Finite Group Databasebeta

← #28617765

Subgroups — #28617765

Order 15104.

874 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×#14273658C211yesmaximal, characteristic
2C2×#16837430C211yesmaximal, characteristic
3#21426720C211yesmaximal, characteristic
4#21426719C211yesmaximal, characteristic
5#21425600C214yesmaximal, complemented
6#21425599C214yesmaximal, complemented
7#21423816C211yesmaximal, characteristic
8#21423812C211yesmaximal, characteristic
9C59×#1736C211yesmaximal, characteristic
10C23×C59⋊C8C2211yescharacteristic
11C2×#14273686C2211yescharacteristic
12C2×#14273686C2211yescharacteristic
13C22×C4⋊Dic59C2211yescharacteristic
14C2×#14273658—24no—
15C2×#14273658C2212yes—
16C2×#14273658C2212yes—
17#16837430—24no—
18C2×Dic59⋊C8C2212yes—
19C2×Dic59⋊C8C2212yes—
20#16837301C2214yes—
21#16837298C2214yes—
22#16837277—22no—
23#16837275—22no—
24#16837272—21no—
25#16837271—21no—
26#16837268—21no—
27#16837267—21no—
28#16837165C2214yes—
29#16837164C2212yes—
30#16837164C2212yes—
31C22×M16×C59C2211yescharacteristic
32C2×#117×C59C2211yescharacteristic
33C22×C4⋊C4×C59C2211yescharacteristic
34C59×#340C2214yes—
35C23×Dic59—21no—
36C22×C59⋊C8—22no—
37C22×C59⋊C8—22no—
38C22×C59⋊C8D414yes—
39C22×C59⋊C8C2311yescharacteristic
40C22×C59⋊C8C2311yescharacteristic
41#14273686—22no—
42#14273686—22no—
43#14273686—22no—
44#14273686Q814yes—
45C2×C4⋊Dic59—22no—
46C2×C4⋊Dic59—22no—
47C2×C4⋊Dic59—21no—
48C2×C4⋊Dic59—24no—
49C2×C4⋊Dic59D411yescharacteristic
50C2×C4⋊Dic59D411yescharacteristic
51C2×C4⋊Dic59C2312yes—
52C2×C4⋊Dic59C2312yes—
53C2×Dic59⋊C4—24no—
54C2×C22⋊Dic59—21no—
55C2×C22⋊Dic59D412yes—
56C2×C4×Dic59—21no—
57C2×C4×Dic59C2312yes—
58#14273658—24no—
59#14273658—24no—
60Dic59⋊C8—24no—
61Dic59⋊C8—24no—
62C59⋊#95—24no—
63C59⋊#95—24no—
64C23×C4×C59—21no—
65C23×C4×C59C2311yescharacteristic
66C2×M16×C59—22no—
67C2×M16×C59—22no—
68C2×M16×C59—22no—
69C22×C8×C59C2312yes—
70#117×C59—22no—
71#117×C59Q814yes—
72C2×C4⋊C4×C59—22no—
73C2×C4⋊C4×C59—22no—
74C2×C4⋊C4×C59—21no—
75C2×C4⋊C4×C59C2312yes—
76C2×C4⋊C4×C59C2312yes—
77C2×C22⋊C4×C59—21no—
78C2×C42×C59—21no—
79C24×C59C2×D411yescharacteristic
80C22×C4×C59—22no—
81C22×C4×C59—22no—
82C22×C4×C59—42no—
83C22×C4×C59—82no—
84C22×C4×C59C2×D411yescharacteristic
85C22×C4×C59C2×D412yes—
86C22×C4×C59C2×Q812yes—
87C22×C4×C59C2×Q812yes—
88C22×C4×C59#4112yes—
89C22×C4×C59#4112yes—
90C22×C4×C59#4112yes—
91C22×C4×C59C2411yescharacteristic
92C22×Dic59—22no—
93C22×Dic59—22no—
94C22×Dic59—42no—
95C22×Dic59—82no—
96C22×Dic59C2×D412yes—
97C22×Dic59C2×D412yes—
98M16×C59—44no—
99C2×C8×C59—22no—
100C2×C8×C59—22no—
101C2×C8×C59—22no—
102C4⋊C4×C59—22no—
103C4⋊C4×C59—22no—
104C4⋊C4×C59—22no—
105C4⋊C4×C59—42no—
106C4⋊C4×C59C2×Q814yes—
107C22⋊C4×C59—24no—
108C42×C59—24no—
109C22⋊Dic59—24no—
110C22⋊Dic59—24no—
111C4⋊Dic59—22no—
112C4⋊Dic59—22no—
113C4⋊Dic59—22no—
114C4⋊Dic59—22no—
115C4⋊Dic59—22no—
116C4⋊Dic59—42no—
117C4⋊Dic59—84no—
118C4⋊Dic59C2×Q814yes—
119Dic59⋊C4—84no—
120C4×Dic59—24no—
121C4×Dic59—22no—
122C4×Dic59—22no—
123C2×C59⋊C8—28no—
124C2×C59⋊C8—21no—
