Finite Group Databasebeta

← #28486348

Subgroups — #28486348

Order 15104.

568 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 103 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#21429063C211yesmaximal, characteristic
2#21422719C212yesmaximal, complemented
3#21422437C211yesmaximal, characteristic, complemented
4#21421951C212yesmaximal, complemented
5C59×#2973C211yesmaximal, characteristic, complemented
6C2×#14273723C411yescharacteristic
7C2×#14273715C2211yescharacteristic
8C2×#14273714C411yescharacteristic
9#16837459C2212yes—
10#16837318—21no—
11#16837318—21no—
12#16837307C411yescharacteristic
13#16837303C411yescharacteristic
14#16837281—21no—
15#16837281—21no—
16C4×C4⋊Dic59C2211yescharacteristic
17#16837141—24no—
18#16837126—21no—
19#16837126—21no—
20C2×C4×D4×C59C2211yescharacteristic
21C59×#435—24no—
22C59×#432C2212yes—
23C59×#407—21no—
24C2×C22⋊C8×C59—21no—
25C22×Dic118C2×C411yescharacteristic
26C22×D236C2×C411yescharacteristic
27C22×C22⋊D59—21no—
28C22×C22⋊D59—21no—
29#14273723—42no—
30#14273715—22no—
31#14273715—22no—
32#14273714—42no—
33C2×C4⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×C4×Dic59C2×C411yescharacteristic
41C2×C4×Dic59C2×C411yescharacteristic
42C4⋊(C59⋊C8)—22no—
43C22⋊(C59⋊C8)—22no—
44#14273638—22no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C22×D4×C59C2×C411yescharacteristic
49C23×C4×C59D411yescharacteristic
50C23×C4×C59D411yescharacteristic
51C2×M16×C59—22no—
52C22×C8×C59—22no—
53C4×D4×C59—22no—
54C4×D4×C59—22no—
55C2×C4⋊C4×C59C2×C411yescharacteristic
56C2×C22⋊C4×C59D411yescharacteristic
57C2×C22⋊C4×C59D411yescharacteristic
58C2×C42×C59C2311yescharacteristic
59C4⋊C8×C59—22no—
60C22⋊C8×C59—22no—
61C22⋊C8×C59—22no—
62C22⋊C8×C59—22no—
63#97×C59D414yescomplemented
64C24×C59C22⋊C411yescharacteristic
65C24×C59C22⋊C411yescharacteristic
66C23×D59—21no—
67C2×D4×C59—22no—
68C2×D4×C59—22no—
69C2×D4×C59—41no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59—81no—
75C22×C4×C59—81no—
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22⋊C411yescharacteristic
78C22×C4×C59C22×C411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C2×C22⋊D59—42no—
82C2×C22⋊D59—42no—
83C2×C22⋊D59—42no—
84C2×C22⋊D59—42no—
85C2×C22⋊D59—81no—
86C2×C22⋊D59—81no—
87C22×Dic59—21no—
88C22×Dic59—21no—
89C22×Dic59—21no—
90C2×D236—22no—
91C2×D236—41no—
92C2×D236—41no—
93C2×Dic118—22no—
94C2×Dic118—41no—
95C2×Dic118—41no—
96M16×C59—82no—
97C2×C8×C59—24no—
98C2×C8×C59—82no—
99C4⋊C4×C59—22no—
100C22⋊C4×C59—22no—
101C22⋊C4×C59—22no—
102C42×C59C2×D412yes—
103C42×C59#4112yes—
104C22⋊Dic59—42no—
105C22⋊Dic59—42no—
106(C2×D59)⋊C4—41no—
107(C2×D59)⋊C4—42no—
108(C2×D59)⋊C4—41no—
109(C2×D59)⋊C4—81no—
110(C2×D59)⋊C4—81no—
111C4⋊Dic59—42no—
112C4×Dic59—41no—
113C4×Dic59—41no—
114C2×C59⋊C8—42no—
115C23×C59—21no—
