Finite Group Databasebeta

← #28507851

Subgroups — #28507851

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×#16837638C211yesmaximal, characteristic
2C2×#16837229C211yesmaximal, characteristic, complemented
3C2×#16837202C211yesmaximal, characteristic, complemented
4C2×#16837202C211yesmaximal, characteristic, complemented
5C2×C22⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
6#21422187C211yesmaximal, characteristic, complemented
7C59×#1808C211yesmaximal, characteristic, complemented
8C22×D472C2211yescharacteristic
9C2×#14273701—21no—
10C22×C59⋊C16C2211yescharacteristic
11C22×C59⋊C16C2211yescharacteristic
12#16837638—22no—
13#16837638—22no—
14C2×#14273661C2211yescharacteristic
15C2×#14273649—21no—
16#16837229—22no—
17#16837202—22no—
18#16837202—22no—
19C22⋊(C59⋊C16)—22no—
20C23×C8×C59C2211yescharacteristic
21C2×#107×C59C2211yescharacteristic
22C2×#107×C59C2211yescharacteristic
23C59×#380—21no—
24C59×#340—21no—
25C22×D236—21no—
26C22×C22⋊D59—41no—
27C2×D472—21no—
28C2×D472—21no—
29C2×D472—21no—
30C2×D472—21no—
31C2×D472—21no—
32C2×D472D412yes—
33#14273701—24no—
34#14273701—42no—
35C2×C4⋊Dic59—21no—
36C2×(C2×D59)⋊C4—41no—
37C2×C59⋊C16—21no—
38C2×C59⋊C16—21no—
39C2×C59⋊C16—21no—
40C2×C59⋊C16—21no—
41C2×C59⋊C16—21no—
42C2×C59⋊C16D412yescomplemented
43#14273661—21no—
44#14273661D412yes—
45#14273649—24no—
46C23×C4×C59D411yescharacteristic
47C22×C8×C59—21no—
48C22×C8×C59—21no—
49C22×C8×C59—21no—
50C22×C8×C59—21no—
51C22×C8×C59—21no—
52C22×C8×C59—21no—
53C22×C8×C59D411yescharacteristic
54C22×C8×C59C2311yescharacteristic
55C2×C4⋊C4×C59—21no—
56C2×C4⋊C4×C59—21no—
57C2×C22⋊C4×C59—41no—
58#107×C59—21no—
59#107×C59—21no—
60#107×C59—21no—
61#107×C59—21no—
62#95×C59—41no—
63C24×C59D811yescharacteristic
64C23×D59—41no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—41no—
72C22×C4×C59—41no—
73C22×C4×C59D811yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C2×C22⋊D59—44no—
76C2×C22⋊D59—82no—
77C2×C22⋊D59—82no—
78C22×Dic59—41no—
79C2×D236—22no—
80C2×D236—41no—
81C2×D236—41no—
82C2×D236—41no—
83C2×D236—41no—
84C2×D236—41no—
85C2×C8×C59—21no—
86C2×C8×C59—21no—
87C2×C8×C59—21no—
88C2×C8×C59—41no—
89C2×C8×C59—41no—
90C2×C8×C59—41no—
91C2×C8×C59—41no—
92C2×C8×C59C2×D411yescharacteristic
93C2×C8×C59C2×D411yescharacteristic
94C2×C8×C59C2×D411yescharacteristic
95C2×C8×C59#4111yescharacteristic
96C2×C8×C59#4111yescharacteristic
97C2×C8×C59#4111yescharacteristic
98C4⋊C4×C59—41no—
99C4⋊C4×C59—41no—
100C4⋊C4×C59—41no—
101C4⋊C4×C59—41no—
102C22⋊C4×C59—82no—
103(C2×D59)⋊C4—44no—
104C4⋊Dic59—22no—
105C4⋊Dic59—41no—
106D472—22no—
107D472—22no—
108D472—41no—
109D472—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×C59SD3212yes—
119C23×C59C2×D811yescharacteristic
120C22×D59—42no—
121C22×D59—81no—
122C22×D59—81no—
123C22×D59—81no—
124C22×D59—81no—
125C22×D59—81no—
126C22×D59—81no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—81no—
135C2×C4×C59—81no—
136C2×C4×C59—81no—
137C2×C4×C59—81no—
138C2×C4×C59—81no—
139C2×C4×C59—81no—
140C2×C4×C59#12011yescharacteristic
141C2×C4×C59#12111yescharacteristic
142C2×C4×C59#12311yescharacteristic
143C2×C4×C59#13511yescharacteristic
144C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C59#13711yescharacteristic
146C22⋊D59—88no—
147C2×Dic59—42no—
148C2×Dic59—81no—
149C2×Dic59—81no—
150D236—42no—
151D236—42no—
152D236—81no—
153D236—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#44711yescharacteristic
168C22×C59#46611yescharacteristic
169C22×C59#48011yescharacteristic
170C22×C59C2×SD3211yescharacteristic
171C22×C59#50811yescharacteristic
172C22×C59#50911yescharacteristic
173C22×C59#50911yescharacteristic
174C2×D59—82no—
175C2×D59—82no—
176C2×D59—82no—
177C2×D59—161no—
178C2×D59—161no—
179C2×D59—161no—
180C2×D59—161no—
181C4×C59—41no—
182C4×C59—161no—
183C4×C59—161no—
184C4×C59#39411yescharacteristic
185C4×C59#45211yescharacteristic
186C4×C59#46811yescharacteristic
187C4×C59#48211yescharacteristic
188Dic59—82no—
189C2×#466—591no—
190C2×#367—591no—
191C2×#357—591no—
192C2×#357—591no—
193C2×C22⋊C16—591no—
194#1808D5911yescharacteristic, complemented
195#1380—591no—
196C2×C59—42no—
197C2×C59#203311yescharacteristic
198C2×C59#218611yescharacteristic
