Finite Group Databasebeta

← #29458016

Subgroups — #29458016

Order 15104.

2014 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×#16837978C211yesmaximal, characteristic
2#21434485C214yesmaximal, complemented
3#21434478C214yesmaximal, complemented
4#21434466C2116yesmaximal, complemented
5#21432613C212yesmaximal
6C2×#16837410C212yesmaximal, complemented
7#21432515C211yesmaximal, characteristic, complemented
8C59×#3458C211yesmaximal, characteristic, complemented
9C22×#11745685C2211yescharacteristic
10C23×Dic118C2211yescharacteristic
11C22×#11745664C2212yes—
12#16837978—24no—
13#16837978C2218yes—
14C2×#14273727C2212yes—
15C2×#14273715C2212yes—
16C2×#14273711—22no—
17C2×#14273707—22no—
18#16837772C22116yescomplemented
19#16837764C22116yescomplemented
20#16837759C22116yescomplemented
21#16837754C22116yescomplemented
22#16837410C2218yes—
23#16837410C2214yes—
24#16837410C2218yescomplemented
25#16837406C2214yes—
26#16837401C2218yes—
27#16837396C2218yes—
28C2×#14273650C2218yescomplemented
29#16837358C2218yescomplemented
30C4×#11745664C2214yes—
31C59×#544—22no—
32C59×#544C2218yescomplemented
33C59×#543—22no—
34C59×#533—24no—
35C59×#533C2214yes—
36C2×#117×C59C2212yes—
37C2×#117×C59C2211yescharacteristic
38C2×#11745685—22no—
39C2×#11745685—28no—
40C2×#11745685C2314yes—
41C22×Dic118—28no—
42C22×Dic118—21no—
43C22×Dic118—22no—
44C22×Dic118C2314yes—
45C22×Dic118C2312yes—
46C22×Dic118C2311yescharacteristic
47C22×D236C2311yescharacteristic
48C22×C22⋊D59—21no—
49C23×Dic59—21no—
50C22×C4×D59—21no—
51C2×#11745664—28no—
52C2×#11745664C2314yes—
53C2×#11745664C2314yes—
54C22×C59⋊C8C2314yes—
55#14273727—24no—
56#14273727C2318yes—
57#14273715—24no—
58#14273715C2318yes—
59#14273711—216no—
60#14273707—216no—
61C2×C4⋊Dic59—24no—
62C2×(C2×D59)⋊C4—24no—
63#14273650C23132yes—
64#14273650C23116yes—
65#14273647C23116yes—
66#14273638C2318yes—
67C4×C59⋊C8C2318yes—
68C23×C4×C59C2311yescharacteristic
69C4⋊Q8×C59—28no—
70C4⋊Q8×C59C2318yes—
71#125×C59—28no—
72#125×C59C2318yes—
73C22⋊Q8×C59—28no—
74C22⋊Q8×C59—216no—
75C4×Q8×C59—28no—
76#117×C59—22no—
77#117×C59—22no—
78#117×C59D414yes—
79#117×C59C2314yes—
80#117×C59C2314yes—
81#117×C59C2314yes—
82C2×C4⋊C4×C59D412yes—
83C2×C4⋊C4×C59C2314yes—
84C2×C22⋊C4×C59—22no—
85C2×C22⋊C4×C59D412yes—
86C2×C42×C59—22no—
87C2×C42×C59C2312yes—
88C24×C59C2×D411yescharacteristic
89C23×D59—21no—
90C2×Q8×C59—28no—
91C22×C4×C59—24no—
92C22×C4×C59C2×D414yes—
93C22×C4×C59C2×D412yes—
94C22×C4×C59C2×D411yescharacteristic
95C22×C4×C59#4118yes—
96C22×C4×C59C2411yescharacteristic
97C22×C4×C59C2412yes—
98C2×C22⋊D59—24no—
99C2×C22⋊D59—44no—
100C22×Dic59—22no—
101C22×Dic59—21no—
102C22×Dic59—24no—
103C22×Dic59—44no—
104#11745685—28no—
105#11745685—28no—
106#11745685—48no—
107C2×D236—24no—
108C2×D236—21no—
109C2×D236C2412yes—
110C2×C4×D59—24no—
111C2×C4×D59—44no—
112C2×Dic118—28no—
113C2×Dic118—28no—
114C2×Dic118—28no—
115C2×Dic118—24no—
116C2×Dic118—24no—
117C2×Dic118—22no—
118C2×Dic118—21no—
119C2×Dic118—44no—
120C2×Dic118—48no—
121C2×Dic118C2414yes—
122C2×Dic118C2412yes—
123C4⋊C4×C59—24no—
124C4⋊C4×C59—28no—
125C4⋊C4×C59—216no—
126C4⋊C4×C59C2×D418yes—
