Finite Group Databasebeta

← #28487647

Subgroups — #28487647

Order 15104.

600 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14273716C211yesmaximal, characteristic
2C4⋊C8×Dic59C211yesmaximal, characteristic, complemented
3C4×Dic59⋊C8C211yesmaximal, characteristic, complemented
4C2×Q8⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
5C2×#16837131C211yesmaximal, characteristic
6C2×#16837129C211yesmaximal, characteristic
7C2×C59×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#14273716C411yescharacteristic
9C2×C4×D236C2211yescharacteristic
10C2×C4⋊(C59⋊C8)C2211yescharacteristic
11C2×Dic59⋊C8—21no—
12C2×C4×C59⋊C8C2211yescharacteristic
13C2×C8×Dic59—21no—
14#16837309C411yescharacteristic
15#16837303C411yescharacteristic
16#16837292—21no—
17#16837292—21no—
18#16837267C411yescharacteristic
19C4×(C2×D59)⋊C4—21no—
20C4×(C2×D59)⋊C4—21no—
21C42×Dic59C2211yescharacteristic
22#16837156—21no—
23#16837156—21no—
24Q8⋊(C59⋊C8)—22no—
25#16837131—22no—
26#16837129—22no—
27C2×C4×Q8×C59C2211yescharacteristic
28C2×C4⋊C8×C59C2211yescharacteristic
29C2×C4×C8×C59C2211yescharacteristic
30C59×Q8⋊C8—22no—
31C22×D236C2×C411yescharacteristic
32C22×C4×D59—21no—
33C22×C59⋊C8—21no—
34C22×C59⋊C8—21no—
35#14273716—22no—
36#14273716C814yescomplemented
37C4×D236—21no—
38C4×D236—21no—
39C4×D236—21no—
40C4×D236—21no—
41C2×C4⋊Dic59C2×C411yescharacteristic
42C2×(C2×D59)⋊C4—21no—
43C2×(C2×D59)⋊C4—21no—
44C2×(C2×D59)⋊C4—21no—
45C2×(C2×D59)⋊C4—21no—
46C2×(C2×D59)⋊C4—21no—
47C2×C4×Dic59C2×C411yescharacteristic
48C2×C4×Dic59C2×C411yescharacteristic
49C2×C4×Dic59D411yescharacteristic
50C2×C4×Dic59D411yescharacteristic
51C2×C4×Dic59D411yescharacteristic
52C2×C4×Dic59D411yescharacteristic
53C4⋊(C59⋊C8)—21no—
54C4⋊(C59⋊C8)D412yes—
55Dic59⋊C8—24no—
56C4×C59⋊C8—21no—
57C4×C59⋊C8—21no—
58C8×Dic59—42no—
59C59⋊#95—21no—
60C59⋊#95—21no—
61C59⋊#95—21no—
62C59⋊#95—21no—
63C22×Q8×C59C2×C411yescharacteristic
64C22×C8×C59—21no—
65C22×C8×C59—21no—
66C4×Q8×C59—21no—
67C4×Q8×C59—21no—
68C4×Q8×C59—21no—
69C4×Q8×C59—21no—
70C2×C4⋊C4×C59—21no—
71C2×C4⋊C4×C59C2×C411yescharacteristic
72C2×C42×C59—21no—
73C2×C42×C59C2311yescharacteristic
74C4⋊C8×C59—21no—
75C4⋊C8×C59D412yes—
76C4×C8×C59—21no—
77C4×C8×C59—21no—
78C23×D59—21no—
79C2×Q8×C59—21no—
80C2×Q8×C59—21no—
81C2×Q8×C59—21no—
82C2×Q8×C59—21no—
83C2×Q8×C59—21no—
84C2×Q8×C59C2×C812yes—
85C22×C4×C59—21no—
86C22×C4×C59—21no—
87C22×C4×C59C22×C411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×Dic59—21no—
91C22×Dic59C22⋊C411yescharacteristic
92C22×Dic59C22⋊C411yescharacteristic
93C22×Dic59C22⋊C411yescharacteristic
94C22×Dic59C22⋊C411yescharacteristic
95C2×D236—21no—
96C2×D236—21no—
97C2×D236—21no—
98C2×D236—21no—
99C2×D236—21no—
100C2×D236C2×C812yes—
101C2×C4×D59—41no—
102C2×C4×D59—41no—
103C2×C4×D59—41no—
104C2×C4×D59—41no—
105C2×C4×D59—41no—
106C2×C4×D59—41no—
107C2×C8×C59—22no—
108C2×C8×C59—22no—
109C2×C8×C59—41no—
110C2×C8×C59—41no—
