Finite Group Databasebeta

← #28507384

Subgroups — #28507384

Order 15104.

670 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273668C211yesmaximal, characteristic, complemented
2C2×#16837638C211yesmaximal, characteristic, complemented
3C2×#16837203C211yesmaximal, characteristic, complemented
4C2×#16837202C211yesmaximal, characteristic, complemented
5C2×C22⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
6#21422187C211yesmaximal, characteristic, complemented
7C2×C59×#466C211yesmaximal, characteristic, complemented
8C22×D472C2211yescharacteristic
9C2×#14273701—21no—
10C2×#14273668—22no—
11C2×#14273668—22no—
12C2×#14273668—22no—
13C22×C59⋊C16C2211yescharacteristic
14C22×C59⋊C16C2211yescharacteristic
15#16837638—22no—
16#16837638—22no—
17C2×#14273661C2211yescharacteristic
18C2×#14273649—21no—
19#16837203—22no—
20#16837202—22no—
21C22⋊(C59⋊C16)—22no—
22C22×D8×C59C2211yescharacteristic
23C23×C8×C59C2211yescharacteristic
24C2×#121×C59—21no—
25C59×#466—22no—
26C59×#466—22no—
27C2×#107×C59C2211yescharacteristic
28C2×#102×C59—21no—
29C22×D236—21no—
30C22×C22⋊D59—41no—
31C2×D472—21no—
32C2×D472—21no—
33C2×D472—21no—
34C2×D472—21no—
35C2×D472—21no—
36C2×D472D412yescomplemented
37#14273701—24no—
38#14273701—42no—
39C2×C4⋊Dic59—21no—
40C2×(C2×D59)⋊C4—41no—
41#14273668—28no—
42C2×C59⋊C16—21no—
43C2×C59⋊C16—21no—
44C2×C59⋊C16—21no—
45C2×C59⋊C16—21no—
46C2×C59⋊C16—21no—
47C2×C59⋊C16D412yescomplemented
48#14273661—21no—
49#14273661D412yescomplemented
50#14273649—24no—
51C22×D4×C59—21no—
52C22×D4×C59—41no—
53C23×C4×C59D411yescharacteristic
54C2×D8×C59—21no—
55C2×D8×C59—21no—
56C2×D8×C59—21no—
57C2×D8×C59—21no—
58C2×D8×C59—21no—
59C2×D8×C59D412yescomplemented
60C22×C8×C59—21no—
61C22×C8×C59—21no—
62C22×C8×C59—21no—
63C22×C8×C59—21no—
64C22×C8×C59—21no—
65C22×C8×C59—21no—
66C22×C8×C59D411yescharacteristic
67C22×C8×C59C2311yescharacteristic
68#121×C59—24no—
69#121×C59—42no—
70C2×C4⋊C4×C59—21no—
71C2×C22⋊C4×C59—41no—
72#107×C59—21no—
73#107×C59D412yescomplemented
74#102×C59—24no—
75C24×C59—41no—
76C24×C59D811yescharacteristic
77C23×D59—41no—
78C2×D4×C59—22no—
79C2×D4×C59—44no—
80C2×D4×C59—41no—
81C2×D4×C59—41no—
82C2×D4×C59—41no—
83C2×D4×C59—41no—
84C2×D4×C59—41no—
85C2×D4×C59—82no—
86C2×D4×C59—82no—
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—41no—
94C22×C4×C59D811yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C2×C22⋊D59—44no—
97C2×C22⋊D59—82no—
98C2×C22⋊D59—82no—
99C22×Dic59—41no—
100C2×D236—22no—
101C2×D236—41no—
102C2×D236—41no—
103C2×D236—41no—
104C2×D236—41no—
105C2×D236—41no—
106D8×C59—22no—
107D8×C59—22no—
108D8×C59—41no—
109D8×C59—41no—
110C2×C8×C59—21no—
111C2×C8×C59—21no—
112C2×C8×C59—21no—
113C2×C8×C59—41no—
114C2×C8×C59—41no—
115C2×C8×C59—41no—
116C2×C8×C59—41no—
117C2×C8×C59C2×D411yescharacteristic
118C2×C8×C59C2×D411yescharacteristic
119C2×C8×C59C2×D411yescharacteristic
120C2×C8×C59C2×D411yescharacteristic
121C2×C8×C59C2×D411yescharacteristic
122C2×C8×C59#4111yescharacteristic
123C4⋊C4×C59—22no—
124C4⋊C4×C59—41no—
125C22⋊C4×C59—44no—
126(C2×D59)⋊C4—44no—
127C4⋊Dic59—22no—
128C4⋊Dic59—41no—
129D472—22no—
130D472—22no—
131D472—41no—
132D472—41no—
133C59⋊C16—22no—
134C59⋊C16—41no—
135C23×C59—21no—
136C23×C59—21no—
137C23×C59—21no—
138C23×C59—21no—
139C23×C59—21no—
140C23×C59—21no—
141C23×C59—42no—
142C23×C59—81no—
143C23×C59—81no—
144C23×C59—81no—
145C23×C59—81no—
146C23×C59—81no—
147C23×C59—81no—
148C23×C59D1612yescomplemented
149C23×C59C2×D811yescharacteristic
150C22×D59—42no—
151C22×D59—81no—
152C22×D59—81no—
153C22×D59—81no—
154C22×D59—81no—
