Finite Group Databasebeta

← #28484909

Subgroups — #28484909

Order 15104.

562 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837679C211yesmaximal, characteristic
2C2×#16837479C211yesmaximal, characteristic
3#21424899C211yesmaximal, characteristic, complemented
4#21424097C211yesmaximal, characteristic
5#21421980C211yesmaximal, characteristic, complemented
6#21421930C211yesmaximal, characteristic, complemented
7C2×C59×#332C211yesmaximal, characteristic, complemented
8C22×#11745659—21no—
9C2×C4×D236C2211yescharacteristic
10#16837679—22no—
11#16837679C414yescomplemented
12#16837479—22no—
13#16837479—22no—
14C2×#14273660—21no—
15C2×C4⋊(C59⋊C8)C2211yescharacteristic
16C2×#14273651—21no—
17C2×#14273651—21no—
18C2×#14273649—21no—
19C2×#14273647—21no—
20C2×(C2×D59)⋊C8—21no—
21C2×C4×C59⋊C8C2211yescharacteristic
22#16837272C2211yescharacteristic
23#16837165—21no—
24C2×C4⋊Q8×C59C2211yescharacteristic
25C2×C4⋊C8×C59C2211yescharacteristic
26C2×C4⋊C8×C59C2211yescharacteristic
27C59×#332—22no—
28C22×D236D411yescharacteristic
29C22×C4×D59—21no—
30C2×#11745659—24no—
31C2×#11745659—42no—
32C2×#11745659—42no—
33C22×C59⋊C8—21no—
34C22×C59⋊C8D411yescharacteristic
35C22×C59⋊C8D411yescharacteristic
36C4×D236—21no—
37C4×D236—21no—
38C4×D236C2×C412yes—
39C4×D236D412yes—
40C2×C4⋊Dic59—21no—
41C2×C4⋊Dic59—21no—
42C2×C4⋊Dic59—21no—
43C2×C4⋊Dic59D411yescharacteristic
44C2×(C2×D59)⋊C4—21no—
45#14273660—24no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)C2×C412yes—
48#14273651—42no—
49#14273651—42no—
50#14273649—42no—
51#14273647—24no—
52(C2×D59)⋊C8—24no—
53C4×C59⋊C8—21no—
54C4×C59⋊C8—21no—
55C22×Q8×C59—21no—
56C22×C8×C59—21no—
57C22×C8×C59—21no—
58C4⋊Q8×C59—21no—
59C4⋊Q8×C59—21no—
60C4⋊Q8×C59—21no—
61C4⋊Q8×C59C2×C412yes—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59—21no—
64C2×C42×C59C2311yescharacteristic
65C4⋊C8×C59—21no—
66C4⋊C8×C59—21no—
67C4⋊C8×C59—21no—
68C4⋊C8×C59D412yes—
69C23×D59—21no—
70C2×Q8×C59—22no—
71C2×Q8×C59—41no—
72C2×Q8×C59—41no—
73C2×Q8×C59—41no—
74C2×Q8×C59—41no—
75C2×Q8×C59—41no—
76C22×C4×C59—41no—
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59#4111yescharacteristic
80C22×Dic59—21no—
81C22×Dic59—41no—
82C2×D236—21no—
83C2×D236—21no—
84C2×D236—21no—
85C2×D236—21no—
86C2×D236—21no—
87C2×D236C22⋊C412yes—
88C2×C4×D59—22no—
89C2×C4×D59—22no—
90C2×C4×D59—41no—
91C2×C4×D59—41no—
92C2×C4×D59—41no—
93C2×C4×D59—41no—
94C2×C8×C59—22no—
95C2×C8×C59—41no—
96C2×C8×C59—41no—
97C2×C8×C59—41no—
98C2×C8×C59—41no—
99C2×C8×C59—41no—
100C4⋊C4×C59—22no—
101C4⋊C4×C59—22no—
102C4⋊C4×C59—41no—
103C4⋊C4×C59—41no—
104C42×C59C22×C411yescharacteristic
105C42×C59C2×D411yescharacteristic
106C42×C59C2×D411yescharacteristic
107C42×C59#4111yescharacteristic
108(C2×D59)⋊C4—22no—
109(C2×D59)⋊C4—22no—
110C4⋊Dic59—21no—
111C4⋊Dic59—41no—
112C4⋊Dic59—41no—
113C4⋊Dic59—41no—
114C4⋊Dic59—41no—
115C4⋊Dic59—81no—
116C4⋊Dic59C22⋊C412yes—
117C2×C59⋊C8—21no—
118C2×C59⋊C8—21no—
119C2×C59⋊C8—22no—
120C2×C59⋊C8—21no—
121C2×C59⋊C8—21no—
122C2×C59⋊C8—21no—
123C2×C59⋊C8—41no—
124C2×C59⋊C8—41no—
125C2×C59⋊C8D812yes—
126#11745659—48no—
127C23×C59#12111yescharacteristic
128C22×D59—22no—
129C22×D59—41no—
130C22×D59—41no—
131C22×D59—41no—
132C22×D59—41no—
133C22×D59—41no—
134C22×D59—41no—
135Q8×C59—42no—
136Q8×C59—42no—
137Q8×C59—81no—
138Q8×C59—81no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—42no—
142C2×C4×C59—81no—
