Finite Group Databasebeta

← #28504680

Subgroups — #28504680

Order 15104.

588 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#16837195C212yesmaximal
2#21424084C212yesmaximal, complemented
3#21422221C212yesmaximal, complemented
4C59×#2328C211yesmaximal, characteristic
5C2×#14273642C2214yes—
6#16837262—22no—
7#16837195—24no—
8#16837169—22no—
9C2×#127×C59C2211yescharacteristic
10C2×C4×D4×C59C411yescharacteristic, complemented
11C2×C4×D4×C59C2211yescharacteristic
12C59×#390C411yescharacteristic, complemented
13C59×#390C2211yescharacteristic
14C59×#386—22no—
15C59×#386—22no—
16C59×#382C411yescharacteristic, complemented
17C59×#376C411yescharacteristic, complemented
18C2×C4⋊Dic59—42no—
19C2×C22⋊Dic59—24no—
20#14273642—24no—
21#14273642—24no—
22C59⋊#95—42no—
23C22×D4×C59—22no—
24C22×D4×C59C2×C411yescharacteristic
25C22×D4×C59C2311yescharacteristic
26C23×C4×C59—21no—
27C23×C4×C59D412yes—
28#127×C59—22no—
29#127×C59D412yes—
30#127×C59Q812yes—
31C4×D4×C59—42no—
32C4×D4×C59—42no—
33C2×C4⋊C4×C59C2×C411yescharacteristic
34C2×C4⋊C4×C59C2×C411yescharacteristic
35C2×C22⋊C4×C59—21no—
36C2×C22⋊C4×C59—22no—
37C2×C22⋊C4×C59—22no—
38C2×C22⋊C4×C59D412yes—
39C2×C42×C59C2×C411yescharacteristic
40C2×C42×C59C2×C411yescharacteristic
41C2×C42×C59C2×C411yescharacteristic
42#95×C59—21no—
43#95×C59—21no—
44#95×C59—22no—
45C24×C59—21no—
46C24×C59C2×D412yes—
47C2×D4×C59—24no—
48C2×D4×C59—24no—
49C2×D4×C59—21no—
50C2×D4×C59—41no—
51C2×D4×C59—41no—
52C2×D4×C59—41no—
53C2×D4×C59—41no—
54C2×D4×C59—82no—
55C2×D4×C59—82no—
56C2×D4×C59C4⋊C412yescomplemented
57C2×D4×C59C2×D411yescharacteristic
58C2×D4×C59C2×Q811yescharacteristic
59C2×D4×C59#4112yes—
60C22×C4×C59—21no—
61C22×C4×C59—22no—
62C22×C4×C59—22no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—42no—
66C22×C4×C59—82no—
67C22×C4×C59—82no—
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C412yes—
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22×C411yescharacteristic
72C22×Dic59—44no—
73C4⋊C4×C59—41no—
74C4⋊C4×C59—41no—
75C22⋊C4×C59—42no—
76C22⋊C4×C59—42no—
77C22⋊C4×C59—82no—
78C22⋊C4×C59—82no—
79C42×C59—22no—
80C42×C59—21no—
81C42×C59—41no—
82C42×C59C4⋊C412yescomplemented
83C22⋊Dic59—44no—
84C22⋊Dic59—44no—
85C4⋊Dic59—84no—
86C23×C59—22no—
87C23×C59—22no—
88C23×C59—22no—
89C23×C59—42no—
90C23×C59—42no—
91C23×C59—42no—
92C23×C59—42no—
93C23×C59—41no—
94C23×C59—41no—
95C23×C59C2×C22⋊C411yescharacteristic
96C23×C59#12112yes—
97C23×C59C22⋊Q812yes—
98D4×C59—41no—
99D4×C59—41no—
100D4×C59—42no—
101D4×C59—42no—
102D4×C59—42no—
103D4×C59—84no—
104D4×C59—84no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—22no—
108C2×C4×C59—21no—
109C2×C4×C59—22no—
110C2×C4×C59—41no—
111C2×C4×C59—44no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—42no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—42no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—82no—
122C2×C4×C59—84no—
123C2×C4×C59#9912yes—
124C2×C4×C59C2×C4⋊C411yescharacteristic
125C2×C4×C59#11711yescharacteristic
126C2×Dic59—84no—
127C2×Dic59—84no—
128C2×Dic59—84no—
129#12799—591nomaximal, Sylow
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—22no—
133C22×C59—22no—
134C22×C59—42no—
135C22×C59—42no—
136C22×C59—44no—
137C22×C59—42no—
138C22×C59—42no—
139C22×C59—42no—
140C22×C59—82no—
141C22×C59—82no—
142C22×C59—81no—
143C22×C59—81no—
144C22×C59#38011yescharacteristic
145C22×C59C2×#9911yescharacteristic
146C22×C59#41011yescharacteristic
147C4×C59—21no—
148C4×C59—21no—
149C4×C59—22no—
150C4×C59—42no—
151C4×C59—42no—
152C4×C59—41no—
153C4×C59—81no—
154C4×C59—81no—
155Dic59—164no—
156#2328D5911yescharacteristic
157C2×#353—592no—
158#1799—592no—
159#1393—592no—
160C2×C59—21no—
161C2×C59—21no—
162C2×C59—42no—