127C4⋊C4×C59C2418yes—
128C22⋊C4×C59—24no—
129C22⋊C4×C59—24no—
130C22⋊C4×C59C2×D418yes—
131C42×C59—24no—
132C42×C59—24no—
133C42×C59C2×D414yes—
134C42×C59C2414yes—
135(C2×D59)⋊C4—216no—
136C4⋊Dic59—216no—
137#11745664—28no—
138#11745664—28no—
139C2×C59⋊C8—28no—
140C2×C59⋊C8C2418yes—
141C23×C59—42no—
142C23×C59#12414yes—
143C23×C59C22×D411yescharacteristic
144C23×C59C22×D412yes—
145C22×D59—22no—
146C22×D59—44no—
147C22×D59—42no—
148Q8×C59—88no—
149C2×C4×C59—28no—
150C2×C4×C59—44no—
151C2×C4×C59—84no—
152C2×C4×C59#12414yes—
153C2×C4×C59#12418yes—
154C2×C4×C59C22×D418yes—
155C2×C4×C59C22×D411yescharacteristic
156C2×C4×C59C22×D412yes—
157C2×C4×C59C2×#4114yes—
158C2×C4×C59C2×#4114yes—
159C2×C4×C59C2×#4114yes—
160C2×C4×C59C2511yescharacteristic
161C22⋊D59—48no—
162C2×Dic59—22no—
163C2×Dic59—24no—
164C2×Dic59—44no—
165C2×Dic59—48no—
166C2×Dic59—42no—
167D236—24no—
168D236—42no—
169C4×D59—48no—
170Dic118—28no—
171Dic118—24no—
172Dic118—44no—
173Dic118—48no—
174Dic118—42no—
175C59⋊C8—28no—
176#60620—591nomaximal, Sylow
177C22×C59—22no—
178C22×C59—24no—
179C22×C59—44no—
180C22×C59C2×#12412yes—
181C22×C59C2×#12414yes—
182C22×C59C23×D411yescharacteristic
183C2×D59—44no—
184C2×D59—42no—
185C2×D59—84no—
186C4×C59—28no—
187C4×C59—48no—
188C4×C59C2×#12412yes—
189C4×C59C2×#12414yes—
190C4×C59C22×#4112yes—
191Dic59—44no—
192Dic59—42no—
193#3458D5911yescharacteristic, complemented
194C2×#536—591no—
195#3140—594no—
196#3139—594no—
197#3137—5916no—
198#2896—592no—
199C2×#419—592no—
200#2873—591no—
201C2×C59—22no—
202C2×C59—24no—
203C2×C59C22×#12411yescharacteristic
204C2×C59#358612yes—
205D59—82no—
206C22×#41—591no—
207C23×Q8—591no—
208C22×M16—592no—
209#544—22no—
210#544C2×D5918yescomplemented
211#543—22no—
212#536—598no—
213#536—1184no—
214#533—24no—
215#533C2×D5914yes—
216C2×C4⋊Q8—592no—
217C2×#124—592no—
218C2×#123—1182no—
219C2×C22⋊Q8—1182no—
220C2×#117C2×D5912yes—
221C2×#117C2×D5911yescharacteristic
222#491—5916no—
223#489—5916no—
224#488—5916no—
225#487—5916no—
226#419—594no—
227#419—598no—
228#419—598no—
229#418—594no—
230#417—598no—
231#416—598no—
232C2×#103—598no—
233#405—598no—
234C4×M16—594no—
235C59#6062011yescharacteristic, complemented, Sylow
236C2×#41—594no—
237C2×#41—1182no—
238C2×#41—1188no—
239C22×Q8—592no—
240C22×Q8—591no—
241C22×Q8—594no—
242C22×Q8—1181no—
243C22×Q8—1182no—
244C22×Q8—1188no—
245C22×D4—591no—
246C22×D4—1181no—
247C23×C4—1181no—
248C23×C4—1181no—
249C23×C4C22×D5911yescharacteristic
250C2×M16—594no—
251C2×M16—594no—
252C2×M16—1188no—
253C22×C8—594no—
254C4⋊Q8—28no—
255C4⋊Q8—598no—
256C4⋊Q8—1184no—
257C4⋊Q8C22×D5918yes—
258#125—28no—
259#125C22×D5918yes—
260#124—598no—
261#124—1184no—
262#123—11816no—
263C22⋊Q8—28no—
264C22⋊Q8—216no—
265C22⋊Q8—11816no—
266C4×Q8—28no—
267#117—22no—
268#117—22no—
269#117C22⋊D5914yes—
270#117C22×D5914yes—
271#117C22×D5914yes—
272#117C22×D5914yes—
273C2×C4⋊C4—1184no—
274C2×C4⋊C4C22⋊D5912yes—
275C2×C4⋊C4C22×D5914yes—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—1184no—
278C2×C22⋊C4C22⋊D5912yes—
279C2×C42—22no—
280C2×C42C22×D5912yes—
281#103—5932no—
282#103—5916no—
283#102—5916no—
284#97—598no—