111C2×C8×C59—41no—
112C2×C8×C59—41no—
113C4⋊C4×C59—21no—
114C4⋊C4×C59—21no—
115C4⋊C4×C59—41no—
116C4⋊C4×C59—41no—
117C42×C59—41no—
118C42×C59—41no—
119C42×C59C2×D411yescharacteristic
120C42×C59C2×D411yescharacteristic
121C42×C59#4111yescharacteristic
122C42×C59#4111yescharacteristic
123(C2×D59)⋊C4—24no—
124(C2×D59)⋊C4—24no—
125(C2×D59)⋊C4—42no—
126(C2×D59)⋊C4—41no—
127(C2×D59)⋊C4—41no—
128(C2×D59)⋊C4—42no—
129C4⋊Dic59—21no—
130C4⋊Dic59—21no—
131C4×Dic59—21no—
132C4×Dic59—21no—
133C4×Dic59—21no—
134C4×Dic59—21no—
135C4×Dic59—21no—
136C4×Dic59—21no—
137C4×Dic59—22no—
138C4×Dic59—41no—
139C4×Dic59C2×C812yes—
140C4×Dic59M1612yes—
141C2×C59⋊C8—22no—
142C2×C59⋊C8—22no—
143C2×C59⋊C8—41no—
144C2×C59⋊C8—41no—
145C2×C59⋊C8—41no—
146C2×C59⋊C8—41no—
147C23×C59C2×C22⋊C411yescharacteristic
148C22×D59—22no—
149C22×D59—41no—
150C22×D59—41no—
151C22×D59—41no—
152C22×D59—41no—
153C22×D59—41no—
154C22×D59—41no—
155Q8×C59—22no—
156Q8×C59—22no—
157Q8×C59—41no—
158Q8×C59—41no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—22no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59C4×D411yescharacteristic
168C2×C4×C59C4×D411yescharacteristic
169C2×C4×C59#12111yescharacteristic
170C2×C4×C59#12111yescharacteristic
171C2×C4×C59#12411yescharacteristic
172C2×C4×C59#12711yescharacteristic
173C2×C4×C59C22×C811yescharacteristic
174C2×C4×C59C2×M1611yescharacteristic
175C2×C4×C59#13111yescharacteristic
176C2×C4×C59#13111yescharacteristic
177C2×C4×C59#13511yescharacteristic
178C2×C4×C59#13511yescharacteristic
179C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C59#13711yescharacteristic
181C2×Dic59—21no—
182C2×Dic59—21no—
183C2×Dic59—21no—
184C2×Dic59—21no—
185C2×Dic59—21no—
186C2×Dic59—21no—
187C2×Dic59—21no—
188C2×Dic59—21no—
189C2×Dic59—21no—
190C2×Dic59—41no—
191C2×Dic59—41no—
192C2×Dic59—41no—
193C2×Dic59C22⋊C812yes—
194C2×Dic59C22⋊C812yes—
195C2×Dic59#10412yes—
196D236—22no—
197D236—22no—
198D236—41no—
199D236—41no—
200C4×D59—81no—
201C4×D59—81no—
202C4×D59—81no—
203C4×D59—81no—
204C8×C59—42no—
205C8×C59—42no—
206C59⋊C8—42no—
207C59⋊C8—42no—
208#9675—591nomaximal, Sylow
209C22×C59#39011yescharacteristic
210C22×C59C2×C22⋊C811yescharacteristic
211C22×C59#40811yescharacteristic
212C22×C59#41611yescharacteristic
213C22×C59#41711yescharacteristic
214C22×C59C2×#10411yescharacteristic
215C22×C59#42111yescharacteristic
216C2×D59—42no—
217C2×D59—42no—
218C2×D59—42no—
219C2×D59—81no—
220C2×D59—81no—
221C2×D59—81no—
222C2×D59—81no—
223C4×C59—21no—
224C4×C59—21no—
225C4×C59—42no—
226C4×C59—81no—
227C4×C59C8×D411yescharacteristic
228C4×C59C8×D411yescharacteristic
229C4×C59#43611yescharacteristic
230C4×C59#43611yescharacteristic
231C4×C59#46311yescharacteristic
232C4×C59#46311yescharacteristic
233C4×C59#46411yescharacteristic
234C4×C59#46411yescharacteristic
235Dic59—22no—
236Dic59—22no—
237Dic59—22no—
238Dic59—41no—
239Dic59—81no—
240C4×#124—591no—
241C4×C4⋊C8—591no—
