋊D5911yescharacteristic
218C4⋊C8—41no—
219C22⋊C8—22no—
220C4×C8—41no—
221#95—592no—
222#95—1181no—
223C24—1182no—
224C24(C2×D59)⋊C411yescharacteristic
225C2×D4—4722no—
226C2×D4—4722no—
227C2×D4—4722no—
228C2×D4—4722no—
229C2×D4—2362no—
230C2×D4—2362no—
231C2×D4—2364no—
232C2×D4—2361no—
233C2×D4—2361no—
234C2×D4—2362no—
235C2×D4—2362no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—1182no—
240C22×C4#1174566012yes—
241C22×C4D47212yes—
242C22×C4(C2×D59)⋊C411yescharacteristic
243C22×C4(C2×D59)⋊C411yescharacteristic
244C22×C4(C2×D59)⋊C411yescharacteristic
245C22×C4C2×C4×D5911yescharacteristic
246C22×C4C2×D23611yescharacteristic
247C22×C4C2×C22⋊D5911yescharacteristic
248M16—1184no—
249M16—2364no—
250M16—2362no—
251C2×C8—42no—
252C2×C8—2362no—
253C2×C8—2364no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—2362no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—4722no—
260C22⋊C4—4722no—
261C22⋊C4—2362no—
262C22⋊C4—2362no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—2362no—
272C23—2362no—
273C23—2362no—
274C23—2362no—
275C23—2362no—
276C23—2362no—
277C23—2362no—
278C23#1427364912yes—
279C23C2×(C2×D59)⋊C411yescharacteristic
280D4—4724no—
281D4—4724no—
282D4—4728no—
283D4—4722no—
284D4—4722no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—2362no—
300C2×C4—2362no—
301C2×C4—2362no—
302C2×C4#1427364111yescharacteristic
303C2×C4#1427364111yescharacteristic
304C2×C4#1427364912yes—
305C2×C4#1427365312yes—
306C2×C4C2×D47211yescharacteristic
307C2×C4C2×#1174566011yescharacteristic
308C8—82no—
309C8—2364no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—4722no—
318C22—4722no—
319C22—4722no—
320C22—4722no—
321C22—4722no—
322C22—4722no—
323C22—4722no—
324C22C2×#1427364111yescharacteristic
325C22C2×#1427364911yescharacteristic
326C22C2×#1427365311yescharacteristic
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—4722no—
334C4#1683713712yes—
335C4#1683713712yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—9442no—
340C2C2×#1683713711yescharacteristic
341C2#2142574511yescharacteristic
342C2#2142578711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1458C2
socleC22×C59C2×#99
prime radical#21429846C2
cosocleC2×C4⋊C4×C59C23
hypercenter#1458D59
hypocenterC59#11699
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #21423835C2semidirect
3. #21422100C2semidirect
4. C59×#1458C2semidirect
167. #1458D59semidirect
191. C59#11699semidirect

Maximal subgroups

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

subgroupindex
#214298462
#214238352
#214221002
C59×#14582
#1169959