Finite Group Databasebeta

← #28502486

Subgroups — #28502486

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
1Q16×D236C211yesmaximal, characteristic
2#21426324C211yesmaximal, characteristic, complemented
3#21426238C211yesmaximal, characteristic, complemented
4#21423486C211yesmaximal, characteristic
5#21422144C211yesmaximal, characteristic, complemented
6#21422052C211yesmaximal, characteristic
7C59×#1332C211yesmaximal, characteristic, complemented
8C2×Q16×D59—21no—
9Q8×D236—21no—
10#16837807C2211yescharacteristic
11#16837782C2211yescharacteristic
12#16837644—21no—
13#16837612—21no—
14#16837608—21no—
15#16837559—22no—
16C8×D236C2211yescharacteristic
17#16837229—21no—
18C4⋊(C59⋊C16)C2211yescharacteristic
19#16837207—22no—
20#16837206—21no—
21#16837149—21no—
22C4×Q16×C59C2211yescharacteristic
23C4×C16×C59C2211yescharacteristic
24C59×#335C2211yescharacteristic
25C2×Q8×D59—22no—
26C2×Dic236—21no—
27C2×Dic236D412yes—
28C2×#11745672—22no—
29Q16×D59—42no—
30Q16×D59—42no—
31C2×#11745660—41no—
32C2×C8×D59—21no—
33#14273727—21no—
34#14273727—21no—
35#14273727—41no—
36#14273707—22no—
37#14273707—42no—
38C4×D236—41no—
39C4×D236D411yescharacteristic
40C2×C59⋊C16—21no—
41#14273661—21no—
42#14273661—21no—
43C4⋊(C59⋊C8)D411yescharacteristic
44#14273652—41no—
45#14273650—22no—
46(C2×D59)⋊C8—21no—
47C2×Q16×C59D411yescharacteristic
48C4×Q8×C59—21no—
49C2×C16×C59—21no—
50#107×C59D411yescharacteristic
51C4⋊C8×C59—21no—
52#103×C59—21no—
53C4×C8×C59C2311yescharacteristic
54C2×Q8×C59—21no—
55Q8×D59—44no—
56Q8×D59—82no—
57C2×D236D811yescharacteristic
58C2×C4×D59—21no—
59C2×C4×D59—42no—
60C2×Dic118—22no—
61C2×Dic118—42no—
62C2×Dic118—41no—
63Q16×C59—21no—
64Q16×C59SD1612yescomplemented
65C2×C8×C59—41no—
66C2×C8×C59C2×D411yescharacteristic
67C2×C8×C59C2×D411yescharacteristic
68C4⋊C4×C59—21no—
69C4⋊C4×C59—41no—
70C42×C59—41no—
71C42×C59C2×D411yescharacteristic
72#11745672—44no—
73(C2×D59)⋊C4—21no—
74(C2×D59)⋊C4—42no—
75C4⋊Dic59—42no—
76C4⋊Dic59—41no—
77C4⋊Dic59—41no—
78C4⋊Dic59—41no—
79C4⋊Dic59—41no—
80C4⋊Dic59D811yescharacteristic
81C2×C59⋊C8—21no—
82Dic236—42no—
83Dic236—41no—
84Dic236—41no—
85#11745660—82no—
86C8×D59—41no—
87C8×D59—41no—
88C16×C59—41no—
89C59⋊C16—41no—
90C22×D59—21no—
91Q8×C59—22no—
92Q8×C59—41no—
93C2×C4×C59—41no—
94C2×C4×C59—41no—
95C2×C4×C59#12011yescharacteristic
96C2×C4×C59C2×D811yescharacteristic
97C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
98C2×Dic59—21no—
99C2×Dic59—42no—
100C2×Dic59—81no—
101D236—21no—
102D236SD3212yes—
103C4×D59—41no—
104C4×D59—41no—
105C4×D59—84no—
106Dic118—42no—
107Dic118—42no—
108Dic118—84no—
109Dic118—81no—
110Dic118—81no—
111C8×C59—21no—
112C8×C59—81no—
113C8×C59C2×SD1611yescharacteristic
114C8×C59AGL(1,ℤ/8)11yescharacteristic
115C59⋊C8—41no—
116#12370—591nomaximal, Sylow
117C22×C59#44711yescharacteristic
118C2×D59—22no—
119C2×D59—41no—
120C2×D59—41no—
121C4×C59—21no—
122C4×C59—42no—
123C4×C59—81no—
124C4×C59#45011yescharacteristic
125C4×C59#45511yescharacteristic
126C4×C59C2×SD3211yescharacteristic
127C4×C59#51011yescharacteristic
128Dic59—41no—
129Dic59—82no—
130Dic59—161no—
131D4×Q16—591no—
132#2241—591no—
133#2225—591no—
134#1673—591no—
135#1362—591no—
136#1332D5911yescharacteristic, complemented
137D4⋊C16—591no—
138C2×C59#163911yescharacteristic
