Finite Group Databasebeta

← #28508283

Subgroups — #28508283

Order 15104.

482 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273668C211yesmaximal, characteristic
2C2×C4⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
3C2×#16837203C211yesmaximal, characteristic, complemented
4C2×#16837202C211yesmaximal, characteristic, complemented
5#21424962C211yesmaximal, characteristic, complemented
6#21422187C211yesmaximal, characteristic, complemented
7C59×#1970C211yesmaximal, characteristic, complemented
8C22×D472C2211yescharacteristic
9C2×#14273668—22no—
10C2×#14273668—22no—
11C2×#14273668—22no—
12C22×C59⋊C16C2211yescharacteristic
13C22×C59⋊C16C2211yescharacteristic
14C2×#14273661C2211yescharacteristic
15C2×#14273649—21no—
16#16837304—21no—
17C4⋊(C59⋊C16)—22no—
18#16837203—22no—
19#16837202—22no—
20C22×D8×C59C2211yescharacteristic
21C2×#107×C59C2211yescharacteristic
22C2×#102×C59—21no—
23C2×C4×C8×C59C2211yescharacteristic
24C59×#390—21no—
25C22×D236—21no—
26C2×D472—21no—
27C2×D472—21no—
28C2×D472—21no—
29C2×D472—21no—
30C2×D472—21no—
31C2×D472D412yes—
32C2×C4⋊Dic59—21no—
33C2×(C2×D59)⋊C4—41no—
34C2×(C2×D59)⋊C4—41no—
35#14273668—28no—
36C2×C59⋊C16—21no—
37C2×C59⋊C16—21no—
38C2×C59⋊C16—21no—
39C2×C59⋊C16—21no—
40C2×C59⋊C16—21no—
41C2×C59⋊C16D412yes—
42#14273661—21no—
43#14273661—21no—
44#14273649—42no—
45C22×D4×C59—21no—
46C2×D8×C59—21no—
47C2×D8×C59—21no—
48C2×D8×C59—21no—
49C2×D8×C59—21no—
50C2×D8×C59—21no—
51C2×D8×C59D412yes—
52C22×C8×C59D411yescharacteristic
53C22×C8×C59C2311yescharacteristic
54C2×C4⋊C4×C59—21no—
55C2×C22⋊C4×C59—41no—
56C2×C22⋊C4×C59—41no—
57C2×C42×C59D411yescharacteristic
58#107×C59—21no—
59#107×C59—21no—
60#102×C59—42no—
61C4×C8×C59—21no—
62C4×C8×C59—21no—
63C24×C59—41no—
64C23×D59—41no—
65C2×D4×C59—22no—
66C2×D4×C59—41no—
67C2×D4×C59—41no—
68C2×D4×C59—41no—
69C2×D4×C59—41no—
70C2×D4×C59—41no—
71C22×C4×C59—41no—
72C22×C4×C59D811yescharacteristic
73C22×C4×C59D811yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×Dic59—41no—
76C2×D236—22no—
77C2×D236—41no—
78C2×D236—41no—
79C2×D236—41no—
80C2×D236—41no—
81C2×D236—41no—
82D8×C59—22no—
83D8×C59—22no—
84D8×C59—41no—
85D8×C59—41no—
86C2×C8×C59—21no—
87C2×C8×C59—21no—
88C2×C8×C59—21no—
89C2×C8×C59C2×D411yescharacteristic
90C2×C8×C59C2×D411yescharacteristic
91C2×C8×C59C2×D411yescharacteristic
92C2×C8×C59#4111yescharacteristic
93C2×C8×C59#4111yescharacteristic
94C2×C8×C59#4111yescharacteristic
95C4⋊C4×C59—41no—
96C4⋊C4×C59—41no—
97C22⋊C4×C59—82no—
98C22⋊C4×C59—82no—
99C42×C59—21no—
100C42×C59—21no—
101(C2×D59)⋊C4—82no—
102(C2×D59)⋊C4—82no—
103C4⋊Dic59—41no—
104C4⋊Dic59—41no—
105D472—22no—
106D472—22no—
107D472—41no—
108D472—41no—
109C59⋊C16—22no—
110C59⋊C16—41no—
111C23×C59—42no—
112C23×C59—81no—
113C23×C59—81no—
114C23×C59—81no—
115C23×C59—81no—
116C23×C59—81no—
117C23×C59—81no—
118C23×C59C2×D811yescharacteristic
119C22×D59—42no—
120C22×D59—81no—
121C22×D59—81no—
122C22×D59—81no—
123C22×D59—81no—
124C22×D59—81no—
125C22×D59—81no—
126D4×C59—42no—
127D4×C59—42no—
128D4×C59—81no—
129D4×C59—81no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—81no—
