Finite Group Databasebeta

← #28504721

Subgroups — #28504721

Order 15104.

420 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 89 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#21432804C211yesmaximal, characteristic
2#21425779C212yesmaximal
3#21424144C212yesmaximal, complemented
4#21422219C211yesmaximal, characteristic, complemented
5C59×#1392C211yesmaximal, characteristic, complemented
6C2×#14273735C411yescharacteristic
7C2×#14273734C2211yescharacteristic
8#16837948C2211yescharacteristic
9#16837477—22no—
10(C22⋊D59)⋊C8—22no—
11C2×#14273662—22no—
12#16837425C411yescharacteristic
13#16837386C2212yes—
14#16837368C411yescharacteristic
15C2×C8×Dic59C411yescharacteristic
16#16837200—22no—
17#16837198—22no—
18C59⋊#338—22no—
19C59×#413C2212yes—
20C59×#408C2211yescharacteristic
21C2×#11745685C2×C411yescharacteristic
22#14273735—42no—
23#14273734—42no—
24C2×#11745664—22no—
25C2×#11745664C2×C411yescharacteristic
26C2×#11745664D412yes—
27C2×#11745659—21no—
28C22×C59⋊C8C2×C411yescharacteristic
29C22×C59⋊C8D412yes—
30C2×C8×D59—21no—
31#14273725D411yescharacteristic
32#14273723Q811yescharacteristic
33#14273716D411yescharacteristic
34#14273714Q811yescharacteristic
35#14273699—22no—
36C4×C22⋊D59—22no—
37C2×C4×Dic59C2×C411yescharacteristic
38#14273662—44no—
39Dic59⋊C8—22no—
40#14273646—22no—
41#14273644—22no—
42(C2×D59)⋊C8—22no—
43C8×Dic59—21no—
44C8×Dic59—21no—
45C2×#41×C59C2311yescharacteristic
46C2×M16×C59—22no—
47C2×M16×C59C2×C411yescharacteristic
48C22×C8×C59C2×C411yescharacteristic
49#101×C59—22no—
50#100×C59—22no—
51C22⋊C8×C59—22no—
52#41×C59—41no—
53#41×C59—41no—
54C2×Q8×C59C2×D411yescharacteristic
55C2×D4×C59C2×Q811yescharacteristic
56C2×D4×C59#4112yes—
57C22×C4×C59C22×C411yescharacteristic
58C22×C4×C59C2×D412yes—
59C2×C22⋊D59—21no—
60C2×C22⋊D59—22no—
61C22×Dic59—21no—
62#11745685—41no—
63#11745685—41no—
64C2×D236C4⋊C411yescharacteristic
65C2×C4×D59—21no—
66C2×Dic118C4⋊C411yescharacteristic
67M16×C59—41no—
68M16×C59—42no—
69M16×C59—42no—
70C2×C8×C59—21no—
71C2×C8×C59—21no—
72C2×C8×C59—41no—
73C2×C8×C59—42no—
74C22⋊Dic59—22no—
75(C2×D59)⋊C4—22no—
76Dic59⋊C4—22no—
77C4×Dic59—21no—
78C4×Dic59C4⋊C411yescharacteristic
79C4×Dic59C4⋊C411yescharacteristic
80#11745664—41no—
81#11745664—42no—
82#11745664—42no—
83#11745664—42no—
84C2×C59⋊C8—24no—
85C2×C59⋊C8—22no—
86C2×C59⋊C8—42no—
87C2×C59⋊C8C22⋊C411yescharacteristic
88C2×C59⋊C8C22⋊C412yes—
89C2×C59⋊C8C22⋊C411yescharacteristic
90#11745659—42no—
91C8×D59—42no—
92C23×C59#11711yescharacteristic
93C23×C59C22⋊Q812yes—
94C22×D59—21no—
95Q8×C59—41no—
96D4×C59—41no—
97D4×C59—42no—
98C2×C4×C59—22no—
99C2×C4×C59—21no—
100C2×C4×C59—22no—
101C2×C4×C59—21no—
102C2×C4×C59C2×C22⋊C411yescharacteristic
103C2×C4×C59C2×C4⋊C411yescharacteristic
104C2×C4×C59#12112yes—
105C22⋊D59—44no—
106C22⋊D59—42no—
107C2×Dic59—21no—
108C2×Dic59—21no—
109C2×Dic59—21no—
110C2×Dic59—42no—
111D236—41no—
112C4×D59—41no—
113C4×D59—41no—
114Dic118—41no—
115C8×C59—41no—
116C8×C59—41no—
117C8×C59—82no—
118C59⋊C8—21no—
119C59⋊C8—22no—
120C59⋊C8—82no—
121C59⋊C8#9912yes—
122#12809—591nomaximal, Sylow
123C22×C59—21no—
124C22×C59—22no—
125C22×C59—41no—
126C22×C59—42no—
127C22×C59#38011yescharacteristic
128C2×D59—42no—
129C2×D59—41no—
130C4×C59—22no—
131C4×C59—21no—
132C4×C59C2×#9911yescharacteristic
133C4×C59#41011yescharacteristic
