Finite Group Databasebeta

← #28507848

Subgroups — #28507848

Order 15104.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837229C211yesmaximal, characteristic, complemented
2C2×#16837203C211yesmaximal, characteristic, complemented
3C2×#16837203C211yesmaximal, characteristic, complemented
4C2×C22⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
5#21424125C211yesmaximal, characteristic, complemented
6#21422187C211yesmaximal, characteristic, complemented
7C2×C59×#466C211yesmaximal, characteristic
8C22×C59⋊C16C2211yescharacteristic
9C22×C59⋊C16C2211yescharacteristic
10C2×#14273661C2211yescharacteristic
11C2×#14273661C2211yescharacteristic
12#16837262—21no—
13#16837229—22no—
14#16837203—22no—
15#16837203—22no—
16C22⋊(C59⋊C16)—22no—
17#16837165—21no—
18C22×D8×C59C2211yescharacteristic
19C23×C8×C59C2211yescharacteristic
20C2×#121×C59—21no—
21C59×#466—22no—
22C59×#466—22no—
23C2×#107×C59C2211yescharacteristic
24C2×#102×C59—21no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59—21no—
27C2×C22⋊Dic59—41no—
28C2×C59⋊C16—21no—
29C2×C59⋊C16—21no—
30C2×C59⋊C16—21no—
31C2×C59⋊C16—21no—
32C2×C59⋊C16—21no—
33C2×C59⋊C16D412yescomplemented
34#14273661—21no—
35#14273661—21no—
36#14273661—21no—
37#14273661—21no—
38C59⋊#95—41no—
39C22×D4×C59—21no—
40C22×D4×C59—41no—
41C23×C4×C59D411yescharacteristic
42C2×D8×C59—21no—
43C2×D8×C59—21no—
44C2×D8×C59—21no—
45C2×D8×C59—21no—
46C2×D8×C59—21no—
47C2×D8×C59D412yes—
48C22×C8×C59—21no—
49C22×C8×C59—21no—
50C22×C8×C59—21no—
51C22×C8×C59—21no—
52C22×C8×C59—21no—
53C22×C8×C59—21no—
54C22×C8×C59D411yescharacteristic
55C22×C8×C59C2311yescharacteristic
56#121×C59—24no—
57#121×C59—42no—
58C2×C4⋊C4×C59—21no—
59C2×C22⋊C4×C59—41no—
60#107×C59—21no—
61#107×C59D412yes—
62#102×C59—24no—
63C24×C59—41no—
64C24×C59D811yescharacteristic
65C2×D4×C59—22no—
66C2×D4×C59—44no—
67C2×D4×C59—41no—
68C2×D4×C59—41no—
69C2×D4×C59—41no—
70C2×D4×C59—41no—
71C2×D4×C59—41no—
72C2×D4×C59—82no—
73C2×D4×C59—82no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—21no—
78C22×C4×C59—21no—
79C22×C4×C59—21no—
80C22×C4×C59—41no—
81C22×C4×C59D811yescharacteristic
82C22×C4×C59C2×D411yescharacteristic
83C22×Dic59—41no—
84C22×Dic59—41no—
85D8×C59—22no—
86D8×C59—22no—
87D8×C59—41no—
88D8×C59—41no—
89C2×C8×C59—21no—
90C2×C8×C59—21no—
91C2×C8×C59—21no—
92C2×C8×C59—41no—
93C2×C8×C59—41no—
94C2×C8×C59—41no—
95C2×C8×C59—41no—
96C2×C8×C59C2×D411yescharacteristic
97C2×C8×C59C2×D411yescharacteristic
98C2×C8×C59C2×D411yescharacteristic
99C2×C8×C59#4111yescharacteristic
100C2×C8×C59#4111yescharacteristic
101C2×C8×C59#4111yescharacteristic
102C4⋊C4×C59—22no—
103C4⋊C4×C59—41no—
104C22⋊C4×C59—44no—
105C22⋊Dic59—82no—
106C4⋊Dic59—41no—
107C4⋊Dic59—41no—
108C4⋊Dic59—41no—
109C4⋊Dic59—41no—
110C59⋊C16—22no—
111C59⋊C16—41no—
112C23×C59—21no—
113C23×C59—21no—
114C23×C59—21no—
115C23×C59—21no—
116C23×C59—21no—
117C23×C59—21no—
118C23×C59—42no—
119C23×C59—81no—
120C23×C59—81no—
121C23×C59—81no—
122C23×C59—81no—
