Finite Group Databasebeta

← #28744263

Subgroups — #28744263

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#21434675C211yesmaximal, characteristic, complemented
2#21426383C211yesmaximal, characteristic, complemented
3#21426363C211yesmaximal, characteristic, complemented
4#21426339C211yesmaximal, characteristic, complemented
5#21426332C211yesmaximal, characteristic, complemented
6#21426211C212yesmaximal, complemented
7#21426197C212yesmaximal, complemented
8C4×#14273670C212yesmaximal, complemented
9C4×#14273668C212yesmaximal, complemented
10C4×#14273667C211yesmaximal, characteristic, complemented
11C59×#3168C211yesmaximal, characteristic, complemented
12#16837952—21no—
13C2×#14273670C2212yes—
14C2×#14273668C2212yes—
15C2×#14273667C2212yes—
16#16837807C2211yescharacteristic
17#16837779C2211yescharacteristic
18#16837749—21no—
19#16837638—21no—
20#16837638C2212yes—
21C4×D472—21no—
22C4×D472C2212yes—
23#16837229C2212yes—
24#16837227C2212yes—
25#16837206C2212yes—
26#16837203C2212yes—
27#16837202C2212yes—
28#16837202C2212yes—
29#16837178C2211yescharacteristic
30#16837178C2211yescharacteristic
31C4×C59⋊C16C2212yes—
32C59×#529—21no—
33C59×#499C2211yescharacteristic
34C59×#495C2211yescharacteristic
35C59×#488—21no—
36C59×#467C2211yescharacteristic
37C59×#466C2211yescharacteristic
38C59×#465—21no—
39C4×Q16×C59C2211yescharacteristic
40C4×SD16×C59—21no—
41C4×D8×C59C2211yescharacteristic
42C2×C4×C8×C59C2211yescharacteristic
43C2×#11745685—41no—
44C2×D472—21no—
45C2×D472C2312yes—
46#14273727—21no—
47#14273724—21no—
48#14273715—41no—
49#14273701—22no—
50#14273701—41no—
51C4×D236—22no—
52C4×D236—41no—
53#14273670—24no—
54#14273668—24no—
55#14273667—24no—
56C2×C59⋊C16C2312yes—
57C2×C59⋊C16C2312yes—
58#14273661—21no—
59#14273661C2312yes—
60#14273649—22no—
61#14273649—22no—
62C2×Q16×C59C2311yescharacteristic
63C2×SD16×C59—21no—
64C2×D8×C59C2311yescharacteristic
65C22×C8×C59D411yescharacteristic
66C22×C8×C59C2311yescharacteristic
67#126×C59—41no—
68#125×C59—21no—
69#124×C59—21no—
70#123×C59—41no—
71C22⋊Q8×C59—21no—
72#121×C59—21no—
73C4×Q8×C59—21no—
74C4×D4×C59—21no—
75C4×D4×C59—41no—
76#117×C59—41no—
77C2×C42×C59D411yescharacteristic
78#107×C59C2311yescharacteristic
79#107×C59C2311yescharacteristic
80#106×C59—21no—
81#103×C59—21no—
82#103×C59—21no—
83#102×C59—21no—
84#102×C59—21no—
85C4×C8×C59D411yescharacteristic
86C4×C8×C59D411yescharacteristic
87C4×C8×C59C2311yescharacteristic
88C4×C8×C59C2311yescharacteristic
89C2×Q8×C59—21no—
90C2×D4×C59—21no—
91C2×D4×C59—41no—
92C22×C4×C59—41no—
93C22×C4×C59D811yescharacteristic
94C22×C4×C59D811yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C2×C22⋊D59—42no—
97#11745685—84no—
98C2×D236—22no—
99C2×D236—41no—
100C2×D236—41no—
101C2×C4×D59—42no—
102C2×Dic118—41no—
103Q16×C59—22no—
104SD16×C59—42no—
105D8×C59—22no—
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—21no—
113C4⋊C4×C59—21no—
114C4⋊C4×C59—41no—
115C4⋊C4×C59—41no—
116C4⋊C4×C59—41no—
117C4⋊C4×C59—41no—
118C22⋊C4×C59—41no—
119C22⋊C4×C59—41no—
120C22⋊C4×C59—41no—
121C22⋊C4×C59—41no—
122C22⋊C4×C59—41no—
123C42×C59—41no—
124C42×C59D811yescharacteristic
125C42×C59D811yescharacteristic
126C42×C59C2×D411yescharacteristic
127C42×C59C2×D411yescharacteristic
128(C2×D59)⋊C4—42no—
129(C2×D59)⋊C4—42no—
130C4⋊Dic59—22no—
131C4⋊Dic59—41no—
132D472—22no—
133D472—22no—
134D472—42no—
135C59⋊C16—22no—
136C59⋊C16#4114yescomplemented
137C23×C59—41no—
138C23×C59C2×D811yescharacteristic
139C22×D59—42no—
140Q8×C59—41no—
141Q8×C59—41no—
142D4×C59—41no—
143D4×C59—41no—
144D4×C59—82no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—22no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—82no—
153C2×C4×C59C2×D811yescharacteristic
154C2×C4×C59C2×D811yescharacteristic
155C2×C4×C59C2×D811yescharacteristic
156C2×C4×C59C2×D811yescharacteristic
157C2×C4×C59C2×D811yescharacteristic
158C2×C4×C59C22×D411yescharacteristic
159C22⋊D59—84no—
160C2×Dic59—42no—
161D236—42no—
162D236—42no—
163D236—82no—
164D236—82no—
165C4×D59—84no—
166Dic118—82no—
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—81no—
176C22×C59—81no—
177C22×C59—81no—
178C22×C59C22×D811yescharacteristic
179C2×D59—82no—
180C2×D59—82no—
181C2×D59—82no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—21no—
186C4×C59—81no—