199C2×C59#218711yescharacteristic
200C2×C59#221711yescharacteristic
201C2×C59#221811yescharacteristic
202C2×C59#223311yescharacteristic
203C2×C59#223411yescharacteristic
204D59—162no—
205C22×D8—591no—
206C23×C8C2×D5911yescharacteristic
207C2×#121—1181no—
208C22×C16—591no—
209C22×C16—591no—
210#466—1182no—
211#466—1182no—
212C2×#107—591no—
213C2×#107C2×D5911yescharacteristic
214C2×#107C2×D5911yescharacteristic
215C2×#102—1181no—
216#380—21no—
217#367—1182no—
218#357—1182no—
219#357—1182no—
220C22⋊C16—1182no—
221#340—21no—
222C59#1338111yescharacteristic, complemented, Sylow
223C22×D4—1181no—
224C22×D4—2361no—
225C23×C4C22⋊D5911yescharacteristic
226C2×D8—592no—
227C2×D8—1181no—
228C2×D8—1181no—
229C2×D8—1181no—
230C2×D8—1181no—
231C2×D8—1181no—
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—1184no—
241#121—2362no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
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—21no—
255#107—21no—
256#107—21no—
257#107—592no—
258#107—1181no—
259#102—1184no—
260#95—41no—
261C24—2361no—
262C24#1174566911yescharacteristic
263C2×D4—1182no—
264C2×D4—4722no—
265C2×D4—4722no—
266C2×D4—2364no—
267C2×D4—2361no—
268C2×D4—2361no—
269C2×D4—2361no—
270C2×D4—2361no—
271C2×D4—2361no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—2361no—
281C22×C4#1174566911yescharacteristic
282C22×C4C2×C22⋊D5911yescharacteristic
283D8—1182no—
284D8—1182no—
285D8—2361no—
286D8—2361no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8#1174568511yescharacteristic
295C2×C8#1174568511yescharacteristic
296C2×C8D4×D5911yescharacteristic
297C2×C8D4×D5911yescharacteristic
298C2×C8D4×D5911yescharacteristic
299C2×C8#1174568711yescharacteristic
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—1182no—
305C4⋊C4—2361no—
306C22⋊C4—82no—
307C22⋊C4—2364no—
308C16—1182no—
309C16—2361no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—4721no—
317C23—4721no—
318C23—4721no—
319C23—4721no—
320C23—4721no—
321C23—4721no—
322C23—2362no—
323C23#1427367012yes—
324C23C2×#1174566911yescharacteristic
325D4—4728no—
326D4—4721no—
327D4—4721no—
328D4—2362no—
329D4—2362no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—4721no—
344C2×C4—4721no—
345C2×C4—2362no—
346C2×C4#1427369511yescharacteristic
347C2×C4#1427369911yescharacteristic
348C2×C4#1427371011yescharacteristic
349C2×C4#1427374711yescharacteristic
350C2×C4#1427375211yescharacteristic
351C2×C4#1427375811yescharacteristic
352C8—41no—
353C8#1427370011yescharacteristic
354C8#1427370011yescharacteristic
355C8#1427370911yescharacteristic
356C8#1427371311yescharacteristic
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—9441no—
365C22—9441no—
366C22—9441no—
367C22—9441no—
368C22—4722no—
369C22—4722no—
370C22—4722no—
371C22#1683753511yescharacteristic
372C22#1683763611yescharacteristic
373C22#1683771511yescharacteristic
374C22C2×#1427367011yescharacteristic
375C22#1683783211yescharacteristic
376C22#1683783711yescharacteristic
377C22#1683783811yescharacteristic
378C4—41no—
379C4—161no—
380C4—161no—
381C4—4722no—
382C4#1683731711yescharacteristic
383C4#1683756211yescharacteristic
384C4#1683765011yescharacteristic
385C4#1683772611yescharacteristic
386C2—42no—
387C2—9442no—
388C2#2142527811yescharacteristic
389C2#2142601711yescharacteristic
390C2#2142602411yescharacteristic
391C2#2142619411yescharacteristic
392C2#2142619711yescharacteristic
393C2#2142628111yescharacteristic
394C2#2142628711yescharacteristic
subgroupG/H
centerC23C2×#11745669
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#1808C2
socleC23×C59C2×D8
prime radicalC2×#16837638C2
hypercenter#1808D59
hypocenterC59#13381
LCSG ▷ C22×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ #1808

Complements

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

NH ≅ G/Ntype
2. C2×#16837229C2semidirect
3. C2×#16837202C2semidirect
4. C2×#16837202C2semidirect
5. C2×C22⋊(C59⋊C16)C2semidirect
6. #21422187C2semidirect
7. C59×#1808C2semidirect
42. C2×C59⋊C16D4semidirect
194. #1808D59semidirect
222. C59#13381semidirect

Maximal subgroups

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

subgroupindex
C2×#168376382
C2×#168372292
C2×#168372022
C2×C22⋊(C59⋊C16)2
#214221872
C59×#18082
#1338159