Finite Group Databasebeta

← #28498309

Subgroups — #28498309

Order 15104.

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

normal subgroups only

This group has 616 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#21429042C211yesmaximal, characteristic
2#21422434C211yesmaximal, characteristic, complemented
3#21422027C214yesmaximal, complemented
4C59×#1459C211yesmaximal, characteristic, complemented
5C2×#14273724C411yescharacteristic
6C2×#14273723C412yes—
7C2×#14273701—21no—
8C2×#14273701—21no—
9#16837366—21no—
10#16837311—21no—
11#16837311—21no—
12#16837303C2212yes—
13C4×C4⋊Dic59C411yescharacteristic
14#16837128C2212yes—
15#16837126—21no—
16C2×C4×D4×C59C2211yescharacteristic
17C2×C22⋊C8×C59—21no—
18C59×#324C2212yes—
19C59×#323—21no—
20C22×D236—21no—
21C22×D236—21no—
22C22×D236C2×C412yes—
23C22×C22⋊D59—22no—
24C22×C22⋊D59—22no—
25C2×#11745664—41no—
26#14273724—41no—
27#14273724—41no—
28#14273723—44no—
29#14273701—44no—
30#14273701—44no—
31C2×C4⋊Dic59—21no—
32C2×(C2×D59)⋊C4—22no—
33C2×(C2×D59)⋊C4—22no—
34C2×C4×Dic59C2×C412yes—
35C22⋊(C59⋊C8)—22no—
36C22⋊(C59⋊C8)—41no—
37C59⋊#95—21no—
38C22×D4×C59C2×C411yescharacteristic
39C23×C4×C59D411yescharacteristic
40C23×C4×C59D411yescharacteristic
41C22×C8×C59—41no—
42C4×D4×C59—41no—
43C4×D4×C59—41no—
44C2×C4⋊C4×C59C2311yescharacteristic
45C2×C22⋊C4×C59D411yescharacteristic
46C2×C22⋊C4×C59D411yescharacteristic
47C2×C42×C59C2×C411yescharacteristic
48C22⋊C8×C59—22no—
49C22⋊C8×C59—41no—
50C24×C59C22⋊C411yescharacteristic
51C24×C59C22⋊C411yescharacteristic
52C23×D59—22no—
53C2×D4×C59—22no—
54C2×D4×C59—22no—
55C2×D4×C59—41no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—42no—
61C22×C4×C59—42no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22×C411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C2×C22⋊D59—44no—
68C2×C22⋊D59—44no—
69C2×C22⋊D59—44no—
70C2×C22⋊D59—44no—
71C2×C22⋊D59—82no—
72C2×C22⋊D59—82no—
73C22×Dic59—22no—
74C2×D236—24no—
75C2×D236—42no—
76C2×D236—42no—
77C2×D236—42no—
78C2×D236—42no—
79C2×D236—42no—
80C2×D236—41no—
81C2×D236—41no—
82C2×D236—82no—
83C2×C8×C59—42no—
84C2×C8×C59—82no—
85C4⋊C4×C59—41no—
86C22⋊C4×C59—41no—
87C22⋊C4×C59—41no—
88C42×C59—21no—
89C42×C59—21no—
90(C2×D59)⋊C4—44no—
91(C2×D59)⋊C4—44no—
92C4⋊Dic59—24no—
93C4×Dic59—42no—
94#11745664—82no—
95C2×C59⋊C8—42no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59#10012yes—
106C23×C59C2×C22⋊C411yescharacteristic
107C22×D59—42no—
108C22×D59—42no—
109C22×D59—42no—
110C22×D59—42no—
111C22×D59—42no—
112C22×D59—42no—
113C22×D59—42no—
114D4×C59—44no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—22no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—42no—
122C2×C4×C59—42no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—81no—
127C2×C4×C59—81no—
128C2×C4×C59#9911yescharacteristic
129C2×C4×C59#9911yescharacteristic
130C2×C4×C59#13112yes—
131C22⋊D59—88no—
132C22⋊D59—88no—
133C2×Dic59—24no—
134C2×Dic59—42no—
135C2×Dic59—42no—
136D236—84no—
137D236—82no—
138D236—84no—
139D236—82no—
140C8×C59—82no—
141C59⋊C8—82no—
142#11563—591nomaximal, Sylow
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—22no—
146C22×C59—42no—
147C22×C59—42no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—81no—
154C22×C59—81no—
155C22×C59#40811yescharacteristic
156C22×C59C2×#9911yescharacteristic
157C22×C59C2×#10011yescharacteristic
158C2×D59—82no—
159C2×D59—82no—
160C2×D59—82no—
161C2×D59—82no—
162C2×D59—82no—
163C2×D59—82no—
164C2×D59—82no—
165C4×C59—21no—
166C4×C59—22no—
167C4×C59—41no—
168C4×C59—41no—
169C4×C59—41no—
170C4×C59#35312yes—
171Dic59—44no—
172#2541—591no—
173#1465—591no—
174#1459D5911yescharacteristic, complemented