123C23×C59—81no—
124C23×C59—81no—
125C23×C59SD3212yes—
126C23×C59C2×D811yescharacteristic
127D4×C59—42no—
128D4×C59—42no—
129D4×C59—88no—
130D4×C59—81no—
131D4×C59—81no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—42no—
140C2×C4×C59—81no—
141C2×C4×C59—81no—
142C2×C4×C59#12011yescharacteristic
143C2×C4×C59#12111yescharacteristic
144C2×C4×C59#12311yescharacteristic
145C2×C4×C59#13511yescharacteristic
146C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C59#13711yescharacteristic
148C2×Dic59—81no—
149C2×Dic59—81no—
150C2×Dic59—81no—
151C2×Dic59—81no—
152C2×Dic59—81no—
153C2×Dic59—81no—
154C8×C59—41no—
155C8×C59#12111yescharacteristic
156C8×C59#12111yescharacteristic
157C8×C59#12311yescharacteristic
158C8×C59#12411yescharacteristic
159#13381—591nomaximal, Sylow
160C22×C59—22no—
161C22×C59—22no—
162C22×C59—22no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—82no—
168C22×C59—82no—
169C22×C59—82no—
170C22×C59—161no—
171C22×C59—161no—
172C22×C59—161no—
173C22×C59—161no—
174C22×C59#44711yescharacteristic
175C22×C59#46611yescharacteristic
176C22×C59#48011yescharacteristic
177C22×C59C2×SD3211yescharacteristic
178C22×C59#50811yescharacteristic
179C22×C59#50911yescharacteristic
180C22×C59#50911yescharacteristic
181C4×C59—41no—
182C4×C59—82no—
183C4×C59#39411yescharacteristic
184C4×C59#45211yescharacteristic
185C4×C59#46811yescharacteristic
186C4×C59#48211yescharacteristic
187Dic59—161no—
188Dic59—161no—
189C2×#466D5911yescharacteristic
190C2×#367—591no—
191C2×#357—591no—
192C2×#357—591no—
193C2×C22⋊C16—591no—
194#1808—591no—
195#1380—591no—
196C2×C59—42no—
197C2×C59—162no—
198C2×C59#203311yescharacteristic
199C2×C59#218611yescharacteristic
200C2×C59#218711yescharacteristic
201C2×C59#221711yescharacteristic
202C2×C59#221811yescharacteristic
203C2×C59#223311yescharacteristic
204C2×C59#223411yescharacteristic
205C22×D8C2×D5911yescharacteristic
206C23×C8C2×D5911yescharacteristic
207C2×#121—21no—
208C22×C16—591no—
209C22×C16—591no—
210#466—22no—
211#466—22no—
212C2×#107—591no—
213C2×#107—591no—
214C2×#107C2×D5911yescharacteristic
215C2×#102—21no—
216#380—1181no—
217#367—1182no—
218#357—1182no—
219#357—1182no—
220C22⋊C16—1182no—
221#340—1181no—
222C59#1338111yescharacteristic, complemented, Sylow
223C22×D4—21no—
224C22×D4—41no—
225C23×C4C22⋊D5911yescharacteristic
226C2×D8—21no—
227C2×D8—21no—
228C2×D8—21no—
229C2×D8—21no—
230C2×D8—21no—
231C2×D8C22⋊D5912yes—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8C22⋊D5911yescharacteristic
239C22×C8C22×D5911yescharacteristic
240#121—24no—
241#121—42no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—1181no—
244C2×C4⋊C4—1181no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—2361no—
247C2×C16—592no—
248C2×C16—1181no—
249C2×C16—1181no—
250C2×C16—1181no—
251C2×C16—1181no—
252C2×C16—1181no—
253#107—21no—
254#107—1181no—
255#107—1181no—
256#107—1181no—
257#107—1181no—
258#107C22⋊D5912yes—
259#102—24no—
260#95—2361no—
261C24—41no—