187C4×C59—81no—
188C4×C59—81no—
189C4×C59#50912yes—
190C4×C59#53211yescharacteristic
191C4×C59C2×#13511yescharacteristic
192Dic59—82no—
193#3169—591no—
194#3168D5911yescharacteristic, complemented
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—161no—
206C2×C59#316911yescharacteristic
207C2×C59C2×#50911yescharacteristic
208C2×C59#341411yescharacteristic
209D59—162no—
210#532—1181no—
211#529—21no—
212C2×SD32—592no—
213C2×D16—592no—
214C2×M32—592no—
215#500—591no—
216#499C2×D5911yescharacteristic
217#495C2×D5911yescharacteristic
218#493—591no—
219#488—21no—
220#486—1181no—
221#467C2×D5911yescharacteristic
222#466—592no—
223#466—1181no—
224#466C2×D5911yescharacteristic
225#465—21no—
226C4×Q16C2×D5911yescharacteristic
227C4×SD16—21no—
228C4×D8—592no—
229C4×D8—1181no—
230C4×D8C2×D5911yescharacteristic
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—2361no—
243C2×Q16C22×D5911yescharacteristic
244C2×SD16—21no—
245C2×D8—592no—
246C2×D8—1181no—
247C2×D8C22×D5911yescharacteristic
248C22×C8C22⋊D5911yescharacteristic
249C22×C8C22×D5911yescharacteristic
250C4⋊Q8—1181no—
251#127—1181no—
252#126—41no—
253#125—21no—
254#124—21no—
255#124—2361no—
256#123—41no—
257C22⋊Q8—21no—
258#121—21no—
259#121—1182no—
260#121—2361no—
261C4×Q8—21no—
262C4×D4—21no—
263C4×D4—41no—
264C4×D4—1182no—
265C4×D4—2361no—
266#117—41no—
267C2×C42C22⋊D5911yescharacteristic
268SD32—1184no—
269D16—1184no—
270M32—1184no—
271C2×C16—592no—
272C2×C16—592no—
273#107—592no—
274#107—1181no—
275#107C22×D5911yescharacteristic
276#107C22×D5911yescharacteristic
277#106—21no—
278#103—21no—
279#103—21no—
280#102—21no—
281#102—21no—
282#102—1182no—
283#102—1182no—
284C4×C8C22⋊D5911yescharacteristic
285C4×C8C22⋊D5911yescharacteristic
286C4×C8C22×D5911yescharacteristic
287C4×C8C22×D5911yescharacteristic
288#41—4724no—
289C2×Q8—21no—
290C2×Q8—2361no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—1182no—
294C2×D4—2362no—
295C2×D4—2361no—
296C2×D4—2361no—
297C22×C4—41no—
298C22×C4—2362no—
299C22×C4#1174566911yescharacteristic
300C22×C4#1174566911yescharacteristic
301C22×C4C2×C22⋊D5911yescharacteristic
302Q16—22no—
303SD16—42no—
304D8—22no—
305D8—1182no—
306D8—1182no—
307D8—2362no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8C2×C22⋊D5911yescharacteristic
311C2×C8C2×C22⋊D5911yescharacteristic
312C2×C8C2×C22⋊D5911yescharacteristic
313C2×C8C23×D5911yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—1182no—
321C4⋊C4—2361no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—2362no—
328C22⋊C4—2362no—
329C42—41no—
330C42#1174566911yescharacteristic
331C42#1174566911yescharacteristic
332C42C2×C22⋊D5911yescharacteristic
333C42C2×C22⋊D5911yescharacteristic
334C16—594no—
335C16—1182no—
336C23—41no—
337C23—2362no—
338C23C2×#1174566911yescharacteristic
339Q8—41no—
340Q8—41no—
341Q8—4722no—
342D4—41no—
343D4—41no—
344D4—82no—
345D4—4722no—
346D4—4722no—
347D4—4724no—
348D4—2362no—
349D4—2362no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—82no—
358C2×C4—4724no—
359C2×C4—2362no—
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—81no—
374C22—81no—
375C22—81no—
376C22—4722no—
377C22—4722no—
378C22—4722no—
379C22C22×#1174566911yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—4722no—
388C4#1683783712yes—
389C4#1683794811yescharacteristic
390C4C2×#1427374711yescharacteristic
391C2—41no—
392C2—161no—
393C2—9442no—
394C2#2143467111yescharacteristic
395C2C2×#1683783711yescharacteristic
396C2#2143723011yescharacteristic
subgroupG/H
centerC2×C4C2×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3168C2
socleC22×C59C22×D8
hypercenter#3168D59
hypocenterC59#33737
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. #21434675C2semidirect
2. #21426383C2semidirect
3. #21426363C2semidirect
4. #21426339C2semidirect
5. #21426332C2semidirect
6. #21426211C2semidirect
7. #21426197C2semidirect
8. C4×#14273670C2semidirect
9. C4×#14273668C2semidirect
10. C4×#14273667C2semidirect
11. C59×#3168C2semidirect
136. C59⋊C16#41semidirect
194. #3168D59semidirect
241. C59#33737semidirect

Maximal subgroups

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

subgroupindex
#214346752
#214263832
#214263632
#214263392
#214263322
#214262112
#214261972
C4×#142736702
C4×#142736682
C4×#142736672
C59×#31682
#3373759