Finite Group Databasebeta

← #28502546

Subgroups — #28502546

Order 15104.

350 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 71 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#21426332C211yesmaximal, characteristic, complemented
2#21426248C211yesmaximal, characteristic, complemented
3#21423609C211yesmaximal, characteristic, complemented
4#21422139C211yesmaximal, characteristic, complemented
5#21422066C211yesmaximal, characteristic, complemented
6#21422054C211yesmaximal, characteristic, complemented
7C59×#3277C211yesmaximal, characteristic
8#16837807C2211yescharacteristic
9#16837607—21no—
10#16837227—21no—
11C4⋊(C59⋊C16)C2211yescharacteristic
12#16837206—21no—
13#16837206—21no—
14#16837203—21no—
15#16837178C2211yescharacteristic
16#16837153C2211yescharacteristic
17#16837147—21no—
18C2×#137×C59—21no—
19C59×#548—21no—
20C59×#495C2211yescharacteristic
21C59×#468—21no—
22C59×#464—21no—
23C59×#451—21no—
24C59×#449—21no—
25C4×Q16×C59C2211yescharacteristic
26C59×#436C2211yescharacteristic
27C2×#11745670—41no—
28#14273727—21no—
29C2×C59⋊C16—21no—
30C2×C59⋊C16—21no—
31#14273661—21no—
32#14273661—21no—
33C4⋊(C59⋊C8)—21no—
34#14273650—41no—
35C2×#41×C59—21no—
36C22×Q8×C59—21no—
37#137×C59—42no—
38#137×C59—42no—
39C2×Q16×C59—21no—
40C2×Q16×C59D411yescharacteristic
41C2×Q16×C59D411yescharacteristic
42C2×SD16×C59—21no—
43C2×SD16×C59—21no—
44C2×D8×C59D411yescharacteristic
45C2×M16×C59—21no—
46#124×C59—21no—
47#124×C59—41no—
48C22⋊Q8×C59—21no—
49C22⋊Q8×C59—41no—
50#121×C59—21no—
51#121×C59—41no—
52C4×Q8×C59—21no—
53C4×D4×C59—41no—
54C4×D4×C59D411yescharacteristic
55#107×C59D411yescharacteristic
56C4⋊C8×C59D411yescharacteristic
57#103×C59—21no—
58#103×C59—21no—
59#102×C59—21no—
60C22⋊C8×C59—21no—
61C4×C8×C59C2311yescharacteristic
62#41×C59—42no—
63#41×C59—81no—
64C2×Q8×C59—21no—
65C2×Q8×C59—21no—
66C2×Q8×C59—41no—
67C2×Q8×C59—42no—
68C2×Q8×C59—81no—
69C2×D4×C59—21no—
70C2×D4×C59—41no—
71C2×D4×C59—41no—
72C2×D4×C59D811yescharacteristic
73C22×C4×C59—21no—
74C22×C4×C59—41no—
75C22×C4×C59—41no—
76C2×Dic118—41no—
77Q16×C59—21no—
78Q16×C59—41no—
79Q16×C59—42no—
80Q16×C59SD1612yescomplemented
81SD16×C59—42no—
82SD16×C59—41no—
83SD16×C59—41no—
84D8×C59—41no—
85M16×C59—41no—
86M16×C59—41no—
87C2×C8×C59—21no—
88C2×C8×C59C2×D411yescharacteristic
89C2×C8×C59C2×D411yescharacteristic
90C4⋊C4×C59—21no—
91C4⋊C4×C59—41no—
92C4⋊C4×C59—41no—
93C4⋊C4×C59D811yescharacteristic
94C22⋊C4×C59—21no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—41no—
97C22⋊C4×C59—41no—
98C22⋊C4×C59—41no—
99C42×C59—41no—
100C42×C59C2×D411yescharacteristic
101#11745670—82no—
102C4⋊Dic59—41no—
103C4⋊Dic59—41no—
104C2×C59⋊C8—41no—
105C59⋊C16—41no—
106C59⋊C16—41no—
107C23×C59—21no—
108C23×C59—41no—
109Q8×C59—22no—
110Q8×C59—41no—
111Q8×C59—41no—
112Q8×C59—41no—
113Q8×C59—82no—
114D4×C59—41no—
115D4×C59—41no—
116D4×C59—41no—
117D4×C59—82no—
118D4×C59—82no—
119C2×C4×C59—21no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—82no—
126C2×C4×C59—82no—
127C2×C4×C59#12011yescharacteristic
128C2×C4×C59C2×D811yescharacteristic
129C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
130C2×Dic59—81no—
131Dic118—81no—
132Dic118—81no—
133C8×C59—21no—
134C8×C59—41no—
135C8×C59C2×SD1611yescharacteristic
136C8×C59AGL(1,ℤ/8)11yescharacteristic
137C59⋊C8—81no—
138#12379—591nomaximal, Sylow
139C22×C59—41no—
140C22×C59—41no—
141C22×C59—41no—
142C22×C59—81no—
143C22×C59—81no—
144C22×C59—81no—
145C22×C59#44711yescharacteristic
146C4×C59—21no—
147C4×C59—21no—
148C4×C59—41no—
149C4×C59—42no—
150C4×C59—81no—
151C4×C59—81no—
152C4×C59#45011yescharacteristic
153C4×C59#45511yescharacteristic
154Dic59—161no—
155#3277D5911yescharacteristic
