Finite Group Databasebeta

← #28497905

Subgroups — #28497905

Order 15104.

310 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 87 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#21432736C211yesmaximal, characteristic
2#21425896C211yesmaximal, characteristic, complemented
3#21425892C211yesmaximal, characteristic, complemented
4#21422037C211yesmaximal, characteristic, complemented
5#21422037C211yesmaximal, characteristic, complemented
6#21422022C211yesmaximal, characteristic
7C59×#1315C211yesmaximal, characteristic
8#41×Dic59C411yescharacteristic
9#16837460—21no—
10#16837460—21no—
11#16837452—21no—
12#16837452—21no—
13#16837430C411yescharacteristic
14#16837428C2211yescharacteristic
15#16837369C411yescharacteristic
16M16×Dic59C411yescharacteristic
17C2×C8×Dic59C2211yescharacteristic
18Dic59⋊C16—21no—
19Dic59⋊C16—21no—
20C22⋊(C59⋊C16)C2211yescharacteristic
21C22⋊(C59⋊C16)C2211yescharacteristic
22#16837177—21no—
23C16×Dic59—21no—
24C59×#408C2211yescharacteristic
25C59×C22⋊C16C2211yescharacteristic
26C59×C22⋊C16C2211yescharacteristic
27C2×#11745664C2×C411yescharacteristic
28C22×C59⋊C8C2×C411yescharacteristic
29Q8×Dic59C811yescharacteristic
30Q8×Dic59C811yescharacteristic
31D4×Dic59—21no—
32D4×Dic59—21no—
33D4×Dic59C811yescharacteristic
34D4×Dic59C811yescharacteristic
35#14273686—21no—
36#14273686C2×C411yescharacteristic
37C2×C4×Dic59—21no—
38C2×C4×Dic59C2×C411yescharacteristic
39C2×C59⋊C16—21no—
40C2×C59⋊C16—21no—
41Dic59⋊C8—21no—
42Dic59⋊C8—21no—
43Dic59⋊C8—21no—
44Dic59⋊C8—21no—
45C22⋊(C59⋊C8)—21no—
46C22⋊(C59⋊C8)—21no—
47#14273637—21no—
48C8×Dic59—21no—
49C8×Dic59D411yescharacteristic
50C8×Dic59D411yescharacteristic
51C8×Dic59D411yescharacteristic
52C8×Dic59D411yescharacteristic
53C2×#41×C59C2×C411yescharacteristic
54C2×M16×C59C2×C411yescharacteristic
55C22×C8×C59C2311yescharacteristic
56C2×C16×C59—21no—
57C2×C16×C59—21no—
58C22⋊C8×C59—21no—
59C22⋊C8×C59—21no—
60#41×C59—41no—
61#41×C59—41no—
62C2×Q8×C59C2×C811yescharacteristic
63C2×D4×C59—21no—
64C2×D4×C59C2×C811yescharacteristic
65C22×C4×C59—21no—
66C22×C4×C59C22×C411yescharacteristic
67C22×Dic59—21no—
68C22×Dic59—21no—
69C22×Dic59M1611yescharacteristic
70C22×Dic59M1611yescharacteristic
71M16×C59—41no—
72C2×C8×C59—21no—
73C2×C8×C59—41no—
74C2×C8×C59C2×D411yescharacteristic
75C2×C8×C59C2×D411yescharacteristic
76C22⋊Dic59—21no—
77C22⋊Dic59—21no—
78C22⋊Dic59—21no—
79C4⋊Dic59—21no—
80C4⋊Dic59—21no—
81C4⋊Dic59C2×C811yescharacteristic
82C4⋊Dic59C2×C811yescharacteristic
83C4×Dic59—21no—
84C4×Dic59—21no—
85C4×Dic59—21no—
86C4×Dic59C22⋊C411yescharacteristic
87C4×Dic59C22⋊C411yescharacteristic
88C4×Dic59C2×C811yescharacteristic
89C4×Dic59C2×C811yescharacteristic
90#11745664—41no—
91C2×C59⋊C8—21no—
92C2×C59⋊C8—41no—
93C2×C59⋊C8C22⋊C411yescharacteristic
94C2×C59⋊C8C22⋊C411yescharacteristic
95C16×C59—41no—
96C16×C59—41no—
97C59⋊C16—41no—
98C59⋊C16—41no—
99C23×C59—21no—
100C23×C59C2×M1611yescharacteristic
101Q8×C59—41no—
102D4×C59—41no—
103D4×C59—42no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59C2×C22⋊C411yescharacteristic
110C2×C4×C59C22×C811yescharacteristic
111C2×Dic59—22no—
112C2×Dic59—21no—
113C2×Dic59—21no—
114C2×Dic59—41no—
115C2×Dic59—42no—
116C2×Dic59—42no—
117C2×Dic59C22⋊C811yescharacteristic
118C2×Dic59C22⋊C811yescharacteristic
119C2×Dic59C22⋊C811yescharacteristic
120C2×Dic59C22⋊C811yescharacteristic
121C8×C59—21no—
122C8×C59—21no—
123C8×C59—41no—
124C59⋊C8—21no—
125C59⋊C8—21no—
126C59⋊C8—41no—
127#11464—591nomaximal, Sylow
128C22×C59—21no—
129C22×C59—21no—
130C22×C59—21no—
131C22×C59—41no—
132C22×C59—41no—
133C22×C59—41no—
134C22×C59C2×C22⋊C811yescharacteristic
