Finite Group Databasebeta

← #28507007

Subgroups — #28507007

Order 15104.

602 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 103 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#21429063C211yesmaximal, characteristic
2C2×#16837196C212yesmaximal, complemented
3C2×#16837194C212yesmaximal
4#21424084C211yesmaximal, characteristic, complemented
5C4×#99×C59C211yesmaximal, characteristic, complemented
6C2×#14273723C2211yescharacteristic
7C2×#14273715C411yescharacteristic, complemented
8C2×#14273714C2211yescharacteristic
9C2×#14273642C2212yes—
10#16837318—21no—
11#16837318—21no—
12#16837307C411yescharacteristic, complemented
13#16837303C411yescharacteristic, complemented
14#16837281—21no—
15#16837281—21no—
16#16837262—21no—
17C4×C4⋊Dic59C411yescharacteristic, complemented
18#16837196—24no—
19#16837194—24no—
20#16837169—21no—
21C2×C4×D4×C59C2211yescharacteristic
22C2×#99×C59C2212yes—
23C4×C22⋊C4×C59—21no—
24C59×#342—21no—
25C22×Dic118C2×C411yescharacteristic
26C22×D236C2×C411yescharacteristic
27C22×C22⋊D59—21no—
28C22×C22⋊D59—21no—
29#14273723—22no—
30#14273723—22no—
31#14273715—41no—
32#14273715—41no—
33#14273714—22no—
34#14273714—22no—
35C2×C4⋊Dic59—21no—
36C2×C4⋊Dic59—41no—
37C2×C22⋊Dic59—21no—
38C2×C22⋊Dic59—21no—
39C2×C22⋊Dic59—22no—
40C2×(C2×D59)⋊C4—21no—
41C2×(C2×D59)⋊C4—21no—
42C2×(C2×D59)⋊C4—21no—
43C2×(C2×D59)⋊C4—21no—
44C2×C4×Dic59C2×C411yescharacteristic
45C2×C4×Dic59C2×C411yescharacteristic
46#14273642—22no—
47#14273642—22no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C59⋊#95—41no—
52C22×D4×C59C2311yescharacteristic
53C23×C4×C59D411yescharacteristic
54C23×C4×C59D411yescharacteristic
55C4×D4×C59—41no—
56C4×D4×C59—41no—
57C2×C4⋊C4×C59C2×C411yescharacteristic
58C2×C22⋊C4×C59—22no—
59C2×C22⋊C4×C59D411yescharacteristic
60C2×C22⋊C4×C59D411yescharacteristic
61C2×C42×C59—41no—
62C2×C42×C59C2×C411yescharacteristic
63#99×C59—22no—
64#99×C59D414yes—
65#95×C59—41no—
66C24×C59C2×D411yescharacteristic
67C24×C59C2×D411yescharacteristic
68C23×D59—21no—
69C2×D4×C59—41no—
70C2×D4×C59—41no—
71C2×D4×C59C2×D412yes—
72C2×D4×C59#4112yes—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—42no—
78C22×C4×C59—81no—
79C22×C4×C59—81no—
80C22×C4×C59C22⋊C411yescharacteristic
81C22×C4×C59C22⋊C411yescharacteristic
82C22×C4×C59C22⋊C411yescharacteristic
83C22×C4×C59C22⋊C411yescharacteristic
84C22×C4×C59C22×C411yescharacteristic
85C2×C22⋊D59—42no—
86C2×C22⋊D59—42no—
87C2×C22⋊D59—42no—
88C2×C22⋊D59—41no—
89C2×C22⋊D59—41no—
90C2×C22⋊D59—81no—
91C2×C22⋊D59—81no—
92C22×Dic59—21no—
93C22×Dic59—21no—
94C22×Dic59—21no—
95C22×Dic59—42no—
96C2×D236—22no—
97C2×D236—41no—
98C2×D236—41no—
99C2×Dic118—22no—
100C2×Dic118—41no—
101C2×Dic118—41no—
102C4⋊C4×C59—41no—
103C22⋊C4×C59—24no—
104C22⋊C4×C59—41no—
105C22⋊C4×C59—41no—
106C22⋊C4×C59—42no—
107C42×C59—21no—
108C42×C59—21no—
109C42×C59—82no—
110C22⋊Dic59—42no—
111C22⋊Dic59—41no—
112C22⋊Dic59—41no—
113C22⋊Dic59—42no—
114C22⋊Dic59—42no—
115(C2×D59)⋊C4—42no—
116(C2×D59)⋊C4—41no—
117(C2×D59)⋊C4—41no—
118(C2×D59)⋊C4—81no—
119(C2×D59)⋊C4—81no—
120C4⋊Dic59—42no—
121C4⋊Dic59—82no—
122C4×Dic59—22no—
123C4×Dic59—22no—
124C23×C59—21no—
125C23×C59—21no—
126C23×C59—21no—
127C23×C59—21no—