156#2243—591no—
157#2227—591no—
158#1693—591no—
159#1360—591no—
160#1334—591no—
161#1330—591no—
162C2×C59—81no—
163C2×C59—161no—
164C2×C59#163911yescharacteristic
165C2×C59#219011yescharacteristic
166C2×C59#219511yescharacteristic
167C2×#137—21no—
168#548—21no—
169#500—591no—
170#495C2×D5911yescharacteristic
171#468—21no—
172#464—21no—
173#461—1181no—
174#451—21no—
175#449—21no—
176C4×Q16C2×D5911yescharacteristic
177#436C2×D5911yescharacteristic
178#366—1181no—
179C4⋊C16—591no—
180#358—1181no—
181#358—1181no—
182#357—1181no—
183#346—591no—
184#334—591no—
185#332—1181no—
186C59#1237911yescharacteristic, complemented, Sylow
187C2×#41—21no—
188C22×Q8—21no—
189#137—42no—
190#137—42no—
191C2×Q16—21no—
192C2×Q16C22⋊D5911yescharacteristic
193C2×Q16C22⋊D5911yescharacteristic
194C2×SD16—21no—
195C2×SD16—21no—
196C2×SD16—2361no—
197C2×D8C22⋊D5911yescharacteristic
198C2×M16—21no—
199C4⋊Q8—1181no—
200#124—21no—
201#124—41no—
202C22⋊Q8—21no—
203C22⋊Q8—41no—
204#121—21no—
205#121—41no—
206C4×Q8—21no—
207C4×D4—41no—
208C4×D4C22⋊D5911yescharacteristic
209C2×C16—1181no—
210C2×C16—1181no—
211#107—1181no—
212#107—1181no—
213#107C22⋊D5911yescharacteristic
214C4⋊C8—1181no—
215C4⋊C8C22⋊D5911yescharacteristic
216#103—21no—
217#103—21no—
218#103—2361no—
219#102—21no—
220C22⋊C8—21no—
221C4×C8C22×D5911yescharacteristic
222#41—42no—
223#41—81no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—41no—
227C2×Q8—42no—
228C2×Q8—81no—
229C2×Q8—2361no—
230C2×D4—21no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4#1174566911yescharacteristic
234C22×C4—21no—
235C22×C4—41no—
236C22×C4—41no—
237Q16—21no—
238Q16—41no—
239Q16—42no—
240Q16#1174567012yescomplemented
241SD16—42no—
242SD16—41no—
243SD16—41no—
244SD16—4722no—
245D8—41no—
246M16—41no—
247M16—41no—
248C2×C8—21no—
249C2×C8—2361no—
250C2×C8C2×C22⋊D5911yescharacteristic
251C2×C8C2×C22⋊D5911yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—2361no—
256C4⋊C4—2361no—
257C4⋊C4#1174566911yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C42—41no—
264C42C2×C22⋊D5911yescharacteristic
265C16—2361no—
266C16—2361no—
267C23—21no—
268C23—41no—
269Q8—22no—
270Q8—41no—
271Q8—41no—
272Q8—41no—
273Q8—82no—
274Q8—4721no—
275Q8—4721no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—82no—
280D4—82no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—4721no—
290C2×C4#1427369611yescharacteristic
291C2×C4C2×#1174566911yescharacteristic
292C2×C4#1427375311yescharacteristic
293C8—21no—
294C8—41no—
295C8—4721no—
296C8C2×#1174567011yescharacteristic
297C8#1427375311yescharacteristic
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22#1683753611yescharacteristic
305C4—21no—
306C4—21no—
307C4—41no—
308C4—42no—
309C4—81no—
310C4—81no—
311C4—9441no—
312C4#1683755311yescharacteristic
313C4#1683758011yescharacteristic
314C2—81no—
315C2—161no—
316C2#2142327511yescharacteristic
317C2#2142604211yescharacteristic
318C2#2142607411yescharacteristic
subgroupG/H
centerC22#16837536
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
Fitting, prime radicalC59×#3277C2
socleC22×C59#447
hypercenter#3277D59
hypocenterC59#12379
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #436 ◁ #3277

Complements

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

NH ≅ G/Ntype
1. #21426332C2semidirect
2. #21426248C2semidirect
3. #21423609C2semidirect
4. #21422139C2semidirect
5. #21422066C2semidirect
6. #21422054C2semidirect
80. Q16×C59SD16semidirect
186. C59#12379semidirect
240. Q16#11745670semidirect

Maximal subgroups

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

subgroupindex
#214263322
#214262482
#214236092
#214221392
#214220662
#214220542
C59×#32772
#1237959