Finite Group Databasebeta

← #28477008

Subgroups — #28477008

Order 15104.

624 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 119 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#21426660C211yesmaximal, characteristic
2#21424211C212yesmaximal, complemented
3#21421917C211yesmaximal, characteristic, complemented
4#21421916C212yesmaximal, complemented
5C59×#1270C211yesmaximal, characteristic, complemented
6C2×#14273723C411yescharacteristic
7C2×D4×Dic59C411yescharacteristic
8C2×#14273689C411yescharacteristic
9C2×(C22⋊D59)⋊C4—22no—
10C2×Dic59⋊C8—22no—
11#16837304C2211yescharacteristic
12#16837302—22no—
13#16837271C411yescharacteristic
14C42×Dic59C2211yescharacteristic
15#16837157C2212yes—
16#16837156—22no—
17C59×#390C2211yescharacteristic
18C59×#336C2212yes—
19C22×D236C2×C411yescharacteristic
20C22×C22⋊D59—22no—
21C23×Dic59—21no—
22C22×C4×D59—21no—
23C22×C59⋊C8—22no—
24#14273723—22no—
25#14273723—22no—
26D4×Dic59—42no—
27#14273689—42no—
28(C22⋊D59)⋊C4—44no—
29(C22⋊D59)⋊C4—44no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×Dic59⋊C4—22no—
32C2×C22⋊Dic59—21no—
33C2×(C2×D59)⋊C4—22no—
34C2×(C2×D59)⋊C4—21no—
35C2×C4×Dic59C2×C411yescharacteristic
36C2×C4×Dic59C2×C411yescharacteristic
37C2×C4×Dic59D412yes—
38C2×C4×Dic59D412yes—
39Dic59⋊C8—44no—
40C22×D4×C59C2×C411yescharacteristic
41C22×C8×C59—22no—
42C2×C4⋊C4×C59C2×C411yescharacteristic
43C2×C22⋊C4×C59—22no—
44C2×C42×C59C2311yescharacteristic
45C24×C59—21no—
46C23×D59—21no—
47C2×D4×C59—21no—
48C2×D4×C59—41no—
49C2×D4×C59—41no—
50C2×D4×C59M1612yes—
51C22×C4×C59—21no—
52C22×C4×C59C22×C411yescharacteristic
53C22×C4×C59C2×D412yes—
54C2×C22⋊D59—28no—
55C2×C22⋊D59—44no—
56C2×C22⋊D59—44no—
57C22×Dic59—21no—
58C22×Dic59—42no—
59C22×Dic59—44no—
60C22×Dic59C22⋊C411yescharacteristic
61C22×Dic59C22⋊C412yes—
62C22×Dic59C22⋊C411yescharacteristic
63C2×D236—21no—
64C2×D236—41no—
65C2×D236—41no—
66C2×D236M1612yes—
67C2×C4×D59—42no—
68C2×C4×D59—44no—
69C2×C8×C59—42no—
70C2×C8×C59—42no—
71C2×C8×C59—42no—
72C4⋊C4×C59—41no—
73C22⋊C4×C59—42no—
74C22⋊C4×C59—42no—
75C42×C59—41no—
76C22⋊Dic59—42no—
77(C2×D59)⋊C4—42no—
78(C2×D59)⋊C4—42no—
79(C2×D59)⋊C4—42no—
80C4⋊Dic59—41no—
81Dic59⋊C4—44no—
82C4×Dic59—21no—
83C4×Dic59—21no—
84C4×Dic59—22no—
85C4×Dic59—42no—
86C4×Dic59D814yes—
87C4×Dic59SD1614yes—
88C2×C59⋊C8—42no—
89C2×C59⋊C8—42no—
90C2×C59⋊C8—42no—
91C23×C59—22no—
92C23×C59—41no—
93C23×C59—41no—
94C23×C59—42no—
95C23×C59—42no—
96C23×C59C2×C22⋊C411yescharacteristic
97C22×D59—22no—
98C22×D59—41no—
99C22×D59—41no—
100C22×D59—42no—
101C22×D59—42no—
102D4×C59—42no—
103D4×C59—42no—
104C2×C4×C59—22no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—22no—
108C2×C4×C59—41no—
109C2×C4×C59—42no—
110C2×C4×C59C2×M1611yescharacteristic
111C2×C4×C59C2×M1611yescharacteristic
112C2×C4×C59C2×D812yes—
113C2×C4×C59C2×SD1612yes—
114C22⋊D59—416no—
115C2×Dic59—22no—
116C2×Dic59—22no—
117C2×Dic59—21no—
118C2×Dic59—22no—
119C2×Dic59—41no—
120C2×Dic59—42no—
121C2×Dic59—84no—
122C2×Dic59#10012yes—
123C2×Dic59#10214yes—
124C2×Dic59#10214yes—
125D236—42no—
126D236—42no—
127C4×D59—84no—
128C8×C59—82no—
129C59⋊C8—82no—
130#7651—591nomaximal, Sylow
131C22×C59—42no—
132C22×C59—44no—
133C22×C59—81no—
134C22×C59—81no—
135C22×C59—82no—
136C22×C59#40711yescharacteristic
137C22×C59#41011yescharacteristic
138C22×C59C2×#10011yescharacteristic
139C22×C59C2×#10212yes—
140C22×C59#42112yes—
141C2×D59—42no—
142C2×D59—44no—
143C2×D59—81no—
144C2×D59—81no—
145C2×D59—82no—
146C4×C59—21no—
147C4×C59—21no—
148C4×C59—22no—
149C4×C59—22no—
150C4×C59—81no—
151Dic59—24no—
152Dic59—41no—
153Dic59—81no—
154Dic59#33114yes—
155C4×#127—591no—
156#1831—592no—
