Finite Group Databasebeta

← #28506998

Subgroups — #28506998

Order 15104.

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

normal subgroups only

This group has 724 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#21429042C211yesmaximal, characteristic
2C2×#16837194C214yesmaximal, complemented
3#21424084C211yesmaximal, characteristic, complemented
4C4×#99×C59C211yesmaximal, characteristic, complemented
5C2×#14273724C411yescharacteristic, complemented
6C2×#14273723C2212yes—
7C2×#14273701—21no—
8C2×#14273701—21no—
9C2×#14273642C2212yes—
10#16837311—21no—
11#16837311—21no—
12#16837303C412yescomplemented
13#16837262—21no—
14C4×C4⋊Dic59C411yescharacteristic, complemented
15#16837194—28no—
16#16837169—21no—
17C2×C4×D4×C59C2211yescharacteristic
18C2×#99×C59C2212yes—
19C4×C22⋊C4×C59—21no—
20C59×#342—21no—
21C22×D236—21no—
22C22×D236—21no—
23C22×D236C2×C412yes—
24C22×C22⋊D59—22no—
25C22×C22⋊D59—22no—
26#14273724—41no—
27#14273724—41no—
28#14273723—24no—
29#14273723—24no—
30#14273701—44no—
31#14273701—44no—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59—41no—
34C2×C22⋊Dic59—22no—
35C2×(C2×D59)⋊C4—22no—
36C2×(C2×D59)⋊C4—22no—
37C2×C4×Dic59C2×C412yes—
38#14273642—24no—
39C59⋊#95—21no—
40C59⋊#95—41no—
41C22×D4×C59C2311yescharacteristic
42C23×C4×C59D411yescharacteristic
43C23×C4×C59D411yescharacteristic
44C4×D4×C59—41no—
45C4×D4×C59—41no—
46C2×C4⋊C4×C59C2×C411yescharacteristic
47C2×C22⋊C4×C59—22no—
48C2×C22⋊C4×C59D411yescharacteristic
49C2×C22⋊C4×C59D411yescharacteristic
50C2×C42×C59—41no—
51C2×C42×C59C2×C411yescharacteristic
52#99×C59—22no—
53#99×C59D414yescomplemented
54#95×C59—41no—
55C24×C59C2×D411yescharacteristic
56C24×C59C2×D411yescharacteristic
57C23×D59—22no—
58C2×D4×C59—41no—
59C2×D4×C59—41no—
60C2×D4×C59C2×D412yes—
61C2×D4×C59#4112yes—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—42no—
67C22×C4×C59—42no—
68C22×C4×C59—42no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22×C411yescharacteristic
74C2×C22⋊D59—44no—
75C2×C22⋊D59—44no—
76C2×C22⋊D59—44no—
77C2×C22⋊D59—42no—
78C2×C22⋊D59—42no—
79C2×C22⋊D59—82no—
80C2×C22⋊D59—82no—
81C22×Dic59—22no—
82C22×Dic59—42no—
83C2×D236—24no—
84C2×D236—42no—
85C2×D236—42no—
86C2×D236—42no—
87C2×D236—42no—
88C2×D236—42no—
89C2×D236—41no—
90C2×D236—41no—
91C2×D236—82no—
92C4⋊C4×C59—41no—
93C22⋊C4×C59—24no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—42no—
97C42×C59—21no—
98C42×C59—21no—
99C42×C59—82no—
100C22⋊Dic59—42no—
101C22⋊Dic59—42no—
102(C2×D59)⋊C4—44no—
103(C2×D59)⋊C4—44no—
104C4⋊Dic59—24no—
105C4⋊Dic59—82no—
106C4×Dic59—24no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59C2×C22⋊C411yescharacteristic
116C23×C59#12112yes—
117C23×C59#12411yescharacteristic
118C23×C59#12711yescharacteristic
119C22×D59—42no—
120C22×D59—42no—
121C22×D59—42no—
122C22×D59—42no—
123C22×D59—42no—
124C22×D59—42no—
125C22×D59—42no—
126D4×C59—42no—
127D4×C59—42no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—22no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—42no—
135C2×C4×C59—42no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—44no—
140C2×C4×C59—81no—
141C2×C4×C59—81no—
142C2×C4×C59—82no—
143C2×C4×C59—82no—
144C2×C4×C59#9911yescharacteristic
145C2×C4×C59#9911yescharacteristic
146C2×C4×C59C4×D412yes—
147C22⋊D59—88no—
148C22⋊D59—84no—
149C22⋊D59—84no—
150C2×Dic59—24no—
151C2×Dic59—42no—
152C2×Dic59—42no—
153C2×Dic59—82no—
154C2×Dic59—82no—
155C2×Dic59—82no—
156D236—84no—
157D236—82no—
158D236—84no—
159D236—82no—
160#13223—591nomaximal, Sylow
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—22no—
165C22×C59—21no—
166C22×C59—42no—
167C22×C59—41no—
168C22×C59—41no—
169C22×C59—41no—
170C22×C59—41no—
171C22×C59—41no—
172C22×C59—41no—
173C22×C59—81no—
174C22×C59—81no—
175C22×C59#39011yescharacteristic
176C22×C59C2×#9911yescharacteristic
177C22×C59#41011yescharacteristic
178C2×D59—82no—
179C2×D59—82no—
180C2×D59—82no—
181C2×D59—82no—
182C2×D59—82no—
183C2×D59—82no—
184C2×D59—82no—
185C4×C59—21no—
186C4×C59—22no—
187C4×C59—41no—
188C4×C59—41no—
189C4×C59—41no—
190C4×C59—84no—
191C4×C59#35312yescomplemented
192Dic59—44no—
193Dic59—162no—
194#2541—591no—
195C2×#353—594no—
196#1799—591no—
197C4×#99D5911yescharacteristic, complemented
198C2×C59—21no—
199C2×C59—21no—
