Finite Group Databasebeta

← #28744256

Subgroups — #28744256

Order 15104.

554 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434666C211yesmaximal, characteristic, complemented
2#21426379C211yesmaximal, characteristic, complemented
3#21426360C211yesmaximal, characteristic, complemented
4#21426337C211yesmaximal, characteristic, complemented
5#21426335C211yesmaximal, characteristic, complemented
6#21426207C212yesmaximal, complemented
7#21426197C212yesmaximal, complemented
8C4×#14273671C212yesmaximal, complemented
9C4×#14273668C212yesmaximal, complemented
10C4×#14273667C211yesmaximal, characteristic, complemented
11C59×#3169C211yesmaximal, characteristic, complemented
12#16837936—21no—
13C2×#14273671C2212yes—
14C2×#14273668C2212yes—
15C2×#14273667C2212yes—
16#16837804C2211yescharacteristic
17#16837786C2211yescharacteristic
18#16837757—21no—
19#16837643C2211yescharacteristic
20#16837638C2211yescharacteristic
21#16837631—21no—
22C4×Dic236C2211yescharacteristic
23C4×#11745660—21no—
24C4×D472C2211yescharacteristic
25#16837229C2212yes—
26#16837227C2212yes—
27#16837205C2212yes—
28#16837203C2212yes—
29#16837203C2212yes—
30#16837202C2212yes—
31#16837178C2211yescharacteristic
32#16837178C2211yescharacteristic
33C4×C59⋊C16C2212yes—
34C59×#532—21no—
35C59×#500C2211yescharacteristic
36C59×#493C2211yescharacteristic
37C59×#486—21no—
38C59×#466—21no—
39C59×#466C2212yes—
40C4×D8×C59—21no—
41C4×D8×C59C2212yes—
42C2×C4×C8×C59C2211yescharacteristic
43C2×Dic236C2311yescharacteristic
44C2×#11745660—21no—
45C2×D472C2311yescharacteristic
46#14273721—41no—
47#14273719—21no—
48#14273715—21no—
49#14273710—41no—
50#14273706—21no—
51#14273701—21no—
52C4×Dic118—21no—
53C4×D236—21no—
54C4×C22⋊D59—41no—
55#14273684—41no—
56#14273671—24no—
57#14273668—24no—
58#14273667—24no—
59C2×C59⋊C16C2312yes—
60C2×C59⋊C16C2312yes—
61#14273661C2311yescharacteristic
62#14273661C2311yescharacteristic
63#14273659—21no—
64#14273652—21no—
65#14273652—21no—
66#14273649—21no—
67#14273649—21no—
68C2×#41×C59—41no—
69C2×D8×C59—21no—
70C2×D8×C59C2312yes—
71C22×C8×C59D411yescharacteristic
72C22×C8×C59C2311yescharacteristic
73C4⋊Q8×C59—21no—
74#127×C59—21no—
75#124×C59—41no—
76#121×C59—22no—
77#121×C59—41no—
78C4×D4×C59—22no—
79C4×D4×C59—41no—
80C2×C42×C59D411yescharacteristic
81#107×C59—21no—
82#107×C59C2312yes—
83#102×C59—22no—
84#102×C59—22no—
85C4×C8×C59D411yescharacteristic
86C4×C8×C59D411yescharacteristic
87C4×C8×C59C2311yescharacteristic
88C4×C8×C59C2311yescharacteristic
89#41×C59—84no—
90C2×Q8×C59—41no—
91C2×D4×C59—22no—
92C2×D4×C59—42no—
93C2×D4×C59—41no—
94C2×D4×C59—41no—
95C22×C4×C59—42no—
96C22×C4×C59D811yescharacteristic
97C22×C4×C59D811yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99C2×C22⋊D59—41no—
100C2×D236—21no—
101C2×C4×D59—41no—
102C2×Dic118—21no—
103D8×C59—22no—
104D8×C59—22no—
105D8×C59—42no—
106C2×C8×C59—22no—
107C2×C8×C59—22no—
108C2×C8×C59C2×D411yescharacteristic
109C2×C8×C59C2×D411yescharacteristic
110C2×C8×C59C2×D411yescharacteristic
111C2×C8×C59C2411yescharacteristic
112C4⋊C4×C59—22no—
113C4⋊C4×C59—41no—
114C22⋊C4×C59—42no—
115C22⋊C4×C59—42no—
116C42×C59D811yescharacteristic