157#1270—591no—
158#1270—592no—
159#1270D5911yescharacteristic, complemented
160C2×C59—82no—
161C2×C59#146611yescharacteristic
162C2×C59#148612yes—
163C2×C59#151712yes—
164C2×C59#153611yescharacteristic
165C2×C59C2×#33111yescharacteristic
166D59—82no—
167C2×#127—591no—
168C2×C4×D4—591no—
169C2×C4×D4—591no—
170C2×C4×D4—1182no—
171C2×C4⋊C8—1182no—
172#390—591no—
173#390—1182no—
174#390C2×D5911yescharacteristic
175#384—591no—
176C43—591no—
177#336—592no—
178#336—1182no—
179#336C2×D5912yes—
180C59#765111yescharacteristic, complemented, Sylow
181C22×D4—591no—
182C22×D4—1182no—
183C22×D4C4×D5911yescharacteristic
184C23×C4—1181no—
185C23×C4—1181no—
186C22×C8—22no—
187C22×C8—1182no—
188#127—1182no—
189#127—1182no—
190C4×D4—2364no—
191C4×D4—2364no—
192C4×D4—2362no—
193C4×D4—2362no—
194C2×C4⋊C4—591no—
195C2×C4⋊C4—1182no—
196C2×C4⋊C4C4×D5911yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—1181no—
199C2×C22⋊C4—1181no—
200C2×C22⋊C4—1182no—
201C2×C42—592no—
202C2×C42—591no—
203C2×C42—591no—
204C2×C42—592no—
205C2×C42C22×D5911yescharacteristic
206C4⋊C8—2364no—
207C24—21no—
208C24—1181no—
209C2×D4—21no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—592no—
213C2×D4—1188no—
214C2×D4—1181no—
215C2×D4—2361no—
216C2×D4—2361no—
217C2×D4—2364no—
218C2×D4—2364no—
219C2×D4#1174565912yes—
220C22×C4—21no—
221C22×C4—591no—
222C22×C4—592no—
223C22×C4—591no—
224C22×C4—1181no—
225C22×C4—2364no—
226C22×C4—2362no—
227C22×C4—2364no—
228C22×C4—2362no—
229C22×C4C2×C4×D5911yescharacteristic
230C22×C4D4×D5912yes—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—2362no—
235C2×C8—2362no—
236C2×C8—2362no—
237C4⋊C4—41no—
238C4⋊C4—2361no—
239C4⋊C4—2364no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—2362no—
243C22⋊C4—2362no—
244C22⋊C4—2362no—
245C22⋊C4—2362no—
246C42—41no—
247C42—594no—
248C42—594no—
249C42—1182no—
250C42—1181no—
251C42—1181no—
252C42—2362no—
253C23—22no—
254C23—41no—
255C23—42no—
256C23—42no—
257C23—41no—
258C23—1182no—
259C23—2361no—
260C23—2362no—
261C23—2362no—
262C23—2361no—
263C23C22⋊C4×D5911yescharacteristic
264D4—42no—
265D4—42no—
266D4—2362no—
267D4—2362no—
268D4—23616no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—592no—
276C2×C4—594no—
277C2×C4—594no—
278C2×C4—1182no—
279C2×C4—1182no—
280C2×C4—1181no—
281C2×C4—1182no—
282C2×C4—4724no—
283C2×C4—4724no—
284C2×C4—2361no—
285C2×C4—2362no—
286C2×C4M16×D5911yescharacteristic
287C2×C4C2×#1174565911yescharacteristic
288C2×C4D8×D5912yes—
289C2×C4SD16×D5912yes—
290C8—82no—
291C8—4722no—
292C22—42no—
293C22—44no—
294C22—82no—
295C22—81no—
296C22—81no—
297C22—4722no—
298C22—4721no—
299C22—4721no—
300C22—2362no—
301C22—2364no—
302C22#1683736411yescharacteristic
303C22#1683737411yescharacteristic
304C22#100×D5911yescharacteristic
305C22#102×D5912yes—
306C22#1683741712yes—
307C4—22no—
308C4—22no—
309C4—21no—
310C4—21no—
311C4—81no—
312C4—594no—
313C4—1184no—
314C4—4721no—
315C4—2361no—
316C2—82no—
317C2—4722no—
318C2#2142243511yescharacteristic
319C2#2142251912yes—
320C2#2142265412yes—
321C2#2142272911yescharacteristic
322C2#331×D5911yescharacteristic
subgroupG/H
centerC23C22⋊C4×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1270C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#14273723C4
cosocleC2×C42×C59C23
hypercenter#1270D59
hypocenterC59#7651
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #21424211C2semidirect
3. #21421917C2semidirect
4. #21421916C2semidirect
5. C59×#1270C2semidirect
159. #1270D59semidirect
180. C59#7651semidirect

Maximal subgroups

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

subgroupindex
#214266602
#214242112
#214219172
#214219162
C59×#12702
#765159