Finite Group Databasebeta

← #28483708

Subgroups — #28483708

Order 15104.

972 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 972 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837319C211yesmaximal, characteristic
2#21424026C212yesmaximal
3C2×#16837128C211yesmaximal, characteristic, complemented
4#21421974C212yesmaximal, complemented
5C59×#2316C211yesmaximal, characteristic, complemented
6C23×D236C2211yescharacteristic
7C22×(C2×D59)⋊C4—21no—
8C22×(C2×D59)⋊C4—21no—
9C2×#14273647—24no—
10C2×C22⋊(C59⋊C8)C2212yes—
11#16837319—22no—
12#16837319C414yescomplemented
13C2×C59⋊#95C2211yescharacteristic
14#16837128—22no—
15C2×#117×C59—21no—
16C22×C4⋊C4×C59C2211yescharacteristic
17C59×#389—24no—
18C59×#380—21no—
19C59×#380C2212yes—
20C24×D59—21no—
21C22×D236—22no—
22C22×D236—22no—
23C22×D236—21no—
24C22×D236—44no—
25C22×D236C2×C412yes—
26C23×Dic59—21no—
27C22×C59⋊C8—22no—
28C2×(C2×D59)⋊C4—24no—
29C2×(C2×D59)⋊C4—24no—
30C2×(C2×D59)⋊C4—44no—
31C2×(C2×D59)⋊C4—42no—
32C2×(C2×D59)⋊C4—42no—
33#14273647—48no—
34C22⋊(C59⋊C8)—22no—
35C22⋊(C59⋊C8)—22no—
36C59⋊#95—21no—
37C59⋊#95C2×C412yes—
38C23×C4×C59D411yescharacteristic
39C23×C4×C59D411yescharacteristic
40C23×C4×C59C2311yescharacteristic
41#117×C59—42no—
42#117×C59—42no—
43C2×C4⋊C4×C59—22no—
44C2×C4⋊C4×C59—22no—
45C2×C4⋊C4×C59—22no—
46C2×C4⋊C4×C59—22no—
47C2×C4⋊C4×C59C2×C412yes—
48C2×C4⋊C4×C59D412yes—
49C2×C4⋊C4×C59D414yes—
50C2×C22⋊C4×C59—22no—
51C2×C42×C59—22no—
52#95×C59—22no—
53#95×C59—22no—
54C24×C59C2×D411yescharacteristic
55C23×D59—24no—
56C23×D59—22no—
57C23×D59—42no—
58C23×D59—42no—
59C23×D59—42no—
60C23×D59—41no—
61C23×D59—41no—
62C23×D59—41no—
63C23×D59—41no—
64C23×D59—41no—
65C23×D59—41no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—24no—
73C22×C4×C59—42no—
74C22×C4×C59—42no—
75C22×C4×C59C22⋊C412yes—
76C22×C4×C59C22⋊C412yes—
77C22×C4×C59D812yes—
78C22×C4×C59SD1612yes—
79C22×C4×C59C22×C411yescharacteristic
80C22×C4×C59C2×D412yes—
81C22×C4×C59C2×D411yescharacteristic
82C22×C4×C59#4112yes—
83C22×Dic59—22no—
84C22×Dic59—42no—
85C22×Dic59—42no—
86C22×Dic59—41no—
87C22×Dic59—41no—
88C2×D236—28no—
89C2×D236—24no—
90C2×D236—44no—
91C2×D236—48no—
92C2×D236—44no—
93C2×D236—44no—
94C2×D236—42no—
95C4⋊C4×C59—24no—
96C4⋊C4×C59—24no—
97C4⋊C4×C59—44no—
98C4⋊C4×C59—44no—
99C4⋊C4×C59—82no—
100C22⋊C4×C59—42no—
101C22⋊C4×C59—42no—
102C42×C59—44no—
103(C2×D59)⋊C4—48no—
104(C2×D59)⋊C4—48no—
105C2×C59⋊C8—44no—
106C2×C59⋊C8—42no—
107C23×C59—22no—
108C23×C59—22no—
109C23×C59C2×C22⋊C411yescharacteristic
110C23×C59#12011yescharacteristic
111C23×C59#12311yescharacteristic
112C23×C59C2×D811yescharacteristic
113C23×C59C2×SD1611yescharacteristic
114C23×C59AGL(1,ℤ/8)11yescharacteristic
115C23×C59#13711yescharacteristic
116C22×D59—24no—
117C22×D59—42no—
118C22×D59—44no—
119C22×D59—44no—
120C22×D59—44no—
121C22×D59—44no—
122C22×D59—44no—
123C22×D59—42no—
124C22×D59—42no—
125C22×D59—42no—
126C22×D59—42no—
127C22×D59—42no—
128C22×D59—44no—
129C22×D59—82no—
130C22×D59—82no—
131C22×D59—82no—
