Finite Group Databasebeta

← #28489958

Subgroups — #28489958

Order 15104.

654 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837807C211yesmaximal, characteristic, complemented
2C2×#16837683C211yesmaximal, characteristic
3#21429045C211yesmaximal, characteristic
4#21425019C211yesmaximal, characteristic, complemented
5C2×#16837147C211yesmaximal, characteristic, complemented
6C2×#16837131C211yesmaximal, characteristic
7C59×#1279C211yesmaximal, characteristic, complemented
8C2×#14273727C2211yescharacteristic
9C2×#14273727C2211yescharacteristic
10C2×#14273707—22no—
11C2×C4×D236C2211yescharacteristic
12#16837807—22no—
13#16837807—22no—
14#16837683—22no—
15#16837683—22no—
16C2×#14273661—22no—
17C2×#14273661—21no—
18C2×C4⋊(C59⋊C8)C2211yescharacteristic
19C2×#14273650—22no—
20C2×#14273649—21no—
21#16837339—22no—
22#16837304—21no—
23#16837272—21no—
24#16837147—22no—
25#16837131—22no—
26C2×C4⋊C8×C59C2211yescharacteristic
27C2×C4×C8×C59C2211yescharacteristic
28C4×C4⋊C4×C59C2211yescharacteristic
29C22×Dic118—21no—
30C22×Dic118—21no—
31C22×D236D411yescharacteristic
32C22×C4×D59—21no—
33C22×C59⋊C8—21no—
34#14273727—21no—
35#14273727—21no—
36#14273727—21no—
37#14273727—21no—
38#14273727—21no—
39#14273727—21no—
40#14273727—21no—
41#14273727D412yes—
42#14273707—44no—
43#14273707—44no—
44C4×D236—21no—
45C4×D236—21no—
46C4×D236—21no—
47C4×D236—21no—
48C2×C4⋊Dic59—21no—
49C2×C4⋊Dic59—21no—
50C2×C4⋊Dic59—24no—
51C2×C4⋊Dic59—21no—
52C2×C4⋊Dic59D412yes—
53C2×C4⋊Dic59D411yescharacteristic
54C2×(C2×D59)⋊C4—24no—
55C2×(C2×D59)⋊C4—21no—
56#14273661—24no—
57#14273661—28no—
58C4⋊(C59⋊C8)—21no—
59C4⋊(C59⋊C8)—21no—
60#14273650—44no—
61#14273649—24no—
62C22×C8×C59—21no—
63C22×C8×C59—21no—
64C2×C4⋊C4×C59D412yes—
65C2×C42×C59—21no—
66C2×C42×C59C2311yescharacteristic
67C4⋊C8×C59—21no—
68C4⋊C8×C59D412yes—
69C4×C8×C59—21no—
70C4×C8×C59D412yes—
71#95×C59—21no—
72C23×D59—21no—
73C22×C4×C59—22no—
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×Dic59—22no—
78C22×Dic59—21no—
79C22×Dic59—41no—
80C2×D236—21no—
81C2×D236—21no—
82C2×D236—21no—
83C2×D236—21no—
84C2×D236—21no—
85C2×D236Q1612yes—
86C2×C4×D59—41no—
87C2×C4×D59—41no—
88C2×C4×D59—42no—
89C2×C4×D59—42no—
90C2×Dic118—22no—
91C2×Dic118—42no—
92C2×Dic118—42no—
93C2×Dic118—41no—
94C2×Dic118—41no—
95C2×Dic118—41no—
96C2×Dic118—42no—
97C2×Dic118—42no—
98C2×C8×C59—24no—
99C2×C8×C59—22no—
100C2×C8×C59—22no—
101C2×C8×C59—41no—
102C2×C8×C59—41no—
103C4⋊C4×C59—22no—
104C4⋊C4×C59D814yes—
105C42×C59—41no—
106C42×C59—41no—
107C42×C59C2×D411yescharacteristic
108C42×C59C2×D411yescharacteristic
109C42×C59C2×D411yescharacteristic
110C42×C59#4111yescharacteristic
111(C2×D59)⋊C4—48no—
112(C2×D59)⋊C4—41no—
113(C2×D59)⋊C4—41no—
114C4⋊Dic59—22no—
115C4⋊Dic59—22no—
116C4⋊Dic59—21no—
117C4⋊Dic59—22no—
118C4⋊Dic59—21no—
119C4⋊Dic59—22no—
120C4⋊Dic59—48no—
121C4⋊Dic59—41no—
122C4⋊Dic59—41no—
123C4⋊Dic59—41no—
124C4⋊Dic59—41no—
125C2×C59⋊C8—41no—
126C2×C59⋊C8—42no—
127C23×C59#12011yescharacteristic
128C22×D59—22no—
129C22×D59—41no—
130C22×D59—42no—
131C22×D59—41no—
132C22×D59—42no—
133C2×C4×C59—22no—
134C2×C4×C59—24no—
