Finite Group Databasebeta

← #28504940

Subgroups — #28504940

Order 15104.

628 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 99 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#21426688C211yesmaximal, characteristic
2#21424014C211yesmaximal, characteristic
3#21422215C214yesmaximal, complemented
4C59×#1785C211yesmaximal, characteristic
5C2×#14273686C411yescharacteristic, complemented
6C22×C4⋊Dic59C411yescharacteristic, complemented
7C2×#14273642—21no—
8C2×#14273642C2212yes—
9#16837309—21no—
10#16837305—21no—
11#16837282C412yescomplemented
12#16837282C2212yes—
13#16837265—21no—
14#16837262—21no—
15#16837262—21no—
16C22×#41×C59C2211yescharacteristic
17C2×#101×C59—21no—
18C2×#100×C59C2212yes—
19C59×#407—21no—
20C23×Dic59C2×C412yes—
21#14273686—42no—
22C2×C4⋊Dic59—21no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59—42no—
26C2×C22⋊Dic59—22no—
27C2×C22⋊Dic59—22no—
28C2×C22⋊Dic59—22no—
29C2×C22⋊Dic59C2×C412yes—
30C2×C4×Dic59—21no—
31#14273642—24no—
32#14273642—42no—
33C59⋊#95—22no—
34C59⋊#95—22no—
35C59⋊#95—42no—
36C2×#41×C59—24no—
37C2×#41×C59—42no—
38C2×#41×C59—42no—
39C22×Q8×C59D411yescharacteristic
40C22×D4×C59C2×C411yescharacteristic
41C22×D4×C59D411yescharacteristic
42C22×D4×C59C2311yescharacteristic
43C23×C4×C59C2×C411yescharacteristic
44C23×C4×C59D411yescharacteristic
45C23×C4×C59D411yescharacteristic
46C2×M16×C59—22no—
47#101×C59—42no—
48#100×C59—24no—
49C22⋊C8×C59—42no—
50C24×C59C22⋊C411yescharacteristic
51C24×C59C22×C411yescharacteristic
52C24×C59C2×D411yescharacteristic
53#41×C59—416no—
54C2×Q8×C59—22no—
55C2×Q8×C59—22no—
56C2×Q8×C59—41no—
57C2×D4×C59—22no—
58C2×D4×C59—22no—
59C2×D4×C59—22no—
60C2×D4×C59—41no—
61C2×D4×C59—41no—
62C2×D4×C59—41no—
63C2×D4×C59—41no—
64C2×D4×C59—41no—
65C2×D4×C59#4114yes—
66C22×C4×C59—22no—
67C22×C4×C59—22no—
68C22×C4×C59—22no—
69C22×C4×C59—22no—
70C22×C4×C59—22no—
71C22×C4×C59—22no—
72C22×C4×C59—22no—
73C22×C4×C59—41no—
74C22×C4×C59—41no—
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22⋊C411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×Dic59—22no—
80C22×Dic59—22no—
81C22×Dic59—42no—
82C22×Dic59—42no—
83C22×Dic59—82no—
84M16×C59—42no—
85M16×C59—42no—
86C2×C8×C59—42no—
87C22⋊Dic59—42no—
88C22⋊Dic59—42no—
89C22⋊Dic59—42no—
90C22⋊Dic59—82no—
91C22⋊Dic59—82no—
92C4⋊Dic59—42no—
93C4⋊Dic59—42no—
94C4⋊Dic59—42no—
95C4×Dic59—42no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59C2×C22⋊C411yescharacteristic
110C23×C59#11712yes—
111C23×C59#12312yes—
112Q8×C59—44no—
113D4×C59—44no—
114D4×C59—44no—
115D4×C59—44no—
116C2×C4×C59—28no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—44no—
127C2×C4×C59—44no—
128C2×C4×C59#9912yes—
129C2×C4×C59#9912yes—
130C2×C4×C59#12312yes—
131C2×Dic59—42no—
132C2×Dic59—42no—
133C2×Dic59—42no—
134C2×Dic59—42no—
135C2×Dic59—42no—
136C2×Dic59—42no—
137C2×Dic59—84no—
138C2×Dic59—82no—
139C2×Dic59—82no—
140C2×Dic59—82no—
141C8×C59—82no—
142#12855—591nomaximal, Sylow
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—22no—
146C22×C59—22no—
147C22×C59—42no—
148C22×C59—42no—
149C22×C59—42no—
150C22×C59—42no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—41no—
158C22×C59—81no—
159C22×C59—81no—
160C22×C59#38111yescharacteristic
161C22×C59C2×#9911yescharacteristic
162C22×C59C2×#9911yescharacteristic
163C4×C59—22no—
164C4×C59—22no—
165C4×C59—22no—
166C4×C59—41no—
