Finite Group Databasebeta

← #28477196

Subgroups — #28477196

Order 15104.

444 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 107 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#21432463C211yesmaximal, characteristic
2#21423842C211yesmaximal, characteristic, complemented
3#21421918C212yesmaximal, complemented
4#21421916C212yesmaximal
5C59×#1745C211yesmaximal, characteristic, complemented
6C2×#14273721—21no—
7C2×C4×C59⋊C8—21no—
8#16837304C411yescharacteristic
9#16837304C411yescharacteristic
10#16837304C2211yescharacteristic
11#16837300C411yescharacteristic
12#16837300C411yescharacteristic
13#16837300C2211yescharacteristic
14#16837293—21no—
15#16837293—21no—
16#16837293—21no—
17#16837157—21no—
18#16837157C2212yes—
19C2×#97×C59—21no—
20C43×C59C2211yescharacteristic
21C59×#336—21no—
22C59×#336C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C59⋊C8—22no—
25#14273721—44no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C4×C59⋊C8—24no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C22×C8×C59—22no—
43C2×C42×C59C2×C411yescharacteristic
44C2×C42×C59C2×C411yescharacteristic
45C2×C42×C59D411yescharacteristic
46C2×C42×C59D411yescharacteristic
47C2×C42×C59D411yescharacteristic
48C2×C42×C59D411yescharacteristic
49C2×C42×C59C2311yescharacteristic
50#97×C59—24no—
51C23×D59—21no—
52C22×C4×C59C22⋊C411yescharacteristic
53C22×C4×C59C22⋊C411yescharacteristic
54C22×C4×C59C22⋊C411yescharacteristic
55C22×C4×C59C22⋊C411yescharacteristic
56C22×C4×C59C22×C411yescharacteristic
57C22×C4×C59C2×D411yescharacteristic
58C22×C4×C59C2×D411yescharacteristic
59C22×Dic59—21no—
60C22×Dic59—21no—
61C22×Dic59—21no—
62C2×D236—22no—
63C2×D236—41no—
64C2×D236—41no—
65C2×C8×C59—24no—
66C2×C8×C59—42no—
67C2×C8×C59—42no—
68C42×C59—22no—
69C42×C59—22no—
70C42×C59—22no—
71C42×C59—22no—
72C42×C59—22no—
73C42×C59—41no—
74C42×C59—41no—
75(C2×D59)⋊C4—42no—
76(C2×D59)⋊C4—42no—
77(C2×D59)⋊C4—42no—
78(C2×D59)⋊C4—42no—
79(C2×D59)⋊C4—42no—
80(C2×D59)⋊C4—42no—
81C4⋊Dic59—22no—
82C4⋊Dic59—41no—
83C4⋊Dic59—41no—
84Dic59⋊C4—42no—
85Dic59⋊C4—42no—
86Dic59⋊C4—42no—
87C2×C59⋊C8—24no—
88C2×C59⋊C8—42no—
89C2×C59⋊C8—42no—
90C23×C59C2×C22⋊C411yescharacteristic
91C22×D59—41no—
92C22×D59—41no—
93C22×D59—41no—
94C22×D59—41no—
95C22×D59—41no—
96C22×D59—41no—
97C22×D59—41no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59#10412yes—
115C2×C4×C59#10412yes—
116C2×C4×C59#13112yes—
117C2×C4×C59#13512yes—
118C2×C4×C59AGL(1,ℤ/8)12yes—
119C2×Dic59—41no—
120C2×Dic59—41no—
121C2×Dic59—41no—
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128D236—82no—
129C8×C59—44no—
130C59⋊C8—44no—
131#7692—591nomaximal, Sylow
132C22×C59#40811yescharacteristic
133C22×C59#41011yescharacteristic
134C22×C59#41311yescharacteristic
135C22×C59#41611yescharacteristic
136C22×C59#41811yescharacteristic
137C22×C59C2×#10411yescharacteristic
138C22×C59C2×#10411yescharacteristic
139C2×D59—81no—
140C2×D59—81no—
141C2×D59—81no—
142C2×D59—81no—
143C2×D59—81no—
144C2×D59—81no—
145C2×D59—81no—
146C4×C59—22no—
147C4×C59—22no—
148C4×C59—22no—
149C4×C59—22no—
150C4×C59—22no—
151C4×C59—41no—
152C4×C59—41no—
153Dic59—81no—
154Dic59—81no—
155Dic59—81no—
156#2860—591no—
157#1745D5911yescharacteristic, complemented
158#1744—591no—
159#1271—592no—
160#1270—592no—
161C2×C59#147311yescharacteristic
162C2×C59#148411yescharacteristic
163C2×C59#149411yescharacteristic
164C2×C59#150011yescharacteristic
165C2×C59#150511yescharacteristic
166C2×C59#152411yescharacteristic
167C2×C59#155311yescharacteristic
168D59—161no—
169C2×#126—1181no—
170C2×#97—21no—