132C22×D59—82no—
133C22×D59—82no—
134C22×D59—81no—
135C22×D59—81no—
136C22×D59—82no—
137C22×D59—81no—
138C22×D59—81no—
139C2×C4×C59—22no—
140C2×C4×C59—22no—
141C2×C4×C59—22no—
142C2×C4×C59—22no—
143C2×C4×C59—24no—
144C2×C4×C59—22no—
145C2×C4×C59—22no—
146C2×C4×C59—22no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—42no—
150C2×C4×C59—42no—
151C2×C4×C59—42no—
152C2×C4×C59—44no—
153C2×C4×C59—44no—
154C2×C4×C59—44no—
155C2×C4×C59—81no—
156C2×C4×C59#10214yescomplemented
157C2×C4×C59C4×D412yes—
158C2×C4×C59#12112yes—
159C2×Dic59—42no—
160C2×Dic59—42no—
161C2×Dic59—42no—
162C2×Dic59—82no—
163C2×Dic59—82no—
164D236—28no—
165D236—48no—
166D236—44no—
167C59⋊C8—82no—
168#9019—591nomaximal, Sylow
169C22×C59—22no—
170C22×C59—22no—
171C22×C59—22no—
172C22×C59—22no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—21no—
176C22×C59—21no—
177C22×C59#38611yescharacteristic
178C22×C59C2×#10211yescharacteristic
179C22×C59#41711yescharacteristic
180C22×C59#44711yescharacteristic
181C22×C59#44811yescharacteristic
182C22×C59#45412yes—
183C22×C59B(4,𝔽2)12yes—
184C22×C59#48011yescharacteristic
185C22×C59#48111yescharacteristic
186C2×D59—44no—
187C2×D59—44no—
188C2×D59—44no—
189C2×D59—82no—
190C2×D59—82no—
191C2×D59—84no—
192C2×D59—84no—
193C2×D59—84no—
194C2×D59—84no—
195C2×D59—82no—
196C2×D59—82no—
197C4×C59—24no—
198C4×C59—22no—
199C4×C59—22no—
200C4×C59—42no—
201C4×C59—84no—
202Dic59—82no—
203C2×#394—591no—
204#2316D5911yescharacteristic, complemented
205#1786—592no—
206C2×#324—591no—
207#1294—592no—
208C2×C59—21no—
209C2×C59—21no—
210C2×C59—21no—
211C2×C59—21no—
212C2×C59#159812yes—
213C2×C59#159912yes—
214C2×C59#187411yescharacteristic
215C2×C59#188811yescharacteristic
216C2×C59#188911yescharacteristic
217D59—84no—
218C23×D4—591no—
219C2×#117—21no—
220C22×C4⋊C4C2×D5911yescharacteristic
221C22×C22⋊C4—1181no—
222C22×C22⋊C4—1181no—
223C2×#102—1184no—
224C2×C22⋊C8—592no—
225#394—594no—
226#394—1182no—
227#389—24no—
228#380—21no—
229#380C2×D5912yes—
230C2×#95—591no—
231#324—1182no—
232C59#901911yescharacteristic, complemented, Sylow
233C25—1181no—
234C22×D4—592no—
235C22×D4—1181no—
236C22×D4—1182no—
237C22×D4—1182no—
238C22×D4—2364no—
239C23×C4—1181no—
240C23×C4C22⋊D5911yescharacteristic
241C23×C4C22⋊D5911yescharacteristic
242C23×C4C22×D5911yescharacteristic
243C22×C8—1182no—
244#117—42no—
245#117—42no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C4×D5912yes—
251C2×C4⋊C4D23612yes—
252C2×C4⋊C4C22⋊D5914yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—1184no—
255C2×C22⋊C4—1184no—
256C2×C22⋊C4—2364no—
257C2×C22⋊C4—2362no—
258C2×C22⋊C4—2362no—
259C2×C42—22no—
260#102—2368no—
261C22⋊C8—1182no—
262C22⋊C8—1182no—
263#95—22no—
264#95—22no—
265#95—592no—
266#95—1181no—
267C24—1184no—
268C24—1182no—
269C24—2362no—
270C24—2362no—
271C24—2362no—
272C24—2361no—
273C24—2361no—
274C24—2361no—
275C24—2361no—
276C24—2361no—
277C24—2361no—
278C24C2×C22⋊D5911yescharacteristic
279C2×D4—1188no—
280C2×D4—1184no—
281C2×D4—2364no—
282C2×D4—2364no—