136C2×C4×C59—81no—
137C2×C4×C59—81no—
138C2×C4×C59D1612yescomplemented
139C2×C4×C59#12111yescharacteristic
140C2×C4×C59#12111yescharacteristic
141C2×C4×C59#12111yescharacteristic
142C2×C4×C59#13511yescharacteristic
143C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
145C2×Dic59—81no—
146C2×Dic59—81no—
147C2×Dic59—81no—
148D236—42no—
149D236—42no—
150D236—81no—
151D236—81no—
152C8×C59—41no—
153C8×C59#12011yescharacteristic
154C8×C59#12311yescharacteristic
155C8×C59#12311yescharacteristic
156C8×C59#12411yescharacteristic
157#13448—591nomaximal, Sylow
158C22×C59—82no—
159C22×C59—82no—
160C22×C59—82no—
161C22×C59—161no—
162C22×C59—161no—
163C22×C59—161no—
164C22×C59—161no—
165C22×C59#45911yescharacteristic
166C22×C59#45911yescharacteristic
167C22×C59#46611yescharacteristic
168C22×C59C2×D1611yescharacteristic
169C22×C59#50811yescharacteristic
170C22×C59#50911yescharacteristic
171C22×C59#50911yescharacteristic
172C2×D59—82no—
173C2×D59—82no—
174C2×D59—82no—
175C2×D59—161no—
176C2×D59—161no—
177C2×D59—161no—
178C2×D59—161no—
179C4×C59—22no—
180C4×C59—41no—
181C4×C59—161no—
182C4×C59#39411yescharacteristic
183C4×C59#46311yescharacteristic
184C4×C59#46311yescharacteristic
185C4×C59#46911yescharacteristic
186Dic59—161no—
187C22×D16—591no—
188C2×C4⋊C16—591no—
189C2×#357—591no—
190C2×#357—591no—
191#1970—591no—
192#1970D5911yescharacteristic, complemented
193#1380—591no—
194C2×C59—162no—
195C2×C59#205411yescharacteristic
196C2×C59#220611yescharacteristic
197C2×C59#220611yescharacteristic
198C2×C59#221011yescharacteristic
199C2×C59#221011yescharacteristic
200C2×C59#221511yescharacteristic
201C2×C59#221811yescharacteristic
202D59—162no—
203C22×D8—591no—
204C22×D8C2×D5911yescharacteristic
205C2×D16—1182no—
206C2×D16—1182no—
207C2×D16—1182no—
208C22×C16—591no—
209C22×C16—591no—
210C2×#107—591no—
211C2×#107C2×D5911yescharacteristic
212C2×#102—21no—
213C2×#102—1181no—
214C2×C4×C8C2×D5911yescharacteristic
215#390—21no—
216#390—1181no—
217C4⋊C16—1182no—
218#357—1182no—
219#357—1182no—
220C59#1344811yescharacteristic, complemented, Sylow
221C22×D4—21no—
222C22×D4—1181no—
223C2×D8—21no—
224C2×D8—21no—
225C2×D8—21no—
226C2×D8—21no—
227C2×D8—21no—
228C2×D8—592no—
229C2×D8—1181no—
230C2×D8—1181no—
231C2×D8—1181no—
232C2×D8—1181no—
233C2×D8—1181no—
234C2×D8D23612yes—
235C22×C8C22⋊D5911yescharacteristic
236C22×C8C22×D5911yescharacteristic
237C2×C4⋊C4—21no—
238C2×C4⋊C4—1181no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—2361no—
242C2×C22⋊C4—2361no—
243C2×C42C22⋊D5911yescharacteristic
244D16—1188no—
245C2×C16—592no—
246C2×C16—1181no—
247C2×C16—1181no—
248C2×C16—1181no—
249C2×C16—1181no—
250C2×C16—1181no—
251#107—21no—
252#107—21no—
253#107—1181no—
254#107—1181no—
255#102—42no—
256#102—2362no—
257C4×C8—21no—
258C4×C8—21no—
259C24—41no—
260C24—2361no—
261C2×D4—22no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—1182no—
268C2×D4—2361no—
269C2×D4—2361no—
270C2×D4—2361no—
271C2×D4—2361no—
272C2×D4—2361no—
273C22×C4—41no—
274C22×C4—2361no—
275C22×C4#1174566911yescharacteristic
276C22×C4#1174566911yescharacteristic