128C23×C59—41no—
129C23×C59—41no—
130C23×C59—41no—
131C23×C59—41no—
132C23×C59C2×C22⋊C411yescharacteristic
133C23×C59#12112yes—
134C23×C59#12411yescharacteristic
135C23×C59#12711yescharacteristic
136C22×D59—41no—
137C22×D59—41no—
138C22×D59—41no—
139C22×D59—41no—
140C22×D59—41no—
141C22×D59—41no—
142C22×D59—41no—
143D4×C59—42no—
144D4×C59—42no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—41no—
151C2×C4×C59—42no—
152C2×C4×C59—42no—
153C2×C4×C59—41no—
154C2×C4×C59—41no—
155C2×C4×C59—41no—
156C2×C4×C59—44no—
157C2×C4×C59—81no—
158C2×C4×C59—81no—
159C2×C4×C59—82no—
160C2×C4×C59—82no—
161C2×C4×C59#9911yescharacteristic
162C2×C4×C59#9911yescharacteristic
163C2×C4×C59C4×D412yes—
164C22⋊D59—84no—
165C22⋊D59—82no—
166C22⋊D59—82no—
167C2×Dic59—22no—
168C2×Dic59—41no—
169C2×Dic59—41no—
170C2×Dic59—41no—
171C2×Dic59—41no—
172C2×Dic59—41no—
173C2×Dic59—41no—
174C2×Dic59—41no—
175C2×Dic59—41no—
176C2×Dic59—82no—
177C2×Dic59—82no—
178C2×Dic59—82no—
179D236—82no—
180Dic118—82no—
181#13225—591nomaximal, Sylow
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—22no—
186C22×C59—21no—
187C22×C59—42no—
188C22×C59—41no—
189C22×C59—41no—
190C22×C59—41no—
191C22×C59—41no—
192C22×C59—41no—
193C22×C59—41no—
194C22×C59—81no—
195C22×C59—81no—
196C22×C59#39011yescharacteristic
197C22×C59C2×#9911yescharacteristic
198C22×C59#41011yescharacteristic
199C2×D59—81no—
200C2×D59—81no—
201C2×D59—81no—
202C2×D59—81no—
203C2×D59—81no—
204C2×D59—81no—
205C2×D59—81no—
206C4×C59—21no—
207C4×C59—21no—
208C4×C59—22no—
209C4×C59—41no—
210C4×C59—41no—
211C4×C59—41no—
212C4×C59—84no—
213Dic59—42no—
214Dic59—81no—
215Dic59—81no—
216Dic59—162no—
217#2544—591no—
218C2×#354—592no—
219C2×#353—592no—
220#1799—591no—
221C4×#99D5911yescharacteristic, complemented
222C2×C59—21no—
223C2×C59—21no—
224C2×C59—41no—
225C2×C59—42no—
226C2×C59—41no—
227C2×C59#196411yescharacteristic
228C2×C59#212611yescharacteristic
229C2×C59#212711yescharacteristic
230D59—161no—
231C2×#127—591no—
232C2×#124—591no—
233C2×#124—591no—
234C2×C4×D4C2×D5911yescharacteristic
235C2×#99—592no—
236C2×#99C2×D5912yes—
237#394—1181no—
238#394—1181no—
239#391—591no—
240#390—591no—
241#386—1181no—
242#386—1181no—
243#380—1181no—
244C4×C4⋊C4—591no—
245C4×C22⋊C4—21no—
246#354—1184no—
247#353—1184no—
248#342—21no—
249#342—1181no—
250C59#1322511yescharacteristic, complemented, Sylow
251C22×Q8—591no—
252C22×D4—591no—
253C22×D4—1181no—
254C22×D4—1181no—
255C22×D4C22×D5911yescharacteristic
256C23×C4D23611yescharacteristic
257C23×C4C22⋊D5911yescharacteristic
258#127—1182no—
259#127—1182no—
260#124—1182no—
261#124—1182no—
262#124—2361no—
263#124—2361no—
264C4×D4—41no—
265C4×D4—41no—
266C2×C4⋊C4—1181no—
267C2×C4⋊C4—2361no—
268C2×C4⋊C4C4×D5911yescharacteristic
269C2×C22⋊C4—22no—
270C2×C22⋊C4—1181no—
271C2×C22⋊C4—1181no—
272C2×C22⋊C4—1181no—
273C2×C22⋊C4—1181no—
274C2×C22⋊C4—1181no—
275C2×C22⋊C4—1181no—
276C2×C22⋊C4—1182no—
277C2×C22⋊C4D23611yescharacteristic
278C2×C22⋊C4C22⋊D5911yescharacteristic
279C2×C42—41no—
280C2×C42—591no—
281C2×C42—591no—
282C2×C42C4×D5911yescharacteristic
283#99—22no—
284#99—1182no—
285#99—1182no—
286#99D23614yes—
287#95—41no—
288#95—1181no—
289#95—1181no—
290#95—1181no—
