Finite Group Databasebeta

← #28505682

Subgroups — #28505682

Order 15104.

572 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 117 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#21433678C211yesmaximal, characteristic
2C2×#106×D59C211yesmaximal, characteristic, complemented
3C2×#16837213C211yesmaximal, characteristic, complemented
4C2×#16837211C211yesmaximal, characteristic, complemented
5#21425706C211yesmaximal, characteristic, complemented
6#21422201C211yesmaximal, characteristic, complemented
7C59×#1386C211yesmaximal, characteristic, complemented
8C2×#14273749C2211yescharacteristic
9C22×C8×D59C2211yescharacteristic
10#16837914—21no—
11C2×C4⋊C4×D59—21no—
12C2×#14273667C2211yescharacteristic
13#16837644C412yescomplemented
14#16837639C412yescomplemented
15#16837632—22no—
16C2×#14273659C2211yescharacteristic
17C2×#14273658—21no—
18#106×D59—22no—
19#106×D59—22no—
20#16837396—21no—
21#16837213—22no—
22#16837211—22no—
23(C2×D59)⋊C16—22no—
24C2×M32×C59C2211yescharacteristic
25C59×#427C2211yescharacteristic
26C2×#106×C59C2211yescharacteristic
27C2×#11745685—21no—
28C22×C4×D59—41no—
29C22×C4×D59D411yescharacteristic
30#14273749—21no—
31#14273749—21no—
32#14273749—21no—
33#14273749—21no—
34C2×Dic236C2×C411yescharacteristic, complemented
35C2×#11745660—21no—
36C2×D472C2×C411yescharacteristic, complemented
37C22×C59⋊C8D411yescharacteristic
38C2×C8×D59—21no—
39C2×C8×D59—21no—
40C2×C8×D59—21no—
41C2×C8×D59—21no—
42C2×C8×D59C2×C412yes—
43C2×C8×D59D412yes—
44#14273711—41no—
45#14273707—22no—
46#14273702—22no—
47#14273697—41no—
48C4×D236—42no—
49C2×C4⋊Dic59—21no—
50C2×Dic59⋊C4—41no—
51C4⋊C4×D59—24no—
52C4⋊C4×D59—42no—
53#14273667—21no—
54#14273667—21no—
55C2×C59⋊C16—21no—
56#14273659—21no—
57#14273659D412yescomplemented
58#14273658—24no—
59#14273650—22no—
60#14273647—22no—
61C22×C8×C59C2311yescharacteristic
62#117×C59—21no—
63C2×C4⋊C4×C59—21no—
64M32×C59—21no—
65M32×C59—21no—
66C2×C16×C59—21no—
67#107×C59C2×C412yescomplemented
68#106×C59—21no—
69#106×C59—21no—
70#106×C59D412yescomplemented
71C23×D59D811yescharacteristic
72C22×C4×C59—41no—
73C22×C4×C59C2×D411yescharacteristic
74C2×C22⋊D59—41no—
75C22×Dic59—41no—
76C22×Dic59D811yescharacteristic
77#11745685—41no—
78#11745685—41no—
79#11745685—41no—
80#11745685—41no—
81C2×D236—21no—
82C2×D236—42no—
83C2×C4×D59—21no—
84C2×C4×D59—21no—
85C2×C4×D59—21no—
86C2×C4×D59—21no—
87C2×C4×D59—44no—
88C2×C4×D59—41no—
89C2×C4×D59—84no—
90C2×C4×D59C22⋊C412yes—
91C2×C4×D59SD1612yes—
92C2×Dic118—21no—
93C2×C8×C59C22×C411yescharacteristic
94C2×C8×C59C2×D411yescharacteristic
95C2×C8×C59C2×D411yescharacteristic
96C2×C8×C59C2×D411yescharacteristic
97C2×C8×C59#4111yescharacteristic
98C2×C8×C59#4111yescharacteristic
99C4⋊C4×C59—22no—
100C4⋊C4×C59—22no—
101C4⋊C4×C59—41no—
102C22⋊C4×C59—41no—
103C42×C59—41no—
104(C2×D59)⋊C4—42no—
105(C2×D59)⋊C4—42no—
106C4⋊Dic59—22no—
107C4⋊Dic59—41no—
108C4⋊Dic59—42no—
109Dic59⋊C4—44no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—21no—
112C2×C59⋊C8C22⋊C412yes—
113Dic236—21no—
114Dic236—21no—
115D472—21no—
116D472—21no—
117#11745660—41no—
118#11745660—41no—
119C8×D59—22no—
120C8×D59—22no—
121C8×D59—22no—
122C8×D59—22no—
123C16×C59—41no—
