Finite Group Databasebeta

← #28486354

Subgroups — #28486354

Order 15104.

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

normal subgroups only

This group has 486 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#21433181C211yesmaximal, characteristic, complemented
2#21422720C212yesmaximal
3#21422440C211yesmaximal, characteristic, complemented
4#21421952C212yesmaximal, complemented
5C59×#2545C211yesmaximal, characteristic
6#16837533—24no—
7#16837459—21no—
8#16837459C2212yes—
9#16837459C2212yes—
10#16837455—21no—
11#16837142—24no—
12#16837128—21no—
13#16837128—21no—
14C2×C4⋊Q8×C59C411yescharacteristic
15C2×#124×C59C411yescharacteristic
16C2×#124×C59C2211yescharacteristic
17C2×C4×Q8×C59C2211yescharacteristic
18C59×#396—21no—
19C59×#396—21no—
20C59×#391C411yescharacteristic
21C59×#390C411yescharacteristic
22C59×#388—21no—
23C59×#388—21no—
24C4×C4⋊C4×C59C2211yescharacteristic
25C4⋊(C59⋊C8)—22no—
26C4⋊(C59⋊C8)—22no—
27C4⋊(C59⋊C8)—22no—
28C4⋊(C59⋊C8)—22no—
29C22⋊(C59⋊C8)—22no—
30C22⋊(C59⋊C8)—22no—
31#14273638—22no—
32#14273638—22no—
33#14273638D414yes—
34C4×C59⋊C8—22no—
35C22×Q8×C59C2×C411yescharacteristic
36C22×Q8×C59C2×C411yescharacteristic
37C22×D4×C59C2×C411yescharacteristic
38C4⋊Q8×C59—42no—
39#124×C59—22no—
40#124×C59—22no—
41#124×C59—42no—
42C4×Q8×C59—22no—
43C4×Q8×C59—22no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59—21no—
46C2×C4⋊C4×C59—21no—
47C2×C4⋊C4×C59C2×C411yescharacteristic
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59—21no—
53C2×C22⋊C4×C59—21no—
54C2×C22⋊C4×C59—21no—
55C2×C22⋊C4×C59—21no—
56C2×C42×C59C2×C411yescharacteristic
57C2×C42×C59C2×C411yescharacteristic
58C2×C42×C59D411yescharacteristic
59C2×C42×C59D411yescharacteristic
60C2×C42×C59C2311yescharacteristic
61#95×C59—21no—
62#95×C59—21no—
63#95×C59—21no—
64C24×C59—21no—
65C2×Q8×C59—22no—
66C2×Q8×C59—22no—
67C2×Q8×C59—22no—
68C2×Q8×C59—41no—
69C2×Q8×C59—41no—
70C2×Q8×C59—41no—
71C2×D4×C59—22no—
72C2×D4×C59—41no—
73C2×D4×C59—41no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59C22⋊C411yescharacteristic
78C22×C4×C59C22⋊C411yescharacteristic
79C22×C4×C59C22⋊C411yescharacteristic
80C22×C4×C59C22⋊C411yescharacteristic
81C22×C4×C59C22×C411yescharacteristic
82C22×C4×C59C2×D411yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84C4⋊C4×C59—22no—
85C4⋊C4×C59—22no—
86C4⋊C4×C59—22no—
87C4⋊C4×C59—42no—
88C4⋊C4×C59—42no—
89C4⋊C4×C59—42no—
90C22⋊C4×C59—42no—
91C22⋊C4×C59—42no—
92C22⋊C4×C59—42no—
93C22⋊C4×C59—42no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—42no—
96C22⋊C4×C59—41no—
97C42×C59—21no—
98C42×C59—21no—
99C42×C59—21no—
100C42×C59—21no—
101C42×C59—41no—
102C42×C59—41no—
103C42×C59C2×D412yes—
104C42×C59#4112yes—
105C2×C59⋊C8—24no—
106C2×C59⋊C8—42no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59C2×C22⋊C411yescharacteristic
115Q8×C59—44no—
116Q8×C59—82no—
117D4×C59—82no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—22no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59#10012yes—
141C2×C4×C59C4×D412yes—
142C2×C4×C59#12011yescharacteristic
143C2×C4×C59#12112yes—
144C2×C4×C59#12311yescharacteristic
145C59⋊C8—44no—
146C59⋊C8—82no—
147#9448—591nomaximal, Sylow
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—81no—
151C22×C59—81no—
152C22×C59—81no—
153C22×C59—81no—
154C22×C59—81no—
155C22×C59—81no—
156C22×C59—81no—
157C22×C59#38611yescharacteristic
158C22×C59C2×#10011yescharacteristic
159C22×C59#41311yescharacteristic
160C4×C59—22no—
161C4×C59—22no—
162C4×C59—22no—
