Finite Group Databasebeta

← #28501027

Subgroups — #28501027

Order 15104.

354 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 59 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
2#21423132C211yesmaximal, characteristic
3#21423098C211yesmaximal, characteristic
4#21422123C211yesmaximal, characteristic, complemented
5#21422102C211yesmaximal, characteristic, complemented
6#21422042C211yesmaximal, characteristic, complemented
7C59×#2794C211yesmaximal, characteristic, complemented
8#16837983C2211yescharacteristic
9#16837720—21no—
10#16837715—21no—
11C2×#14273655—21no—
12#16837401—21no—
13#16837171C2211yescharacteristic
14#16837135—21no—
15#16837135—21no—
16#16837128C2211yescharacteristic
17#16837128C2211yescharacteristic
18C59×#400—21no—
19C59×#400—21no—
20C59×#400C2211yescharacteristic
21C59×#398C2211yescharacteristic
22C59×#387—21no—
23C59×#387—21no—
24C59×#387—21no—
25C59×#387—21no—
26C59×#384C2211yescharacteristic
27C2×#11745685—21no—
28C2×#11745664—21no—
29#14273707—41no—
30#14273702—41no—
31#14273701—21no—
32#14273701—21no—
33#14273660—41no—
34#14273658—41no—
35#14273655—42no—
36#14273650—41no—
37#14273647—41no—
38#14273647—41no—
39#14273647—41no—
40C22⋊(C59⋊C8)—21no—
41C22⋊(C59⋊C8)—21no—
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59—21no—
46C2×C4⋊C4×C59D411yescharacteristic
47C2×C4⋊C4×C59D411yescharacteristic
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C4⋊C4×C59C2311yescharacteristic
50C2×C42×C59—21no—
51C2×C42×C59D411yescharacteristic
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—21no—
55#95×C59—21no—
56#95×C59—21no—
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60#95×C59—21no—
61#95×C59D411yescharacteristic
62#95×C59D411yescharacteristic
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C2×C22⋊D59—41no—
73#11745685—41no—
74#11745685—41no—
75#11745685—41no—
76#11745685—41no—
77C2×D236—21no—
78C2×D236—21no—
79C2×D236—21no—
80C2×C4×D59—41no—
81C2×Dic118—21no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—42no—
84C4⋊C4×C59—42no—
85C4⋊C4×C59—42no—
86C4⋊C4×C59—41no—
87C4⋊C4×C59—41no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—41no—
90C4⋊C4×C59—41no—
91C42×C59—21no—
92C42×C59—21no—
93C42×C59—81no—
94(C2×D59)⋊C4—41no—
95(C2×D59)⋊C4—41no—
96C4⋊Dic59—21no—
97C4⋊Dic59—21no—
98#11745664—41no—
99#11745664—41no—
100C2×C59⋊C8—41no—
101C2×C59⋊C8—41no—
102C2×C59⋊C8—41no—
103C23×C59#12011yescharacteristic
104C22×D59—41no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—22no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—42no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59#13511yescharacteristic
129C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
130C22⋊D59—81no—
131C22⋊D59—81no—
132C2×Dic59—41no—
133D236—41no—
134D236—41no—
135D236—81no—
136D236—81no—
137C4×D59—81no—
138C4×D59—81no—
139Dic118—41no—
140Dic118—41no—
141C59⋊C8—81no—
142C59⋊C8—81no—
143C59⋊C8—81no—
144#12142—591nomaximal, Sylow
145C22×C59—21no—
146C22×C59—21no—
147C22×C59#44911yescharacteristic
148C22×C59#45411yescharacteristic
149C22×C59B(4,𝔽2)11yescharacteristic
150C2×D59—81no—
151C2×D59—81no—
152C2×D59—81no—
153C4×C59—21no—
154C4×C59—21no—
155C4×C59—42no—
156C4×C59—41no—
157C4×C59—41no—
158C4×C59—81no—
159C4×C59—81no—
160C4×C59—81no—
161C4×C59—81no—
162C4×C59—81no—
