Finite Group Databasebeta

← #28489773

Subgroups — #28489773

Order 15104.

702 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837750C211yesmaximal, characteristic, complemented
2C2×#16837597C211yesmaximal, characteristic
3#21425023C211yesmaximal, characteristic, complemented
4C2×#16837148C211yesmaximal, characteristic, complemented
5C2×D4⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
6#21421937C211yesmaximal, characteristic, complemented
7C59×#2547C211yesmaximal, characteristic, complemented
8C22×#11745669—21no—
9C2×#14273724C2211yescharacteristic
10#16837750—22no—
11#16837750—22no—
12#16837597—22no—
13#16837597—22no—
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15C2×C4⋊(C59⋊C8)C2211yescharacteristic
16C2×#14273647—22no—
17C2×#14273647—21no—
18C2×#14273647—22no—
19C2×C4×C59⋊C8C2211yescharacteristic
20#16837148—22no—
21D4⋊(C59⋊C8)—22no—
22C2×C4⋊Q8×C59C2211yescharacteristic
23C2×#124×C59—21no—
24C2×C4×D4×C59C2211yescharacteristic
25C59×#397—22no—
26C59×#391—21no—
27C59×#380—22no—
28C4×C4⋊C4×C59C2211yescharacteristic
29C22×D236—21no—
30C22×D236—21no—
31C22×D236—21no—
32C2×#11745669—24no—
33C2×#11745669—42no—
34C2×#11745669—42no—
35C22×C59⋊C8—21no—
36C22×C59⋊C8—21no—
37C22×C59⋊C8—21no—
38#14273724—21no—
39#14273724—21no—
40#14273724—21no—
41#14273724D412yes—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)D412yes—
46#14273647—28no—
47#14273647—24no—
48#14273647—44no—
49C4×C59⋊C8—21no—
50C4×C59⋊C8D412yes—
51C22×Q8×C59—21no—
52C22×D4×C59D411yescharacteristic
53C23×C4×C59—21no—
54C4⋊Q8×C59—21no—
55C4⋊Q8×C59—21no—
56C4⋊Q8×C59—21no—
57C4⋊Q8×C59—21no—
58#124×C59—42no—
59#124×C59—42no—
60C4×D4×C59—21no—
61C4×D4×C59—21no—
62C4×D4×C59—21no—
63C4×D4×C59—21no—
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59D412yes—
66C2×C4⋊C4×C59D411yescharacteristic
67C2×C4⋊C4×C59D412yes—
68C2×C22⋊C4×C59—24no—
69C2×C22⋊C4×C59—21no—
70C2×C42×C59—21no—
71C2×C42×C59C2311yescharacteristic
72#95×C59—24no—
73#95×C59—21no—
74C24×C59—21no—
75C23×D59—41no—
76C2×Q8×C59—41no—
77C2×Q8×C59—41no—
78C2×Q8×C59—41no—
79C2×Q8×C59—41no—
80C2×Q8×C59—41no—
81C2×Q8×C59—41no—
82C2×D4×C59—21no—
83C2×D4×C59—21no—
84C2×D4×C59—21no—
85C2×D4×C59—21no—
86C2×D4×C59—21no—
87C2×D4×C59D812yes—
88C22×C4×C59—21no—
89C22×C4×C59—22no—
90C22×C4×C59—22no—
91C22×C4×C59—41no—
92C22×C4×C59—41no—
93C22×C4×C59—42no—
94C22×C4×C59—42no—
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59C2×D411yescharacteristic
97C22×C4×C59C2×D411yescharacteristic
98C2×D236—22no—
99C2×D236—22no—
100C2×D236—22no—
101C2×D236—44no—
102C2×D236—41no—
103C2×D236—41no—
104C2×D236—42no—
105C2×D236—42no—
106C2×D236—41no—
107C2×D236—41no—
108C2×D236—41no—
109C2×D236—41no—
110C2×D236—41no—
111C4⋊C4×C59—21no—
112C4⋊C4×C59—22no—
113C4⋊C4×C59—21no—
114C4⋊C4×C59—22no—
115C4⋊C4×C59—22no—
116C4⋊C4×C59—22no—
117C4⋊C4×C59—41no—
118C4⋊C4×C59—41no—
119C22⋊C4×C59—48no—
120C22⋊C4×C59—41no—
121C22⋊C4×C59—41no—
122C42×C59—41no—
123C42×C59—41no—
124C42×C59C2×D411yescharacteristic
125C42×C59C2×D411yescharacteristic
126C42×C59C2×D411yescharacteristic
127C42×C59#4111yescharacteristic
128#11745669—48no—
129C2×C59⋊C8—22no—
130C2×C59⋊C8—22no—
131C2×C59⋊C8—42no—
132C2×C59⋊C8—41no—
133C2×C59⋊C8—41no—
134C2×C59⋊C8—41no—
135C2×C59⋊C8—42no—
136C23×C59—22no—
137C23×C59—41no—