285C4×C8—598no—
286C24—1181no—
287C24C2×C22⋊D5911yescharacteristic
288#41—1188no—
289#41—1188no—
290#41—2368no—
291C2×Q8—28no—
292C2×Q8—594no—
293C2×Q8—592no—
294C2×Q8—1184no—
295C2×Q8—1182no—
296C2×Q8—1181no—
297C2×Q8—1188no—
298C2×Q8—1188no—
299C2×Q8—1188no—
300C2×Q8—1184no—
301C2×Q8—2364no—
302C2×Q8—2368no—
303C2×D4—592no—
304C2×D4—1184no—
305C2×D4—1181no—
306C2×D4—1184no—
307C2×D4—2364no—
308C22×C4—24no—
309C22×C4—1182no—
310C22×C4—1181no—
311C22×C4—1184no—
312C22×C4—1184no—
313C22×C4—2364no—
314C22×C4—2364no—
315C22×C4#1174568918yes—
316C22×C4C2×C22⋊D5914yes—
317C22×C4C2×C22⋊D5912yes—
318C22×C4C2×C22⋊D5911yescharacteristic
319C22×C4C23×D5911yescharacteristic
320C22×C4C23×D5912yes—
321M16—1188no—
322M16—1188no—
323C2×C8—598no—
324C2×C8—1188no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—216no—
328C4⋊C4—11816no—
329C4⋊C4C2×C22⋊D5918yes—
330C4⋊C4C23×D5918yes—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—11816no—
334C22⋊C4C2×C22⋊D5918yes—
335C42—24no—
336C42—24no—
337C42C2×C22⋊D5914yes—
338C42C23×D5914yes—
339C23—42no—
340C23—1182no—
341C23—2364no—
342C23—2362no—
343C23#1427371614yes—
344C23C22×C22⋊D5911yescharacteristic
345C23C22×C22⋊D5912yes—
346Q8—88no—
347Q8—1188no—
348Q8—1184no—
349Q8—2368no—
350Q8—2364no—
351Q8—2362no—
352D4—1184no—
353D4—2368no—
354D4—2362no—
355C2×C4—28no—
356C2×C4—44no—
357C2×C4—84no—
358C2×C4—1184no—
359C2×C4—1182no—
360C2×C4—2362no—
361C2×C4—2364no—
362C2×C4—2368no—
363C2×C4—2368no—
364C2×C4#1427371614yes—
365C2×C4#1427371618yes—
366C2×C4C22×C22⋊D5918yes—
367C2×C4C22×C22⋊D5911yescharacteristic
368C2×C4C22×C22⋊D5912yes—
369C2×C4C2×#1174568914yes—
370C2×C4C2×#1174568914yes—
371C2×C4C2×#1174568914yes—
372C2×C4C24×D5911yescharacteristic
373C8—1188no—
374C22—22no—
375C22—24no—
376C22—44no—
377C22—4724no—
378C22—2362no—
379C22—2364no—
380C22C2×#1427371612yes—
381C22C2×#1427371614yes—
382C22C23×C22⋊D5911yescharacteristic
383C4—28no—
384C4—48no—
385C4—2362no—
386C4—2364no—
387C4C2×#1427371614yes—
388C4C2×#1427371612yes—
389C4C22×#1174568912yes—
390C2—22no—
391C2—24no—
392C2—4722no—
393C2C22×#1427371611yescharacteristic
394C2#2143888212yes—
subgroupG/H
centerC22C23×C22⋊D59
commutator, cosocleC2×C4×C59C25
FrattiniC2×C4C24×D59
FittingC59×#3458C2
socleC22×C59C23×D4
prime radicalC22×#11745685C22
hypercenter#3458D59
hypocenterC59#60620
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×#117 ◁ #3458

Complements

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

NH ≅ G/Ntype
2. #21434485C2semidirect
3. #21434478C2semidirect
4. #21434466C2semidirect
6. C2×#16837410C2semidirect
7. #21432515C2semidirect
8. C59×#3458C2semidirect
18. #16837772C22semidirect
19. #16837764C22semidirect
20. #16837759C22semidirect
21. #16837754C22semidirect
24. #16837410C22semidirect
28. C2×#14273650C22semidirect
29. #16837358C22semidirect
32. C59×#544C22semidirect
193. #3458D59semidirect
210. #544C2×D59semidirect
235. C59#60620semidirect

Maximal subgroups

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

subgroupindex
C2×#168379782
#214344852
#214344782
#214344662
#214326132
C2×#168374102
#214325152
C59×#34582
#6062059