Finite Group Databasebeta

← #28501114

Subgroups — #28501114

Order 15104.

416 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 63 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#21425320C211yesmaximal, characteristic, complemented
2#21423098C211yesmaximal, characteristic, complemented
3#21423030C211yesmaximal, characteristic
4#21422102C211yesmaximal, characteristic, complemented
5#21422099C211yesmaximal, characteristic, complemented
6#21422042C211yesmaximal, characteristic, complemented
7C59×#2823C211yesmaximal, characteristic, complemented
8#16837983C2211yescharacteristic
9#16837715—21no—
10#16837536—21no—
11C2×#14273655—21no—
12#16837401—21no—
13#16837171C2211yescharacteristic
14#16837135—21no—
15#16837135—21no—
16#16837128C2211yescharacteristic
17#16837128C2211yescharacteristic
18C2×#126×C59—21no—
19C59×#398—21no—
20C59×#398C2211yescharacteristic
21C59×#396—21no—
22C59×#396—21no—
23C59×#396—21no—
24C59×#394—21no—
25C59×#394C2211yescharacteristic
26C59×#384C2211yescharacteristic
27C2×#11745685—21no—
28C2×#11745669—41no—
29C2×#11745664—21no—
30#14273707—41no—
31#14273702—41no—
32#14273701—21no—
33#14273701—21no—
34#14273660—41no—
35#14273655—42no—
36#14273650—41no—
37#14273648—41no—
38#14273647—41no—
39#14273647—41no—
40C22⋊(C59⋊C8)—21no—
41C22⋊(C59⋊C8)—21no—
42C22×D4×C59D411yescharacteristic
43#126×C59—42no—
44#126×C59—42no—
45C2×C4⋊C4×C59—21no—
46C2×C4⋊C4×C59—21no—
47C2×C4⋊C4×C59—21no—
48C2×C4⋊C4×C59—21no—
49C2×C4⋊C4×C59—21no—
50C2×C4⋊C4×C59—21no—
51C2×C4⋊C4×C59D411yescharacteristic
52C2×C4⋊C4×C59D411yescharacteristic
53C2×C4⋊C4×C59C2311yescharacteristic
54C2×C22⋊C4×C59—21no—
55C2×C22⋊C4×C59—21no—
56C2×C22⋊C4×C59—21no—
57C2×C22⋊C4×C59—21no—
58C2×C22⋊C4×C59—21no—
59C2×C22⋊C4×C59—21no—
60C2×C42×C59D411yescharacteristic
61#95×C59—21no—
62#95×C59—21no—
63#95×C59D411yescharacteristic
64#95×C59D411yescharacteristic
65C24×C59—21no—
66C2×D4×C59—41no—
67C2×D4×C59—81no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C2×C22⋊D59—41no—
77#11745685—41no—
78#11745685—41no—
79#11745685—41no—
80#11745685—41no—
81C2×D236—21no—
82C2×D236—21no—
83C2×D236—21no—
84C2×C4×D59—41no—
85C2×Dic118—21no—
86C4⋊C4×C59—41no—
87C4⋊C4×C59—42no—
88C4⋊C4×C59—42no—
89C4⋊C4×C59—41no—
90C4⋊C4×C59—41no—
91C4⋊C4×C59—41no—
92C4⋊C4×C59—41no—
93C4⋊C4×C59—81no—
94C4⋊C4×C59—81no—
95C4⋊C4×C59—81no—
96C22⋊C4×C59—42no—
97C22⋊C4×C59—42no—
98C22⋊C4×C59—41no—
99C22⋊C4×C59—41no—
100C22⋊C4×C59—81no—
101C22⋊C4×C59—81no—
102C22⋊C4×C59—81no—
103C42×C59—21no—
104C42×C59D812yescomplemented
105#11745669—82no—
106(C2×D59)⋊C4—41no—
107(C2×D59)⋊C4—41no—
108C4⋊Dic59—21no—
109C4⋊Dic59—21no—
110#11745664—41no—
111#11745664—41no—
112C2×C59⋊C8—41no—
113C2×C59⋊C8—41no—
114C2×C59⋊C8—41no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59—41no—
121C23×C59—41no—
122C23×C59#12011yescharacteristic
123C22×D59—41no—
124D4×C59—42no—
125D4×C59—81no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—22no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
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—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59C2×D811yescharacteristic
148C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
149C22⋊D59—81no—
150C22⋊D59—81no—
151C2×Dic59—41no—
152D236—41no—
153D236—41no—
154D236—81no—
155D236—81no—
156C4×D59—81no—
157C4×D59—81no—
158Dic118—41no—
159Dic118—41no—
160C59⋊C8—81no—
161C59⋊C8—81no—
162C59⋊C8—81no—
163#12155—591nomaximal, Sylow
164C22×C59—21no—
165C22×C59—21no—
166C22×C59—81no—
167C22×C59—81no—
168C22×C59—81no—
169C22×C59—81no—
170C22×C59—81no—
171C22×C59—81no—
172C22×C59—81no—
173C22×C59#44711yescharacteristic
174C22×C59#45411yescharacteristic
175C22×C59B(4,𝔽2)11yescharacteristic
176C2×D59—81no—
177C2×D59—81no—
178C2×D59—81no—
179C4×C59—21no—
180C4×C59—21no—
181C4×C59—42no—
182C4×C59—41no—