143C2×C4×C59—81no—
144C2×C4×C59C2×C22⋊C411yescharacteristic
145C2×C4×C59C4×D411yescharacteristic
146C2×C4×C59C4×D411yescharacteristic
147C2×C4×C59#12011yescharacteristic
148C2×C4×C59#12111yescharacteristic
149C2×C4×C59#12311yescharacteristic
150C2×C4×C59C2×D811yescharacteristic
151C2×C4×C59#13511yescharacteristic
152C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C59#13711yescharacteristic
156C2×C4×C59#13711yescharacteristic
157C2×C4×C59#13711yescharacteristic
158C2×Dic59—22no—
159C2×Dic59—41no—
160C2×Dic59—41no—
161C2×Dic59—81no—
162C2×Dic59—81no—
163C2×Dic59—81no—
164D236—22no—
165D236—22no—
166D236—41no—
167D236—41no—
168C4×D59—42no—
169C4×D59—42no—
170C4×D59—42no—
171C4×D59—42no—
172C8×C59—42no—
173C8×C59—81no—
174C59⋊C8—22no—
175C59⋊C8—42no—
176C59⋊C8—41no—
177#9227—591nomaximal, Sylow
178C22×C59#38611yescharacteristic
179C22×C59#45911yescharacteristic
180C22×C59#46311yescharacteristic
181C22×C59#46811yescharacteristic
182C22×C59#47011yescharacteristic
183C22×C59#47111yescharacteristic
184C22×C59#47311yescharacteristic
185C2×D59—42no—
186C2×D59—42no—
187C2×D59—42no—
188C2×D59—81no—
189C2×D59—81no—
190C2×D59—81no—
191C2×D59—81no—
192C4×C59—21no—
193C4×C59—21no—
194C4×C59—82no—
195C4×C59#41711yescharacteristic
196C4×C59#41911yescharacteristic
197C4×C59C4×D811yescharacteristic
198C4×C59#44211yescharacteristic
199C4×C59#44711yescharacteristic
200C4×C59#44911yescharacteristic
201C4×C59#46811yescharacteristic
202C4×C59#47011yescharacteristic
203Dic59—42no—
204Dic59—161no—
205C2×#474—591no—
206C2×#436—591no—
207#1956—591no—
208#1802—591no—
209C2×#332D5911yescharacteristic, complemented
210#1296—591no—
211#1277—591no—
212C2×C59#168511yescharacteristic
213C2×C59#168911yescharacteristic
214C2×C59#169911yescharacteristic
215C2×C59#170311yescharacteristic
216C2×C59#187911yescharacteristic
217C2×C59#189411yescharacteristic
218C2×C59#190811yescharacteristic
219D59—82no—
220C22×M16—1181no—
221C2×C4⋊Q8C2×D5911yescharacteristic
222C2×C4×D4—591no—
223#474—594no—
224#474—1182no—
225#436—1182no—
226#436—1182no—
227C2×#107—1181no—
228C2×C4⋊C8—591no—
229C2×C4⋊C8C2×D5911yescharacteristic
230C2×C4⋊C8C2×D5911yescharacteristic
231C2×#103—1181no—
232C2×#103—1181no—
233C2×#102—1181no—
234C2×#102—1181no—
235C2×C22⋊C8—1181no—
236C2×C4×C8—591no—
237#384—591no—
238#340—1181no—
239#332—22no—
240C59#922711yescharacteristic, complemented, Sylow
241C22×Q8—21no—
242C22×D4—591no—
243C23×C4—1181no—
244C2×M16—1184no—
245C2×M16—2362no—
246C2×M16—2362no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—591no—
250C22×C8—591no—
251C22×C8—1181no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8C4×D5912yes—
256C4×D4—592no—
257C4×D4—592no—
258C4×D4—1181no—
259C4×D4—1181no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—591no—
263C2×C4⋊C4—1181no—
264C2×C4⋊C4—1181no—
265C2×C4⋊C4—1181no—
266C2×C22⋊C4—1181no—
267C2×C42C22×D5911yescharacteristic
268#107—1184no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—592no—
273C4⋊C8—1181no—
274C4⋊C8C22⋊D5912yes—
275#103—2362no—
276#103—2362no—
277#102—1184no—
278#102—2362no—
279C22⋊C8—1184no—
280C4×C8—1181no—
281C4×C8—1181no—
282C24—1181no—
283C2×Q8—22no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×D4—592no—
290C2×D4—1181no—
291C2×D4—1181no—
292C2×D4—1181no—
293C2×D4—1181no—
294C2×D4—1181no—
295C22×C4—41no—
296C22×C4—1181no—
297C22×C4—1182no—
298C22×C4—1182no—
