Finite Group Databasebeta

← #28504677

Subgroups — #28504677

Order 15104.

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

normal subgroups only

This group has 588 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#21426818C211yesmaximal, characteristic
2C2×#16837194C212yesmaximal
3#21424084C211yesmaximal, characteristic, complemented
4#21422220C212yesmaximal, complemented
5C59×#1799C211yesmaximal, characteristic, complemented
6C2×#14273723C2211yescharacteristic
7C2×C4×D236C411yescharacteristic, complemented
8C2×#14273642C2212yes—
9#16837305C2211yescharacteristic
10#16837303C411yescharacteristic, complemented
11#16837281—21no—
12#16837281—21no—
13#16837278—21no—
14#16837278—21no—
15#16837267C411yescharacteristic, complemented
16#16837262—21no—
17#16837248C411yescharacteristic, complemented
18#16837194—24no—
19#16837169—21no—
20C2×C4×D4×C59C2211yescharacteristic
21C2×#99×C59C2212yes—
22C59×#380—21no—
23C59×#342—21no—
24C22×D236C2×C411yescharacteristic
25C22×C22⋊D59—21no—
26C22×C22⋊D59—21no—
27C22×C4×D59—21no—
28#14273723—21no—
29#14273723—21no—
30#14273723D412yes—
31#14273723Q812yes—
32C4×D236—41no—
33C4×D236—41no—
34C2×C4⋊Dic59—41no—
35C2×C4⋊Dic59C2×C411yescharacteristic
36C2×C22⋊Dic59—22no—
37C2×C22⋊Dic59—21no—
38C2×C22⋊Dic59—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×(C2×D59)⋊C4—21no—
41C2×(C2×D59)⋊C4—21no—
42C2×C4×Dic59C2×C411yescharacteristic
43C2×C4×Dic59C2×C411yescharacteristic
44#14273642—22no—
45#14273642—22no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95—41no—
51C22×D4×C59C2311yescharacteristic
52C23×C4×C59D411yescharacteristic
53C23×C4×C59D411yescharacteristic
54C4×D4×C59—41no—
55C4×D4×C59—41no—
56C2×C4⋊C4×C59—41no—
57C2×C4⋊C4×C59C2×C411yescharacteristic
58C2×C22⋊C4×C59—22no—
59C2×C22⋊C4×C59D411yescharacteristic
60C2×C22⋊C4×C59D411yescharacteristic
61C2×C42×C59C2×C411yescharacteristic
62#99×C59—22no—
63#99×C59—22no—
64#95×C59—41no—
65C24×C59C2×D411yescharacteristic
66C24×C59C2×D411yescharacteristic
67C23×D59—21no—
68C2×D4×C59—41no—
69C2×D4×C59—41no—
70C2×D4×C59C2×D411yescharacteristic
71C2×D4×C59C2×Q811yescharacteristic
72C2×D4×C59#4111yescharacteristic
73C2×D4×C59#4111yescharacteristic
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—21no—
78C22×C4×C59—42no—
79C22×C4×C59—81no—
80C22×C4×C59—81no—
81C22×C4×C59C22⋊C411yescharacteristic
82C22×C4×C59C22⋊C411yescharacteristic
83C22×C4×C59C22⋊C411yescharacteristic
84C22×C4×C59C22⋊C411yescharacteristic
85C22×C4×C59C22×C411yescharacteristic
86C2×C22⋊D59—22no—
87C2×C22⋊D59—22no—
88C2×C22⋊D59—22no—
89C2×C22⋊D59—22no—
90C2×C22⋊D59—81no—
91C2×C22⋊D59—81no—
92C2×C22⋊D59—81no—
93C2×C22⋊D59—81no—
94C22×Dic59—21no—
95C22×Dic59—21no—
96C22×Dic59—21no—
97C22×Dic59—42no—
98C2×D236—21no—
99C2×D236—41no—
100C2×D236—41no—
101C2×D236C4⋊C412yescomplemented
102C2×C4×D59—42no—
103C2×C4×D59—81no—
104C2×C4×D59—81no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59—82no—
107C22⋊C4×C59—41no—
108C22⋊C4×C59—41no—
109C22⋊C4×C59—42no—
110C22⋊C4×C59—42no—
111C42×C59—21no—
112C42×C59—21no—
113C22⋊Dic59—42no—
114C22⋊Dic59—42no—
115C22⋊Dic59—81no—
116C22⋊Dic59—81no—
117(C2×D59)⋊C4—41no—
118(C2×D59)⋊C4—41no—
119(C2×D59)⋊C4—81no—
120(C2×D59)⋊C4—81no—
121C4⋊Dic59—41no—
122C4⋊Dic59—82no—
123C4×Dic59—21no—
124C4×Dic59—41no—
125C4×Dic59C4⋊C412yescomplemented
126C23×C59—21no—
127C23×C59—21no—
128C23×C59—21no—
129C23×C59—21no—
130C23×C59—41no—
