Finite Group Databasebeta

← #28507002

Subgroups — #28507002

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#21429088C211yesmaximal, characteristic
2C2×#16837196C214yesmaximal
3#21424084C211yesmaximal, characteristic
4C4×#99×C59C211yesmaximal, characteristic
5C2×#14273727C411yescharacteristic, complemented
6C2×#14273714C2212yes—
7C2×#14273642C2212yes—
8#16837327—21no—
9#16837327—21no—
10#16837307C412yescomplemented
11#16837262—21no—
12#16837262—21no—
13#16837262—21no—
14C4×C4⋊Dic59C411yescharacteristic, complemented
15#16837196—28no—
16#16837169—21no—
17C2×C4×D4×C59C2211yescharacteristic
18C2×#99×C59C2212yes—
19C4×C22⋊C4×C59—21no—
20C59×#342—21no—
21C22×Dic118C2×C412yes—
22#14273727—41no—
23#14273727—41no—
24#14273714—24no—
25#14273714—24no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59—21no—
29C2×C4⋊Dic59—41no—
30C2×C22⋊Dic59—22no—
31C2×C22⋊Dic59—22no—
32C2×C22⋊Dic59—22no—
33C2×C4×Dic59C2×C412yes—
34#14273642—24no—
35C59⋊#95—21no—
36C59⋊#95—22no—
37C59⋊#95—22no—
38C59⋊#95—41no—
39C22×D4×C59C2311yescharacteristic
40C23×C4×C59D411yescharacteristic
41C23×C4×C59D411yescharacteristic
42C4×D4×C59—41no—
43C4×D4×C59—41no—
44C2×C4⋊C4×C59C2×C411yescharacteristic
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59D411yescharacteristic
47C2×C22⋊C4×C59D411yescharacteristic
48C2×C42×C59—41no—
49C2×C42×C59C2×C411yescharacteristic
50#99×C59—22no—
51#99×C59D414yes—
52#95×C59—41no—
53C24×C59C2×D411yescharacteristic
54C24×C59C2×D411yescharacteristic
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C2×D4×C59C2×D412yes—
58C2×D4×C59#4112yes—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—42no—
64C22×C4×C59—42no—
65C22×C4×C59—42no—
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22×C411yescharacteristic
71C22×Dic59—22no—
72C22×Dic59—22no—
73C22×Dic59—42no—
74C2×Dic118—24no—
75C2×Dic118—42no—
76C2×Dic118—42no—
77C4⋊C4×C59—41no—
78C22⋊C4×C59—24no—
79C22⋊C4×C59—41no—
80C22⋊C4×C59—41no—
81C22⋊C4×C59—42no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59—82no—
85C22⋊Dic59—44no—
86C22⋊Dic59—42no—
87C22⋊Dic59—42no—
88C22⋊Dic59—42no—
89C22⋊Dic59—42no—
90C4⋊Dic59—42no—
91C4⋊Dic59—42no—
92C4⋊Dic59—41no—
93C4⋊Dic59—41no—
94C4⋊Dic59—82no—
95C4×Dic59—24no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—41no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59C2×C22⋊C411yescharacteristic
105C23×C59#12112yes—
106C23×C59#12411yescharacteristic
107C23×C59#12711yescharacteristic
108D4×C59—42no—
109D4×C59—42no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—22no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—44no—
122C2×C4×C59—81no—
123C2×C4×C59—81no—
124C2×C4×C59—82no—
125C2×C4×C59—82no—
126C2×C4×C59#9911yescharacteristic
127C2×C4×C59#9911yescharacteristic
128C2×C4×C59C4×D412yes—
129C2×Dic59—42no—
130C2×Dic59—42no—
131C2×Dic59—42no—
132C2×Dic59—42no—
133C2×Dic59—42no—
134C2×Dic59—42no—
135C2×Dic59—82no—
136C2×Dic59—82no—
137C2×Dic59—82no—
138Dic118—84no—
139#13224—591nomaximal, Sylow
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—21no—
143C22×C59—22no—
144C22×C59—21no—
145C22×C59—42no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—81no—
153C22×C59—81no—
154C22×C59#39011yescharacteristic
155C22×C59C2×#9911yescharacteristic
156C22×C59#41011yescharacteristic
157C4×C59—21no—
158C4×C59—22no—
159C4×C59—41no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—84no—
163C4×C59#35412yescomplemented
164Dic59—82no—
165Dic59—82no—
166Dic59—162no—
167#2547—591no—
168C2×#354—594no—
169#1799—591no—
170C4×#99D5911yescharacteristic
171C2×C59—21no—
172C2×C59—21no—
173C2×C59—41no—
