Finite Group Databasebeta

← #28507008

Subgroups — #28507008

Order 15104.

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

normal subgroups only

This group has 602 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
1C2×#16837197C212yesmaximal, complemented
2C2×#16837195C212yesmaximal
3#21424084C211yesmaximal, characteristic, complemented
4C4×#14273642C211yesmaximal, characteristic, complemented
5C59×#2544C211yesmaximal, characteristic
6C2×#14273642C2212yes—
7C2×#14273642C2212yes—
8#16837262—21no—
9C4×C22⋊Dic59—21no—
10#16837197—24no—
11#16837195—24no—
12#16837169—21no—
13#16837169—21no—
14C2×#127×C59C2211yescharacteristic
15C2×#124×C59C411yescharacteristic, complemented
16C2×#124×C59C2211yescharacteristic
17C2×C4×D4×C59C2211yescharacteristic
18C59×#394—21no—
19C59×#394—21no—
20C59×#391C411yescharacteristic, complemented
21C59×#390C411yescharacteristic, complemented
22C59×#386—21no—
23C59×#386—21no—
24C4×C4⋊C4×C59C411yescharacteristic, complemented
25C2×C4⋊Dic59—41no—
26C2×C22⋊Dic59—22no—
27C2×C22⋊Dic59—22no—
28C2×C4×Dic59—41no—
29#14273642—22no—
30#14273642—22no—
31#14273642—22no—
32#14273642D414yes—
33C59⋊#95—41no—
34C59⋊#95—41no—
35C22×Q8×C59C2×C411yescharacteristic
36C22×D4×C59—21no—
37C22×D4×C59—21no—
38C22×D4×C59C2×C411yescharacteristic
39C22×D4×C59C2311yescharacteristic
40C23×C4×C59D411yescharacteristic
41C23×C4×C59D411yescharacteristic
42#127×C59—22no—
43#127×C59—22no—
44#124×C59—22no—
45#124×C59—22no—
46#124×C59—41no—
47#124×C59—41no—
48C4×D4×C59—41no—
49C4×D4×C59—41no—
50C2×C4⋊C4×C59—21no—
51C2×C4⋊C4×C59C2×C411yescharacteristic
52C2×C22⋊C4×C59—21no—
53C2×C22⋊C4×C59—21no—
54C2×C22⋊C4×C59—21no—
55C2×C22⋊C4×C59—21no—
56C2×C22⋊C4×C59—21no—
57C2×C22⋊C4×C59—21no—
58C2×C22⋊C4×C59D411yescharacteristic
59C2×C22⋊C4×C59D411yescharacteristic
60C2×C42×C59C2×C411yescharacteristic
61C2×C42×C59C2×C411yescharacteristic
62C2×C42×C59C2×C411yescharacteristic
63#95×C59—21no—
64#95×C59—21no—
65#95×C59—21no—
66C24×C59—21no—
67C24×C59C2×D411yescharacteristic
68C24×C59C2×D411yescharacteristic
69C2×Q8×C59—22no—
70C2×Q8×C59—41no—
71C2×Q8×C59—41no—
72C2×D4×C59—22no—
73C2×D4×C59—41no—
74C2×D4×C59—41no—
75C2×D4×C59—42no—
76C2×D4×C59—42no—
77C2×D4×C59—41no—
78C2×D4×C59—41no—
79C2×D4×C59—42no—
80C2×D4×C59—41no—
81C2×D4×C59—41no—
82C2×D4×C59—81no—
83C2×D4×C59—81no—
84C2×D4×C59C2×D412yes—
85C2×D4×C59#4112yes—
86C22×C4×C59—21no—
87C22×C4×C59—21no—
88C22×C4×C59—21no—
89C22×C4×C59—21no—
90C22×C4×C59—21no—
91C22×C4×C59—21no—
92C22×C4×C59—21no—
93C22×C4×C59—81no—
94C22×C4×C59—81no—
95C22×C4×C59C22⋊C411yescharacteristic
96C22×C4×C59C22⋊C411yescharacteristic
97C22×C4×C59C22⋊C411yescharacteristic
98C22×C4×C59C22⋊C411yescharacteristic
99C22×C4×C59C22×C411yescharacteristic
100C22×Dic59—42no—
101C22×Dic59—42no—
102C4⋊C4×C59—41no—
103C4⋊C4×C59—42no—
104C22⋊C4×C59—41no—
105C22⋊C4×C59—41no—
106C22⋊C4×C59—42no—
107C22⋊C4×C59—42no—
108C22⋊C4×C59—41no—
109C22⋊C4×C59—41no—
110C22⋊C4×C59—41no—
111C22⋊C4×C59—41no—
112C22⋊C4×C59—81no—