131C23×C59—41no—
132C23×C59—41no—
133C23×C59—41no—
134C23×C59C2×C22⋊C411yescharacteristic
135C23×C59#12111yescharacteristic
136C23×C59#12111yescharacteristic
137C23×C59C22⋊Q811yescharacteristic
138C23×C59C22⋊Q811yescharacteristic
139C22×D59—22no—
140C22×D59—41no—
141C22×D59—41no—
142C22×D59—41no—
143C22×D59—41no—
144C22×D59—41no—
145C22×D59—41no—
146D4×C59—41no—
147D4×C59—41no—
148D4×C59—41no—
149D4×C59—41no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—41no—
156C2×C4×C59—42no—
157C2×C4×C59—42no—
158C2×C4×C59—41no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—81no—
162C2×C4×C59—81no—
163C2×C4×C59—82no—
164C2×C4×C59—82no—
165C2×C4×C59—82no—
166C2×C4×C59#9911yescharacteristic
167C2×C4×C59#9911yescharacteristic
168C2×C4×C59C2×C4⋊C411yescharacteristic
169C2×C4×C59#11711yescharacteristic
170C22⋊D59—82no—
171C22⋊D59—82no—
172C22⋊D59—82no—
173C22⋊D59—82no—
174C2×Dic59—22no—
175C2×Dic59—41no—
176C2×Dic59—41no—
177C2×Dic59—41no—
178C2×Dic59—41no—
179C2×Dic59—41no—
180C2×Dic59—41no—
181C2×Dic59—41no—
182C2×Dic59—41no—
183C2×Dic59—82no—
184C2×Dic59—82no—
185C2×Dic59—82no—
186D236—42no—
187D236—42no—
188C4×D59—84no—
189#12799—591nomaximal, Sylow
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—21no—
193C22×C59—21no—
194C22×C59—21no—
195C22×C59—21no—
196C22×C59—41no—
197C22×C59—41no—
198C22×C59—41no—
199C22×C59—41no—
200C22×C59—41no—
201C22×C59—41no—
202C22×C59—41no—
203C22×C59—41no—
204C22×C59—81no—
205C22×C59—81no—
206C22×C59#38011yescharacteristic
207C22×C59C2×#9911yescharacteristic
208C22×C59#41011yescharacteristic
209C2×D59—42no—
210C2×D59—42no—
211C2×D59—42no—
212C2×D59—81no—
213C2×D59—81no—
214C2×D59—81no—
215C2×D59—81no—
216C4×C59—21no—
217C4×C59—21no—
218C4×C59—21no—
219C4×C59—21no—
220C4×C59—41no—
221C4×C59—41no—
222C4×C59—41no—
223C4×C59—162no—
224Dic59—42no—
225Dic59—81no—
226Dic59—81no—
227Dic59—162no—
228#2328—591no—
229C2×#353—592no—
230#1799—591no—
231#1799D5911yescharacteristic, complemented
232#1393—592no—
233C2×C59—21no—
234C2×C59—21no—
235C2×C59—41no—
236C2×C59—41no—
237C2×C59—41no—
238C2×C59—41no—
239C2×C59#179911yescharacteristic
240C2×C59#212711yescharacteristic
241C2×C59#212811yescharacteristic
242D59—82no—
243C2×#127—591no—
244C2×C4×D4—591no—
245C2×C4×D4C2×D5911yescharacteristic
246C2×#99—592no—
247C2×#99C2×D5912yes—
248#390—591no—
249#390—591no—
250#386—1181no—
251#386—1181no—
252#386—1181no—
253#386—1181no—
254#382—591no—
255#380—21no—
256#380—1181no—
257#376—591no—
258#353—1184no—
259#342—21no—
260#342—1181no—
261C59#1279911yescharacteristic, complemented, Sylow
262C22×D4—591no—
263C22×D4—1181no—
264C22×D4—1181no—
265C22×D4C22×D5911yescharacteristic
266C23×C4—1181no—
267C23×C4D23611yescharacteristic
268C23×C4C22⋊D5911yescharacteristic
269#127—592no—
270#127—592no—
271#127—1181no—
272#127—1181no—
273C4×D4—41no—
274C4×D4—41no—
275C4×D4—2361no—
276C4×D4—2361no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—591no—
279C2×C4⋊C4—2361no—
280C2×C4⋊C4C4×D5911yescharacteristic
281C2×C22⋊C4—22no—
282C2×C22⋊C4—1181no—
283C2×C22⋊C4—1181no—
284C2×C22⋊C4—1181no—
285C2×C22⋊C4—1181no—
286C2×C22⋊C4—1181no—
287C2×C22⋊C4—1182no—
288C2×C22⋊C4D23611yescharacteristic
289C2×C22⋊C4C22⋊D5911yescharacteristic
290C2×C42—591no—
291C2×C42—591no—
292C2×C42C4×D5911yescharacteristic
293#99—22no—
294#99—22no—
295#99—1182no—
296#99—1182no—
297#95—41no—
298#95—1181no—
299#95—1181no—
