Finite Group Databasebeta

← #28499362

Subgroups — #28499362

Order 15104.

502 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 79 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#21429846C211yesmaximal, characteristic
2#21422430C211yesmaximal, characteristic, complemented
3#21422092C214yesmaximal, complemented
4C59×#1820C211yesmaximal, characteristic, complemented
5C2×#14273724C411yescharacteristic, complemented
6C2×#14273701—21no—
7#16837459—21no—
8#16837319—21no—
9#16837319C412yescomplemented
10#16837311—21no—
11#16837311C2212yes—
12#16837304—21no—
13#16837268C411yescharacteristic, complemented
14#16837128—21no—
15#16837126C2212yes—
16C2×#117×C59C2211yescharacteristic
17C59×#381—21no—
18C59×#342—21no—
19C59×#342C2212yes—
20C22×D236—21no—
21C22×D236—21no—
22C22×D236C2×C412yes—
23C22×C22⋊D59—22no—
24#14273724—41no—
25#14273724—41no—
26#14273701—44no—
27C2×C4⋊Dic59—21no—
28C2×(C2×D59)⋊C4—22no—
29C2×(C2×D59)⋊C4—22no—
30C2×(C2×D59)⋊C4—22no—
31C4⋊(C59⋊C8)—41no—
32C22⋊(C59⋊C8)—22no—
33#14273638—41no—
34C59⋊#95—21no—
35C59⋊#95C2×C412yes—
36C23×C4×C59D411yescharacteristic
37#117×C59—41no—
38#117×C59—41no—
39C2×C4⋊C4×C59C2×C411yescharacteristic
40C2×C4⋊C4×C59D411yescharacteristic
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59D411yescharacteristic
43C2×C22⋊C4×C59C2311yescharacteristic
44C2×C42×C59C2×C411yescharacteristic
45C2×C42×C59D411yescharacteristic
46#95×C59—41no—
47#95×C59—41no—
48C24×C59C2×D411yescharacteristic
49C23×D59—22no—
50C22×C4×C59—22no—
51C22×C4×C59—21no—
52C22×C4×C59—21no—
53C22×C4×C59—41no—
54C22×C4×C59—42no—
55C22×C4×C59C22⋊C411yescharacteristic
56C22×C4×C59C22⋊C411yescharacteristic
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59C22×C411yescharacteristic
60C22×C4×C59C2×D411yescharacteristic
61C2×C22⋊D59—44no—
62C2×C22⋊D59—82no—
63C2×C22⋊D59—82no—
64C22×Dic59—22no—
65C2×D236—42no—
66C2×D236—42no—
67C2×D236—42no—
68C2×D236—42no—
69C2×D236—41no—
70C2×D236—41no—
71C2×D236—82no—
72C2×D236—82no—
73C4⋊C4×C59—41no—
74C4⋊C4×C59—41no—
75C22⋊C4×C59—41no—
76C22⋊C4×C59—41no—
77C22⋊C4×C59—42no—
78C22⋊C4×C59—42no—
79C42×C59—21no—
80C42×C59—21no—
81C42×C59—21no—
82C42×C59—21no—
83(C2×D59)⋊C4—44no—
84(C2×D59)⋊C4—82no—
85(C2×D59)⋊C4—82no—
86C4⋊Dic59—42no—
87C2×C59⋊C8—42no—
88C23×C59—21no—
89C23×C59—21no—
90C23×C59—41no—
91C23×C59—41no—
92C23×C59C2×C22⋊C411yescharacteristic
93C23×C59AGL(1,ℤ/8)12yes—
94C22×D59—42no—
95C22×D59—42no—
96C22×D59—42no—
97C22×D59—42no—
98C22×D59—42no—
99C22×D59—42no—
100C22×D59—42no—
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—42no—
111C2×C4×C59—42no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—82no—
117C2×C4×C59—82no—
118C2×C4×C59—82no—
119C2×C4×C59#9911yescharacteristic
120C2×C4×C59#9911yescharacteristic
121C22⋊D59—88no—
122C2×Dic59—42no—
123C2×Dic59—42no—
124C2×Dic59—42no—
125D236—82no—
126D236—82no—
127D236—84no—
128D236—84no—
129C59⋊C8—82no—
130#11800—591nomaximal, Sylow
131C22×C59—21no—
132C22×C59—21no—
133C22×C59—22no—
134C22×C59—42no—
135C22×C59—41no—
136C22×C59—41no—
137C22×C59—81no—
138C22×C59C2×#9911yescharacteristic
139C22×C59#41811yescharacteristic
140C22×C59#42111yescharacteristic
141C2×D59—82no—
142C2×D59—82no—
143C2×D59—82no—
144C2×D59—82no—
145C2×D59—82no—
146C2×D59—82no—
147C2×D59—82no—
148C4×C59—21no—
149C4×C59—21no—
150C4×C59—41no—
151C4×C59—41no—
152C4×C59—41no—
153C4×C59—41no—
154C4×C59—41no—
155C4×C59—162no—
156Dic59—82no—
157#2619—591no—
158#1820D5911yescharacteristic, complemented
159#1464—591no—