113C22⋊C4×C59—81no—
114C42×C59—21no—
115C42×C59—21no—
116C42×C59—22no—
117C42×C59—22no—
118C22⋊Dic59—24no—
119C22⋊Dic59—42no—
120C22⋊Dic59—42no—
121C22⋊Dic59—42no—
122C4⋊Dic59—82no—
123C4×Dic59—82no—
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×C59—41no—
133C23×C59—41no—
134C23×C59—41no—
135C23×C59—41no—
136C23×C59—41no—
137C23×C59—41no—
138C23×C59—41no—
139C23×C59C2×C22⋊C411yescharacteristic
140C23×C59#12112yes—
141C23×C59#12411yescharacteristic
142C23×C59#12711yescharacteristic
143Q8×C59—82no—
144D4×C59—42no—
145D4×C59—42no—
146D4×C59—82no—
147D4×C59—84no—
148D4×C59—82no—
149D4×C59—82no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—22no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—41no—
157C2×C4×C59—42no—
158C2×C4×C59—42no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—41no—
168C2×C4×C59—41no—
169C2×C4×C59—41no—
170C2×C4×C59—81no—
171C2×C4×C59—81no—
172C2×C4×C59#9911yescharacteristic
173C2×C4×C59#9911yescharacteristic
174C2×C4×C59C4×D412yes—
175C2×Dic59—44no—
176C2×Dic59—82no—
177C2×Dic59—82no—
178C2×Dic59—82no—
179C2×Dic59—82no—
180C2×Dic59—82no—
181#13225—591nomaximal, Sylow
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—22no—
185C22×C59—21no—
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—81no—
197C22×C59—81no—
198C22×C59—81no—
199C22×C59—81no—
200C22×C59—81no—
201C22×C59—81no—
202C22×C59—81no—
203C22×C59#39011yescharacteristic
204C22×C59C2×#9911yescharacteristic
205C22×C59#41011yescharacteristic
206C4×C59—21no—
207C4×C59—21no—
208C4×C59—22no—
209C4×C59—42no—
210C4×C59—41no—
211C4×C59—41no—
212C4×C59—41no—
213C4×C59—81no—
214C4×C59—81no—
215Dic59—84no—
216Dic59—162no—
217#2544D5911yescharacteristic
218C2×#354—592no—
219C2×#353—592no—
220#1799—591no—
221C4×#99—591no—
222C2×C59—21no—
223C2×C59—21no—
224C2×C59—42no—
225C2×C59—41no—
226C2×C59—41no—
227C2×C59—161no—
228C2×C59#196411yescharacteristic
229C2×C59#212611yescharacteristic
230C2×C59#212711yescharacteristic
231C2×#127C2×D5911yescharacteristic
232C2×#124Dic5911yescharacteristic, complemented
233C2×#124C2×D5911yescharacteristic
234C2×C4×D4C2×D5911yescharacteristic
235C2×#99—592no—
236C2×#99—592no—
237#394—21no—
238#394—21no—
239#391Dic5911yescharacteristic, complemented
240#390Dic5911yescharacteristic, complemented
241#386—21no—
242#386—21no—
243#380—1181no—
244C4×C4⋊C4Dic5911yescharacteristic, complemented
245C4×C22⋊C4—1181no—
246#354—1184no—
247#353—1184no—
248#342—1181no—
249#342—1181no—
250C59#1322511yescharacteristic, complemented, Sylow
251C22×Q8C2×Dic5911yescharacteristic
252C22×D4—21no—
253C22×D4—21no—
254C22×D4C2×Dic5911yescharacteristic
255C22×D4C22×D5911yescharacteristic
256C23×C4C22⋊D5911yescharacteristic
257C23×C4C22⋊D5911yescharacteristic
258#127—22no—
259#127—22no—
260#124—22no—
261#124—22no—
262#124—41no—
263#124—41no—
264C4×D4—41no—
265C4×D4—41no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—2361no—
268C2×C4⋊C4C2×Dic5911yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—1182no—
276C2×C22⋊C4—1182no—