116C23×C59—21no—
117C23×C59—21no—
118C23×C59—21no—
119C23×C59—21no—
120C23×C59—41no—
121C23×C59—41no—
122C23×C59—41no—
123C23×C59—41no—
124C23×C59#10012yes—
125C23×C59C2×C22⋊C411yescharacteristic
126C22×D59—41no—
127C22×D59—41no—
128C22×D59—41no—
129C22×D59—41no—
130C22×D59—41no—
131C22×D59—41no—
132C22×D59—41no—
133D4×C59—44no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—42no—
141C2×C4×C59—42no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—81no—
145C2×C4×C59—81no—
146C2×C4×C59C4×D412yes—
147C2×C4×C59#12011yescharacteristic
148C2×C4×C59#12112yes—
149C2×C4×C59#12311yescharacteristic
150C22⋊D59—84no—
151C22⋊D59—84no—
152C2×Dic59—22no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—41no—
160C2×Dic59—41no—
161D236—82no—
162Dic118—82no—
163C8×C59—44no—
164C59⋊C8—82no—
165#9447—591nomaximal, Sylow
166C22×C59—21no—
167C22×C59—21no—
168C22×C59—22no—
169C22×C59—42no—
170C22×C59—42no—
171C22×C59—41no—
172C22×C59—41no—
173C22×C59—41no—
174C22×C59—41no—
175C22×C59—41no—
176C22×C59—81no—
177C22×C59—81no—
178C22×C59#38611yescharacteristic
179C22×C59C2×#10011yescharacteristic
180C22×C59#41311yescharacteristic
181C2×D59—81no—
182C2×D59—81no—
183C2×D59—81no—
184C2×D59—81no—
185C2×D59—81no—
186C2×D59—81no—
187C2×D59—81no—
188C4×C59—22no—
189C4×C59—22no—
190C4×C59—21no—
191C4×C59—21no—
192C4×C59—41no—
193C4×C59—41no—
194Dic59—42no—
195Dic59—81no—
196Dic59—81no—
197#2973D5911yescharacteristic, complemented
198#2544—591no—
199#1533—592no—
200#1466—591no—
201#1285—592no—
202C2×C59—21no—
203C2×C59—21no—
204C2×C59—42no—
205C2×C59—81no—
206C2×C59#188411yescharacteristic
207C2×C59#190011yescharacteristic
208C2×C59#191111yescharacteristic
209D59—161no—
210C2×#127—591no—
211C2×#124—591no—
212C2×#124—591no—
213C2×C4×D4C2×D5911yescharacteristic
214#435—24no—
215#432—592no—
216#432C2×D5912yes—
217#407—21no—
218C2×C22⋊C8—21no—
219#394—1181no—
220#394—1181no—
221#391—591no—
222#390—591no—
223#386—1181no—
224#386—1181no—
225C4×C4⋊C4—591no—
226#329—1184no—
227#323—1181no—
228#323—1181no—
229C59#944711yescharacteristic, complemented, Sylow
230C22×Q8—591no—
231C22×D4—591no—
232C22×D4—1181no—
233C22×D4—1181no—
234C22×D4C4×D5911yescharacteristic
235C23×C4C22⋊D5911yescharacteristic
236C23×C4C22⋊D5911yescharacteristic
237C2×M16—22no—
238C22×C8—22no—
239#127—2362no—
240#124—1182no—
241#124—1182no—
242#124—2362no—
243C4×D4—22no—
244C4×D4—22no—
245C2×C4⋊C4—1181no—
246C2×C4⋊C4C4×D5911yescharacteristic
247C2×C22⋊C4—1181no—
248C2×C22⋊C4—1181no—
249C2×C22⋊C4—1181no—
250C2×C22⋊C4—1181no—
251C2×C22⋊C4—1181no—
252C2×C22⋊C4—1181no—
253C2×C22⋊C4D23611yescharacteristic
254C2×C22⋊C4D23611yescharacteristic
255C2×C42—591no—
256C2×C42—591no—
257C2×C42C22×D5911yescharacteristic
258C4⋊C8—22no—
259C4⋊C8—1182no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—1182no—
