Finite Group Databasebeta

← #28499513

Subgroups — #28499513

Order 15104.

382 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 79 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#21429956C211yesmaximal, characteristic
2#21422440C211yesmaximal, characteristic, complemented
3#21422125C212yesmaximal, complemented
4#21422098C212yesmaximal
5C59×#1801C211yesmaximal, characteristic, complemented
6C2×#14273715C411yescharacteristic
7#16837459—21no—
8#16837337—21no—
9#16837322C411yescharacteristic
10#16837322C2211yescharacteristic
11#16837319C411yescharacteristic
12#16837319C2211yescharacteristic
13#16837313—21no—
14#16837293—21no—
15#16837293—21no—
16#16837270C411yescharacteristic
17#16837128—21no—
18#16837128C2212yes—
19C2×C4×Q8×C59C2211yescharacteristic
20C59×#423—21no—
21C2×#101×C59—21no—
22C59×#343C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C22×D236C2×C411yescharacteristic
25#14273715—41no—
26#14273715—41no—
27C2×Dic59⋊C4—21no—
28C2×Dic59⋊C4—21no—
29C2×(C2×D59)⋊C4—21no—
30C2×(C2×D59)⋊C4—21no—
31C2×(C2×D59)⋊C4—21no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C4⋊(C59⋊C8)—41no—
36C22⋊(C59⋊C8)—22no—
37#14273638—41no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C22×Q8×C59D411yescharacteristic
45C2×M16×C59—22no—
46C4×Q8×C59—41no—
47C4×Q8×C59—41no—
48C2×C4⋊C4×C59C2×C411yescharacteristic
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C4⋊C4×C59C2311yescharacteristic
51C2×C42×C59C2×C411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C4⋊C8×C59—41no—
55C4⋊C8×C59—41no—
56#101×C59—42no—
57C23×D59—21no—
58C2×Q8×C59—21no—
59C2×Q8×C59—21no—
60C2×Q8×C59—41no—
61C2×Q8×C59—41no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22×C411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×Dic59—21no—
70C22×Dic59—21no—
71C22×Dic59—21no—
72C2×D236—41no—
73C2×D236—81no—
74C2×Dic118—41no—
75C2×Dic118—81no—
76M16×C59—42no—
77M16×C59—42no—
78C2×C8×C59—42no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59—21no—
86C42×C59—21no—
87C42×C59—21no—
88(C2×D59)⋊C4—42no—
89(C2×D59)⋊C4—42no—
90(C2×D59)⋊C4—41no—
91(C2×D59)⋊C4—41no—
92(C2×D59)⋊C4—41no—
93(C2×D59)⋊C4—41no—
94(C2×D59)⋊C4—81no—
95(C2×D59)⋊C4—81no—
96Dic59⋊C4—81no—
97Dic59⋊C4—81no—
98C2×C59⋊C8—42no—
99C23×C59C2×C22⋊C411yescharacteristic
100C22×D59—41no—
101C22×D59—41no—
102C22×D59—41no—
103C22×D59—41no—
104C22×D59—41no—
105C22×D59—41no—
106C22×D59—41no—
107Q8×C59—42no—
108Q8×C59—42no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59#9911yescharacteristic
125C2×C4×C59#9911yescharacteristic
126C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C59#13711yescharacteristic
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137D236—82no—
138Dic118—82no—
139C8×C59—82no—
140C59⋊C8—82no—
141#11833—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59C2×#9911yescharacteristic
145C22×C59#41711yescharacteristic
146C22×C59#42111yescharacteristic
147C2×D59—81no—
148C2×D59—81no—
149C2×D59—81no—
150C2×D59—81no—
151C2×D59—81no—
152C2×D59—81no—
153C2×D59—81no—
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—21no—
157C4×C59—21no—
158C4×C59—41no—
159C4×C59—41no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—41no—
163Dic59—81no—
164Dic59—81no—
165Dic59—81no—
166#2632—591no—
167#1801D5911yescharacteristic, complemented
168#1467—591no—
169#1354—592no—
170#1345—592no—