135C2×C4×C59—42no—
136C2×C4×C59—42no—
137C2×C4×C59#12011yescharacteristic
138C2×C4×C59#12011yescharacteristic
139C2×C4×C59#12111yescharacteristic
140C2×C4×C59#12111yescharacteristic
141C2×C4×C59#12111yescharacteristic
142C2×C4×C59#12711yescharacteristic
143C2×C4×C59C2×D812yes—
144C2×C4×C59C2×Q1611yescharacteristic
145C2×C4×C59#13511yescharacteristic
146C2×C4×C59AGL(1,ℤ/8)12yes—
147C2×C4×C59#13711yescharacteristic
148C2×C4×C59#13711yescharacteristic
149C2×Dic59—42no—
150C2×Dic59—42no—
151C2×Dic59—41no—
152C2×Dic59—42no—
153C2×Dic59—44no—
154C2×Dic59—82no—
155D236—22no—
156D236—22no—
157D236—41no—
158D236—41no—
159C4×D59—82no—
160C4×D59—81no—
161C4×D59—81no—
162Dic118—44no—
163Dic118—82no—
164Dic118—81no—
165Dic118—81no—
166Dic118—81no—
167Dic118—81no—
168C8×C59—24no—
169C8×C59—42no—
170C59⋊C8—81no—
171#10076—591nomaximal, Sylow
172C22×C59#39211yescharacteristic
173C22×C59#44712yes—
174C22×C59#45111yescharacteristic
175C22×C59#45211yescharacteristic
176C22×C59#45612yes—
177C2×D59—44no—
178C2×D59—42no—
179C2×D59—82no—
180C2×D59—81no—
181C2×D59—81no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—44no—
185C4×C59#45111yescharacteristic
186C4×C59#45211yescharacteristic
187C4×C59#45912yes—
188C4×C59#46711yescharacteristic
189C4×C59#47011yescharacteristic
190C4×C59#49311yescharacteristic
191C4×C59#49611yescharacteristic
192Dic59—82no—
193Dic59—82no—
194Dic59—81no—
195C2×#500—591no—
196C2×#474—591no—
197#2542—591no—
198#1981—591no—
199C2×#332—591no—
200C2×D4⋊C8—591no—
201#1279D5911yescharacteristic, complemented
202C2×C59#164712yes—
203C2×C59#164912yes—
204C2×C59#200111yescharacteristic
205C2×C59#200511yescharacteristic
206C2×C59#201111yescharacteristic
207D59—82no—
208C2×C4⋊Q8—591no—
209C2×C4⋊Q8—591no—
210C2×C22⋊Q8—1182no—
211C2×C4×D4—591no—
212#500—1182no—
213#500—1182no—
214#474—1182no—
215#474—1182no—
216C2×#107—1182no—
217C2×#107—1181no—
218C2×C4⋊C8—591no—
219C2×C4⋊C8C2×D5911yescharacteristic
220C2×#103—1182no—
221C2×#102—1181no—
222C2×C4×C8C2×D5911yescharacteristic
223#399—1182no—
224#390—1181no—
225#384—1181no—
226C4×C4⋊C4C2×D5911yescharacteristic
227#332—1182no—
228D4⋊C8—1182no—
229C59#1007611yescharacteristic, complemented, Sylow
230C22×Q8—1181no—
231C22×Q8—1181no—
232C22×D4—591no—
233C23×C4—1181no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—1181no—
237C4⋊Q8—592no—
238C4⋊Q8—1181no—
239C4⋊Q8—1181no—
240C4⋊Q8—1181no—
241C4⋊Q8—1181no—
242C4⋊Q8—1181no—
243C4⋊Q8—1181no—
244C4⋊Q8—1181no—
245C22⋊Q8—2364no—
246C22⋊Q8—2364no—
247C4×D4—1181no—
248C4×D4—1181no—
249C4×D4—1181no—
250C4×D4—1181no—
251C2×C4⋊C4—591no—
252C2×C4⋊C4—592no—
253C2×C4⋊C4—1181no—
254C2×C4⋊C4—1181no—
255C2×C4⋊C4—1184no—
256C2×C4⋊C4—1181no—
257C2×C4⋊C4D23612yes—
258C2×C22⋊C4—1184no—
259C2×C22⋊C4—1181no—
260C2×C42—21no—
261C2×C42C22×D5911yescharacteristic
262#107—1184no—
263#107—1188no—
264C4⋊C8—21no—
265C4⋊C8—1181no—
266C4⋊C8—1181no—
267C4⋊C8D23612yes—
268#103—2364no—
269#102—1184no—
270C4×C8—21no—
271C4×C8D23612yes—
272#95—21no—
273C24—1181no—
274C2×Q8—1182no—
275C2×Q8—2362no—
276C2×Q8—2362no—
277C2×Q8—2361no—
278C2×Q8—2361no—