299C22×C4—2361no—
300C22×C4—2361no—
301C22×C4—2361no—
302C22×C4—2361no—
303C22×C4—2361no—
304C22×C4#1174568511yescharacteristic
305C22×C4D4×D5911yescharacteristic
306C22×C4D4×D5911yescharacteristic
307M16—2368no—
308C2×C8—22no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—592no—
315C2×C8—1181no—
316C2×C8—1181no—
317C2×C8—1182no—
318C2×C8—1181no—
319C2×C8—1181no—
320C2×C8—1181no—
321C2×C8—2361no—
322C2×C8—2361no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—592no—
328C4⋊C4—1181no—
329C4⋊C4—4721no—
330C4⋊C4—2361no—
331C4⋊C4—2361no—
332C4⋊C4—2361no—
333C4⋊C4—2361no—
334C22⋊C4—1182no—
335C22⋊C4—1182no—
336C42C2×C4×D5911yescharacteristic
337C42D4×D5911yescharacteristic
338C42#1174568711yescharacteristic
339C42C2×C22⋊D5911yescharacteristic
340C23—1182no—
341C23—2361no—
342C23—2361no—
343C23—2361no—
344C23—2361no—
345C23—2361no—
346C23—2361no—
347C23#1427370011yescharacteristic
348Q8—42no—
349Q8—42no—
350Q8—81no—
351Q8—81no—
352D4—1182no—
353D4—1182no—
354D4—2361no—
355D4—2361no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—1182no—
362C2×C4—4721no—
363C2×C4—4721no—
364C2×C4—4721no—
365C2×C4—2361no—
366C2×C4—2361no—
367C2×C4—2362no—
368C2×C4—2362no—
369C2×C4—2362no—
370C2×C4—2362no—
371C2×C4C22⋊C4×D5911yescharacteristic
372C2×C4(C22⋊D59)⋊C411yescharacteristic
373C2×C4C4×C22⋊D5911yescharacteristic
374C2×C4#1427369511yescharacteristic
375C2×C4#1427369811yescharacteristic
376C2×C4#1427370911yescharacteristic
377C2×C4D8×D5911yescharacteristic
378C2×C4#1427374811yescharacteristic
379C2×C4#1427375111yescharacteristic
380C2×C4#1427375411yescharacteristic
381C2×C4#1427375411yescharacteristic
382C2×C4#1427375611yescharacteristic
383C2×C4#1427375911yescharacteristic
384C2×C4#1427375911yescharacteristic
385C8—42no—
386C8—81no—
387C8—1182no—
388C8—2362no—
389C8—2361no—
390C22—4721no—
391C22—4721no—
392C22—4721no—
393C22—4721no—
394C22—2362no—
395C22—2362no—
396C22—2362no—
397C22#1683727811yescharacteristic
398C22#1683759611yescharacteristic
399C22#1683761611yescharacteristic
400C22#1683765111yescharacteristic
401C22#1683765911yescharacteristic
402C22#1683766811yescharacteristic
403C22#1683767611yescharacteristic
404C4—21no—
405C4—21no—
406C4—82no—
407C4—9441no—
408C4—2362no—
409C4#1683740211yescharacteristic
410C4#1683740911yescharacteristic
411C4#1683748211yescharacteristic
412C4#1683751111yescharacteristic
413C4#1683753411yescharacteristic
414C4#1683754611yescharacteristic
415C4#1683764511yescharacteristic
416C4#1683765711yescharacteristic
417C2—4722no—
418C2#2142356411yescharacteristic
419C2#2142358411yescharacteristic
420C2#2142364211yescharacteristic
421C2#2142367011yescharacteristic
422C2#2142444111yescharacteristic
423C2#2142452611yescharacteristic
424C2#2142460811yescharacteristic
subgroupG/H
centerC23#14273700
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC2×C59×#332C2
socleC23×C59#121
prime radicalC22×D236D4
cosocleC2×C42×C59C23
hypercenterC2×#332D59
hypocenterC59#9227
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. #21424899C2semidirect
5. #21421980C2semidirect
6. #21421930C2semidirect
7. C2×C59×#332C2semidirect
11. #16837679C4semidirect
209. C2×#332D59semidirect
240. C59#9227semidirect

Maximal subgroups

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

subgroupindex
C2×#168376792
C2×#168374792
#214248992
#214240972
#214219802
#214219302
C2×C59×#3322
#922759