125C2×C59⋊C8—21no—
126C2×C59⋊C8—21no—
127C2×C59⋊C8—21no—
128C2×C59⋊C8C2×D412yes—
129C2×C59⋊C8C2×D412yes—
130C23×C59—22no—
131C23×C59—22no—
132C23×C59#12112yes—
133C23×C59C22⋊Q812yes—
134C23×C59C22⋊Q812yes—
135C23×C59C22×D411yescharacteristic
136C2×C4×C59—24no—
137C2×C4×C59—42no—
138C2×C4×C59—42no—
139C2×C4×C59—42no—
140C2×C4×C59—44no—
141C2×C4×C59—84no—
142C2×C4×C59#12112yes—
143C2×C4×C59C22⋊Q812yes—
144C2×C4×C59C22⋊Q812yes—
145C2×C4×C59#12314yes—
146C2×C4×C59#12514yes—
147C2×C4×C59#12514yes—
148C2×C4×C59C4⋊Q814yes—
149C2×C4×C59C22×D411yescharacteristic
150C2×C4×C59C22×Q811yescharacteristic
151C2×C4×C59C22×Q811yescharacteristic
152C2×C4×C59C2×#4111yescharacteristic
153C2×C4×C59C2×#4111yescharacteristic
154C2×C4×C59C2×#4111yescharacteristic
155C2×Dic59—22no—
156C2×Dic59—22no—
157C2×Dic59—24no—
158C2×Dic59—24no—
159C2×Dic59—24no—
160C2×Dic59—42no—
161C2×Dic59—42no—
162C2×Dic59—42no—
163C2×Dic59—44no—
164C2×Dic59—84no—
165C8×C59—42no—
166C59⋊C8—22no—
167C59⋊C8—22no—
168#23668—591nomaximal, Sylow
169C22×C59—24no—
170C22×C59—24no—
171C22×C59—22no—
172C22×C59—22no—
173C22×C59#39814yes—
174C22×C59C2×#12111yescharacteristic
175C22×C59C2×C22⋊Q811yescharacteristic
176C22×C59C2×C22⋊Q811yescharacteristic
177C22×C59#57712yes—
178C22×C59#57712yes—
179C4×C59—44no—
180C4×C59—82no—
181C4×C59#39814yes—
182C4×C59C2×#12311yescharacteristic
183C4×C59C2×#12511yescharacteristic
184C4×C59C2×#12511yescharacteristic
185C4×C59C2×C4⋊Q811yescharacteristic
186Dic59—44no—
187Dic59—42no—
188Dic59—42no—
189Dic59—82no—
190C22×#106—591no—
191C2×#424—591no—
192#2317—591no—
193#2317—591no—
194#2088—594no—
195#2088—594no—
196#1736—591no—
197#1736D5911yescharacteristic
198#1735—591no—
199C2×C59—24no—
200C2×C59C2×#39811yescharacteristic
201C2×C59#305711yescharacteristic
202C2×C59#305711yescharacteristic
203C2×C59#308412yes—
204C2×C59#308412yes—
205C22×M16C2×D5911yescharacteristic
206C23×C8—591no—
207C2×#117—591no—
208C2×#117—591no—
209C2×#117C2×D5911yescharacteristic
210C22×C4⋊C4—591no—
211C22×C4⋊C4C2×D5911yescharacteristic
212C2×#106—592no—
213C2×#106—592no—
214C2×#106—1184no—
215#424—1184no—
216C2×C4⋊C8—592no—
217C2×C4⋊C8—592no—
218#389—594no—
219#389—594no—
220#385—1182no—
221#385—1182no—
222#384—1181no—
223#384—1181no—
224#382—1181no—
225#382—1181no—
226#340—594no—
227#340—592no—
228#340—592no—
229#340C2×D5914yes—
230C59#2366811yescharacteristic, complemented, Sylow
231C23×C4—21no—
232C23×C4—1181no—
233C23×C4C22×D5911yescharacteristic
234C2×M16—22no—
235C2×M16—22no—
236C2×M16—22no—
237C22×C8—594no—
238C22×C8—591no—
239C22×C8—591no—
240C22×C8—1182no—
241C22×C8—1182no—
242C22×C8C22×D5912yes—
243#117—22no—
244#117—594no—
245#117—1182no—
246#117—1182no—
247#117—1182no—
248#117Dic11814yes—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—592no—
253C2×C4⋊C4—591no—
254C2×C4⋊C4—591no—
255C2×C4⋊C4—592no—
256C2×C4⋊C4—1182no—
257C2×C4⋊C4—1182no—
258C2×C4⋊C4—1184no—
259C2×C4⋊C4—1184no—
260C2×C4⋊C4—1181no—
261C2×C4⋊C4C22×D5912yes—
262C2×C4⋊C4C22×D5912yes—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—592no—