291#95—2361no—
292C24—1181no—
293C24C2×D23611yescharacteristic
294C24C2×C22⋊D5911yescharacteristic
295C2×Q8—1182no—
296C2×Q8—2361no—
297C2×Q8—2361no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—1182no—
301C2×D4—4721no—
302C2×D4—4721no—
303C2×D4—2361no—
304C2×D4—2361no—
305C2×D4—2362no—
306C2×D4—2362no—
307C2×D4—2362no—
308C2×D4—2361no—
309C2×D4—2361no—
310C2×D4C2×D23612yes—
311C2×D4#1174568512yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—42no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—1181no—
320C22×C4—1181no—
321C22×C4—1181no—
322C22×C4—2362no—
323C22×C4(C2×D59)⋊C411yescharacteristic
324C22×C4(C2×D59)⋊C411yescharacteristic
325C22×C4(C2×D59)⋊C411yescharacteristic
326C22×C4(C2×D59)⋊C411yescharacteristic
327C22×C4C2×C4×D5911yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—4722no—
330C4⋊C4—2362no—
331C22⋊C4—24no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—42no—
335C22⋊C4—4721no—
336C22⋊C4—4721no—
337C22⋊C4—2362no—
338C22⋊C4—2362no—
339C22⋊C4—2362no—
340C22⋊C4—2362no—
341C22⋊C4—2361no—
342C22⋊C4—2361no—
343C22⋊C4—2361no—
344C22⋊C4—2361no—
345C42—21no—
346C42—21no—
347C42—82no—
348C42—1182no—
349C42—1182no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—2361no—
359C23—2361no—
360C23—2361no—
361C23—2361no—
362C23—2361no—
363C23—2361no—
364C23—2361no—
365C23C2×(C2×D59)⋊C411yescharacteristic
366C23#1427370112yes—
367C23#1427371511yescharacteristic
368C23#1427372411yescharacteristic
369Q8—4722no—
370D4—42no—
371D4—42no—
372D4—4722no—
373D4—4724no—
374D4—4722no—
375D4—4722no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—1182no—
393C2×C4—4722no—
394C2×C4—4722no—
395C2×C4—4722no—
396C2×C4—2361no—
397C2×C4—2361no—
398C2×C4—2361no—
399C2×C4—2361no—
400C2×C4—2361no—
401C2×C4—2361no—
402C2×C4—2361no—
403C2×C4—2361no—
404C2×C4#1427364111yescharacteristic
405C2×C4#1427364111yescharacteristic
406C2×C4C4×D23612yes—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—21no—
411C22—21no—
412C22—42no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—81no—
420C22—81no—
421C22—4721no—
422C22—4721no—
423C22—4721no—
424C22—4721no—
425C22—4721no—
426C22—4721no—
427C22—4721no—
428C22#1683730411yescharacteristic
429C22C2×#1427364111yescharacteristic
430C22#1683737511yescharacteristic
431C4—22no—
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—84no—
438C4—9442no—
439C4—4721no—
440C4—4721no—
441C4—2362no—
442C2—21no—
443C2—21no—
444C2—42no—
445C2—41no—
446C2—41no—
447C2—9441no—
448C2#2142493211yescharacteristic
449C2#2142575911yescharacteristic
450C2#2142576311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC4×#99×C59C2
socleC22×C59C2×#99
prime radicalC2×#14273723C22
cosocleC22×D4×C59C23
hypercenterC4×#99D59
hypocenterC59#13225
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×#16837196C2semidirect
4. #21424084C2semidirect
5. C4×#99×C59C2semidirect
7. C2×#14273715C4semidirect
12. #16837307C4semidirect
13. #16837303C4semidirect
17. C4×C4⋊Dic59C4semidirect
221. C4×#99D59semidirect
250. C59#13225semidirect

Maximal subgroups

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

subgroupindex
#214290632
C2×#168371962
C2×#168371942
#214240842
C4×#99×C592
#1322559