171C2×C59—21no—
172C2×C59—21no—
173C2×C59#151311yescharacteristic
174C2×C59#212211yescharacteristic
175C2×C59#213311yescharacteristic
176D59—161no—
177C2×#124—591no—
178C2×C4×Q8C2×D5911yescharacteristic
179#432—1181no—
180#423—21no—
181C2×#101—21no—
182#399—1181no—
183#395—591no—
184#395—591no—
185#394—591no—
186#394—591no—
187#393—1181no—
188#388—1181no—
189#388—1181no—
190#383—591no—
191#343C2×D5912yes—
192#324—592no—
193#324—1181no—
194C59#1183311yescharacteristic, complemented, Sylow
195C22×Q8—591no—
196C22×Q8C22⋊D5911yescharacteristic
197C22×D4—591no—
198C2×M16—22no—
199#124—2361no—
200#124—2361no—
201C4×Q8—41no—
202C4×Q8—41no—
203C2×C4⋊C4—1181no—
204C2×C4⋊C4—1181no—
205C2×C4⋊C4C4×D5911yescharacteristic
206C2×C4⋊C4D23611yescharacteristic
207C2×C4⋊C4C22×D5911yescharacteristic
208C2×C22⋊C4—1181no—
209C2×C22⋊C4—1181no—
210C2×C22⋊C4—1181no—
211C2×C22⋊C4—1181no—
212C2×C22⋊C4—1181no—
213C2×C22⋊C4—1181no—
214C2×C42C4×D5911yescharacteristic
215C2×C42D23611yescharacteristic
216C2×C42C22⋊D5911yescharacteristic
217C4⋊C8—41no—
218C4⋊C8—41no—
219C4⋊C8—2361no—
220#101—42no—
221C22⋊C8—1182no—
222#97—2361no—
223#95—591no—
224#95—591no—
225#95—1181no—
226#95—1181no—
227#95—1181no—
228#95—1181no—
229C24—1181no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—4721no—
235C2×Q8—2361no—
236C2×D4—4721no—
237C2×D4—2361no—
238C22×C4—1181no—
239C22×C4—1181no—
240C22×C4—1181no—
241C22×C4(C2×D59)⋊C411yescharacteristic
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—42no—
249M16—42no—
250C2×C8—42no—
251C2×C8—2362no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—4721no—
256C4⋊C4—4721no—
257C22⋊C4—4721no—
258C22⋊C4—4721no—
259C22⋊C4—2362no—
260C22⋊C4—2362no—
261C22⋊C4—2361no—
262C22⋊C4—2361no—
263C22⋊C4—2361no—
264C22⋊C4—2361no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C23—2361no—
272C23—2361no—
273C23—2361no—
274C23—2361no—
275C23—2361no—
276C23—2361no—
277C23—2361no—
278C23C2×(C2×D59)⋊C411yescharacteristic
279Q8—42no—
280Q8—42no—
281Q8—4722no—
282D4—4722no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—2361no—
299C2×C4—2361no—
300C2×C4—2361no—
301C2×C4—2361no—
302C2×C4—2361no—
303C2×C4—2361no—
304C2×C4—2361no—
305C2×C4—2361no—
306C2×C4—2361no—
307C2×C4#1427364111yescharacteristic
308C2×C4#1427364111yescharacteristic
309C2×C4#1427375311yescharacteristic
310C2×C4#1427375811yescharacteristic
311C8—82no—
312C8—4722no—
313C22—21no—
314C22—21no—
315C22—4721no—
316C22—4721no—
317C22—4721no—
318C22—4721no—
319C22—4721no—
320C22—4721no—
321C22—4721no—
322C22C2×#1427364111yescharacteristic
323C22#1683740111yescharacteristic
324C22#1683742011yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—4721no—
335C4—4721no—
336C4—4721no—
337C2—21no—
338C2—21no—
339C2—9441no—
340C2#2142263711yescharacteristic
341C2#2142574711yescharacteristic
342C2#2142578311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1801C2
socleC22×C59C2×#99
prime radicalC22×D236C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1801D59
hypocenterC59#11833
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #21422440C2semidirect
3. #21422125C2semidirect
5. C59×#1801C2semidirect
167. #1801D59semidirect
194. C59#11833semidirect

Maximal subgroups

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

subgroupindex
#214299562
#214224402
#214221252
#214220982
C59×#18012
#1183359