277C2×C22⋊C4C22⋊D5911yescharacteristic
278C2×C22⋊C4C22⋊D5911yescharacteristic
279C2×C42—2361no—
280C2×C42C2×Dic5911yescharacteristic
281C2×C42C2×Dic5911yescharacteristic
282C2×C42C2×Dic5911yescharacteristic
283#99—594no—
284#99—1182no—
285#99—1182no—
286#99—1182no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—2361no—
291#95—2361no—
292C24—21no—
293C24C2×C22⋊D5911yescharacteristic
294C24C2×C22⋊D5911yescharacteristic
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×D4—22no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—81no—
309C2×D4—81no—
310C2×D4D4×D5912yes—
311C2×D4#1174568712yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—2362no—
322C22×C4—2362no—
323C22×C4C22⋊Dic5911yescharacteristic
324C22×C4C22⋊Dic5911yescharacteristic
325C22×C4C22⋊Dic5911yescharacteristic
326C22×C4C22⋊Dic5911yescharacteristic
327C22×C4C22×Dic5911yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—4722no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—81no—
340C22⋊C4—81no—
341C22⋊C4—1184no—
342C22⋊C4—2362no—
343C22⋊C4—2362no—
344C22⋊C4—2362no—
345C42—21no—
346C42—21no—
347C42—22no—
348C42—22no—
349C42—4722no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23C2×C22⋊Dic5911yescharacteristic
366C23#1427369812yes—
367C23#1427371411yescharacteristic
368C23#1427372311yescharacteristic
369Q8—82no—
370D4—42no—
371D4—42no—
372D4—82no—
373D4—84no—
374D4—82no—
375D4—82no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—4722no—
399C2×C4—4722no—
400C2×C4—4722no—
401C2×C4—4722no—
402C2×C4—4722no—
403C2×C4—2364no—
404C2×C4#1427364211yescharacteristic
405C2×C4#1427364211yescharacteristic
406C2×C4D4×Dic5912yes—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—21no—
412C22—42no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22#1683730511yescharacteristic
429C22C2×#1427364211yescharacteristic
430C22#1683737611yescharacteristic
431C4—22no—
432C4—21no—
433C4—21no—
434C4—42no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—81no—
439C4—81no—
440C4—9442no—
441C4—4724no—
442C2—21no—
443C2—21no—
444C2—41no—
445C2—42no—
446C2—41no—
447C2—161no—
448C2#2142493311yescharacteristic
449C2#2142576111yescharacteristic
450C2#2142576511yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#2544C2
socleC22×C59C2×#99
prime radicalC2×#127×C59C22
cosocleC22×D4×C59C23
hypercenter#2544D59
hypocenterC59#13225
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2544

Complements

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

NH ≅ G/Ntype
1. C2×#16837197C2semidirect
3. #21424084C2semidirect
4. C4×#14273642C2semidirect
15. C2×#124×C59C4semidirect
20. C59×#391C4semidirect
21. C59×#390C4semidirect
24. C4×C4⋊C4×C59C4semidirect
232. C2×#124Dic59semidirect
239. #391Dic59semidirect
240. #390Dic59semidirect
244. C4×C4⋊C4Dic59semidirect
250. C59#13225semidirect

Maximal subgroups

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

subgroupindex
C2×#168371972
C2×#168371952
#214240842
C4×#142736422
C59×#25442
#1322559