265C2×C22⋊C4—1181no—
266C2×C42—21no—
267C2×C42—592no—
268C2×C42—1181no—
269#106—1184no—
270#106—1184no—
271C4⋊C8—1184no—
272C4⋊C8—1184no—
273#95—1184no—
274#95—1184no—
275C24D4×D5911yescharacteristic
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—82no—
280C22×C4—592no—
281C22×C4—592no—
282C22×C4—1182no—
283C22×C4—1182no—
284C22×C4—4722no—
285C22×C4—2362no—
286C22×C4C2×Dic11812yes—
287C22×C4#1174568512yes—
288C22×C4#1174568512yes—
289C22×C4D4×D5911yescharacteristic
290C22×C4D4×D5912yes—
291C22×C4#1174568712yes—
292C22×C4Q8×D5912yes—
293C22×C4C23×D5911yescharacteristic
294M16—44no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—592no—
299C2×C8—592no—
300C2×C8—1188no—
301C2×C8—1181no—
302C2×C8—1181no—
303C2×C8—1181no—
304C2×C8—1181no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—42no—
309C4⋊C4—594no—
310C4⋊C4—1182no—
311C4⋊C4—1182no—
312C4⋊C4—1182no—
313C4⋊C4—1182no—
314C4⋊C4—1182no—
315C4⋊C4—4724no—
316C4⋊C4—4724no—
317C4⋊C4—2362no—
318C4⋊C4C2×Dic11814yes—
319C22⋊C4—24no—
320C22⋊C4—1184no—
321C22⋊C4—1184no—
322C42—24no—
323C42—1184no—
324C42—1182no—
325C42—1182no—
326C23—22no—
327C23—22no—
328C23#1427370012yes—
329C23#1427370412yes—
330C23#1427370512yes—
331C23C2×D4×D5911yescharacteristic
332C2×C4—24no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—84no—
338C2×C4—1182no—
339C2×C4—1182no—
340C2×C4—1184no—
341C2×C4—1184no—
342C2×C4—1184no—
343C2×C4—4724no—
344C2×C4—2362no—
345C2×C4—2364no—
346C2×C4—2362no—
347C2×C4—2362no—
348C2×C4#1427370012yes—
349C2×C4#1427370412yes—
350C2×C4#1427370512yes—
351C2×C4#1427370914yes—
352C2×C4#1427371714yes—
353C2×C4#1427371914yes—
354C2×C4#1427372614yes—
355C2×C4C2×D4×D5911yescharacteristic
356C2×C4C2×Q8×D5911yescharacteristic
357C2×C4C22×Dic11811yescharacteristic
358C2×C4C2×#1174568711yescharacteristic
359C2×C4C2×#1174568511yescharacteristic
360C2×C4C2×#1174568511yescharacteristic
361C8—42no—
362C8—1182no—
363C8—1182no—
364C22—24no—
365C22—24no—
366C22—22no—
367C22—22no—
368C22#1683733614yes—
369C22C2×#1427370011yescharacteristic
370C22C2×#1427370411yescharacteristic
371C22C2×#1427370511yescharacteristic
372C22#1683821112yes—
373C22#1683821212yes—
374C4—44no—
375C4—82no—
376C4—4722no—
377C4—2364no—
378C4—2362no—
379C4—2362no—
380C4#1683733614yes—
381C4C2×#1427370911yescharacteristic
382C4C2×#1427371711yescharacteristic
383C4C2×#1427371911yescharacteristic
384C4C2×#1427372611yescharacteristic
385C2—24no—
386C2C2×#1683733611yescharacteristic
387C2#2143373411yescharacteristic
388C2#2143373811yescharacteristic
389C2#2143395712yes—
390C2#2143396212yes—
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1736C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1736D59
hypocenterC59#23668
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
5. #21425600C2semidirect
6. #21425599C2semidirect
230. C59#23668semidirect

Maximal subgroups

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

subgroupindex
C22×#142736582
C2×#168374302
#214267202
#214267192
#214256002
#214255992
#214238162
#214238122
C59×#17362
#2366859