174C2×C59—42no—
175C2×C59—41no—
176C2×C59#196411yescharacteristic
177C2×C59C2×#35411yescharacteristic
178C2×C59#212911yescharacteristic
179C2×C4⋊Q8—591no—
180C2×#124—592no—
181C2×C4×D4C2×D5911yescharacteristic
182C2×#99—592no—
183C2×#99C2×D5912yes—
184#397—1181no—
185#397—1181no—
186#391—592no—
187#380—1181no—
188#380—1181no—
189#380—1181no—
190C4×C4⋊C4—591no—
191C4×C22⋊C4—21no—
192#354—1188no—
193#342—21no—
194#342—1181no—
195C59#1322411yescharacteristic, complemented, Sylow
196C22×Q8—592no—
197C22×D4C22×D5911yescharacteristic
198C23×C4D23611yescharacteristic
199C23×C4C22⋊D5911yescharacteristic
200C4⋊Q8—2361no—
201C4⋊Q8—2361no—
202#124—1184no—
203#124—1184no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—1181no—
207C2×C4⋊C4—1181no—
208C2×C4⋊C4—1181no—
209C2×C4⋊C4—2361no—
210C2×C4⋊C4C4×D5911yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—1182no—
213C2×C22⋊C4—1182no—
214C2×C22⋊C4—1182no—
215C2×C22⋊C4D23611yescharacteristic
216C2×C22⋊C4C22⋊D5911yescharacteristic
217C2×C42—41no—
218C2×C42—592no—
219C2×C42C4×D5911yescharacteristic
220#99—22no—
221#99—1184no—
222#99D23614yes—
223#95—41no—
224#95—1181no—
225#95—1182no—
226#95—1182no—
227#95—2361no—
228C24C2×D23611yescharacteristic
229C24C2×C22⋊D5911yescharacteristic
230C2×Q8—1184no—
231C2×Q8—2362no—
232C2×Q8—2362no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4C2×D23612yes—
236C2×D4#1174568512yes—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—1182no—
245C22×C4—1182no—
246C22×C4—2362no—
247C22×C4(C2×D59)⋊C411yescharacteristic
248C22×C4(C2×D59)⋊C411yescharacteristic
249C22×C4(C2×D59)⋊C411yescharacteristic
250C22×C4(C2×D59)⋊C411yescharacteristic
251C22×C4C2×C4×D5911yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—4722no—
254C4⋊C4—2362no—
255C4⋊C4—2362no—
256C4⋊C4—2361no—
257C4⋊C4—2361no—
258C22⋊C4—24no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—42no—
262C22⋊C4—2362no—
263C22⋊C4—2362no—
264C22⋊C4—2364no—
265C22⋊C4—2362no—
266C22⋊C4—2362no—
267C42—21no—
268C42—21no—
269C42—82no—
270C42—1184no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23C2×(C2×D59)⋊C411yescharacteristic
280C23#1427370112yes—
281C23#1427371511yescharacteristic
282C23#1427372411yescharacteristic
283Q8—4724no—
284D4—42no—
285D4—42no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—44no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—4722no—
303C2×C4—4722no—
304C2×C4—4722no—
305C2×C4—2362no—
306C2×C4—2362no—
307C2×C4—2362no—
308C2×C4—2362no—
309C2×C4—2362no—
310C2×C4—2362no—
311C2×C4#1427364111yescharacteristic
312C2×C4#1427364111yescharacteristic
313C2×C4C4×D23612yes—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—81no—
328C22#1683730411yescharacteristic
329C22C2×#1427364111yescharacteristic
330C22#1683737511yescharacteristic
331C4—22no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—84no—
337C4—9442no—
338C4—4722no—
339C4—4722no—
340C4#1683719612yescomplemented
341C2—21no—
342C2—21no—
343C2—42no—
344C2—41no—
345C2—41no—
346C2#2142493211yescharacteristic
347C2C2×#1683719611yescharacteristic
348C2#2142577011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC4×#99×C59C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenterC4×#99D59
hypocenterC59#13224
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
5. C2×#14273727C4semidirect
10. #16837307C4semidirect
14. C4×C4⋊Dic59C4semidirect
163. C4×C59#354semidirect
195. C59#13224semidirect
340. C4#16837196semidirect

Maximal subgroups

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

subgroupindex
#214290882
C2×#168371962
#214240842
C4×#99×C592
#1322459