279C2×Q8—2361no—
280C2×Q8—2362no—
281C2×Q8—2362no—
282C2×D4—592no—
283C2×D4—1181no—
284C2×D4—1181no—
285C2×D4—1181no—
286C2×D4—1181no—
287C2×D4—1181no—
288C22×C4—22no—
289C22×C4—1182no—
290C22×C4—1181no—
291C22×C4—2361no—
292C22×C4—2361no—
293C22×C4—2361no—
294C22×C4—2362no—
295C22×C4—2362no—
296C22×C4C2×D23611yescharacteristic
297C22×C4D4×D5911yescharacteristic
298C22×C4D4×D5911yescharacteristic
299C2×C8—24no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—2361no—
305C2×C8—2362no—
306C4⋊C4—22no—
307C4⋊C4—1182no—
308C4⋊C4—1181no—
309C4⋊C4—1182no—
310C4⋊C4—1182no—
311C4⋊C4—1181no—
312C4⋊C4—1182no—
313C4⋊C4—2368no—
314C4⋊C4—2361no—
315C4⋊C4—2361no—
316C4⋊C4—2361no—
317C4⋊C4—2361no—
318C4⋊C4D47214yes—
319C22⋊C4—2368no—
320C22⋊C4—2361no—
321C22⋊C4—2361no—
322C42—41no—
323C42—41no—
324C42C2×D23611yescharacteristic
325C42C2×D23611yescharacteristic
326C42D4×D5911yescharacteristic
327C42#1174568911yescharacteristic
328C23—1182no—
329C23—2362no—
330C23—2361no—
331C23—2361no—
332C23—2362no—
333C23#1427369711yescharacteristic
334Q8—4721no—
335Q8—4721no—
336Q8—4721no—
337Q8—4721no—
338Q8—4722no—
339Q8—2364no—
340D4—1182no—
341D4—1182no—
342D4—2361no—
343D4—2361no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—4722no—
349C2×C4—4721no—
350C2×C4—4721no—
351C2×C4—4722no—
352C2×C4—2364no—
353C2×C4—2362no—
354C2×C4—2361no—
355C2×C4—2362no—
356C2×C4—2362no—
357C2×C4#1427369711yescharacteristic
358C2×C4#1427369711yescharacteristic
359C2×C4#1427370211yescharacteristic
360C2×C4#1427370211yescharacteristic
361C2×C4#1427370211yescharacteristic
362C2×C4#1427372411yescharacteristic
363C2×C4C2×D47212yes—
364C2×C4Q16×D5911yescharacteristic
365C2×C4#1427374611yescharacteristic
366C2×C4#1427375512yes—
367C2×C4#1427375611yescharacteristic
368C2×C4#1427375611yescharacteristic
369C8—24no—
370C8—42no—
371C8—4721no—
372C22—4721no—
373C22—4722no—
374C22—4721no—
375C22—2362no—
376C22—2364no—
377C22#1683731211yescharacteristic
378C22#1683753812yes—
379C22#1683755911yescharacteristic
380C22#1683756511yescharacteristic
381C22#1683758612yes—
382C4—21no—
383C4—21no—
384C4—44no—
385C4—4721no—
386C4—4722no—
387C4—4722no—
388C4#1683755911yescharacteristic
389C4#1683756511yescharacteristic
390C4#1683759812yes—
391C4#1683764411yescharacteristic
392C4#1683766111yescharacteristic
393C4#1683777911yescharacteristic
394C4#1683779111yescharacteristic
395C2—4722no—
396C2#2142333312yes—
397C2#2142334712yes—
398C2#2142510711yescharacteristic
399C2#2142513511yescharacteristic
400C2#2142516911yescharacteristic
subgroupG/H
centerC23#14273697
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1279C2
socleC23×C59#120
prime radicalC22×D236D4
hypercenter#1279D59
hypocenterC59#10076
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
1. C2×#16837807C2semidirect
4. #21425019C2semidirect
5. C2×#16837147C2semidirect
7. C59×#1279C2semidirect
201. #1279D59semidirect
229. C59#10076semidirect

Maximal subgroups

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

subgroupindex
C2×#168378072
C2×#168376832
#214290452
#214250192
C2×#168371472
C2×#168371312
C59×#12792
#1007659