Finite Group Databasebeta

← #28499721

Subgroups — #28499721

Order 15104.

644 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 99 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#21422414C211yesmaximal, characteristic, complemented
3#21422092C214yesmaximal, complemented
4C59×#1799C211yesmaximal, characteristic, complemented
5C2×#14273724C411yescharacteristic, complemented
6C2×#14273723—21no—
7C2×#14273701—21no—
8C2×#14273701—21no—
9C2×C22⋊(C59⋊C8)—21no—
10#16837311C412yescomplemented
11#16837311C2212yes—
12#16837303—21no—
13C4×C4⋊Dic59C411yescharacteristic, complemented
14#16837128—21no—
15#16837126C2212yes—
16C2×C4×D4×C59C2211yescharacteristic
17C2×#99×C59—21no—
18C59×#380—21no—
19C59×#342C2212yes—
20C22×D236—21no—
21C22×D236—21no—
22C22×D236C2×C412yes—
23C22×C22⋊D59—22no—
24C22×C22⋊D59—22no—
25C22×C59⋊C8—41no—
26#14273724—41no—
27#14273724—41no—
28#14273723—42no—
29#14273723—42no—
30#14273701—44no—
31#14273701—44no—
32C2×C4⋊Dic59—21no—
33C2×(C2×D59)⋊C4—22no—
34C2×(C2×D59)⋊C4—22no—
35C2×C4×Dic59—21no—
36C22⋊(C59⋊C8)—22no—
37C22⋊(C59⋊C8)—41no—
38C59⋊#95C2×C412yes—
39C22×D4×C59D411yescharacteristic
40C23×C4×C59D411yescharacteristic
41C23×C4×C59D411yescharacteristic
42C4×D4×C59—41no—
43C4×D4×C59—41no—
44C2×C4⋊C4×C59—41no—
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C22⋊C4×C59—22no—
47C2×C22⋊C4×C59C2×C411yescharacteristic
48C2×C22⋊C4×C59C2311yescharacteristic
49C2×C42×C59C2×C411yescharacteristic
50#99×C59—42no—
51#95×C59—41no—
52C24×C59C22⋊C411yescharacteristic
53C24×C59C2×D411yescharacteristic
54C23×D59—22no—
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C2×D4×C59D812yes—
58C2×D4×C59SD1612yes—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—42no—
64C22×C4×C59—42no—
65C22×C4×C59—42no—
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22×C411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C2×C22⋊D59—44no—
72C2×C22⋊D59—44no—
73C2×C22⋊D59—44no—
74C2×C22⋊D59—42no—
75C2×C22⋊D59—42no—
76C2×C22⋊D59—82no—
77C2×C22⋊D59—82no—
78C22×Dic59—22no—
79C2×D236—42no—
80C2×D236—42no—
81C2×D236—42no—
82C2×D236—42no—
83C2×D236—42no—
84C2×D236—41no—
85C2×D236—41no—
86C2×D236—42no—
87C2×D236—82no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—82no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—41no—
92C22⋊C4×C59—42no—
93C22⋊C4×C59—42no—
94C42×C59—21no—
95C42×C59—21no—
96(C2×D59)⋊C4—44no—
97(C2×D59)⋊C4—44no—
98C4⋊Dic59—24no—
99C4×Dic59—42no—
100C2×C59⋊C8—42no—
101C2×C59⋊C8—82no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59#10212yes—
111C23×C59C2×C22⋊C411yescharacteristic
112C23×C59C2×D811yescharacteristic
113C23×C59C2×SD1611yescharacteristic
114C22×D59—42no—
115C22×D59—42no—
116C22×D59—42no—
117C22×D59—42no—
118C22×D59—42no—
119C22×D59—42no—
120C22×D59—42no—
121D4×C59—42no—
122D4×C59—42no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—22no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—42no—
130C2×C4×C59—42no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—81no—
135C2×C4×C59—81no—
136C2×C4×C59—82no—
137C2×C4×C59—82no—
138C2×C4×C59—82no—
139C2×C4×C59#9911yescharacteristic
140C2×C4×C59#9911yescharacteristic
141C2×C4×C59#10212yes—
142C22⋊D59—88no—
143C22⋊D59—84no—
144C22⋊D59—84no—
145C2×Dic59—24no—
146C2×Dic59—42no—
147C2×Dic59—42no—
148D236—82no—
149D236—84no—
150D236—84no—
151D236—82no—
152C59⋊C8—82no—
153#11877—591nomaximal, Sylow
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—22no—
158C22×C59—21no—
159C22×C59—42no—
160C22×C59—41no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—81no—
167C22×C59—81no—
168C22×C59C2×#9911yescharacteristic
169C22×C59C2×#10211yescharacteristic
170C22×C59#42111yescharacteristic
171C2×D59—82no—
172C2×D59—82no—
173C2×D59—82no—
174C2×D59—82no—
175C2×D59—82no—
176C2×D59—82no—
177C2×D59—82no—
178C4×C59—21no—
179C4×C59—22no—
180C4×C59—41no—
181C4×C59—41no—
182C4×C59—41no—
183C4×C59—162no—
184C4×C59#35312yes—
185Dic59—44no—
186#2541—591no—
187#1799D5911yescharacteristic, complemented