155C22×D59—81no—
156C22×D59—81no—
157D4×C59—42no—
158D4×C59—42no—
159D4×C59—88no—
160D4×C59—81no—
161D4×C59—81no—
162C2×C4×C59—21no—
163C2×C4×C59—21no—
164C2×C4×C59—21no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—41no—
168C2×C4×C59—41no—
169C2×C4×C59—42no—
170C2×C4×C59—81no—
171C2×C4×C59—81no—
172C2×C4×C59#12011yescharacteristic
173C2×C4×C59#12011yescharacteristic
174C2×C4×C59#12111yescharacteristic
175C2×C4×C59#13511yescharacteristic
176C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
178C22⋊D59—88no—
179C2×Dic59—42no—
180C2×Dic59—81no—
181C2×Dic59—81no—
182D236—42no—
183D236—42no—
184D236—81no—
185D236—81no—
186C8×C59—41no—
187C8×C59#12011yescharacteristic
188C8×C59#12111yescharacteristic
189C8×C59#12111yescharacteristic
190C8×C59#12711yescharacteristic
191#13304—591nomaximal, Sylow
192C22×C59—22no—
193C22×C59—22no—
194C22×C59—22no—
195C22×C59—41no—
196C22×C59—41no—
197C22×C59—41no—
198C22×C59—41no—
199C22×C59—82no—
200C22×C59—82no—
201C22×C59—82no—
202C22×C59—161no—
203C22×C59—161no—
204C22×C59—161no—
205C22×C59—161no—
206C22×C59#44711yescharacteristic
207C22×C59#44711yescharacteristic
208C22×C59#46611yescharacteristic
209C22×C59C2×D1611yescharacteristic
210C22×C59#50811yescharacteristic
211C22×C59#50911yescharacteristic
212C22×C59#50911yescharacteristic
213C2×D59—82no—
214C2×D59—82no—
215C2×D59—82no—
216C2×D59—161no—
217C2×D59—161no—
218C2×D59—161no—
219C2×D59—161no—
220C4×C59—41no—
221C4×C59—82no—
222C4×C59#39211yescharacteristic
223C4×C59#44911yescharacteristic
224C4×C59#44911yescharacteristic
225C4×C59#46911yescharacteristic
226Dic59—82no—
227C22×D16—591no—
228C2×#466—591no—
229C2×#466D5911yescharacteristic, complemented
230C2×#357—591no—
231C2×#357—591no—
232C2×C22⋊C16—591no—
233#1380—591no—
234C2×C59—42no—
235C2×C59—162no—
236C2×C59#199911yescharacteristic
237C2×C59#218411yescharacteristic
238C2×C59#218411yescharacteristic
239C2×C59#218611yescharacteristic
240C2×C59#218611yescharacteristic
241C2×C59#221511yescharacteristic
242C2×C59#221811yescharacteristic
243D59—162no—
244C22×D8—591no—
245C22×D8C2×D5911yescharacteristic
246C23×C8C2×D5911yescharacteristic
247C2×#121—21no—
248C2×#121—1181no—
249C2×D16—1182no—
250C2×D16—1182no—
251C2×D16—1182no—
252C22×C16—591no—
253C22×C16—591no—
254#466—22no—
255#466—22no—
256#466—1182no—
257#466—1182no—
258C2×#107—591no—
259C2×#107C2×D5911yescharacteristic
260C2×#102—21no—
261C2×#102—1181no—
262#357—1182no—
263#357—1182no—
264C22⋊C16—1182no—
265C59#1330411yescharacteristic, complemented, Sylow
266C22×D4—21no—
267C22×D4—41no—
268C22×D4—1181no—
269C22×D4—2361no—
270C23×C4C22⋊D5911yescharacteristic
271C2×D8—21no—
272C2×D8—21no—
273C2×D8—21no—
274C2×D8—21no—
275C2×D8—21no—
276C2×D8—592no—
277C2×D8—1181no—
278C2×D8—1181no—
279C2×D8—1181no—
280C2×D8—1181no—
281C2×D8—1181no—
282C2×D8C22⋊D5912yescomplemented
283C22×C8—21no—
284C22×C8—21no—
285C22×C8—21no—
286C22×C8—21no—
287C22×C8—21no—
288C22×C8—21no—
289C22×C8C22⋊D5911yescharacteristic
290C22×C8C22×D5911yescharacteristic
291#121—24no—
292#121—42no—
293#121—1184no—
294#121—2362no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—1181no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—2361no—
299D16—1188no—
300C2×C16—592no—
301C2×C16—1181no—
302C2×C16—1181no—
303C2×C16—1181no—
304C2×C16—1181no—
305C2×C16—1181no—
306#107—21no—
307#107—592no—
308#107—1181no—
309#107C22⋊D5912yescomplemented
310#102—24no—
311#102—1184no—
312C24—41no—
313C24—2361no—
314C24#1174566911yescharacteristic
315C2×D4—22no—
316C2×D4—44no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—82no—