163Dic59—81no—
164#2794D5911yescharacteristic, complemented
165#2039—591no—
166#1616—591no—
167#1610—591no—
168#1353—591no—
169#1346—591no—
170#1325—591no—
171C2×C59—21no—
172C2×C59—21no—
173C2×C59#159711yescharacteristic
174C2×C59#219811yescharacteristic
175C2×C59#220311yescharacteristic
176D59—161no—
177#537—591no—
178#481—1181no—
179#480—1181no—
180C2×#104—1181no—
181#417—1181no—
182#400—21no—
183#400—21no—
184#400C2×D5911yescharacteristic
185#398C2×D5911yescharacteristic
186#387—21no—
187#387—21no—
188#387—21no—
189#387—21no—
190#384C2×D5911yescharacteristic
191#343—591no—
192#327—1181no—
193#327—1181no—
194#324—591no—
195#324—591no—
196C59#1214211yescharacteristic, complemented, Sylow
197C2×#41—1181no—
198C2×M16—1181no—
199C22⋊Q8—2361no—
200#121—1181no—
201#121—1181no—
202#121—2361no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C22⋊D5911yescharacteristic
208C2×C4⋊C4C22⋊D5911yescharacteristic
209C2×C4⋊C4C22⋊D5911yescharacteristic
210C2×C4⋊C4C22×D5911yescharacteristic
211C2×C42—21no—
212C2×C42C22⋊D5911yescharacteristic
213#107—2361no—
214#106—2361no—
215#104—2362no—
216#103—2361no—
217#102—2361no—
218#102—2361no—
219#102—2361no—
220C22⋊C8—1181no—
221C22⋊C8—1181no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95C22⋊D5911yescharacteristic
232#95C22⋊D5911yescharacteristic
233#41—2361no—
234#41—2361no—
235#41—2361no—
236#41—2361no—
237C2×Q8—1181no—
238C2×D4—1181no—
239C2×D4—1181no—
240C2×D4—1181no—
241C2×D4—2361no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—2361no—
249C22×C4C2×C22⋊D5911yescharacteristic
250C22×C4C2×C22⋊D5911yescharacteristic
251C22×C4C2×C22⋊D5911yescharacteristic
252M16—2361no—
253M16—2361no—
254C2×C8—2361no—
255C2×C8—2361no—
256C2×C8—2361no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—1181no—
267C4⋊C4—1181no—
268C22⋊C4—2361no—
269C22⋊C4—2361no—
270C42—21no—
271C42—21no—
272C42—81no—
273C23—2361no—
274C23#1427369611yescharacteristic
275Q8—2361no—
276Q8—2361no—
277D4—4721no—
278D4—4721no—
279D4—4721no—
280D4—4721no—
281D4—2361no—
282D4—2361no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—4721no—
307C2×C4—4721no—
308C2×C4—2361no—
309C2×C4#1427374711yescharacteristic
310C2×C4#1427375211yescharacteristic
311C8—4721no—
312C8—4721no—
313C8—4721no—
314C22—21no—
315C22—21no—
316C22—4721no—
317C22—4721no—
318C22—4721no—
319C22#1683754711yescharacteristic
320C22#1683757511yescharacteristic
321C22#1683758911yescharacteristic
322C4—21no—
323C4—21no—
324C4—42no—
325C4—41no—
326C4—41no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4—4721no—
333C2—21no—
334C2—21no—
335C2—9441no—
336C2#2142302511yescharacteristic
337C2#2142609211yescharacteristic
338C2#2142611711yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2794C2
socleC22×C59B(4,𝔽2)
prime radical#16837983C22
hypercenter#2794D59
hypocenterC59#12142
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2794

Complements

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

NH ≅ G/Ntype
4. #21422123C2semidirect
5. #21422102C2semidirect
6. #21422042C2semidirect
7. C59×#2794C2semidirect
164. #2794D59semidirect
196. C59#12142semidirect

Maximal subgroups

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

subgroupindex
#214253202
#214231322
#214230982
#214221232
#214221022
#214220422
C59×#27942
#1214259