117C42×C59D811yescharacteristic
118C42×C59C2×D411yescharacteristic
119C42×C59C2×D411yescharacteristic
120C22⋊Dic59—41no—
121(C2×D59)⋊C4—41no—
122(C2×D59)⋊C4—41no—
123(C2×D59)⋊C4—41no—
124(C2×D59)⋊C4—41no—
125C4⋊Dic59—21no—
126C4⋊Dic59—21no—
127Dic59⋊C4—41no—
128Dic59⋊C4—41no—
129Dic59⋊C4—41no—
130Dic59⋊C4—41no—
131C4×Dic59—41no—
132Dic236—22no—
133D472—22no—
134#11745660—42no—
135C59⋊C16—22no—
136C59⋊C16#4114yescomplemented
137C23×C59—42no—
138C23×C59C2×D811yescharacteristic
139C22×D59—41no—
140Q8×C59—82no—
141D4×C59—42no—
142D4×C59—42no—
143D4×C59—82no—
144D4×C59—82no—
145D4×C59—84no—
146C2×C4×C59—21no—
147C2×C4×C59—21no—
148C2×C4×C59—22no—
149C2×C4×C59—41no—
150C2×C4×C59—42no—
151C2×C4×C59—84no—
152C2×C4×C59C2×D811yescharacteristic
153C2×C4×C59C2×D811yescharacteristic
154C2×C4×C59C2×D811yescharacteristic
155C2×C4×C59C2×D811yescharacteristic
156C2×C4×C59C2×D811yescharacteristic
157C2×C4×C59C22×D411yescharacteristic
158C22⋊D59—82no—
159C2×Dic59—41no—
160C2×Dic59—41no—
161C2×Dic59—41no—
162D236—41no—
163D236—41no—
164C4×D59—82no—
165Dic118—41no—
166Dic118—41no—
167C8×C59—21no—
168C8×C59—21no—
169C8×C59#13512yes—
170C8×C59C2×#4112yes—
171#33737—591nomaximal, Sylow
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—82no—
176C22×C59—82no—
177C22×C59—82no—
178C22×C59C22×D811yescharacteristic
179C2×D59—81no—
180C2×D59—81no—
181C2×D59—81no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—21no—
186C4×C59—82no—
187C4×C59#50912yes—
188C4×C59#53211yescharacteristic
189C4×C59C2×#13511yescharacteristic
190Dic59—81no—
191Dic59—81no—
192Dic59—81no—
193#3169D5911yescharacteristic, complemented
194#3168—591no—
195#2255—591no—
196#2250—591no—
197#2244—591no—
198#2243—591no—
199#2220—592no—
200#2218—592no—
201C4×SD32—592no—
202C4×D16—592no—
203C4×M32—591no—
204C2×C59—41no—
205C2×C59—162no—
206C2×C59#316911yescharacteristic
207C2×C59C2×#50911yescharacteristic
208C2×C59#341411yescharacteristic
209D59—161no—
210#532—21no—
211#529—1181no—
212C2×SD32—592no—
213C2×D16—592no—
214C2×M32—592no—
215#500C2×D5911yescharacteristic
216#499—591no—
217#495—591no—
218#493C2×D5911yescharacteristic
219#488—1181no—
220#486—21no—
221#467—591no—
222#466—21no—
223#466—591no—
224#466C2×D5912yes—
225#465—1181no—
226C4×Q16—591no—
227C4×SD16—1181no—
228C4×D8—21no—
229C4×D8—591no—
230C4×D8C2×D5912yes—
231C2×C4×C8C2×D5911yescharacteristic
232#367—592no—
233#366—592no—
234#358—592no—
235#357—592no—
236#357—592no—
237#357—592no—
238#346—591no—
239#346—591no—
240C4×C16—592no—
241C59#3373711yescharacteristic, complemented, Sylow
242C2×#41—41no—
243C2×Q16—591no—
244C2×SD16—1181no—
245C2×D8—21no—
246C2×D8—591no—
247C2×D8C22×D5912yes—
248C22×C8C22⋊D5911yescharacteristic
249C22×C8C22×D5911yescharacteristic
250C4⋊Q8—21no—
251#127—21no—
252#126—2361no—
253#125—1181no—
254#124—41no—
255#124—1181no—
256#123—2361no—
257C22⋊Q8—1181no—
258#121—22no—
259#121—41no—
260#121—1181no—
261C4×Q8—1181no—
262C4×D4—22no—
263C4×D4—41no—
264C4×D4—1181no—
265C4×D4—2361no—
266#117—2361no—
267C2×C42C22⋊D5911yescharacteristic
268SD32—1184no—
269D16—1184no—
270M32—1184no—
271C2×C16—592no—
272C2×C16—592no—
273#107—21no—
274#107—591no—
275#107—591no—
276#107C22×D5912yes—
