Finite Group Databasebeta

← #28508287

Subgroups — #28508287

Order 15104.

446 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273670C211yesmaximal, characteristic
2C2×Dic59⋊C16C211yesmaximal, characteristic, complemented
3C2×#16837204C211yesmaximal, characteristic
4#21424960C211yesmaximal, characteristic
5#21424958C211yesmaximal, characteristic, complemented
6#21422187C211yesmaximal, characteristic, complemented
7C2×C59×#358C211yesmaximal, characteristic, complemented
8C22×D472C2211yescharacteristic
9C2×#14273670—22no—
10C2×#14273670—22no—
11C2×#14273670—22no—
12C22×C59⋊C16C2211yescharacteristic
13C2×#14273661C2211yescharacteristic
14C2×#14273651—21no—
15C2×#14273647—21no—
16C2×C8×Dic59C2211yescharacteristic
17#16837309—21no—
18#16837303—21no—
19Dic59⋊C16—22no—
20#16837204—22no—
21C22×Q16×C59C2211yescharacteristic
22C22×C16×C59C2211yescharacteristic
23C2×#107×C59C2211yescharacteristic
24C59×#358—22no—
25C22×D236—21no—
26C2×D472—21no—
27C2×D472—21no—
28C2×D472—21no—
29C2×D472—21no—
30C2×D472—21no—
31C2×D472D412yes—
32C22×C59⋊C8D411yescharacteristic
33C2×C4⋊Dic59—21no—
34C2×(C2×D59)⋊C4—41no—
35C2×(C2×D59)⋊C4—41no—
36C2×C4×Dic59D411yescharacteristic
37#14273670—28no—
38C2×C59⋊C16—21no—
39C2×C59⋊C16—21no—
40C2×C59⋊C16D412yes—
41#14273661—21no—
42#14273661—21no—
43#14273651—42no—
44#14273647—42no—
45C8×Dic59—21no—
46C8×Dic59—21no—
47C59⋊#95—41no—
48C59⋊#95—41no—
49C22×Q8×C59—21no—
50C2×Q16×C59—21no—
51C2×Q16×C59—21no—
52C2×Q16×C59—21no—
53C2×Q16×C59—21no—
54C2×Q16×C59—21no—
55C2×Q16×C59D412yes—
56C22×C8×C59C2311yescharacteristic
57C2×C4⋊C4×C59—21no—
58C2×C16×C59—21no—
59C2×C16×C59—21no—
60C2×C16×C59—21no—
61#107×C59—21no—
62#107×C59—21no—
63C23×D59—41no—
64C2×Q8×C59—22no—
65C2×Q8×C59—41no—
66C2×Q8×C59—41no—
67C2×Q8×C59—41no—
68C2×Q8×C59—41no—
69C2×Q8×C59—41no—
70C22×C4×C59—41no—
71C22×C4×C59—41no—
72C22×C4×C59C2×D411yescharacteristic
73C22×Dic59—41no—
74C22×Dic59D811yescharacteristic
75C22×Dic59D811yescharacteristic
76C2×D236—22no—
77C2×D236—41no—
78C2×D236—41no—
79C2×D236—41no—
80C2×D236—41no—
81C2×D236—41no—
82Q16×C59—22no—
83Q16×C59—22no—
84Q16×C59—41no—
85Q16×C59—41no—
86C2×C8×C59C2×D411yescharacteristic
87C2×C8×C59C2×D411yescharacteristic
88C2×C8×C59C2×D411yescharacteristic
89C2×C8×C59#4111yescharacteristic
90C2×C8×C59#4111yescharacteristic
91C2×C8×C59#4111yescharacteristic
92C4⋊C4×C59—41no—
93C4⋊C4×C59—41no—
94(C2×D59)⋊C4—82no—
95(C2×D59)⋊C4—82no—
96C4⋊Dic59—41no—
97C4⋊Dic59—41no—
98C4×Dic59—21no—
99C4×Dic59—21no—
100C2×C59⋊C8—21no—
101C2×C59⋊C8—21no—
102C2×C59⋊C8—21no—
103D472—22no—
104D472—22no—
105D472—41no—
106D472—41no—
107C16×C59—41no—
108C59⋊C16—22no—
109C23×C59C2×D811yescharacteristic
110C22×D59—42no—
111C22×D59—81no—
112C22×D59—81no—
113C22×D59—81no—
114C22×D59—81no—
115C22×D59—81no—
116C22×D59—81no—
117Q8×C59—42no—
118Q8×C59—42no—
119Q8×C59—81no—
120Q8×C59—81no—
121C2×C4×C59—42no—
122C2×C4×C59—81no—
123C2×C4×C59—81no—