262C24#1174566911yescharacteristic
263C2×D4—22no—
264C2×D4—44no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—82no—
271C2×D4—82no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—2361no—
280C22×C4—2361no—
281C22×C4#1174566911yescharacteristic
282C22×C4C2×C22⋊D5911yescharacteristic
283D8—22no—
284D8—22no—
285D8—41no—
286D8—41no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8D4×D5911yescharacteristic
295C2×C8#1174568711yescharacteristic
296C2×C8#1174568711yescharacteristic
297C2×C8#1174568711yescharacteristic
298C2×C8C2×C22⋊D5911yescharacteristic
299C2×C8C2×C22⋊D5911yescharacteristic
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—2361no—
303C4⋊C4—2361no—
304C4⋊C4—2361no—
305C4⋊C4—2361no—
306C22⋊C4—44no—
307C22⋊C4—4722no—
308C16—1182no—
309C16—2361no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—42no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23#1427367112yes—
324C23C2×#1174566911yescharacteristic
325D4—42no—
326D4—42no—
327D4—88no—
328D4—81no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—4721no—
341C2×C4—4721no—
342C2×C4—4721no—
343C2×C4—4721no—
344C2×C4—4721no—
345C2×C4—4721no—
346C2×C4#1427369611yescharacteristic
347C2×C4#1427369911yescharacteristic
348C2×C4#1427370811yescharacteristic
349C2×C4#1427374711yescharacteristic
350C2×C4#1427375311yescharacteristic
351C2×C4#1427375711yescharacteristic
352C8—41no—
353C8#1427369811yescharacteristic
354C8#1427369911yescharacteristic
355C8#1427370811yescharacteristic
356C8#1427371411yescharacteristic
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22—161no—
368C22—161no—
369C22—161no—
370C22—161no—
371C22#1683753611yescharacteristic
372C22#1683763611yescharacteristic
373C22#1683771311yescharacteristic
374C22C2×#1427367111yescharacteristic
375C22#1683783211yescharacteristic
376C22#1683783711yescharacteristic
377C22#1683783811yescharacteristic
378C4—41no—
379C4—82no—
380C4—9441no—
381C4—9441no—
382C4#1683731611yescharacteristic
383C4#1683756411yescharacteristic
384C4#1683764611yescharacteristic
385C4#1683772311yescharacteristic
386C2—42no—
387C2—162no—
388C2#2142527511yescharacteristic
389C2#2142601911yescharacteristic
390C2#2142602511yescharacteristic
391C2#2142619011yescharacteristic
392C2#2142619711yescharacteristic
393C2#2142627911yescharacteristic
394C2#2142628411yescharacteristic
subgroupG/H
centerC23C2×#11745669
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
Fitting, prime radicalC2×C59×#466C2
socleC23×C59C2×D8
hypercenterC2×#466D59
hypocenterC59#13381
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. C2×#16837229C2semidirect
2. C2×#16837203C2semidirect
3. C2×#16837203C2semidirect
4. C2×C22⋊(C59⋊C16)C2semidirect
5. #21424125C2semidirect
6. #21422187C2semidirect
33. C2×C59⋊C16D4semidirect
222. C59#13381semidirect

Maximal subgroups

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

subgroupindex
C2×#168372292
C2×#168372032
C2×C22⋊(C59⋊C16)2
#214241252
#214221872
C2×C59×#4662
#1338159