163C2×C59—42no—
164C2×C59—82no—
165C2×C59#179911yescharacteristic
166C2×C59#212711yescharacteristic
167C2×C59#212811yescharacteristic
168C2×#127C2×D5911yescharacteristic
169C2×C4×D4Dic5911yescharacteristic, complemented
170C2×C4×D4C2×D5911yescharacteristic
171C2×#99—594no—
172#390Dic5911yescharacteristic, complemented
173#390C2×D5911yescharacteristic
174#386—22no—
175#386—22no—
176#382Dic5911yescharacteristic, complemented
177#380—1182no—
178#376Dic5911yescharacteristic, complemented
179#353—1184no—
180#342—1182no—
181C59#1279911yescharacteristic, complemented, Sylow
182C22×D4—22no—
183C22×D4C2×Dic5911yescharacteristic
184C22×D4C22×D5911yescharacteristic
185C23×C4—21no—
186C23×C4C22⋊D5912yes—
187#127—22no—
188#127Dic11812yes—
189#127D23612yes—
190C4×D4—42no—
191C4×D4—42no—
192C2×C4⋊C4—2362no—
193C2×C4⋊C4C2×Dic5911yescharacteristic
194C2×C4⋊C4C2×Dic5911yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—1184no—
199C2×C22⋊C4C22⋊D5912yes—
200C2×C42C2×Dic5911yescharacteristic
201C2×C42C2×Dic5911yescharacteristic
202C2×C42C2×Dic5911yescharacteristic
203#99—1184no—
204#99—1184no—
205#95—21no—
206#95—22no—
207#95—21no—
208#95—2362no—
209C24—21no—
210C24C2×C22⋊D5912yes—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—21no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—82no—
219C2×D4—82no—
220C2×D4C4⋊Dic5912yescomplemented
221C2×D4C2×Dic11811yescharacteristic
222C2×D4C2×D23611yescharacteristic
223C2×D4#1174568512yes—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—2364no—
233C22×C4C22⋊Dic5911yescharacteristic
234C22×C4C22⋊Dic5912yes—
235C22×C4C22⋊Dic5911yescharacteristic
236C22×C4C22×Dic5911yescharacteristic
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—4724no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—82no—
243C22⋊C4—82no—
244C22⋊C4—2364no—
245C22⋊C4—2364no—
246C42—22no—
247C42—21no—
248C42—41no—
249C42C4⋊Dic5912yescomplemented
250C23—22no—
251C23—22no—
252C23—22no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—41no—
258C23—41no—
259C23C2×C22⋊Dic5911yescharacteristic
260C23#1427370112yes—
261C23#1427370612yes—
262D4—41no—
263D4—41no—
264D4—42no—
265D4—42no—
266D4—42no—
267D4—84no—
268D4—84no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—82no—
286C2×C4—84no—
287C2×C4—4724no—
288C2×C4—4724no—
289C2×C4—4724no—
290C2×C4#1427364212yes—
291C2×C4C2×C4⋊Dic5911yescharacteristic
292C2×C4#1427368611yescharacteristic
293C22—21no—
294C22—21no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—82no—
304C22—82no—
305C22—81no—
306C22—81no—
307C22#1683726211yescharacteristic
308C22C2×#1427364211yescharacteristic
309C22#1683737611yescharacteristic
310C4—22no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—42no—
315C4—41no—
316C4—81no—
317C4—81no—
318C4—9444no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2—42no—
323C2—82no—
324C2#2142408411yescharacteristic
325C2#2142576511yescharacteristic
326C2#2142576811yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#2328C2
socleC22×C59C2×#99
prime radicalC2×#127×C59C22
cosocleC22×D4×C59C23
hypercenter#2328D59
hypocenterC59#12799
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. #21424084C2semidirect
3. #21422221C2semidirect
10. C2×C4×D4×C59C4semidirect
12. C59×#390C4semidirect
16. C59×#382C4semidirect
17. C59×#376C4semidirect
56. C2×D4×C59C4⋊C4semidirect
82. C42×C59C4⋊C4semidirect
169. C2×C4×D4Dic59semidirect
172. #390Dic59semidirect
176. #382Dic59semidirect
178. #376Dic59semidirect
181. C59#12799semidirect
220. C2×D4C4⋊Dic59semidirect
249. C42C4⋊Dic59semidirect

Maximal subgroups

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

subgroupindex
C2×#168371952
#214240842
#214222212
C59×#23282
#1279959