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—42no—
166C4×C59—41no—
167C4×C59—41no—
168C4×C59—41no—
169C4×C59—81no—
170C4×C59—81no—
171#2983—591no—
172#2545D5911yescharacteristic
173#1533—592no—
174#1467—591no—
175#1285—592no—
176C2×C59—21no—
177C2×C59—21no—
178C2×C59—161no—
179C2×C59#188411yescharacteristic
180C2×C59#190011yescharacteristic
181C2×C59#191211yescharacteristic
182C2×C4⋊Q8Dic5911yescharacteristic
183C2×#124Dic5911yescharacteristic
184C2×#124C2×D5911yescharacteristic
185C2×C4×Q8C2×D5911yescharacteristic
186#446—1184no—
187#432—592no—
188#432—592no—
189#432—1181no—
190#431—1181no—
191#396—21no—
192#396—21no—
193#391Dic5911yescharacteristic
194#390Dic5911yescharacteristic
195#388—21no—
196#388—21no—
197C4×C4⋊C4C2×D5911yescharacteristic
198#329—1184no—
199#324—1181no—
200#324—1181no—
201C59#944811yescharacteristic, complemented, Sylow
202C22×Q8C2×Dic5911yescharacteristic
203C22×Q8C2×Dic5911yescharacteristic
204C22×D4C2×Dic5911yescharacteristic
205C4⋊Q8—42no—
206#124—22no—
207#124—22no—
208#124—42no—
209C4×Q8—22no—
210C4×Q8—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic5911yescharacteristic
215C2×C4⋊C4C22⋊D5911yescharacteristic
216C2×C4⋊C4C22⋊D5911yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C42C2×Dic5911yescharacteristic
224C2×C42C2×Dic5911yescharacteristic
225C2×C42C22⋊D5911yescharacteristic
226C2×C42C22⋊D5911yescharacteristic
227C2×C42C22×D5911yescharacteristic
228C4⋊C8—1182no—
229C4⋊C8—1182no—
230C4⋊C8—1182no—
231C4⋊C8—1182no—
232C22⋊C8—1182no—
233C22⋊C8—1182no—
234#97—594no—
235#97—1182no—
236#97—1182no—
237C4×C8—1182no—
238#95—21no—
239#95—21no—
240#95—21no—
241C24—21no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4C22⋊Dic5911yescharacteristic
255C22×C4C22⋊Dic5911yescharacteristic
256C22×C4C22⋊Dic5911yescharacteristic
257C22×C4C22⋊Dic5911yescharacteristic
258C22×C4C22×Dic5911yescharacteristic
259C22×C4C2×C22⋊D5911yescharacteristic
260C22×C4C2×C22⋊D5911yescharacteristic
261C2×C8—1184no—
262C2×C8—2362no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—42no—
275C22⋊C4—41no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—41no—
281C42—41no—
282C42D4×D5912yes—
283C42#1174568712yes—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C2×C22⋊Dic5911yescharacteristic
292Q8—44no—
293Q8—82no—
294D4—82no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4#1427364412yes—
318C2×C4D4×Dic5912yes—
319C2×C4#1427369511yescharacteristic
320C2×C4#1427369912yes—
321C2×C4#1427370811yescharacteristic
322C8—4722no—
323C8—2364no—
324C22—21no—
325C22—21no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22#1683728211yescharacteristic
334C22C2×#1427364411yescharacteristic
335C22#1683738611yescharacteristic
336C4—22no—
337C4—22no—
338C4—22no—
339C4—21no—
340C4—21no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C2—21no—
348C2—21no—
349C2—161no—
350C2#2142448011yescharacteristic
351C2#2142456011yescharacteristic
352C2#2142463411yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2545C2
socleC22×C59#386
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenter#2545D59
hypocenterC59#9448
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2545

Complements

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

NH ≅ G/Ntype
1. #21433181C2semidirect
3. #21422440C2semidirect
4. #21421952C2semidirect
201. C59#9448semidirect

Maximal subgroups

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

subgroupindex
#214331812
#214227202
#214224402
#214219522
C59×#25452
#944859