160#1343—594no—
161C2×C59—21no—
162C2×C59—21no—
163C2×C59—42no—
164C2×C59#151011yescharacteristic
165C2×C59#212211yescharacteristic
166C2×C59#212811yescharacteristic
167D59—162no—
168C2×#127—591no—
169C2×#121—1181no—
170C2×#117C2×D5911yescharacteristic
171#432—1181no—
172#394—592no—
173#394—1181no—
174#392—592no—
175#392—1181no—
176#390—1181no—
177#382—591no—
178#381—21no—
179#342—21no—
180#342C2×D5912yes—
181#324—1181no—
182#323—592no—
183C59#1180011yescharacteristic, complemented, Sylow
184C22×D4—592no—
185C22×D4—1182no—
186C22×D4—1181no—
187C22×D4—1181no—
188C23×C4C22⋊D5911yescharacteristic
189#127—2361no—
190#127—2361no—
191#121—2364no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—1181no—
195C2×C4⋊C4C4×D5911yescharacteristic
196C2×C4⋊C4D23611yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—1182no—
199C2×C22⋊C4—1182no—
200C2×C22⋊C4—1182no—
201C2×C22⋊C4C22⋊D5911yescharacteristic
202C2×C22⋊C4C22×D5911yescharacteristic
203C2×C42C4×D5911yescharacteristic
204C2×C42D23611yescharacteristic
205C4⋊C8—2361no—
206C22⋊C8—1182no—
207#97—2361no—
208#95—41no—
209#95—41no—
210#95—592no—
211#95—1181no—
212C24—1182no—
213C24C2×C22⋊D5911yescharacteristic
214C2×D4—4722no—
215C2×D4—4722no—
216C2×D4—4722no—
217C2×D4—4722no—
218C2×D4—2362no—
219C2×D4—2362no—
220C2×D4—2362no—
221C2×D4—2362no—
222C2×D4—2364no—
223C2×D4—2361no—
224C2×D4—2361no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—1182no—
231C22×C4(C2×D59)⋊C411yescharacteristic
232C22×C4(C2×D59)⋊C411yescharacteristic
233C22×C4(C2×D59)⋊C411yescharacteristic
234C22×C4(C2×D59)⋊C411yescharacteristic
235C22×C4C2×C4×D5911yescharacteristic
236C22×C4C2×D23611yescharacteristic
237C2×C8—2362no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—2362no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—4722no—
246C22⋊C4—4722no—
247C22⋊C4—2364no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—2362no—
257C23—2362no—
258C23—2362no—
259C23—2362no—
260C23—2362no—
261C23—2362no—
262C23—2362no—
263C23C2×(C2×D59)⋊C411yescharacteristic
264C23#1427375212yes—
265D4—4722no—
266D4—4722no—
267D4—4724no—
268D4—4724no—
269D4—4728no—
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—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—2362no—
289C2×C4—2362no—
290C2×C4—2362no—
291C2×C4#1427364111yescharacteristic
292C2×C4#1427364111yescharacteristic
293C8—4722no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—81no—
301C22—4722no—
302C22—4722no—
303C22—4722no—
304C22—4722no—
305C22—4722no—
306C22—4722no—
307C22—4722no—
308C22C2×#1427364111yescharacteristic
309C22#1683740611yescharacteristic
310C22#1683742011yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—162no—
319C4—4722no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—9442no—
324C2#2142262411yescharacteristic
325C2#2142574511yescharacteristic
326C2#2142576711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1820C2
socleC22×C59C2×#99
prime radical#21429846C2
cosocleC2×C22⋊C4×C59C23
hypercenter#1820D59
hypocenterC59#11800
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

Complements

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

NH ≅ G/Ntype
2. #21422430C2semidirect
3. #21422092C2semidirect
4. C59×#1820C2semidirect
5. C2×#14273724C4semidirect
9. #16837319C4semidirect
13. #16837268C4semidirect
158. #1820D59semidirect
183. C59#11800semidirect

Maximal subgroups

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

subgroupindex
#214298462
#214224302
#214220922
C59×#18202
#1180059