Finite Group Databasebeta

← #28483092

Subgroups — #28483092

Order 15104.

392 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 99 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#21424929C211yesmaximal, characteristic
2#21422864C212yesmaximal, complemented
3#21422688C211yesmaximal, characteristic
4#21421988C212yesmaximal, complemented
5C59×#2980C211yesmaximal, characteristic
6#16837426C2211yescharacteristic
7C2×#14273654—21no—
8C2×#14273654—21no—
9C4×#11745664C2212yes—
10#16837248C2211yescharacteristic
11#16837154—22no—
12#16837152—22no—
13#16837142C411yescharacteristic, complemented
14#16837142C412yescomplemented
15#16837142C411yescharacteristic, complemented
16C59×#532C2211yescharacteristic
17C59×#435—22no—
18C59×#435—22no—
19C59×#432C2212yes—
20C59×#408—21no—
21C59×#408—21no—
22C2×#11745664—22no—
23C2×#11745664—21no—
24C2×C4×Dic59—21no—
25C4⋊(C59⋊C8)—22no—
26#14273654—42no—
27#14273654—42no—
28#14273638C2×C412yes—
29#14273638C2×C412yes—
30C4×C59⋊C8—22no—
31C59⋊#95—22no—
32C2×#41×C59D411yescharacteristic
33C2×#41×C59D411yescharacteristic
34C2×M16×C59—21no—
35C2×M16×C59—21no—
36C22×C8×C59—21no—
37C22×C8×C59—21no—
38C4⋊Q8×C59Q811yescharacteristic
39#127×C59Q811yescharacteristic
40#124×C59C2×C411yescharacteristic
41#124×C59C2×C411yescharacteristic
42#121×C59—22no—
43C4×D4×C59—22no—
44C2×C42×C59C2311yescharacteristic
45C4⋊C8×C59D412yes—
46C4⋊C8×C59D412yes—
47C22⋊C8×C59—22no—
48C22⋊C8×C59—22no—
49C22⋊C8×C59—22no—
50#97×C59—22no—
51#41×C59—41no—
52#41×C59—41no—
53#41×C59—41no—
54#41×C59—41no—
55C2×Q8×C59C4⋊C411yescharacteristic
56C2×Q8×C59C4⋊C411yescharacteristic
57C2×D4×C59—21no—
58C2×D4×C59—21no—
59C2×D4×C59—22no—
60C2×D4×C59C4⋊C411yescharacteristic
61C2×D4×C59C4⋊C411yescharacteristic
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59C2×D411yescharacteristic
65C22×C4×C59#4112yes—
66C22×Dic59—41no—
67M16×C59—81no—
68M16×C59—81no—
69C2×C8×C59—22no—
70C2×C8×C59—22no—
71C2×C8×C59—81no—
72C2×C8×C59—81no—
73C4⋊C4×C59—22no—
74C22⋊C4×C59—22no—
75C22⋊C4×C59—22no—
76C42×C59C22×C411yescharacteristic
77C42×C59C2×D412yes—
78C42×C59C2×Q811yescharacteristic
79C4×Dic59—22no—
80C4×Dic59—41no—
81#11745664—44no—
82#11745664—41no—
83#11745664—41no—
84C2×C59⋊C8—22no—
85C2×C59⋊C8—22no—
86C2×C59⋊C8—41no—
87C23×C59—21no—
88C23×C59—21no—
89C23×C59#12311yescharacteristic
90Q8×C59—41no—
91Q8×C59—41no—
92D4×C59—41no—
93D4×C59—41no—
94D4×C59—82no—
95D4×C59—81no—
96D4×C59—81no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—42no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59C2×C4⋊C411yescharacteristic
107C2×C4×C59C4×D412yes—
108C2×C4×C59#12011yescharacteristic
109C2×C4×C59C22⋊Q812yes—
110C2×Dic59—42no—
111C2×Dic59—82no—
112C8×C59—42no—
113C8×C59—42no—
114C59⋊C8—24no—
115C59⋊C8—42no—
116C59⋊C8—81no—
117#8918—591nomaximal, Sylow
118C22×C59—21no—
119C22×C59—22no—
120C22×C59—41no—
121C22×C59—41no—
122C22×C59—42no—
123C22×C59—42no—
124C22×C59#38511yescharacteristic
125C4×C59—21no—
126C4×C59—22no—
127C4×C59—22no—
128C4×C59—41no—
129C4×C59—41no—
130C4×C59#45311yescharacteristic
131C4×C59#45511yescharacteristic
132Dic59—82no—
133#2980D5911yescharacteristic
134#1963—591no—
135#1569—592no—
136#1525—591no—
137#1300—592no—