138C23×C59—42no—
139C23×C59—41no—
140C23×C59—42no—
141C23×C59#12011yescharacteristic
142C22×D59—42no—
143C22×D59—81no—
144C22×D59—81no—
145C22×D59—82no—
146C22×D59—82no—
147Q8×C59—81no—
148Q8×C59—81no—
149Q8×C59—81no—
150Q8×C59—81no—
151D4×C59—22no—
152D4×C59—22no—
153D4×C59—41no—
154D4×C59—41no—
155C2×C4×C59—22no—
156C2×C4×C59—41no—
157C2×C4×C59—42no—
158C2×C4×C59—42no—
159C2×C4×C59—42no—
160C2×C4×C59—42no—
161C2×C4×C59—42no—
162C2×C4×C59—42no—
163C2×C4×C59—42no—
164C2×C4×C59—82no—
165C2×C4×C59—81no—
166C2×C4×C59—81no—
167C2×C4×C59#12011yescharacteristic
168C2×C4×C59#12011yescharacteristic
169C2×C4×C59#12111yescharacteristic
170C2×C4×C59#12111yescharacteristic
171C2×C4×C59#12111yescharacteristic
172C2×C4×C59#12711yescharacteristic
173C2×C4×C59C2×D811yescharacteristic
174C2×C4×C59#13511yescharacteristic
175C2×C4×C59#13512yes—
176C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C59AGL(1,ℤ/8)12yes—
178C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
179D236—44no—
180D236—42no—
181D236—42no—
182D236—44no—
183D236—82no—
184D236—81no—
185D236—81no—
186D236—81no—
187D236—81no—
188C59⋊C8—42no—
189C59⋊C8—42no—
190C59⋊C8—81no—
191#10050—591nomaximal, Sylow
192C22×C59—44no—
193C22×C59—42no—
194C22×C59—82no—
195C22×C59—81no—
196C22×C59—81no—
197C22×C59#39211yescharacteristic
198C22×C59#44711yescharacteristic
199C22×C59#44912yes—
200C22×C59#44911yescharacteristic
201C22×C59#45412yes—
202C2×D59—82no—
203C2×D59—82no—
204C2×D59—82no—
205C2×D59—162no—
206C2×D59—162no—
207C4×C59—21no—
208C4×C59—21no—
209C4×C59—82no—
210C4×C59—82no—
211C4×C59—81no—
212C4×C59#44711yescharacteristic
213C4×C59#44911yescharacteristic
214C4×C59#46312yes—
215C4×C59#46611yescharacteristic
216C4×C59#46911yescharacteristic
217C4×C59#49511yescharacteristic
218C4×C59#49611yescharacteristic
219C2×#486—591no—
220C2×#459—591no—
221#2547D5911yescharacteristic, complemented
222#1982—591no—
223C2×#332—591no—
224C2×D4⋊C8—591no—
225#1279—591no—
226C2×C59—82no—
227C2×C59#162512yes—
228C2×C59#162812yes—
229C2×C59#199911yescharacteristic
230C2×C59#200311yescharacteristic
231C2×C59#201211yescharacteristic
232D59—162no—
233C22×D8—1181no—
234C2×C4⋊Q8C2×D5911yescharacteristic
235C2×#127—591no—
236C2×#124—21no—
237C2×C4×D4C2×D5911yescharacteristic
238#486—1182no—
239#486—1182no—
240#459—1182no—
241#459—1182no—
242C2×C4⋊C8—591no—
243C2×C4⋊C8—591no—
244C2×#102—1182no—
245C2×#102—1181no—
246C2×#102—1182no—
247C2×C4×C8—591no—
248#397—22no—
249#391—21no—
250#380—22no—
251C4×C4⋊C4C2×D5911yescharacteristic
252#332—1182no—
253D4⋊C8—1182no—
254C59#1005011yescharacteristic, complemented, Sylow
255C22×Q8—21no—
256C22×D4—1181no—
257C22×D4—1181no—
258C22×D4—1181no—
259C22×D4C22⋊D5911yescharacteristic
260C23×C4—21no—
261C2×D8—1184no—
262C2×D8—2362no—
263C2×D8—2362no—
264C22×C8—1181no—
265C22×C8—1181no—
266C22×C8—1181no—
267C4⋊Q8—21no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271#127—592no—
272#127—1181no—
273#127—1181no—
274#127—1181no—
275#124—42no—
276#124—42no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4C22⋊D5911yescharacteristic
283C2×C4⋊C4C22⋊D5912yes—
284C2×C4⋊C4C22⋊D5912yes—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—21no—
287C2×C42—21no—
288C2×C42C22×D5911yescharacteristic
289C4⋊C8—592no—
290C4⋊C8—1181no—
291C4⋊C8—1181no—