323C2×D4—82no—
324C2×D4—1182no—
325C2×D4—4722no—
326C2×D4—4722no—
327C2×D4—2364no—
328C2×D4—2361no—
329C2×D4—2361no—
330C2×D4—2361no—
331C2×D4—2361no—
332C2×D4—2361no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—2361no—
341C22×C4#1174566911yescharacteristic
342C22×C4C2×C22⋊D5911yescharacteristic
343D8—22no—
344D8—22no—
345D8—41no—
346D8—41no—
347D8—1182no—
348D8—1182no—
349D8—2361no—
350D8—2361no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8D4×D5911yescharacteristic
359C2×C8D4×D5911yescharacteristic
360C2×C8D4×D5911yescharacteristic
361C2×C8#1174568711yescharacteristic
362C2×C8C2×C22⋊D5911yescharacteristic
363C2×C8C2×C22⋊D5911yescharacteristic
364C4⋊C4—22no—
365C4⋊C4—41no—
366C4⋊C4—1182no—
367C4⋊C4—2361no—
368C22⋊C4—44no—
369C22⋊C4—2364no—
370C16—1182no—
371C16—2361no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—42no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—4721no—
386C23—4721no—
387C23—4721no—
388C23—4721no—
389C23—4721no—
390C23—4721no—
391C23—2362no—
392C23#1427366812yescomplemented
393C23C2×#1174566911yescharacteristic
394D4—42no—
395D4—42no—
396D4—88no—
397D4—81no—
398D4—81no—
399D4—4728no—
400D4—4721no—
401D4—4721no—
402D4—2362no—
403D4—2362no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—4721no—
415C2×C4—4721no—
416C2×C4—2362no—
417C2×C4#1427369511yescharacteristic
418C2×C4#1427369611yescharacteristic
419C2×C4#1427369911yescharacteristic
420C2×C4#1427374711yescharacteristic
421C2×C4#1427375211yescharacteristic
422C2×C4#1427375311yescharacteristic
423C8—41no—
424C8#1427369511yescharacteristic
425C8#1427369811yescharacteristic
426C8#1427369911yescharacteristic
427C8#1427372311yescharacteristic
428C22—22no—
429C22—22no—
430C22—22no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—82no—
436C22—82no—
437C22—82no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—9441no—
443C22—9441no—
444C22—9441no—
445C22—9441no—
446C22—4722no—
447C22—4722no—
448C22—4722no—
449C22#1683753511yescharacteristic
450C22#1683753611yescharacteristic
451C22#1683763611yescharacteristic
452C22C2×#1427366811yescharacteristic
453C22#1683783211yescharacteristic
454C22#1683783711yescharacteristic
455C22#1683783811yescharacteristic
456C4—41no—
457C4—82no—
458C4—4722no—
459C4#1683731011yescharacteristic
460C4#1683754511yescharacteristic
461C4#1683754711yescharacteristic
462C4#1683765311yescharacteristic
463C2—42no—
464C2—162no—
465C2—9442no—
466C2#2142509211yescharacteristic
467C2#2142600711yescharacteristic
468C2#2142600811yescharacteristic
469C2#2142601711yescharacteristic
470C2#2142601911yescharacteristic
471C2#2142618011yescharacteristic
472C2#2142619711yescharacteristic
subgroupG/H
centerC23C2×#11745669
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC2×C59×#466C2
socleC23×C59C2×D8
hypercenterC2×#466D59
hypocenterC59#13304
LCSG ▷ C22×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C22×#14273668C2semidirect
2. C2×#16837638C2semidirect
3. C2×#16837203C2semidirect
4. C2×#16837202C2semidirect
5. C2×C22⋊(C59⋊C16)C2semidirect
6. #21422187C2semidirect
7. C2×C59×#466C2semidirect
36. C2×D472D4semidirect
47. C2×C59⋊C16D4semidirect
49. #14273661D4semidirect
59. C2×D8×C59D4semidirect
73. #107×C59D4semidirect
148. C23×C59D16semidirect
229. C2×#466D59semidirect
265. C59#13304semidirect
282. C2×D8C22⋊D59semidirect
309. #107C22⋊D59semidirect
392. C23#14273668semidirect

Maximal subgroups

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

subgroupindex
C22×#142736682
C2×#168376382
C2×#168372032
C2×#168372022
C2×C22⋊(C59⋊C16)2
#214221872
C2×C59×#4662
#1330459