138C2×C59—41no—
139C2×C59—81no—
140C2×C59—81no—
141C2×C59#185611yescharacteristic
142C2×C59#186112yes—
143#532C2×D5911yescharacteristic
144#435—22no—
145#435—22no—
146#432C2×D5912yes—
147#423—591no—
148C2×#104—1181no—
149C2×#104—1181no—
150#408—21no—
151#408—21no—
152C4×M16—592no—
153#376—591no—
154#335—1182no—
155#334—1182no—
156#329—591no—
157#329—592no—
158#329—591no—
159C59#891811yescharacteristic, complemented, Sylow
160C2×#41C22⋊D5911yescharacteristic
161C2×#41C22⋊D5911yescharacteristic
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—1182no—
165C2×M16—1181no—
166C22×C8—21no—
167C22×C8—21no—
168C4⋊Q8Dic11811yescharacteristic
169#127Dic11811yescharacteristic
170#124C4×D5911yescharacteristic
171#124C4×D5911yescharacteristic
172#121—22no—
173C4×D4—22no—
174C2×C42—1181no—
175C2×C42C22×D5911yescharacteristic
176C4⋊C8—1182no—
177C4⋊C8C22⋊D5912yes—
178C4⋊C8C22⋊D5912yes—
179#104—2362no—
180#104—2362no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184#97—22no—
185#97—592no—
186#97—592no—
187C4×C8—1182no—
188#95—1182no—
189#41—41no—
190#41—41no—
191#41—41no—
192#41—41no—
193C2×Q8Dic59⋊C411yescharacteristic
194C2×Q8Dic59⋊C411yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4Dic59⋊C411yescharacteristic
199C2×D4Dic59⋊C411yescharacteristic
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—2361no—
203C22×C4#1174568512yes—
204C22×C4C2×C22⋊D5911yescharacteristic
205M16—81no—
206M16—81no—
207M16—2364no—
208M16—2361no—
209M16—2361no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—81no—
213C2×C8—81no—
214C2×C8—1182no—
215C2×C8—1182no—
216C2×C8—2361no—
217C4⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C42—1182no—
221C42—2361no—
222C42C2×Dic11811yescharacteristic
223C42C2×C4×D5911yescharacteristic
224C42C2×C22⋊D5912yes—
225C23—21no—
226C23—21no—
227C23#1427371011yescharacteristic
228Q8—41no—
229Q8—41no—
230D4—41no—
231D4—41no—
232D4—82no—
233D4—81no—
234D4—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—4722no—
245C2×C4—2362no—
246C2×C4C2×Dic59⋊C411yescharacteristic
247C2×C4C4×C22⋊D5912yes—
248C2×C4#1427369611yescharacteristic
249C2×C4#1427370612yes—
250C8—42no—
251C8—42no—
252C8—1184no—
253C8—4721no—
254C8—2362no—
255C22—21no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—42no—
260C22—42no—
261C22#1683727511yescharacteristic
262C4—21no—
263C4—22no—
264C4—22no—
265C4—41no—
266C4—41no—
267C4—4722no—
268C4#1683757011yescharacteristic
269C4#1683758011yescharacteristic
270C2—41no—
271C2—81no—
272C2—81no—
273C2#2142430611yescharacteristic
274C2#2142433612yes—
subgroupG/H
centerC2×C4#14273696
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2980C2
socleC22×C59#385
prime radicalC59×#532C22
cosocleC2×C42×C59C23
hypercenter#2980D59
hypocenterC59#8918
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2980

Complements

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

NH ≅ G/Ntype
2. #21422864C2semidirect
4. #21421988C2semidirect
13. #16837142C4semidirect
14. #16837142C4semidirect
15. #16837142C4semidirect
159. C59#8918semidirect

Maximal subgroups

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

subgroupindex
#214249292
#214228642
#214226882
#214219882
C59×#29802
#891859