171C2×C4×C8—1181no—
172#390—591no—
173#390—591no—
174#390—591no—
175#389—591no—
176#389—591no—
177#389—591no—
178#388—1181no—
179#388—1181no—
180#388—1181no—
181C43C2×D5911yescharacteristic
182#336—21no—
183#336—592no—
184#336—1181no—
185#336C2×D5912yes—
186C59#769211yescharacteristic, complemented, Sylow
187C22×D4—591no—
188C22×C8—22no—
189C22×C8—1182no—
190#126—2364no—
191C2×C4⋊C4—591no—
192C2×C4⋊C4—591no—
193C2×C4⋊C4—591no—
194C2×C4⋊C4—1181no—
195C2×C4⋊C4—1181no—
196C2×C4⋊C4—1181no—
197C2×C22⋊C4—1181no—
198C2×C22⋊C4—1181no—
199C2×C22⋊C4—1181no—
200C2×C22⋊C4—1181no—
201C2×C22⋊C4—1181no—
202C2×C22⋊C4—1181no—
203C2×C42C4×D5911yescharacteristic
204C2×C42C4×D5911yescharacteristic
205C2×C42D23611yescharacteristic
206C2×C42D23611yescharacteristic
207C2×C42C22⋊D5911yescharacteristic
208C2×C42C22⋊D5911yescharacteristic
209C2×C42C22×D5911yescharacteristic
210#97—24no—
211C4×C8—1184no—
212#95—1181no—
213#95—1181no—
214#95—1181no—
215C24—1181no—
216C2×D4—1182no—
217C2×D4—2361no—
218C2×D4—2361no—
219C22×C4—1181no—
220C22×C4—1181no—
221C22×C4—1181no—
222C22×C4(C2×D59)⋊C411yescharacteristic
223C22×C4(C2×D59)⋊C411yescharacteristic
224C22×C4(C2×D59)⋊C411yescharacteristic
225C22×C4(C2×D59)⋊C411yescharacteristic
226C22×C4C2×C4×D5911yescharacteristic
227C22×C4C2×D23611yescharacteristic
228C22×C4C2×C22⋊D5911yescharacteristic
229C2×C8—24no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—1184no—
233C2×C8—2362no—
234C2×C8—2362no—
235C4⋊C4—1182no—
236C4⋊C4—2361no—
237C4⋊C4—2361no—
238C4⋊C4—2362no—
239C4⋊C4—2362no—
240C4⋊C4—2362no—
241C22⋊C4—2362no—
242C22⋊C4—2362no—
243C22⋊C4—2362no—
244C22⋊C4—2362no—
245C22⋊C4—2362no—
246C22⋊C4—2362no—
247C42—22no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—41no—
253C42—41no—
254C23—2361no—
255C23—2361no—
256C23—2361no—
257C23—2361no—
258C23—2361no—
259C23—2361no—
260C23—2361no—
261C23C2×(C2×D59)⋊C411yescharacteristic
262D4—4722no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—2361no—
280C2×C4—2361no—
281C2×C4—2361no—
282C2×C4—2361no—
283C2×C4—2361no—
284C2×C4—2361no—
285C2×C4—2361no—
286C2×C4—2361no—
287C2×C4—2361no—
288C2×C4#1427365312yes—
289C2×C4#1427365512yes—
290C2×C4#1427373512yes—
291C2×C4#1427374712yes—
292C2×C4#1427375512yes—
293C8—44no—
294C8—2364no—
295C22—4721no—
296C22—4721no—
297C22—4721no—
298C22—4721no—
299C22—4721no—
300C22—4721no—
301C22—4721no—
302C22#1683737011yescharacteristic
303C22#1683737511yescharacteristic
304C22#1683738411yescharacteristic
305C22#1683739611yescharacteristic
306C22#1683740811yescharacteristic
307C22C2×#1427365311yescharacteristic
308C22C2×#1427365511yescharacteristic
309C4—22no—
310C4—22no—
311C4—22no—
312C4—22no—
313C4—22no—
314C4—41no—
315C4—41no—
316C4—4721no—
317C4—4721no—
318C4—4721no—
319C2—9441no—
320C2#2142246611yescharacteristic
321C2#2142251411yescharacteristic
322C2#2142255411yescharacteristic
323C2#2142258011yescharacteristic
324C2#2142260511yescharacteristic
325C2#2142268511yescharacteristic
326C2#2142280211yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1745C2
socleC23×C59C2×C22⋊C4
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#1745D59
hypocenterC59#7692
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C43 ◁ #1745

Complements

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

NH ≅ G/Ntype
2. #21423842C2semidirect
3. #21421918C2semidirect
5. C59×#1745C2semidirect
157. #1745D59semidirect
186. C59#7692semidirect

Maximal subgroups

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

subgroupindex
#214324632
#214238422
#214219182
#214219162
C59×#17452
#769259