283C2×D4—2368no—
284C2×D4—2364no—
285C2×D4—2362no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—24no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—1182no—
296C22×C4—2362no—
297C22×C4—2362no—
298C22×C4—2361no—
299C22×C4—2361no—
300C22×C4(C2×D59)⋊C412yes—
301C22×C4(C2×D59)⋊C412yes—
302C22×C4#1174566912yes—
303C22×C4#1174567112yes—
304C22×C4C2×C4×D5911yescharacteristic
305C22×C4C2×D23611yescharacteristic
306C22×C4#1174568512yes—
307C22×C4C2×C22⋊D5912yes—
308C2×C8—2364no—
309C2×C8—2362no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—82no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—2368no—
318C22⋊C4—2368no—
319C42—44no—
320C23—22no—
321C23—22no—
322C23—1184no—
323C23—4722no—
324C23—4722no—
325C23—4722no—
326C23—4722no—
327C23—4722no—
328C23—4721no—
329C23—4721no—
330C23—4721no—
331C23—4721no—
332C23—4722no—
333C23—2362no—
334C23—2364no—
335C23—2362no—
336C23—2362no—
337C23—2364no—
338C23—2364no—
339C23—2364no—
340C23—2364no—
341C23—2362no—
342C23—2362no—
343C23—2362no—
344C23—2364no—
345C23C2×(C2×D59)⋊C411yescharacteristic
346C23#1427369611yescharacteristic
347C23#1427371011yescharacteristic
348C23C2×#1174566911yescharacteristic
349C23C2×#1174567111yescharacteristic
350C23#1427375311yescharacteristic
351C23#1427375811yescharacteristic
352D4—1188no—
353D4—2368no—
354D4—2364no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—81no—
372C2×C4—4722no—
373C2×C4—4722no—
374C2×C4—2362no—
375C2×C4—2362no—
376C2×C4—2362no—
377C2×C4#1427364714yescomplemented
378C2×C4C4×C22⋊D5912yes—
379C2×C4#1427370112yes—
380C8—4722no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—4722no—
390C22—4724no—
391C22—4724no—
392C22—4722no—
393C22—4722no—
394C22—4722no—
395C22—4724no—
396C22—4724no—
397C22—2364no—
398C22—2364no—
399C22—2364no—
400C22#1683728111yescharacteristic
401C22C2×#1427364711yescharacteristic
402C22#1683740111yescharacteristic
403C22#1683753611yescharacteristic
404C22#1683754111yescharacteristic
405C22#1683757512yes—
406C22#1683758912yes—
407C22#1683771511yescharacteristic
408C22#1683772011yescharacteristic
409C4—24no—
410C4—22no—
411C4—22no—
412C4—42no—
413C4—84no—
414C4—4722no—
415C2—21no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2—4724no—
420C2#2142303112yes—
421C2#2142303612yes—
422C2#2142441311yescharacteristic
423C2#2142449911yescharacteristic
424C2#2142450411yescharacteristic
subgroupG/H
centerC23#14273696
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2316C2
socleC23×C59#120
prime radicalC23×D236C22
cosocleC23×C4×C59C23
hypercenter#2316D59
hypocenterC59#9019
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
3. C2×#16837128C2semidirect
4. #21421974C2semidirect
5. C59×#2316C2semidirect
12. #16837319C4semidirect
156. C2×C4×C59#102semidirect
204. #2316D59semidirect
232. C59#9019semidirect
377. C2×C4#14273647semidirect

Maximal subgroups

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

subgroupindex
C2×#168373192
#214240262
C2×#168371282
#214219742
C59×#23162
#901959