Finite Group Databasebeta

← #28743925

Subgroups — #28743925

Order 15104.

520 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 131 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#21433007C211yesmaximal, characteristic
2#21426313C212yesmaximal, complemented
3#21426264C212yesmaximal, complemented
4#21426174C212yesmaximal, complemented
5#21426046C212yesmaximal
6#21425985C214yesmaximal, complemented
7#21425908C211yesmaximal, characteristic, complemented
8C59×#3185C211yesmaximal, characteristic, complemented
9#16838202C2211yescharacteristic
10#16838168—21no—
11#16837843C2214yes—
12#16837525—21no—
13#16837525C2212yes—
14#16837520—21no—
15#16837520C2212yes—
16#16837451C2211yescharacteristic
17C2×#14273655—21no—
18#16837224C2214yescomplemented
19#16837223C2212yescomplemented
20#16837218C2214yescomplemented
21#16837211C2214yes—
22#16837184C2212yes—
23#16837180C2212yescomplemented
24C59×#533—21no—
25C59×#498C2211yescharacteristic, complemented
26C59×#494C2211yescharacteristic, complemented
27C59×#487—21no—
28C59×#470—21no—
29C59×#470C2212yes—
30C59×#441—21no—
31C59×#441C2212yescomplemented
32C4×M16×C59C2211yescharacteristic
33C2×#11745685—21no—
34#14273760—22no—
35#14273755—22no—
36#14273749—21no—
37#14273749C2312yes—
38#14273735—42no—
39#14273733—22no—
40#14273733—22no—
41C2×#11745664—21no—
42M16×D59—42no—
43#14273673—24no—
44#14273670—24no—
45#14273667C2314yes—
46#14273663—21no—
47#14273663C2312yes—
48#14273655—22no—
49#14273655—22no—
50C2×Q16×C59—21no—
51C2×Q16×C59C2312yes—
52C2×M16×C59D411yescharacteristic
53C2×M16×C59C2311yescharacteristic
54C4⋊Q8×C59—21no—
55C4⋊Q8×C59—21no—
56#125×C59—41no—
57C22⋊Q8×C59—22no—
58C22⋊Q8×C59—41no—
59C4×Q8×C59—22no—
60C4×Q8×C59—41no—
61#117×C59—41no—
62C2×C42×C59D411yescharacteristic
63#106×C59—21no—
64#106×C59C2312yes—
65#103×C59—22no—
66#103×C59—22no—
67#97×C59D411yescharacteristic
68#97×C59C2311yescharacteristic
69C4×C8×C59D411yescharacteristic
70C4×C8×C59C2311yescharacteristic
71C2×Q8×C59—22no—
72C2×Q8×C59—41no—
73C22×C4×C59—21no—
74C22×C4×C59C2×D411yescharacteristic
75C2×C22⋊D59—41no—
76#11745685—22no—
77#11745685—22no—
78#11745685—41no—
79#11745685—41no—
80C2×D236—21no—
81C2×C4×D59—41no—
82C2×Dic118—21no—
83Q16×C59—24no—
84Q16×C59—42no—
85M16×C59—22no—
86M16×C59—22no—
87C2×C8×C59C2×D411yescharacteristic
88C2×C8×C59C2×D411yescharacteristic
89C2×C8×C59C2×D411yescharacteristic
90C2×C8×C59C2411yescharacteristic
91C4⋊C4×C59—22no—
92C4⋊C4×C59—41no—
93C4⋊C4×C59—41no—
94C4⋊C4×C59—42no—
95C4⋊C4×C59—41no—
96C4⋊C4×C59—42no—
97C22⋊C4×C59—42no—
98C42×C59—21no—
99C42×C59—42no—
100C42×C59C2×D411yescharacteristic
101C42×C59C2×D411yescharacteristic
102#11745664—22no—
103#11745664—41no—
104C2×C59⋊C8—41no—
105Dic236—21no—
106Dic236#4112yes—
107D472—21no—
108D472#4112yes—
109#11745660—22no—
110#11745660—22no—
111#11745659—42no—
112#11745659—42no—
113C8×D59—42no—
114C8×D59—42no—
115C59⋊C16—24no—
116C23×C59AGL(1,ℤ/8)11yescharacteristic
117C22×D59—41no—
118Q8×C59—42no—
119Q8×C59—42no—
120Q8×C59—82no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—42no—
126C2×C4×C59—42no—
127C2×C4×C59—42no—
128C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C59C22×D411yescharacteristic
130C22⋊D59—42no—
131C22⋊D59—42no—
132C2×Dic59—41no—
133D236—22no—
134D236—41no—
135C4×D59—42no—
136C4×D59—42no—
137Dic118—22no—
138Dic118—41no—
139C8×C59—21no—
140C8×C59—21no—
141C8×C59AGL(1,ℤ/8)12yes—
142C8×C59C2×#4112yes—
143C59⋊C8—42no—
144#33689—591nomaximal, Sylow
145C22×C59—21no—
146C22×C59—41no—
147C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
148C2×D59—42no—
149C2×D59—161no—
150C4×C59—22no—
151C4×C59—21no—
152C4×C59—41no—
153C4×C59—82no—
154C4×C59—82no—
155C4×C59#52511yescharacteristic
156C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
157Dic59—42no—
158#3185D5911yescharacteristic, complemented
159#2956—591no—
160#2239—592no—
161#2230—592no—
162#2214—592no—
163#2191—592no—
164#2180—594no—