135C4×C59—21no—
136C4×C59—41no—
137C4×C59#41011yescharacteristic
138C4×C59#41311yescharacteristic
139Dic59—21no—
140Dic59—21no—
141Dic59—21no—
142Dic59—41no—
143Dic59—41no—
144Dic59#35012yes—
145#2922—591no—
146#2163—591no—
147#2162—591no—
148#1323—591no—
149#1323—591no—
150#1315—591no—
151#1315D5911yescharacteristic
152C2×C59—41no—
153C2×C59—81no—
154C2×C59#145811yescharacteristic
155C2×C59C2×#35011yescharacteristic
156C2×C59#211711yescharacteristic
157C4×#41—591no—
158#433—1181no—
159#433—1181no—
160#431—1181no—
161#431—1181no—
162#424—591no—
163#424—591no—
164#408—591no—
165#408C2×D5911yescharacteristic
166C4×M16—591no—
167C2×C4×C8—591no—
168C4⋊C16—1181no—
169C4⋊C16—1181no—
170C22⋊C16—591no—
171C22⋊C16—591no—
172C22⋊C16C2×D5911yescharacteristic
173C22⋊C16C2×D5911yescharacteristic
174#346—1181no—
175C4×C16—1181no—
176C59#1146411yescharacteristic, complemented, Sylow
177C2×#41C4×D5911yescharacteristic
178C2×M16—591no—
179C2×M16C4×D5911yescharacteristic
180C22×C8—591no—
181C22×C8C22×D5911yescharacteristic
182C4×Q8—591no—
183C4×Q8—591no—
184C4×D4—591no—
185C4×D4—591no—
186C4×D4—1181no—
187C4×D4—1181no—
188#117—591no—
189#117—1181no—
190C2×C42—591no—
191C2×C42—1181no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—1181no—
195C2×C16—1181no—
196C4⋊C8—1181no—
197C4⋊C8—1181no—
198C4⋊C8—1181no—
199C4⋊C8—1181no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—1181no—
203C22⋊C8—1181no—
204#97—1181no—
205C4×C8—591no—
206C4×C8—591no—
207C4×C8—591no—
208C4×C8—591no—
209C4×C8—1181no—
210#41—41no—
211#41—41no—
212C2×Q8C8×D5911yescharacteristic
213C2×D4—21no—
214C2×D4C8×D5911yescharacteristic
215C22×C4—21no—
216C22×C4—591no—
217C22×C4—591no—
218C22×C4—1181no—
219C22×C4—1181no—
220C22×C4C2×C4×D5911yescharacteristic
221M16—41no—
222M16—2361no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—591no—
226C2×C8—591no—
227C2×C8—1181no—
228C2×C8—2361no—
229C2×C8D4×D5911yescharacteristic
230C2×C8D4×D5911yescharacteristic
231C4⋊C4—591no—
232C4⋊C4—591no—
233C4⋊C4—1181no—
234C4⋊C4—1181no—
235C22⋊C4—1181no—
236C22⋊C4—1181no—
237C22⋊C4—1181no—
238C42—591no—
239C42—591no—
240C42—591no—
241C42—591no—
242C42—1181no—
243C42—1181no—
244C42—1181no—
245C16—41no—
246C16—41no—
247C16—2361no—
248C16—2361no—
249C23—21no—
250C23M16×D5911yescharacteristic
251Q8—41no—
252D4—41no—
253D4—42no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—591no—
260C2×C4—591no—
261C2×C4—591no—
262C2×C4—591no—
263C2×C4—1182no—
264C2×C4—1181no—
265C2×C4—1181no—
266C2×C4—2362no—
267C2×C4—2361no—
268C2×C4—2362no—
269C2×C4C22⋊C4×D5911yescharacteristic
270C2×C4C2×C8×D5911yescharacteristic
271C8—21no—
272C8—21no—
273C8—41no—
274C8—1181no—
275C8—1181no—
276C8—2361no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22C22⋊C8×D5911yescharacteristic
284C4—21no—
285C4—41no—
286C4—592no—
287C4—1181no—
288C4—1181no—
289C4—1181no—
290C4—2361no—
291C4—2361no—
292C4#1683737411yescharacteristic
293C4#1683738311yescharacteristic
294C2—41no—
295C2—81no—
296C2#2142240711yescharacteristic
297C2#350×D5911yescharacteristic
298C2#2142572211yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D59
commutatorC2×C4×C59C22×C8
FrattiniC22×C8C22×D59
FittingC59×#1315C2
socleC22×C59C2×C22⋊C8
prime radicalC2×#41×C59C2×C4
cosocleC22×C8×C59C23
hypercenter#1315D59
hypocenterC59#11464
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #21425896C2semidirect
3. #21425892C2semidirect
4. #21422037C2semidirect
5. #21422037C2semidirect
176. C59#11464semidirect

Maximal subgroups

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

subgroupindex
#214327362
#214258962
#214258922
#214220372
#214220222
C59×#13152
#1146459