134Dic59—22no—
135Dic59—41no—
136Dic59—41no—
137#2931—591no—
138#2132—592no—
139#1813—592no—
140#1392—591no—
141#1392D5911yescharacteristic, complemented
142C2×C59—21no—
143C2×C59—41no—
144C2×C59—42no—
145C2×C59#179911yescharacteristic
146D59—81no—
147C2×#131—591no—
148C2×#131—591no—
149#532—591no—
150#436—1182no—
151C8×D4—1182no—
152C2×#108—1182no—
153#423—591no—
154#413—592no—
155#413C2×D5912yes—
156#408—591no—
157#408C2×D5911yescharacteristic
158C2×C4×C8—591no—
159#356—1182no—
160#355—1182no—
161#338—1182no—
162C59#1280911yescharacteristic, complemented, Sylow
163C2×#41—591no—
164C2×#41C22×D5911yescharacteristic
165#131—2362no—
166#131—2362no—
167C2×M16—22no—
168C2×M16—591no—
169C2×M16—592no—
170C2×M16—1181no—
171C2×M16—1182no—
172C2×M16C4×D5911yescharacteristic
173C22×C8—591no—
174C22×C8—592no—
175C22×C8—1181no—
176C22×C8C4×D5911yescharacteristic
177C4⋊Q8—591no—
178#127—591no—
179#124—591no—
180#124—591no—
181#121—1182no—
182C4×D4—1182no—
183C2×C42—591no—
184#108—2364no—
185C4⋊C8—1182no—
186#101—22no—
187#101—1182no—
188#100—22no—
189#100—1182no—
190C22⋊C8—22no—
191C22⋊C8—1182no—
192C4×C8—1181no—
193C4×C8—1181no—
194#41—41no—
195#41—41no—
196#41—2361no—
197#41—2361no—
198C2×Q8—591no—
199C2×Q8D4×D5911yescharacteristic
200C2×D4—591no—
201C2×D4—1181no—
202C2×D4—1182no—
203C2×D4#1174568512yes—
204C2×D4Q8×D5911yescharacteristic
205C22×C4—1181no—
206C22×C4—1181no—
207C22×C4C2×C4×D5911yescharacteristic
208C22×C4D4×D5912yes—
209M16—41no—
210M16—42no—
211M16—42no—
212M16—2362no—
213M16—2361no—
214M16—2362no—
215M16—2362no—
216M16—2362no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—41no—
220C2×C8—42no—
221C2×C8—591no—
222C2×C8—591no—
223C2×C8—592no—
224C2×C8—1184no—
225C2×C8—1182no—
226C2×C8—2362no—
227C2×C8—2362no—
228C4⋊C4—1182no—
229C22⋊C4—1182no—
230C22⋊C4—1182no—
231C42—591no—
232C42—591no—
233C42—1181no—
234C23—1181no—
235C23#1427368411yescharacteristic
236C23#1427370512yes—
237Q8—41no—
238Q8—2361no—
239D4—42no—
240D4—41no—
241D4—2361no—
242D4—2364no—
243D4—2362no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—1181no—
249C2×C4—1181no—
250C2×C4—1181no—
251C2×C4—2362no—
252C2×C4—2361no—
253C2×C4—2361no—
254C2×C4C22⋊C4×D5911yescharacteristic
255C2×C4C4⋊C4×D5911yescharacteristic
256C2×C4#1427370012yes—
257C8—41no—
258C8—41no—
259C8—82no—
260C8—592no—
261C8—1182no—
262C8—1181no—
263C8—4722no—
264C22—21no—
265C22—22no—
266C22—41no—
267C22—42no—
268C22—2362no—
269C22—2361no—
270C22#1683726111yescharacteristic
271C4—22no—
272C4—21no—
273C4—1182no—
274C4—2361no—
275C4—2361no—
276C4#99×D5911yescharacteristic
277C4#1683737411yescharacteristic
278C2—21no—
279C2—41no—
280C2—42no—
281C2—4721no—
282C2#2142408311yescharacteristic
subgroupG/H
centerC4#16837374
commutatorC22×C4×C59C22×C4
FrattiniC2×#41C22×D59
FittingC59×#1392C2
socleC2×C59#1799
prime radical#16837948C22
cosocleC2×#41×C59C23
hypercenter#1392D59
hypocenterC59#12809
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1392

Complements

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

NH ≅ G/Ntype
3. #21424144C2semidirect
4. #21422219C2semidirect
5. C59×#1392C2semidirect
141. #1392D59semidirect
162. C59#12809semidirect

Maximal subgroups

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

subgroupindex
#214328042
#214257792
#214241442
#214222192
C59×#13922
#1280959