242C4×C4⋊C8—591no—
243C2×Q8⋊C8—591no—
244C2×Q8⋊C8D5911yescharacteristic, complemented
245C2×D4⋊C8—591no—
246C2×D4⋊C8—591no—
247C2×C59#147611yescharacteristic
248C2×C59#147711yescharacteristic
249C2×C59#148211yescharacteristic
250C2×C59#148311yescharacteristic
251C2×C59#195411yescharacteristic
252C2×C59#198011yescharacteristic
253C2×C59#198311yescharacteristic
254D59—82no—
255C2×#124—591no—
256C2×C4×Q8C2×D5911yescharacteristic
257C2×C4×D4—591no—
258C2×C4⋊C8—591no—
259C2×C4⋊C8—1181no—
260C2×C4⋊C8C2×D5911yescharacteristic
261C2×C4×C8—591no—
262C2×C4×C8—1181no—
263C2×C4×C8C2×D5911yescharacteristic
264#391—591no—
265#390—591no—
266#388—1181no—
267#388—1181no—
268#382—591no—
269C4×C22⋊C4—1181no—
270C4×C22⋊C4—1181no—
271C43—591no—
272#336—1181no—
273#336—1181no—
274Q8⋊C8—22no—
275Q8⋊C8—1182no—
276D4⋊C8—1182no—
277D4⋊C8—1182no—
278C59#967511yescharacteristic, complemented, Sylow
279C22×Q8C4×D5911yescharacteristic
280C22×D4—591no—
281C23×C4—1181no—
282C22×C8—21no—
283C22×C8—21no—
284C22×C8—1181no—
285C22×C8—1181no—
286#124—594no—
287#124—1182no—
288C4×Q8—21no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×D4—1181no—
293C4×D4—1181no—
294C4×D4—1181no—
295C4×D4—1181no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—591no—
298C2×C4⋊C4C4×D5911yescharacteristic
299C2×C22⋊C4—1181no—
300C2×C22⋊C4—1181no—
301C2×C22⋊C4—1181no—
302C2×C22⋊C4—1181no—
303C2×C22⋊C4—1181no—
304C2×C42—21no—
305C2×C42—591no—
306C2×C42—591no—
307C2×C42—591no—
308C2×C42—591no—
309C2×C42—591no—
310C2×C42—591no—
311C2×C42C22×D5911yescharacteristic
312C4⋊C8—21no—
313C4⋊C8—592no—
314C4⋊C8—1184no—
315C4⋊C8—1181no—
316C4⋊C8C22⋊D5912yes—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—1181no—
320C4×C8—1181no—
321C4×C8—2362no—
322#95—1181no—
323#95—1181no—
324#95—1181no—
325#95—1181no—
326C24—1181no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8C8×D5912yes—
333C2×D4—592no—
334C2×D4—1181no—
335C2×D4—1181no—
336C2×D4—1181no—
337C2×D4—1181no—
338C2×D4—1181no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—591no—
342C22×C4—591no—
343C22×C4—591no—
344C22×C4—591no—
345C22×C4—1181no—
346C22×C4—2361no—
347C22×C4—2361no—
348C22×C4—2361no—
349C22×C4—2361no—
350C22×C4—2361no—
351C22×C4—2361no—
352C22×C4C2×C4×D5911yescharacteristic
353C22×C4D4×D5911yescharacteristic
354C22×C4D4×D5911yescharacteristic
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—1182no—
362C2×C8—1182no—
363C2×C8—2361no—
364C2×C8—2361no—
365C2×C8—2361no—
366C2×C8—2361no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—1181no—
372C4⋊C4—1181no—
373C22⋊C4—1184no—
374C22⋊C4—1184no—
375C22⋊C4—2362no—
376C22⋊C4—2362no—
377C22⋊C4—2361no—
378C22⋊C4—2361no—
379C42—41no—
380C42—41no—
381C42—592no—
382C42—592no—
383C42—1182no—
384C42—1181no—
385C42—1181no—
386C42—1181no—
387C42—1181no—
388C42—1181no—
389C42—1181no—
390C42—2361no—
391C42#1174568511yescharacteristic
392C42D4×D5911yescharacteristic