188#1459—591no—
189#1343—594no—
190C2×C59—21no—
191C2×C59—21no—
192C2×C59—41no—
193C2×C59—42no—
194C2×C59—41no—
195C2×C59#151911yescharacteristic
196C2×C59#212211yescharacteristic
197C2×C59C2×#35311yescharacteristic
198D59—162no—
199C2×#127—591no—
200C2×#127—1181no—
201C2×#121—1181no—
202C2×#121—1181no—
203C2×C4×D4C2×D5911yescharacteristic
204C2×#99—21no—
205C2×C22⋊C8—1181no—
206#392—592no—
207#392—592no—
208#390—1181no—
209#380—21no—
210C4×C4⋊C4—591no—
211#342C2×D5912yes—
212#324—1181no—
213#323—592no—
214C59#1187711yescharacteristic, complemented, Sylow
215C22×D4—592no—
216C22×D4—1182no—
217C22×D4—1182no—
218C22×D4—1181no—
219C22×D4—1181no—
220C22×D4C22⋊D5911yescharacteristic
221C23×C4D23611yescharacteristic
222C23×C4C22⋊D5911yescharacteristic
223C22×C8—2361no—
224#127—2361no—
225#127—2361no—
226#127—2362no—
227#127—2362no—
228#121—2364no—
229#121—2364no—
230C4×D4—41no—
231C4×D4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—1181no—
234C2×C4⋊C4D23611yescharacteristic
235C2×C22⋊C4—22no—
236C2×C22⋊C4—1182no—
237C2×C22⋊C4—1182no—
238C2×C22⋊C4C4×D5911yescharacteristic
239C2×C22⋊C4C22×D5911yescharacteristic
240C2×C42—1181no—
241C2×C42C4×D5911yescharacteristic
242#99—42no—
243C22⋊C8—1182no—
244C22⋊C8—2361no—
245#95—41no—
246#95—592no—
247C24—1182no—
248C24(C2×D59)⋊C411yescharacteristic
249C24C2×C22⋊D5911yescharacteristic
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—4722no—
253C2×D4—4722no—
254C2×D4—4722no—
255C2×D4—2362no—
256C2×D4—2362no—
257C2×D4—2362no—
258C2×D4—2362no—
259C2×D4—2362no—
260C2×D4—2364no—
261C2×D4—2364no—
262C2×D4—2364no—
263C2×D4—2362no—
264C2×D4—2362no—
265C2×D4—2361no—
266C2×D4—2361no—
267C2×D4—2362no—
268C2×D4#1174566912yes—
269C2×D4#1174567112yes—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—1182no—
278C22×C4(C2×D59)⋊C411yescharacteristic
279C22×C4(C2×D59)⋊C411yescharacteristic
280C22×C4(C2×D59)⋊C411yescharacteristic
281C22×C4C2×C4×D5911yescharacteristic
282C22×C4C2×D23611yescharacteristic
283C2×C8—4722no—
284C2×C8—2362no—
285C4⋊C4—41no—
286C4⋊C4—82no—
287C4⋊C4—1184no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—2364no—
293C22⋊C4—2364no—
294C42—21no—
295C42—21no—
296C42—2362no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—2362no—
306C23—2362no—
307C23—2362no—
308C23—2362no—
309C23—2362no—
310C23—2362no—
311C23—2362no—
312C23#1427364712yes—
313C23C2×(C2×D59)⋊C411yescharacteristic
314C23C2×#1174566911yescharacteristic
315C23C2×#1174567111yescharacteristic
316D4—42no—
317D4—42no—
318D4—4724no—
319D4—4722no—
320D4—4724no—
321D4—4728no—
322D4—4724no—
323D4—4724no—
324D4—4722no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—1184no—
342C2×C4—2362no—
343C2×C4—2362no—
344C2×C4#1427364111yescharacteristic
345C2×C4#1427364111yescharacteristic
346C2×C4#1427364712yes—
347C8—4722no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—21no—
353C22—42no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22—4722no—
363C22—4722no—
364C22—4722no—
365C22—4722no—
366C22—4722no—
367C22—4722no—
368C22—4722no—
369C22C2×#1427364111yescharacteristic
370C22C2×#1427364711yescharacteristic
371C22#1683742011yescharacteristic
372C4—22no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—162no—
378C4—2364no—
379C4#1683719412yes—
380C2—21no—
381C2—21no—
382C2—41no—
383C2—42no—
384C2—41no—
385C2—9442no—
386C2#2142266311yescharacteristic
387C2#2142574511yescharacteristic
388C2C2×#1683719411yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1799C2
socleC22×C59C2×#99
prime radical#21429042C2
cosocleC2×C22⋊C4×C59C23
hypercenter#1799D59
hypocenterC59#11877
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #21422414C2semidirect
3. #21422092C2semidirect
4. C59×#1799C2semidirect
5. C2×#14273724C4semidirect
10. #16837311C4semidirect
13. C4×C4⋊Dic59C4semidirect
187. #1799D59semidirect
214. C59#11877semidirect

Maximal subgroups

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

subgroupindex
#214290422
#214224142
#214220922
C59×#17992
#1187759