124C2×C4×C59—81no—
125C2×C4×C59—81no—
126C2×C4×C59—81no—
127C2×C4×C59#12111yescharacteristic
128C2×C4×C59#12111yescharacteristic
129C2×C4×C59#12111yescharacteristic
130C2×C4×C59#13511yescharacteristic
131C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
133C2×Dic59—21no—
134C2×Dic59—21no—
135C2×Dic59—21no—
136C2×Dic59—21no—
137C2×Dic59—21no—
138C2×Dic59—81no—
139C2×Dic59—81no—
140C2×Dic59—81no—
141C2×Dic59SD3212yescomplemented
142D236—42no—
143D236—42no—
144D236—81no—
145D236—81no—
146C8×C59#12011yescharacteristic
147C8×C59#12311yescharacteristic
148C8×C59#12311yescharacteristic
149C8×C59#12411yescharacteristic
150C59⋊C8—41no—
151#13449—591nomaximal, Sylow
152C22×C59#45911yescharacteristic
153C22×C59#45911yescharacteristic
154C22×C59#46611yescharacteristic
155C22×C59C2×SD3211yescharacteristic
156C22×C59#50811yescharacteristic
157C22×C59#50911yescharacteristic
158C22×C59#51011yescharacteristic
159C2×D59—82no—
160C2×D59—82no—
161C2×D59—82no—
162C2×D59—161no—
163C2×D59—161no—
164C2×D59—161no—
165C2×D59—161no—
166C4×C59—82no—
167C4×C59—161no—
168C4×C59#39411yescharacteristic
169C4×C59#46311yescharacteristic
170C4×C59#46311yescharacteristic
171C4×C59#46911yescharacteristic
172Dic59—22no—
173Dic59—41no—
174Dic59—161no—
175C22×SD32—591no—
176C2×C4⋊C16—591no—
177C2×#358D5911yescharacteristic, complemented
178C2×#357—591no—
179#1970—591no—
180#1969—591no—
181#1380—591no—
182C2×C59#205411yescharacteristic
183C2×C59#220711yescharacteristic
184C2×C59#220811yescharacteristic
185C2×C59#221011yescharacteristic
186C2×C59#221111yescharacteristic
187C2×C59#221711yescharacteristic
188C2×C59#222011yescharacteristic
189D59—162no—
190C22×Q16C2×D5911yescharacteristic
191C22×D8—591no—
192C2×SD32—1182no—
193C2×SD32—1182no—
194C2×SD32—1182no—
195C22×C16—591no—
196C22×C16C2×D5911yescharacteristic
197C2×#107—591no—
198C2×#107C2×D5911yescharacteristic
199C2×#103—1181no—
200C2×#102—1181no—
201C2×C4×C8—591no—
202#391—1181no—
203#390—1181no—
204C4⋊C16—1182no—
205#358—22no—
206#357—1182no—
207C59#1344911yescharacteristic, complemented, Sylow
208C22×Q8—21no—
209C22×D4—1181no—
210C2×Q16—21no—
211C2×Q16—21no—
212C2×Q16—21no—
213C2×Q16—21no—
214C2×Q16—21no—
215C2×Q16C22⋊D5912yes—
216C2×D8—592no—
217C2×D8—1181no—
218C2×D8—1181no—
219C2×D8—1181no—
220C2×D8—1181no—
221C2×D8—1181no—
222C22×C8—591no—
223C22×C8C22×D5911yescharacteristic
224C2×C4⋊C4—21no—
225C2×C4⋊C4—1181no—
226C2×C22⋊C4—2361no—
227C2×C22⋊C4—2361no—
228C2×C42—591no—
229SD32—1188no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—592no—
234C2×C16—1181no—
235C2×C16—1181no—
236#107—21no—
237#107—21no—
238#107—1181no—
239#107—1181no—
240#103—2362no—
241#102—2362no—
242C4×C8—1181no—
243C4×C8—1181no—
244#95—2361no—
245#95—2361no—
246C24—2361no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×D4—1182no—
254C2×D4—2361no—
255C2×D4—2361no—
256C2×D4—2361no—
257C2×D4—2361no—