292C4⋊C8—1181no—
293#102—1184no—
294#102—1188no—
295#102—2364no—
296C4×C8—592no—
297C4×C8—1181no—
298#95—24no—
299#95—21no—
300C24—21no—
301C24—2361no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—1182no—
314C2×D4—1182no—
315C2×D4—1182no—
316C2×D4—2364no—
317C2×D4—2361no—
318C2×D4—2361no—
319C2×D4—2361no—
320C2×D4—2361no—
321C2×D4—2361no—
322C2×D4—2362no—
323C2×D4—2362no—
324C2×D4—2361no—
325C2×D4—2361no—
326C2×D4#1174566912yes—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4C2×C22⋊D5911yescharacteristic
335C22×C4C2×C22⋊D5911yescharacteristic
336C22×C4C2×C22⋊D5911yescharacteristic
337D8—2368no—
338C2×C8—1182no—
339C2×C8—1182no—
340C2×C8—2361no—
341C2×C8—2362no—
342C2×C8—2362no—
343C2×C8—2361no—
344C2×C8—2361no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C22⋊C4—48no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C42—41no—
357C42—41no—
358C42D4×D5911yescharacteristic
359C42D4×D5911yescharacteristic
360C42D4×D5911yescharacteristic
361C42#1174568711yescharacteristic
362C23—22no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—4721no—
368C23—4722no—
369C23—4721no—
370C23—4722no—
371C23—2362no—
372C23#1427369611yescharacteristic
373Q8—81no—
374Q8—81no—
375Q8—81no—
376Q8—81no—
377D4—22no—
378D4—22no—
379D4—41no—
380D4—41no—
381D4—4722no—
382D4—4721no—
383D4—4721no—
384D4—4721no—
385D4—4721no—
386D4—2364no—
387D4—2362no—
388D4—2362no—
389D4—2364no—
390C2×C4—22no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4#1427369511yescharacteristic
403C2×C4#1427369511yescharacteristic
404C2×C4#1427369811yescharacteristic
405C2×C4#1427369911yescharacteristic
406C2×C4#1427369911yescharacteristic
407C2×C4#1427372311yescharacteristic
408C2×C4C2×#1174566911yescharacteristic
409C2×C4#1427374711yescharacteristic
410C2×C4#1427374712yes—
411C2×C4#1427375211yescharacteristic
412C2×C4#1427375212yes—
413C2×C4#1427375311yescharacteristic
414C8—4721no—
415C8—2362no—
416C8—2362no—
417C22—44no—
418C22—42no—
419C22—81no—
420C22—82no—
421C22—81no—
422C22—9442no—
423C22—9442no—
424C22—4722no—
425C22—4722no—
426C22—4722no—
427C22#1683731011yescharacteristic
428C22#1683753611yescharacteristic
429C22#1683754712yes—
430C22#1683754711yescharacteristic
431C22#1683757512yes—
432C4—21no—
433C4—21no—
434C4—81no—
435C4—82no—
436C4—82no—
437C4#1683753511yescharacteristic
438C4#1683754511yescharacteristic
439C4#1683761412yes—
440C4#1683763611yescharacteristic
441C4#1683765311yescharacteristic
442C4#1683778411yescharacteristic
443C4#1683779211yescharacteristic
444C2—82no—
445C2—9442no—
446C2#2142318512yes—
447C2#2142320612yes—
448C2#2142509211yescharacteristic
449C2#2142511611yescharacteristic
450C2#2142517311yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2547C2
socleC23×C59#120
prime radicalC2×#16837597C2
hypercenter#2547D59
hypocenterC59#10050
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
1. C2×#16837750C2semidirect
3. #21425023C2semidirect
4. C2×#16837148C2semidirect
5. C2×D4⋊(C59⋊C8)C2semidirect
6. #21421937C2semidirect
7. C59×#2547C2semidirect
221. #2547D59semidirect
254. C59#10050semidirect

Maximal subgroups

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

subgroupindex
C2×#168377502
C2×#168375972
#214250232
C2×#168371482
C2×D4⋊(C59⋊C8)2
#214219372
C59×#25472
#1005059