124C59⋊C16—41no—
125C23×C59C2×D811yescharacteristic
126C22×D59—21no—
127C22×D59—21no—
128C22×D59—24no—
129C22×D59—41no—
130C22×D59#10212yes—
131C2×C4×C59—42no—
132C2×C4×C59—42no—
133C2×C4×C59—82no—
134C2×C4×C59C2×C22⋊C411yescharacteristic
135C2×C4×C59#12011yescharacteristic
136C2×C4×C59#12311yescharacteristic
137C2×C4×C59C2×SD1611yescharacteristic
138C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C59#13711yescharacteristic
140C22⋊D59—81no—
141C22⋊D59—81no—
142C2×Dic59—21no—
143C2×Dic59—21no—
144C2×Dic59—41no—
145C2×Dic59—42no—
146C2×Dic59—82no—
147C2×Dic59#10212yes—
148D236—41no—
149D236—41no—
150D236—84no—
151C4×D59—22no—
152C4×D59—22no—
153C4×D59—22no—
154C4×D59—22no—
155C4×D59—88no—
156C4×D59—81no—
157C4×D59—81no—
158Dic118—41no—
159Dic118—41no—
160C8×C59C4×D411yescharacteristic
161C8×C59C4×D411yescharacteristic
162C8×C59#12111yescharacteristic
163C8×C59#12111yescharacteristic
164C59⋊C8—22no—
165#13001—591nomaximal, Sylow
166C22×C59—22no—
167C22×C59C2×#10211yescharacteristic
168C22×C59#44711yescharacteristic
169C22×C59#48011yescharacteristic
170C2×D59—24no—
171C2×D59—24no—
172C2×D59—22no—
173C2×D59—22no—
174C2×D59—81no—
175C2×D59—82no—
176C2×D59#36014yescomplemented
177C4×C59—82no—
178C4×C59—82no—
179C4×C59#38611yescharacteristic
180C4×C59#41711yescharacteristic
181C4×C59#44811yescharacteristic
182C4×C59#48111yescharacteristic
183Dic59—22no—
184Dic59—82no—
185Dic59—81no—
186#3050—591no—
187C22×#106—591no—
188C2×#360—591no—
189C2×#360—591no—
190#2112—591no—
191#1386—591no—
192#1386D5911yescharacteristic, complemented
193C2×C59—21no—
194C2×C59—21no—
195C2×C59#187411yescharacteristic
196C2×C59C2×#36012yes—
197D59—24no—
198D59—161no—
199C2×#135—591no—
200C23×C8—591no—
201#525—1181no—
202C22×C4⋊C4—1181no—
203C2×M32—591no—
204C2×M32C2×D5911yescharacteristic
205#467—592no—
206#466—592no—
207#465—1182no—
208#427C2×D5911yescharacteristic
209C2×#106—591no—
210C2×#106—1182no—
211C2×#106—1181no—
212C2×#106—1182no—
213C2×#106C2×D5911yescharacteristic
214#416—1181no—
215#360—1182no—
216#360—1182no—
217C22⋊C16—1182no—
218C59#1300111yescharacteristic, complemented, Sylow
219C2×#41—1181no—
220C23×C4—591no—
221C23×C4—2361no—
222#135—1181no—
223#135—1181no—
224#135—1181no—
225#135—1181no—
226C2×Q16—591no—
227C2×SD16—1181no—
228C2×D8—591no—
229C22×C8—591no—
230C22×C8—592no—
231C22×C8—592no—
232C22×C8—1181no—
233C22×C8—1181no—
234C22×C8—1181no—
235C22×C8—1181no—
236C22×C8C22×D5911yescharacteristic
237#123—2361no—
238C22⋊Q8—1182no—
239#121—1182no—
240#120—2361no—
241C4×D4—2362no—
242#117—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—1184no—
245C2×C4⋊C4—1181no—
246C2×C4⋊C4—2362no—
247C2×C4⋊C4—2361no—
248M32—21no—
249M32—21no—
250M32—1181no—
251M32—1181no—
252C2×C16—21no—
253C2×C16—1181no—
254#107C4×D5912yescomplemented
255#106—21no—
256#106—21no—
257#106—592no—
258#106—1184no—
259#106—1181no—
260#106D23612yescomplemented
261#103—1182no—
262#102—1182no—
263C24—591no—
264#41—2361no—
265#41—2361no—
266#41—2361no—
267#41—2361no—
268C2×Q8—1181no—
269C2×D4—1181no—
270C2×D4—2362no—
271C2×D4—2361no—
272C22×C4—41no—
273C22×C4—591no—
274C22×C4—592no—
275C22×C4—592no—
276C22×C4—1181no—
277C22×C4—1181no—