258C2×D4—2361no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—591no—
262C22×C4—591no—
263C22×C4—2361no—
264C22×C4D4×D5911yescharacteristic
265Q16—22no—
266Q16—22no—
267Q16—41no—
268Q16—41no—
269D8—1182no—
270D8—1182no—
271D8—2361no—
272D8—2361no—
273C2×C8—1181no—
274C2×C8—1181no—
275C2×C8—1181no—
276C2×C8#1174568511yescharacteristic
277C2×C8#1174568511yescharacteristic
278C2×C8D4×D5911yescharacteristic
279C2×C8D4×D5911yescharacteristic
280C2×C8#1174568711yescharacteristic
281C2×C8C2×C22⋊D5911yescharacteristic
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—2361no—
285C4⋊C4—2361no—
286C22⋊C4—4722no—
287C22⋊C4—4722no—
288C42—1181no—
289C42—1181no—
290C16—41no—
291C16—1182no—
292C23—4721no—
293C23—4721no—
294C23—4721no—
295C23—4721no—
296C23—4721no—
297C23—4721no—
298C23—2362no—
299C23D8×D5911yescharacteristic
300Q8—42no—
301Q8—42no—
302Q8—81no—
303Q8—81no—
304D4—4721no—
305D4—4721no—
306D4—2362no—
307D4—2362no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—592no—
315C2×C4—1181no—
316C2×C4—1181no—
317C2×C4—1181no—
318C2×C4—1181no—
319C2×C4—1181no—
320C2×C4—4721no—
321C2×C4—4721no—
322C2×C4—4721no—
323C2×C4#1427369911yescharacteristic
324C2×C4#1427370011yescharacteristic
325C2×C4#1427370011yescharacteristic
326C2×C4#1427374811yescharacteristic
327C2×C4#1427375111yescharacteristic
328C2×C4#1427375411yescharacteristic
329C8—2361no—
330C8#1427369511yescharacteristic
331C8#1427370911yescharacteristic
332C8#1427371011yescharacteristic
333C8#1427371311yescharacteristic
334C22—9441no—
335C22—9441no—
336C22—9441no—
337C22—9441no—
338C22—4722no—
339C22—4722no—
340C22—4722no—
341C22#1683759511yescharacteristic
342C22#1683759611yescharacteristic
343C22#1683763711yescharacteristic
344C22SD32×D5911yescharacteristic
345C22#1683783511yescharacteristic
346C22#1683783911yescharacteristic
347C22#1683784411yescharacteristic
348C4—82no—
349C4—161no—
350C4—1182no—
351C4—9441no—
352C4—2361no—
353C4#1683731711yescharacteristic
354C4#1683761611yescharacteristic
355C4#1683761911yescharacteristic
356C4#1683765411yescharacteristic
357C2—9442no—
358C2#2142540511yescharacteristic
359C2#2142613411yescharacteristic
360C2#2142614011yescharacteristic
361C2#2142615411yescharacteristic
362C2#2142616011yescharacteristic
363C2#2142619511yescharacteristic
364C2#2142621211yescharacteristic
subgroupG/H
centerC23D8×D59
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC2×C59×#358C2
socleC23×C59C2×D8
prime radicalC22×D472C22
hypercenterC2×#358D59
hypocenterC59#13449
LCSG ▷ C22×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
2. C2×Dic59⋊C16C2semidirect
5. #21424958C2semidirect
6. #21422187C2semidirect
7. C2×C59×#358C2semidirect
141. C2×Dic59SD32semidirect
177. C2×#358D59semidirect
207. C59#13449semidirect

Maximal subgroups

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

subgroupindex
C22×#142736702
C2×Dic59⋊C162
C2×#168372042
#214249602
#214249582
#214221872
C2×C59×#3582
#1344959