277#106—1181no—
278#103—1181no—
279#103—1181no—
280#102—22no—
281#102—22no—
282#102—1181no—
283#102—1181no—
284C4×C8C22⋊D5911yescharacteristic
285C4×C8C22⋊D5911yescharacteristic
286C4×C8C22×D5911yescharacteristic
287C4×C8C22×D5911yescharacteristic
288#41—84no—
289C2×Q8—41no—
290C2×Q8—1181no—
291C2×D4—22no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—1181no—
296C2×D4—2361no—
297C22×C4—42no—
298C22×C4—2361no—
299C22×C4#1174566911yescharacteristic
300C22×C4#1174566911yescharacteristic
301C22×C4C2×C22⋊D5911yescharacteristic
302Q16—1182no—
303SD16—2362no—
304D8—22no—
305D8—22no—
306D8—42no—
307D8—1182no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8C2×C22⋊D5911yescharacteristic
311C2×C8C2×C22⋊D5911yescharacteristic
312C2×C8C2×C22⋊D5911yescharacteristic
313C2×C8C23×D5911yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—1181no—
317C4⋊C4—1181no—
318C4⋊C4—2361no—
319C4⋊C4—2361no—
320C4⋊C4—2361no—
321C4⋊C4—2361no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—2361no—
325C22⋊C4—2361no—
326C22⋊C4—2361no—
327C22⋊C4—2361no—
328C22⋊C4—2361no—
329C42—2361no—
330C42#1174566911yescharacteristic
331C42#1174566911yescharacteristic
332C42C2×C22⋊D5911yescharacteristic
333C42C2×C22⋊D5911yescharacteristic
334C16—594no—
335C16—1182no—
336C23—42no—
337C23—2361no—
338C23C2×#1174566911yescharacteristic
339Q8—82no—
340Q8—2361no—
341Q8—2361no—
342D4—42no—
343D4—42no—
344D4—82no—
345D4—82no—
346D4—84no—
347D4—4722no—
348D4—2361no—
349D4—2361no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—84no—
356C2×C4—4722no—
357C2×C4—2361no—
358C2×C4—2361no—
359C2×C4—2361no—
360C2×C4C2×#1174566911yescharacteristic
361C2×C4C2×#1174566911yescharacteristic
362C2×C4C2×#1174566911yescharacteristic
363C2×C4C2×#1174566911yescharacteristic
364C2×C4C2×#1174566911yescharacteristic
365C2×C4C22×C22⋊D5911yescharacteristic
366C8—21no—
367C8—21no—
368C8#1427374712yes—
369C8#41×D5912yes—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22—4721no—
377C22—4721no—
378C22—4721no—
379C22C22×#1174566911yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—4721no—
386C4—4721no—
387C4—4721no—
388C4#1683783812yes—
389C4#1683794811yescharacteristic
390C4C2×#1427374711yescharacteristic
391C2—41no—
392C2—162no—
393C2—9441no—
394C2#2143467111yescharacteristic
395C2C2×#1683783811yescharacteristic
396C2#2143723011yescharacteristic
subgroupG/H
centerC2×C4C2×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3169C2
socleC22×C59C22×D8
hypercenter#3169D59
hypocenterC59#33737
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. #21434666C2semidirect
2. #21426379C2semidirect
3. #21426360C2semidirect
4. #21426337C2semidirect
5. #21426335C2semidirect
6. #21426207C2semidirect
7. #21426197C2semidirect
8. C4×#14273671C2semidirect
9. C4×#14273668C2semidirect
10. C4×#14273667C2semidirect
11. C59×#3169C2semidirect
136. C59⋊C16#41semidirect
193. #3169D59semidirect
241. C59#33737semidirect

Maximal subgroups

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

subgroupindex
#214346662
#214263792
#214263602
#214263372
#214263352
#214262072
#214261972
C4×#142736712
C4×#142736682
C4×#142736672
C59×#31692
#3373759