175#1318—594no—
176C2×C59—21no—
177C2×C59—21no—
178C2×C59—42no—
179C2×C59—81no—
180C2×C59#147111yescharacteristic
181C2×C59C2×#35311yescharacteristic
182C2×C59#213411yescharacteristic
183D59—162no—
184C2×#127—592no—
185C2×#127—591no—
186C2×#121—1181no—
187C2×#121—1181no—
188C2×C4×D4C2×D5911yescharacteristic
189#407—1181no—
190C2×C22⋊C8—21no—
191#392—1181no—
192#392—1181no—
193#390—592no—
194C4×C4⋊C4—591no—
195#324—592no—
196#324C2×D5912yes—
197#323—21no—
198#323—1181no—
199C59#1156311yescharacteristic, complemented, Sylow
200C22×D4—592no—
201C22×D4—1182no—
202C22×D4—1182no—
203C22×D4—1181no—
204C22×D4—1181no—
205C22×D4C4×D5911yescharacteristic
206C23×C4D23611yescharacteristic
207C23×C4C22⋊D5911yescharacteristic
208C2×M16—2361no—
209C22×C8—41no—
210#127—2361no—
211#127—2361no—
212#127—2364no—
213#121—2364no—
214#121—2364no—
215C4×D4—41no—
216C4×D4—41no—
217C2×C4⋊C4—1181no—
218C2×C4⋊C4C22×D5911yescharacteristic
219C2×C22⋊C4—1182no—
220C2×C22⋊C4—1182no—
221C2×C22⋊C4D23611yescharacteristic
222C2×C22⋊C4C22⋊D5911yescharacteristic
223C2×C42—592no—
224C2×C42C4×D5911yescharacteristic
225C22⋊C8—22no—
226C22⋊C8—41no—
227C22⋊C8—1182no—
228C22⋊C8—2361no—
229#95—1181no—
230C24—1182no—
231C24(C2×D59)⋊C411yescharacteristic
232C24(C2×D59)⋊C411yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—41no—
236C2×D4—1184no—
237C2×D4—4722no—
238C2×D4—4722no—
239C2×D4—4722no—
240C2×D4—2362no—
241C2×D4—2362no—
242C2×D4—2362no—
243C2×D4—2362no—
244C2×D4—2362no—
245C2×D4—2364no—
246C2×D4—2364no—
247C2×D4—2364no—
248C2×D4—2364no—
249C2×D4—2361no—
250C2×D4—2361no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—1182no—
258C22×C4(C2×D59)⋊C411yescharacteristic
259C22×C4(C2×D59)⋊C411yescharacteristic
260C22×C4C2×C4×D5911yescharacteristic
261C22×C4C2×D23611yescharacteristic
262C22×C4C2×C22⋊D5911yescharacteristic
263M16—4722no—
264C2×C8—42no—
265C2×C8—82no—
266C2×C8—2362no—
267C4⋊C4—41no—
268C4⋊C4—1184no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—2364no—
272C22⋊C4—2364no—
273C42—21no—
274C42—21no—
275C42—2362no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—2362no—
286C23—2362no—
287C23—2362no—
288C23—2362no—
289C23—2362no—
290C23—2362no—
291C23—2362no—
292C23#1427364312yes—
293C23C2×(C2×D59)⋊C411yescharacteristic
294D4—44no—
295D4—4724no—
296D4—4722no—
297D4—4724no—
298D4—4728no—
299D4—4728no—
300D4—4722no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—1184no—
315C2×C4—2362no—
316C2×C4—2362no—
317C2×C4#1427364111yescharacteristic
318C2×C4#1427364111yescharacteristic
319C2×C4#1427373312yes—
320C8—82no—
321C8—4722no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—42no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—4722no—
335C22—4722no—
336C22—4722no—
337C22—4722no—
338C22—4722no—
339C22—4722no—
340C22—4722no—
341C22#1683736811yescharacteristic
342C22C2×#1427364111yescharacteristic
343C22C2×#1427364311yescharacteristic
344C4—21no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—2364no—
350C4#1683719412yes—
351C2—21no—
352C2—21no—
353C2—42no—
354C2—81no—
355C2—9442no—
356C2#2142245511yescharacteristic
357C2C2×#1683719411yescharacteristic
358C2#2142578711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1459C2
socleC22×C59C2×#99
prime radical#21429042C2
cosocleC2×C4⋊C4×C59C23
hypercenter#1459D59
hypocenterC59#11563
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. #21422434C2semidirect
3. #21422027C2semidirect
4. C59×#1459C2semidirect
174. #1459D59semidirect
199. C59#11563semidirect

Maximal subgroups

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

subgroupindex
#214290422
#214224342
#214220272
C59×#14592
#1156359