Finite Group Databasebeta

← #28485101

Subgroups — #28485101

Order 15104.

534 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837602C211yesmaximal, characteristic
2#21424936C211yesmaximal, characteristic
3#21422765C211yesmaximal, characteristic
4#21422525C211yesmaximal, characteristic, complemented
5#21421970C211yesmaximal, characteristic, complemented
6#21421947C211yesmaximal, characteristic, complemented
7Q8×M16×C59C211yesmaximal, characteristic, complemented
8C22×#11745671C412yes—
9C2×#14273724C2211yescharacteristic
10#16837602—21no—
11#16837602—21no—
12#16837602—21no—
13#16837602—21no—
14#16837455C2211yescharacteristic
15#16837426C2211yescharacteristic
16C2×C4⋊(C59⋊C8)C2211yescharacteristic
17C2×#14273647C412yes—
18C2×#14273643—21no—
19C2×#14273643—21no—
20#16837145—22no—
21Q8⋊(C59⋊C8)—21no—
22Q8⋊(C59⋊C8)—21no—
23C2×C4×Q8×C59C2211yescharacteristic
24C59×#446—21no—
25C59×#446—22no—
26C8×Q8×C59—21no—
27C59×#423—21no—
28C59×#423C2211yescharacteristic
29C4×M16×C59—21no—
30C4×M16×C59C2211yescharacteristic
31C22×D236—21no—
32C22×D236—21no—
33C22×D236C2×C412yes—
34C2×#11745671—22no—
35C2×#11745671—22no—
36C2×#11745671—44no—
37C2×#11745664—21no—
38C22×C59⋊C8C2×C412yes—
39#14273724—21no—
40#14273724—21no—
41#14273724—21no—
42#14273724—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)—21no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)—21no—
48#14273647—22no—
49#14273647—22no—
50#14273643—42no—
51#14273643—42no—
52C22⋊(C59⋊C8)—21no—
53C4×C59⋊C8—21no—
54C22×Q8×C59C2×C411yescharacteristic
55C2×M16×C59—21no—
56C2×M16×C59D411yescharacteristic
57C2×M16×C59D411yescharacteristic
58C4×Q8×C59—21no—
59C4×Q8×C59—21no—
60C4×Q8×C59D411yescharacteristic
61C4×Q8×C59D411yescharacteristic
62C4×Q8×C59D411yescharacteristic
63C4×Q8×C59D411yescharacteristic
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59C2×C411yescharacteristic
66C2×C42×C59—21no—
67C2×C42×C59C2311yescharacteristic
68C4⋊C8×C59—22no—
69C4⋊C8×C59—22no—
70C4⋊C8×C59—21no—
71C4⋊C8×C59D412yes—
72#97×C59—22no—
73#97×C59—21no—
74C4×C8×C59—22no—
75C4×C8×C59—21no—
76C23×D59—22no—
77C2×Q8×C59—21no—
78C2×Q8×C59—22no—
79C2×Q8×C59—41no—
80C2×Q8×C59C22⋊C411yescharacteristic
81C2×Q8×C59C22⋊C411yescharacteristic
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59C22×C411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59#4111yescharacteristic
87C2×D236—22no—
88C2×D236—22no—
89C2×D236—42no—
90C2×D236—42no—
91C2×D236—41no—
92C2×D236—41no—
93C2×D236—41no—
94C2×D236—41no—
95C2×D236—41no—
96C2×D236—41no—
97C2×D236—41no—
98C2×D236—41no—
99C2×D236—82no—
100M16×C59—21no—
101M16×C59—21no—
102M16×C59—21no—
103M16×C59—41no—
104M16×C59—41no—
105M16×C59SD1612yescomplemented
106C2×C8×C59—21no—
107C2×C8×C59—21no—
108C2×C8×C59—22no—
109C4⋊C4×C59—21no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—21no—
112C4⋊C4×C59—21no—
113C4⋊C4×C59—21no—
114C4⋊C4×C59C22⋊C411yescharacteristic
115C4⋊C4×C59C22⋊C411yescharacteristic
116C42×C59—21no—
117C42×C59—21no—
118C42×C59—21no—
119C42×C59—21no—
120C42×C59C2×D411yescharacteristic
121C42×C59C2×D411yescharacteristic
122C42×C59C2×D411yescharacteristic
123C42×C59#4111yescharacteristic
124#11745671—44no—
125#11745671—44no—
126#11745664—41no—
127#11745664—41no—
128C2×C59⋊C8—22no—
129C2×C59⋊C8—42no—
130C2×C59⋊C8—41no—
131C2×C59⋊C8—41no—
132C23×C59C4×D411yescharacteristic
133C22×D59—42no—
134C22×D59—42no—
135C22×D59—42no—
136C22×D59—82no—
137C22×D59—82no—
138Q8×C59—21no—
139Q8×C59—21no—
140Q8×C59—21no—
141Q8×C59—42no—
142Q8×C59#10012yescomplemented
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—41no—
150C2×C4×C59—42no—
151C2×C4×C59—81no—
152C2×C4×C59C2×C22⋊C411yescharacteristic
153C2×C4×C59C4×D411yescharacteristic
154C2×C4×C59#12011yescharacteristic
155C2×C4×C59#12111yescharacteristic
156C2×C4×C59#12111yescharacteristic
157C2×C4×C59#12311yescharacteristic
158C2×C4×C59C2×SD1611yescharacteristic
159C2×C4×C59#13511yescharacteristic
160C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C59#13711yescharacteristic