183C4×C59—41no—
184C4×C59—81no—
185C4×C59—81no—
186C4×C59—81no—
187C4×C59—81no—
188Dic59—81no—
189#2823D5911yescharacteristic, complemented
190#2039—591no—
191#1610—591no—
192#1598—591no—
193#1346—591no—
194#1345—591no—
195#1325—591no—
196C2×C59—21no—
197C2×C59—21no—
198C2×C59—161no—
199C2×C59#159811yescharacteristic
200C2×C59#220011yescharacteristic
201C2×C59#220311yescharacteristic
202D59—161no—
203#537—591no—
204C2×#126—21no—
205#480—1181no—
206#447—1181no—
207C2×#104—1181no—
208#417—1181no—
209#398—21no—
210#398C2×D5911yescharacteristic
211#396—21no—
212#396—21no—
213#396—21no—
214#394—21no—
215#394C2×D5911yescharacteristic
216#384C2×D5911yescharacteristic
217#343—591no—
218#327—1181no—
219#327—1181no—
220#324—591no—
221#324—591no—
222C59#1215511yescharacteristic, complemented, Sylow
223C2×#41—1181no—
224C22×D4C22⋊D5911yescharacteristic
225C2×D8—2361no—
226C2×M16—1181no—
227#126—42no—
228#126—42no—
229C22⋊Q8—2361no—
230#121—1181no—
231#121—1181no—
232#121—2361no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C22⋊D5911yescharacteristic
240C2×C4⋊C4C22⋊D5911yescharacteristic
241C2×C4⋊C4C22×D5911yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C42C22⋊D5911yescharacteristic
249#107—2361no—
250#104—2362no—
251#103—2361no—
252#102—2361no—
253#102—2361no—
254#102—2361no—
255C22⋊C8—1181no—
256C22⋊C8—1181no—
257#95—21no—
258#95—21no—
259#95C22⋊D5911yescharacteristic
260#95C22⋊D5911yescharacteristic
261C24—21no—
262#41—2361no—
263#41—2361no—
264#41—2361no—
265#41—2361no—
266C2×Q8—1181no—
267C2×D4—41no—
268C2×D4—81no—
269C2×D4—1181no—
270C2×D4—1181no—
271C2×D4—1181no—
272C2×D4—2361no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—2361no—
279C22×C4C2×C22⋊D5911yescharacteristic
280C22×C4C2×C22⋊D5911yescharacteristic
281C22×C4C2×C22⋊D5911yescharacteristic
282D8—4722no—
283M16—2361no—
284M16—2361no—
285C2×C8—2361no—
286C2×C8—2361no—
287C2×C8—2361no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—81no—
296C4⋊C4—81no—
297C4⋊C4—81no—
298C4⋊C4—1181no—
299C4⋊C4—1181no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C22⋊C4—81no—
307C22⋊C4—2361no—
308C22⋊C4—2361no—
309C42—21no—
310C42#1174566912yescomplemented
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—2361no—
319C23#1427369611yescharacteristic
320Q8—2361no—
321Q8—2361no—
322D4—42no—
323D4—81no—
324D4—4721no—
325D4—4721no—
326D4—4721no—
327D4—4721no—
328D4—2361no—
329D4—2361no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—4721no—
352C2×C4—4721no—
353C2×C4—2361no—
354C2×C4C2×#1174566911yescharacteristic
355C2×C4#1427375311yescharacteristic
356C8—4721no—
357C8—4721no—
358C8—4721no—
359C22—21no—
360C22—21no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—4721no—
369C22—4721no—
370C22—4721no—
371C22#1683753611yescharacteristic
372C22#1683757511yescharacteristic
373C22#1683758911yescharacteristic
374C4—21no—
375C4—21no—
376C4—42no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—4721no—
384C2—21no—
385C2—21no—
386C2—161no—
387C2—9441no—
388C2#2142303111yescharacteristic
389C2#2142610311yescharacteristic
390C2#2142611711yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2823C2
socleC22×C59B(4,𝔽2)
prime radical#21423030C2
hypercenter#2823D59
hypocenterC59#12155
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2823

Complements

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

NH ≅ G/Ntype
1. #21425320C2semidirect
2. #21423098C2semidirect
4. #21422102C2semidirect
5. #21422099C2semidirect
6. #21422042C2semidirect
7. C59×#2823C2semidirect
104. C42×C59D8semidirect
189. #2823D59semidirect
222. C59#12155semidirect
310. C42#11745669semidirect

Maximal subgroups

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

subgroupindex
#214253202
#214230982
#214230302
#214221022
#214220992
#214220422
C59×#28232
#1215559