264#97—1182no—
265#97C22⋊D5914yescomplemented
266#95—1181no—
267#95—1181no—
268#95—1181no—
269C24—1181no—
270C24(C2×D59)⋊C411yescharacteristic
271C24(C2×D59)⋊C411yescharacteristic
272C2×Q8—1182no—
273C2×Q8—2361no—
274C2×Q8—2361no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—41no—
278C2×D4—1182no—
279C2×D4—4721no—
280C2×D4—4721no—
281C2×D4—2361no—
282C2×D4—2361no—
283C2×D4—2362no—
284C2×D4—2362no—
285C2×D4—2362no—
286C2×D4—2362no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—1181no—
294C22×C4—1181no—
295C22×C4—1181no—
296C22×C4(C2×D59)⋊C411yescharacteristic
297C22×C4(C2×D59)⋊C411yescharacteristic
298C22×C4C2×C4×D5911yescharacteristic
299C22×C4C2×D23611yescharacteristic
300C22×C4C2×C22⋊D5911yescharacteristic
301M16—82no—
302C2×C8—24no—
303C2×C8—82no—
304C2×C8—2362no—
305C4⋊C4—22no—
306C4⋊C4—2362no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—4721no—
310C22⋊C4—4721no—
311C22⋊C4—2362no—
312C22⋊C4—2362no—
313C22⋊C4—2361no—
314C22⋊C4—2362no—
315C22⋊C4—2361no—
316C42—2361no—
317C42—2361no—
318C42#1174568512yes—
319C42C2×C22⋊D5912yes—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—2361no—
330C23—2361no—
331C23—2361no—
332C23—2361no—
333C23—2361no—
334C23—2361no—
335C23—2361no—
336C23#1427364312yes—
337C23C2×(C2×D59)⋊C411yescharacteristic
338Q8—4722no—
339D4—44no—
340D4—4722no—
341D4—4724no—
342D4—4724no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—1182no—
356C2×C4—2361no—
357C2×C4—2361no—
358C2×C4—2361no—
359C2×C4—2361no—
360C2×C4—2361no—
361C2×C4—2361no—
362C2×C4—2361no—
363C2×C4—2361no—
364C2×C4C4×C22⋊D5912yes—
365C2×C4#1427369611yescharacteristic
366C2×C4#1427370112yes—
367C2×C4#1427371011yescharacteristic
368C8—44no—
369C8—4722no—
370C22—21no—
371C22—21no—
372C22—22no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—4721no—
383C22—4721no—
384C22—4721no—
385C22—4721no—
386C22—4721no—
387C22—4721no—
388C22—4721no—
389C22#1683728111yescharacteristic
390C22C2×#1427364311yescharacteristic
391C22#1683738411yescharacteristic
392C4—22no—
393C4—22no—
394C4—21no—
395C4—21no—
396C4—41no—
397C4—41no—
398C4—4721no—
399C4—4721no—
400C4—2362no—
401C2—21no—
402C2—21no—
403C2—42no—
404C2—81no—
405C2—9441no—
406C2#2142447911yescharacteristic
407C2#2142455911yescharacteristic
408C2#2142462811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2973C2
socleC22×C59#386
prime radicalC2×#14273723C4
cosocleC2×C42×C59C23
hypercenter#2973D59
hypocenterC59#9447
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #21422719C2semidirect
3. #21422437C2semidirect
4. #21421951C2semidirect
5. C59×#2973C2semidirect
63. #97×C59D4semidirect
197. #2973D59semidirect
229. C59#9447semidirect
265. #97C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214290632
#214227192
#214224372
#214219512
C59×#29732
#944759