300#95—1181no—
301#95—1181no—
302#95—2361no—
303C24—1181no—
304C24C2×D23611yescharacteristic
305C24C2×C22⋊D5911yescharacteristic
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—592no—
309C2×D4—1182no—
310C2×D4—1182no—
311C2×D4—1182no—
312C2×D4—1182no—
313C2×D4—1181no—
314C2×D4—4721no—
315C2×D4—4721no—
316C2×D4—4721no—
317C2×D4—4721no—
318C2×D4—2361no—
319C2×D4—2361no—
320C2×D4D4×D5911yescharacteristic
321C2×D4#1174568711yescharacteristic
322C2×D4Q8×D5911yescharacteristic
323C2×D4#1174568911yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—42no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4—1181no—
332C22×C4—1181no—
333C22×C4—1181no—
334C22×C4—4721no—
335C22×C4—4721no—
336C22×C4—2362no—
337C22×C4—2362no—
338C22×C4(C2×D59)⋊C411yescharacteristic
339C22×C4(C2×D59)⋊C411yescharacteristic
340C22×C4(C2×D59)⋊C411yescharacteristic
341C22×C4(C2×D59)⋊C411yescharacteristic
342C22×C4C2×C4×D5911yescharacteristic
343C4⋊C4—41no—
344C4⋊C4—82no—
345C4⋊C4—4722no—
346C4⋊C4—2361no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—4721no—
352C22⋊C4—4721no—
353C22⋊C4—4721no—
354C22⋊C4—4721no—
355C22⋊C4—2362no—
356C22⋊C4—2362no—
357C22⋊C4—2361no—
358C22⋊C4—2361no—
359C42—21no—
360C42—21no—
361C42—592no—
362C42—1181no—
363C42—2361no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—1182no—
373C23—2361no—
374C23—2361no—
375C23—2361no—
376C23—2361no—
377C23—2361no—
378C23—2361no—
379C23C2×(C2×D59)⋊C411yescharacteristic
380C23#1427369811yescharacteristic
381C23#1427370211yescharacteristic
382C23#1427370311yescharacteristic
383C23#1427370711yescharacteristic
384D4—41no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—4722no—
389D4—4722no—
390D4—4722no—
391D4—4722no—
392D4—2362no—
393D4—2362no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—1182no—
411C2×C4—4724no—
412C2×C4—4722no—
413C2×C4—4722no—
414C2×C4—4722no—
415C2×C4—2361no—
416C2×C4—2361no—
417C2×C4—2361no—
418C2×C4—2361no—
419C2×C4—2361no—
420C2×C4—2361no—
421C2×C4—2361no—
422C2×C4—2361no—
423C2×C4#1427364111yescharacteristic
424C2×C4#1427364111yescharacteristic
425C2×C4C4⋊C4×D5911yescharacteristic
426C2×C4#1427368511yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—81no—
442C22—81no—
443C22—4721no—
444C22—4721no—
445C22—4721no—
446C22—4721no—
447C22—2362no—
448C22—2362no—
449C22—2362no—
450C22#1683725911yescharacteristic
451C22C2×#1427364111yescharacteristic
452C22#1683737511yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—162no—
461C4—9442no—
462C4—4721no—
463C4—4721no—
464C4—2362no—
465C2—21no—
466C2—21no—
467C2—41no—
468C2—41no—
469C2—41no—
470C2—41no—
471C2—4722no—
472C2#2142408111yescharacteristic
473C2#2142576311yescharacteristic
474C2#2142576711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#1799C2
socleC22×C59C2×#99
prime radicalC2×#14273723C22
cosocleC22×D4×C59C23
hypercenter#1799D59
hypocenterC59#12799
LCSG ▷ C22×C4×C59 ▷ C22×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
3. #21424084C2semidirect
4. #21422220C2semidirect
5. C59×#1799C2semidirect
7. C2×C4×D236C4semidirect
10. #16837303C4semidirect
15. #16837267C4semidirect
17. #16837248C4semidirect
101. C2×D236C4⋊C4semidirect
125. C4×Dic59C4⋊C4semidirect
231. #1799D59semidirect
261. C59#12799semidirect

Maximal subgroups

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

subgroupindex
#214268182
C2×#168371942
#214240842
#214222202
C59×#17992
#1279959