393C42#1174568711yescharacteristic
394C42C2×C22⋊D5911yescharacteristic
395C23—1182no—
396C23—2361no—
397C23—2361no—
398C23—2361no—
399C23—2361no—
400C23—2361no—
401C23—2361no—
402C23C22⋊C4×D5911yescharacteristic
403Q8—22no—
404Q8—22no—
405Q8—41no—
406Q8—41no—
407D4—1182no—
408D4—1182no—
409D4—2361no—
410D4—2361no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—592no—
420C2×C4—592no—
421C2×C4—592no—
422C2×C4—1181no—
423C2×C4—1181no—
424C2×C4—1181no—
425C2×C4—1181no—
426C2×C4—1181no—
427C2×C4—1181no—
428C2×C4—1181no—
429C2×C4—1181no—
430C2×C4—1181no—
431C2×C4—4721no—
432C2×C4—4721no—
433C2×C4—4721no—
434C2×C4—4721no—
435C2×C4—2361no—
436C2×C4—2361no—
437C2×C4—2361no—
438C2×C4(C22⋊D59)⋊C411yescharacteristic
439C2×C4C4×C22⋊D5911yescharacteristic
440C2×C4#1427369911yescharacteristic
441C2×C4#1427370011yescharacteristic
442C2×C4#1427371311yescharacteristic
443C2×C4#1427372311yescharacteristic
444C2×C4C2×C8×D5911yescharacteristic
445C2×C4M16×D5911yescharacteristic
446C2×C4#1427373311yescharacteristic
447C2×C4#1427373511yescharacteristic
448C2×C4#1427374811yescharacteristic
449C2×C4#1427375011yescharacteristic
450C2×C4#1427375411yescharacteristic
451C2×C4#1427375911yescharacteristic
452C8—42no—
453C8—42no—
454C8—2362no—
455C8—2362no—
456C22—4721no—
457C22—4721no—
458C22—4721no—
459C22—4721no—
460C22—2362no—
461C22—2362no—
462C22—2362no—
463C22#1683730211yescharacteristic
464C22C22⋊C8×D5911yescharacteristic
465C22#1683736711yescharacteristic
466C22#1683739711yescharacteristic
467C22#1683740211yescharacteristic
468C22#104×D5911yescharacteristic
469C22#1683741711yescharacteristic
470C4—21no—
471C4—21no—
472C4—42no—
473C4—81no—
474C4—1182no—
475C4—1182no—
476C4—1182no—
477C4—4721no—
478C4—2361no—
479C4(C22⋊D59)⋊C811yescharacteristic
480C4C8×C22⋊D5911yescharacteristic
481C4#1683747711yescharacteristic
482C4#1683747811yescharacteristic
483C4#1683761911yescharacteristic
484C4#1683762011yescharacteristic
485C4#1683762611yescharacteristic
486C4#1683762711yescharacteristic
487C2—4722no—
488C2#2142247711yescharacteristic
489C2#2142248211yescharacteristic
490C2#2142250311yescharacteristic
491C2#2142250811yescharacteristic
492C2#2142488811yescharacteristic
493C2#2142501311yescharacteristic
494C2#2142502511yescharacteristic
subgroupG/H
centerC22×C4D4×D59
commutatorC2×C4×C59C22×C8
FrattiniC2×C42C22×D59
FittingC2×C59×Q8⋊C8C2
socleC23×C59C2×C22⋊C4
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenterC2×Q8⋊C8D59
hypocenterC59#9675
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. C4⋊C8×Dic59C2semidirect
3. C4×Dic59⋊C8C2semidirect
4. C2×Q8⋊(C59⋊C8)C2semidirect
7. C2×C59×Q8⋊C8C2semidirect
36. #14273716C8semidirect
244. C2×Q8⋊C8D59semidirect
278. C59#9675semidirect

Maximal subgroups

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

subgroupindex
C4×#142737162
C4⋊C8×Dic592
C4×Dic59⋊C82
C2×Q8⋊(C59⋊C8)2
C2×#168371312
C2×#168371292
C2×C59×Q8⋊C82
#967559