278C22×C4—1181no—
279C22×C4—1181no—
280C22×C4—4724no—
281C22×C4—2364no—
282C22×C4—2361no—
283C22×C4—2361no—
284C22×C4D4×D5911yescharacteristic
285Q16—1181no—
286Q16—1181no—
287SD16—2361no—
288SD16—2361no—
289D8—1181no—
290D8—1181no—
291C2×C8—592no—
292C2×C8—1181no—
293C2×C8—1181no—
294C2×C8—1182no—
295C2×C8—1182no—
296C2×C8—1182no—
297C2×C8—1182no—
298C2×C8C2×C4×D5911yescharacteristic
299C2×C8C2×D23611yescharacteristic
300C2×C8#1174568511yescharacteristic
301C2×C8D4×D5911yescharacteristic
302C2×C8D4×D5911yescharacteristic
303C2×C8#1174568911yescharacteristic
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—41no—
307C4⋊C4—1182no—
308C4⋊C4—2362no—
309C4⋊C4—2364no—
310C4⋊C4—2361no—
311C22⋊C4—41no—
312C22⋊C4—2362no—
313C22⋊C4—2362no—
314C42—41no—
315C16—41no—
316C16—2361no—
317C23—592no—
318C23—1184no—
319C23—1181no—
320C23—1181no—
321C23—2361no—
322C23D8×D5911yescharacteristic
323Q8—2361no—
324Q8—2361no—
325D4—4724no—
326D4—4721no—
327D4—4721no—
328D4—2361no—
329D4—2361no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—82no—
333C2×C4—592no—
334C2×C4—1181no—
335C2×C4—1181no—
336C2×C4—1182no—
337C2×C4—1182no—
338C2×C4—1182no—
339C2×C4—1182no—
340C2×C4—4728no—
341C2×C4—4722no—
342C2×C4—4721no—
343C2×C4—4721no—
344C2×C4—2361no—
345C2×C4—2362no—
346C2×C4C22⋊C4×D5911yescharacteristic
347C2×C4#1427369711yescharacteristic
348C2×C4#1427371211yescharacteristic
349C2×C4SD16×D5911yescharacteristic
350C2×C4#1427375411yescharacteristic
351C2×C4#1427375911yescharacteristic
352C8—1182no—
353C8#1427368911yescharacteristic
354C8C4×D23611yescharacteristic
355C8#1427370011yescharacteristic
356C8#1427370211yescharacteristic
357C22—22no—
358C22—594no—
359C22—1184no—
360C22—1184no—
361C22—1182no—
362C22—1182no—
363C22—4721no—
364C22—4722no—
365C22#102×D5911yescharacteristic
366C22#1683753711yescharacteristic
367C22#1683771711yescharacteristic
368C4—82no—
369C4—82no—
370C4—1182no—
371C4—4722no—
372C4—4721no—
373C4#1683728011yescharacteristic
374C4#1683740211yescharacteristic
375C4#1683754211yescharacteristic
376C4#1683772211yescharacteristic
377C2—21no—
378C2—21no—
379C2—1184no—
380C2—9441no—
381C2#2142441211yescharacteristic
382C2#360×D5912yes—
subgroupG/H
centerC22#102×D59
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#1386C2
socleC22×C59C2×#102
prime radical#21433678C2
cosocleC22×C8×C59C23
hypercenter#1386D59
hypocenterC59#13001
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1386

Complements

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

NH ≅ G/Ntype
2. C2×#106×D59C2semidirect
3. C2×#16837213C2semidirect
4. C2×#16837211C2semidirect
5. #21425706C2semidirect
6. #21422201C2semidirect
7. C59×#1386C2semidirect
13. #16837644C4semidirect
14. #16837639C4semidirect
34. C2×Dic236C2×C4semidirect
36. C2×D472C2×C4semidirect
57. #14273659D4semidirect
67. #107×C59C2×C4semidirect
70. #106×C59D4semidirect
176. C2×D59#360semidirect
192. #1386D59semidirect
218. C59#13001semidirect
254. #107C4×D59semidirect
260. #106D236semidirect

Maximal subgroups

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

subgroupindex
#214336782
C2×#106×D592
C2×#168372132
C2×#168372112
#214257062
#214222012
C59×#13862
#1300159