167Dic59—82no—
168Dic59—82no—
169Dic59—162no—
170#2311—591no—
171#1785D5911yescharacteristic
172#1782—591no—
173#1390—594no—
174C2×C59—21no—
175C2×C59—21no—
176C2×C59—42no—
177C2×C59—42no—
178C2×C59—81no—
179C2×C59#181911yescharacteristic
180C2×C59#212211yescharacteristic
181C2×C59#212311yescharacteristic
182C22×#41C2×D5911yescharacteristic
183C2×#117—591no—
184C22×C4⋊C4—591no—
185C2×#101—21no—
186C2×#100C2×D5912yes—
187C2×#99—592no—
188C2×#99—1181no—
189#407—21no—
190#391—1181no—
191#390—1181no—
192#386—592no—
193#386—592no—
194#381—1181no—
195#380—1181no—
196#380—1181no—
197C59#1285511yescharacteristic, complemented, Sylow
198C2×#41—24no—
199C2×#41—42no—
200C2×#41—42no—
201C22×Q8C22⋊D5911yescharacteristic
202C22×D4C4×D5911yescharacteristic
203C22×D4D23611yescharacteristic
204C22×D4C22×D5911yescharacteristic
205C23×C4—592no—
206C23×C4C4×D5911yescharacteristic
207C23×C4D23611yescharacteristic
208C23×C4C22⋊D5911yescharacteristic
209C2×M16—22no—
210#117—2362no—
211C2×C4⋊C4—1181no—
212C2×C4⋊C4—1181no—
213C2×C4⋊C4—1181no—
214C2×C4⋊C4—2362no—
215C2×C22⋊C4—592no—
216C2×C22⋊C4—1182no—
217C2×C22⋊C4—1182no—
218C2×C22⋊C4—1182no—
219C2×C42—1181no—
220#101—42no—
221#100—24no—
222#99—1184no—
223#99—2362no—
224C22⋊C8—42no—
225#95—1182no—
226#95—1182no—
227#95—2362no—
228C24(C2×D59)⋊C411yescharacteristic
229C24C2×C4×D5911yescharacteristic
230C24C2×D23611yescharacteristic
231#41—416no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4#1174568714yes—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—1182no—
254C22×C4—1182no—
255C22×C4—4722no—
256C22×C4—2362no—
257C22×C4—2362no—
258C22×C4(C2×D59)⋊C411yescharacteristic
259C22×C4(C2×D59)⋊C411yescharacteristic
260C22×C4(C2×D59)⋊C411yescharacteristic
261C22×C4C2×C22⋊D5911yescharacteristic
262M16—42no—
263M16—42no—
264C2×C8—42no—
265C4⋊C4—2362no—
266C4⋊C4—2362no—
267C4⋊C4—2362no—
268C22⋊C4—4722no—
269C22⋊C4—4722no—
270C22⋊C4—2362no—
271C22⋊C4—2362no—
272C22⋊C4—2362no—
273C42—2362no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23C2×(C2×D59)⋊C411yescharacteristic
288C23#1427368312yes—
289C23#1427371112yes—
290Q8—44no—
291D4—44no—
292D4—44no—
293D4—44no—
294C2×C4—28no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—4722no—
307C2×C4—4722no—
308C2×C4—4722no—
309C2×C4—4724no—
310C2×C4—2362no—
311C2×C4—2362no—
312C2×C4—2362no—
313C2×C4—2362no—
314C2×C4—2362no—
315C2×C4—2362no—
316C2×C4#1427364112yes—
317C2×C4#1427364112yes—
318C2×C4#1427370812yes—
319C8—82no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22#1683726311yescharacteristic
338C22C2×#1427364111yescharacteristic
339C22C2×#1427364111yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—9442no—
345C4—4722no—
346C4—4722no—
347C2—21no—
348C2—21no—
349C2—42no—
350C2—42no—
351C2—81no—
352C2#2142416811yescharacteristic
353C2#2142574711yescharacteristic
354C2#2142575111yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#1785C2
socleC22×C59C2×#99
prime radicalC22×#41×C59C22
cosocleC22×D4×C59C23
hypercenter#1785D59
hypocenterC59#12855
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
3. #21422215C2semidirect
5. C2×#14273686C4semidirect
6. C22×C4⋊Dic59C4semidirect
11. #16837282C4semidirect
197. C59#12855semidirect

Maximal subgroups

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

subgroupindex
#214266882
#214240142
#214222152
C59×#17852
#1285559