165#2165—591no—
166C2×C59—21no—
167C2×C59—41no—
168C2×C59#310911yescharacteristic
169D59—82no—
170#575—591no—
171#568—1181no—
172#533—21no—
173#510—594no—
174#498C2×D5911yescharacteristic, complemented
175#494C2×D5911yescharacteristic, complemented
176#487—21no—
177#470—21no—
178#470C2×D5912yes—
179#444—592no—
180#444—1181no—
181#443—592no—
182#443—1181no—
183#441—21no—
184#441C2×D5912yescomplemented
185#430—591no—
186C2×#104—1181no—
187C4×M16C2×D5911yescharacteristic
188#365—594no—
189#364—592no—
190#362—594no—
191#360—594no—
192#349—592no—
193#347—592no—
194C59#3368911yescharacteristic, complemented, Sylow
195C2×#41—1181no—
196#137—1182no—
197AGL(1,ℤ/8)—1182no—
198#135—592no—
199#135—1181no—
200C2×Q16—21no—
201C2×Q16C22×D5912yes—
202#131—1182no—
203#131—1182no—
204#131—2362no—
205C2×M16—1181no—
206C2×M16—2362no—
207C2×M16C22⋊D5911yescharacteristic
208C2×M16C22×D5911yescharacteristic
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211#125—41no—
212C22⋊Q8—22no—
213C22⋊Q8—41no—
214C4×Q8—22no—
215C4×Q8—41no—
216#117—41no—
217C2×C42C22⋊D5911yescharacteristic
218Q32—1184no—
219SD32—1184no—
220M32—594no—
221#108—592no—
222#108—1181no—
223#106—21no—
224#106C22×D5912yes—
225#104—1182no—
226#104—1182no—
227#103—22no—
228#103—22no—
229#97C22⋊D5911yescharacteristic
230#97C22×D5911yescharacteristic
231C4×C8C22⋊D5911yescharacteristic
232C4×C8C22×D5911yescharacteristic
233#41—1182no—
234#41—1182no—
235#41—2361no—
236#41—2361no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—1181no—
240C2×D4—1181no—
241C2×D4—2361no—
242C22×C4—21no—
243C22×C4—2361no—
244C22×C4C2×C22⋊D5911yescharacteristic
245Q16—24no—
246Q16—42no—
247Q16—592no—
248Q16—1181no—
249SD16—1182no—
250SD16—1182no—
251D8—592no—
252D8—1181no—
253M16—22no—
254M16—22no—
255M16—1182no—
256M16—2362no—
257M16—2361no—
258M16—2362no—
259C2×C8—2362no—
260C2×C8—2361no—
261C2×C8—2362no—
262C2×C8C2×C22⋊D5911yescharacteristic
263C2×C8C2×C22⋊D5911yescharacteristic
264C2×C8C2×C22⋊D5911yescharacteristic
265C2×C8C23×D5911yescharacteristic
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C22⋊C4—42no—
273C42—21no—
274C42—42no—
275C42C2×C22⋊D5911yescharacteristic
276C42C2×C22⋊D5911yescharacteristic
277C16—1184no—
278C23—2361no—
279C23#1427375311yescharacteristic
280Q8—42no—
281Q8—42no—
282Q8—82no—
283Q8—1182no—
284Q8—2361no—
285D4—1182no—
286D4—2361no—
287D4—2362no—
288D4—2362no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—2362no—
297C2×C4—2361no—
298C2×C4—2362no—
299C2×C4#1427375311yescharacteristic
300C2×C4C22×C22⋊D5911yescharacteristic
301C8—21no—
302C8—21no—
303C8—2362no—
304C8#1427375212yes—
305C8#41×D5912yes—
306C22—21no—
307C22—41no—
308C22—9441no—
309C22—2362no—
310C22C2×#1427375311yescharacteristic
311C4—22no—
312C4—21no—
313C4—41no—
314C4—82no—
315C4—82no—
316C4—2362no—
317C4#1683790911yescharacteristic
318C4C2×#1427375211yescharacteristic
319C2—21no—
320C2—41no—
321C2—4722no—
322C2#2143418311yescharacteristic
subgroupG/H
centerC2#21434183
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3185C2
socleC2×C59#3109
prime radical#16838202C22
hypercenter#3185D59
hypocenterC59#33689
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3185

Complements

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

NH ≅ G/Ntype
2. #21426313C2semidirect
3. #21426264C2semidirect
4. #21426174C2semidirect
6. #21425985C2semidirect
7. #21425908C2semidirect
8. C59×#3185C2semidirect
18. #16837224C22semidirect
19. #16837223C22semidirect
20. #16837218C22semidirect
23. #16837180C22semidirect
25. C59×#498C22semidirect
26. C59×#494C22semidirect
31. C59×#441C22semidirect
158. #3185D59semidirect
174. #498C2×D59semidirect
175. #494C2×D59semidirect
184. #441C2×D59semidirect
194. C59#33689semidirect

Maximal subgroups

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

subgroupindex
#214330072
#214263132
#214262642
#214261742
#214260462
#214259852
#214259082
C59×#31852
#3368959