200C2×C59—41no—
201C2×C59—42no—
202C2×C59—41no—
203C2×C59#196411yescharacteristic
204C2×C59C2×#35311yescharacteristic
205C2×C59#212811yescharacteristic
206D59—162no—
207C2×#127—592no—
208C2×#127—591no—
209C2×#121—1181no—
210C2×#121—1181no—
211C2×C4×D4C2×D5911yescharacteristic
212C2×#99—592no—
213C2×#99C2×D5912yes—
214#392—1181no—
215#392—1181no—
216#390—592no—
217#380—1181no—
218C4×C4⋊C4—591no—
219C4×C22⋊C4—21no—
220#353—1188no—
221#342—21no—
222#342—1181no—
223C59#1322311yescharacteristic, complemented, Sylow
224C22×D4—592no—
225C22×D4—1182no—
226C22×D4—1182no—
227C22×D4—1181no—
228C22×D4—1181no—
229C22×D4C22×D5911yescharacteristic
230C23×C4D23611yescharacteristic
231C23×C4C22⋊D5911yescharacteristic
232#127—1184no—
233#127—1184no—
234#127—2361no—
235#127—2361no—
236#121—2364no—
237#121—2364no—
238C4×D4—41no—
239C4×D4—41no—
240C2×C4⋊C4—1181no—
241C2×C4⋊C4—2361no—
242C2×C4⋊C4C4×D5911yescharacteristic
243C2×C22⋊C4—22no—
244C2×C22⋊C4—1182no—
245C2×C22⋊C4—1182no—
246C2×C22⋊C4—1182no—
247C2×C22⋊C4D23611yescharacteristic
248C2×C22⋊C4C22⋊D5911yescharacteristic
249C2×C42—41no—
250C2×C42—592no—
251C2×C42C4×D5911yescharacteristic
252#99—22no—
253#99—1184no—
254#99D23614yescomplemented
255#95—41no—
256#95—1181no—
257#95—2361no—
258C24—1182no—
259C24C2×D23611yescharacteristic
260C24C2×C22⋊D5911yescharacteristic
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—1184no—
264C2×D4—4722no—
265C2×D4—4722no—
266C2×D4—4722no—
267C2×D4—2362no—
268C2×D4—2362no—
269C2×D4—2362no—
270C2×D4—2362no—
271C2×D4—2362no—
272C2×D4—2364no—
273C2×D4—2364no—
274C2×D4—2364no—
275C2×D4—2362no—
276C2×D4—2362no—
277C2×D4—2361no—
278C2×D4—2361no—
279C2×D4C2×D23612yes—
280C2×D4#1174568512yes—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—1182no—
289C22×C4—2362no—
290C22×C4(C2×D59)⋊C411yescharacteristic
291C22×C4(C2×D59)⋊C411yescharacteristic
292C22×C4(C2×D59)⋊C411yescharacteristic
293C22×C4(C2×D59)⋊C411yescharacteristic
294C22×C4C2×C4×D5911yescharacteristic
295C4⋊C4—41no—
296C4⋊C4—1184no—
297C4⋊C4—4722no—
298C22⋊C4—24no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—42no—
302C22⋊C4—2364no—
303C22⋊C4—2364no—
304C22⋊C4—2362no—
305C22⋊C4—2362no—
306C42—21no—
307C42—21no—
308C42—82no—
309C42—1184no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—2362no—
319C23—2362no—
320C23—2362no—
321C23—2362no—
322C23—2362no—
323C23—2362no—
324C23—2362no—
325C23C2×(C2×D59)⋊C411yescharacteristic
326C23#1427370112yes—
327C23#1427371511yescharacteristic
328C23#1427372411yescharacteristic
329D4—42no—
330D4—42no—
331D4—4724no—
332D4—4722no—
333D4—4724no—
334D4—4728no—
335D4—4724no—
336D4—4724no—
337D4—4722no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—1184no—
355C2×C4—4722no—
356C2×C4—4722no—
357C2×C4—4722no—
358C2×C4—2362no—
359C2×C4—2362no—
360C2×C4#1427364111yescharacteristic
361C2×C4#1427364111yescharacteristic
362C2×C4C4×D23612yes—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—21no—
367C22—21no—
368C22—42no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—81no—
376C22—81no—
377C22—4722no—
378C22—4722no—
379C22—4722no—
380C22—4722no—
381C22—4722no—
382C22—4722no—
383C22—4722no—
384C22#1683730411yescharacteristic
385C22C2×#1427364111yescharacteristic
386C22#1683737511yescharacteristic
387C4—22no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—84no—
393C4—9442no—
394C4—2364no—
395C4#1683719412yescomplemented
396C2—21no—
397C2—21no—
398C2—42no—
399C2—41no—
400C2—41no—
401C2—9442no—
402C2#2142493211yescharacteristic
403C2C2×#1683719411yescharacteristic
404C2#2142576711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC4×#99×C59C2
socleC22×C59C2×#99
prime radical#21429042C2
cosocleC22×D4×C59C23
hypercenterC4×#99D59
hypocenterC59#13223
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. C2×#16837194C2semidirect
3. #21424084C2semidirect
4. C4×#99×C59C2semidirect
5. C2×#14273724C4semidirect
12. #16837303C4semidirect
14. C4×C4⋊Dic59C4semidirect
53. #99×C59D4semidirect
191. C4×C59#353semidirect
197. C4×#99D59semidirect
223. C59#13223semidirect
254. #99D236semidirect
395. C4#16837194semidirect

Maximal subgroups

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

subgroupindex
#214290422
C2×#168371942
#214240842
C4×#99×C592
#1322359