139C2×C59#218511yescharacteristic
140C2×C59#219511yescharacteristic
141D59—42no—
142C22×Q16—1181no—
143D4×Q8—1181no—
144#500—591no—
145#494—591no—
146#467—1181no—
147#462—1181no—
148#461—1181no—
149#451—1182no—
150C4×Q16C2×D5911yescharacteristic
151C8×D4—591no—
152#367—1181no—
153C4⋊C16—591no—
154#358—1182no—
155#358—1181no—
156C4×C16C2×D5911yescharacteristic
157#335C2×D5911yescharacteristic
158#332—1181no—
159C59#1237011yescharacteristic, complemented, Sylow
160C22×Q8—1182no—
161C2×Q16—592no—
162C2×Q16—1182no—
163C2×Q16—1181no—
164C2×Q16—2362no—
165C2×Q16—2362no—
166C2×Q16D23611yescharacteristic
167C2×SD16—2361no—
168C22×C8—1181no—
169C4⋊Q8—1181no—
170C4⋊Q8—1181no—
171C4⋊Q8—2361no—
172C22⋊Q8—1182no—
173C22⋊Q8—2362no—
174C4×Q8—21no—
175C4×D4—591no—
176C4×D4—2361no—
177C2×C16—21no—
178C2×C16—1181no—
179#107—1181no—
180#107—1181no—
181#107D23611yescharacteristic
182C4⋊C8—21no—
183C4⋊C8—591no—
184#103—21no—
185#103—1182no—
186#103—2361no—
187C22⋊C8—1181no—
188C4×C8C22×D5911yescharacteristic
189C2×Q8—21no—
190C2×Q8—1182no—
191C2×Q8—4722no—
192C2×Q8—2362no—
193C2×Q8—2364no—
194C2×Q8—2361no—
195C2×D4—591no—
196C22×C4—1181no—
197C22×C4—2362no—
198Q16—21no—
199Q16—2362no—
200Q16—2364no—
201Q16—2361no—
202Q16—2361no—
203Q16#1174566012yescomplemented
204SD16—4722no—
205C2×C8—41no—
206C2×C8—1181no—
207C2×C8—2361no—
208C2×C8—2361no—
209C2×C8C2×D23611yescharacteristic
210C2×C8D4×D5911yescharacteristic
211C4⋊C4—21no—
212C4⋊C4—41no—
213C4⋊C4—591no—
214C4⋊C4—2362no—
215C4⋊C4—2361no—
216C4⋊C4—2361no—
217C4⋊C4—2361no—
218C4⋊C4—2361no—
219C22⋊C4—1181no—
220C22⋊C4—2362no—
221C42—41no—
222C42D4×D5911yescharacteristic
223C16—41no—
224C16—2361no—
225C23—1181no—
226Q8—22no—
227Q8—41no—
228Q8—4724no—
229Q8—4721no—
230Q8—4721no—
231Q8—2362no—
232Q8—2362no—
233D4—592no—
234D4—1181no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—1181no—
238C2×C4—4724no—
239C2×C4—4721no—
240C2×C4—2362no—
241C2×C4—2361no—
242C2×C4—2361no—
243C2×C4#1427369711yescharacteristic
244C2×C4D8×D5911yescharacteristic
245C2×C4#1427375411yescharacteristic
246C8—21no—
247C8—81no—
248C8—2361no—
249C8C2×#1174566011yescharacteristic
250C8#1427375511yescharacteristic
251C22—1182no—
252C22—2361no—
253C22—2361no—
254C22#1683753711yescharacteristic
255C4—21no—
256C4—42no—
257C4—81no—
258C4—9441no—
259C4—4722no—
260C4—2361no—
261C4#1683755511yescharacteristic
262C4#1683758111yescharacteristic
263C4SD32×D5911yescharacteristic
264C4#1683784411yescharacteristic
265C2—2362no—
266C2#2142327811yescharacteristic
267C2#2142601411yescharacteristic
268C2#2142607511yescharacteristic
subgroupG/H
centerC22#16837537
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1332C2
socleC22×C59#447
prime radicalC2×D236D8
hypercenter#1332D59
hypocenterC59#12370
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1332

Complements

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

NH ≅ G/Ntype
2. #21426324C2semidirect
3. #21426238C2semidirect
5. #21422144C2semidirect
7. C59×#1332C2semidirect
64. Q16×C59SD16semidirect
136. #1332D59semidirect
159. C59#12370semidirect
203. Q16#11745660semidirect

Maximal subgroups

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

subgroupindex
Q16×D2362
#214263242
#214262382
#214234862
#214221442
#214220522
C59×#13322
#1237059