277C22×C4C2×C22⋊D5911yescharacteristic
278D8—22no—
279D8—22no—
280D8—41no—
281D8—41no—
282D8—1182no—
283D8—1182no—
284D8—2361no—
285D8—2361no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8C2×D23611yescharacteristic
290C2×C8#1174568511yescharacteristic
291C2×C8D4×D5911yescharacteristic
292C2×C8D4×D5911yescharacteristic
293C2×C8#1174568711yescharacteristic
294C2×C8#1174568711yescharacteristic
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—2361no—
298C4⋊C4—2361no—
299C22⋊C4—82no—
300C22⋊C4—82no—
301C22⋊C4—4722no—
302C22⋊C4—4722no—
303C42—21no—
304C42—21no—
305C16—1182no—
306C16—2361no—
307C23—42no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—4721no—
315C23—4721no—
316C23—4721no—
317C23—4721no—
318C23—4721no—
319C23—4721no—
320C23—2362no—
321C23C2×#1174566911yescharacteristic
322D4—42no—
323D4—42no—
324D4—81no—
325D4—81no—
326D4—4721no—
327D4—4721no—
328D4—2362no—
329D4—2362no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—4721no—
339C2×C4—4721no—
340C2×C4—4721no—
341C2×C4#1427366812yescomplemented
342C2×C4#1427369811yescharacteristic
343C2×C4#1427369911yescharacteristic
344C2×C4#1427370111yescharacteristic
345C2×C4#1427374711yescharacteristic
346C2×C4#1427375211yescharacteristic
347C2×C4#1427375311yescharacteristic
348C8—41no—
349C8#1427369711yescharacteristic
350C8#1427370911yescharacteristic
351C8#1427371111yescharacteristic
352C8#1427371311yescharacteristic
353C22—82no—
354C22—82no—
355C22—82no—
356C22—161no—
357C22—161no—
358C22—161no—
359C22—161no—
360C22—9441no—
361C22—9441no—
362C22—9441no—
363C22—9441no—
364C22—4722no—
365C22—4722no—
366C22—4722no—
367C22#1683759411yescharacteristic
368C22#1683759711yescharacteristic
369C22#1683763611yescharacteristic
370C22C2×#1427366811yescharacteristic
371C22#1683783211yescharacteristic
372C22#1683783711yescharacteristic
373C22#1683783811yescharacteristic
374C4—22no—
375C4—41no—
376C4—161no—
377C4—9441no—
378C4#1683731811yescharacteristic
379C4#1683761411yescharacteristic
380C4#1683761711yescharacteristic
381C4#1683765311yescharacteristic
382C2—162no—
383C2—9442no—
384C2#2142540611yescharacteristic
385C2#2142612811yescharacteristic
386C2#2142613111yescharacteristic
387C2#2142614811yescharacteristic
388C2#2142615111yescharacteristic
389C2#2142618011yescharacteristic
390C2#2142619711yescharacteristic
subgroupG/H
centerC23C2×#11745669
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#1970C2
socleC23×C59C2×D8
prime radicalC22×#14273668C2
hypercenter#1970D59
hypocenterC59#13448
LCSG ▷ C22×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1970

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(C59⋊C16)C2semidirect
3. C2×#16837203C2semidirect
4. C2×#16837202C2semidirect
5. #21424962C2semidirect
6. #21422187C2semidirect
7. C59×#1970C2semidirect
138. C2×C4×C59D16semidirect
192. #1970D59semidirect
220. C59#13448semidirect
341. C2×C4#14273668semidirect

Maximal subgroups

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

subgroupindex
C22×#142736682
C2×C4⋊(C59⋊C16)2
C2×#168372032
C2×#168372022
#214249622
#214221872
C59×#19702
#1344859