162D236—42no—
163D236—42no—
164D236—82no—
165D236—82no—
166D236—82no—
167D236—81no—
168D236—81no—
169D236—81no—
170D236—81no—
171C8×C59—22no—
172C8×C59—41no—
173C8×C59—42no—
174C59⋊C8—42no—
175C59⋊C8—81no—
176C59⋊C8—81no—
177#9254—591nomaximal, Sylow
178C22×C59—21no—
179C22×C59—21no—
180C22×C59#38611yescharacteristic
181C22×C59C4×SD1611yescharacteristic
182C22×C59#44011yescharacteristic
183C2×D59—82no—
184C2×D59—82no—
185C2×D59—82no—
186C2×D59—162no—
187C2×D59—162no—
188C4×C59—21no—
189C4×C59—22no—
190C4×C59—21no—
191C4×C59—21no—
192C4×C59—41no—
193C4×C59—41no—
194C4×C59—41no—
195C4×C59C2×#10011yescharacteristic
196C4×C59#41311yescharacteristic
197C4×C59#44811yescharacteristic
198C4×C59#44911yescharacteristic
199C4×C59#46011yescharacteristic
200C4×C59#46411yescharacteristic
201C2×#460—591no—
202Q8×M16D5911yescharacteristic, complemented
203#1965—591no—
204#1545—591no—
205#1487—591no—
206#1292—591no—
207#1283—591no—
208C2×C59—21no—
209C2×C59—21no—
210C2×C59#188011yescharacteristic
211C2×C59#190011yescharacteristic
212C2×C59#190311yescharacteristic
213D59—162no—
214C22×SD16—592no—
215C2×#127—591no—
216C2×C4×Q8C2×D5911yescharacteristic
217#460—1181no—
218#460—1181no—
219#460—1181no—
220#460—1181no—
221#446—21no—
222#446—22no—
223C8×Q8—21no—
224#431—591no—
225#423—21no—
226#423—591no—
227#423C2×D5911yescharacteristic
228C2×C4⋊C8—591no—
229C2×#102—592no—
230C2×#100—1181no—
231C2×#100—1181no—
232C4×M16—21no—
233C4×M16C2×D5911yescharacteristic
234#331—1182no—
235Q8⋊C8—1181no—
236Q8⋊C8—1181no—
237C59#925411yescharacteristic, complemented, Sylow
238C22×Q8C4×D5911yescharacteristic
239C22×D4—592no—
240C22×D4—1181no—
241C22×D4—1181no—
242C2×SD16—1182no—
243C2×SD16—1182no—
244C2×SD16—2364no—
245C2×M16—21no—
246C2×M16—1181no—
247C2×M16C22⋊D5911yescharacteristic
248C2×M16C22⋊D5911yescharacteristic
249C22×C8—592no—
250#127—1181no—
251#127—1181no—
252#127—1181no—
253#127—1181no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8D23611yescharacteristic
257C4×Q8D23611yescharacteristic
258C4×Q8C22⋊D5911yescharacteristic
259C4×Q8C22⋊D5911yescharacteristic
260C2×C4⋊C4—21no—
261C2×C4⋊C4C4×D5911yescharacteristic
262C2×C42—21no—
263C2×C42C22×D5911yescharacteristic
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—21no—
267C4⋊C8—1181no—
268C4⋊C8—1181no—
269C4⋊C8—1181no—
270C4⋊C8—1181no—
271C4⋊C8—1181no—
272C4⋊C8C22⋊D5912yes—
273#102—1182no—
274#102—1182no—
275#100—2362no—
276#100—2362no—
277C22⋊C8—1181no—
278#97—22no—
279#97—21no—
280C4×C8—22no—
281C4×C8—21no—
282C4×C8—1181no—
283C24—1182no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8(C2×D59)⋊C411yescharacteristic
288C2×Q8(C2×D59)⋊C411yescharacteristic
289C2×D4—1182no—
290C2×D4—1182no—
291C2×D4—4722no—
292C2×D4—2362no—
293C2×D4—2362no—
294C2×D4—2361no—
295C2×D4—2361no—
296C2×D4—2361no—
297C2×D4—2361no—
298C2×D4—2361no—
299C2×D4—2361no—
300C2×D4—2361no—
301C2×D4—2361no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4C2×C4×D5911yescharacteristic
305C22×C4#1174568511yescharacteristic
306C22×C4C2×C22⋊D5911yescharacteristic
307SD16—2364no—
308SD16—2364no—
309M16—21no—
310M16—21no—
311M16—21no—
312M16—41no—
313M16—41no—
314M16—2361no—
315M16—2361no—
316M16#1174567112yescomplemented
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—1182no—
321C2×C8—2361no—
322C2×C8—2361no—
323C2×C8—2362no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4(C2×D59)⋊C411yescharacteristic
330C4⋊C4(C2×D59)⋊C411yescharacteristic
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42C2×D23611yescharacteristic
336C42#1174568511yescharacteristic
337C42C2×C22⋊D5911yescharacteristic
338C42C2×C22⋊D5911yescharacteristic
339C23—4722no—
340C23—4722no—
341C23—2362no—
342C23—2362no—
343C23—2362no—
344C23C4×C22⋊D5911yescharacteristic
345Q8—21no—
346Q8—21no—
347Q8—21no—
348Q8—42no—
349Q8#1427364312yescomplemented
350D4—4722no—
351D4—4722no—
352D4—4722no—
353D4—4721no—
354D4—4721no—
355D4—4721no—
356D4—4721no—
357D4—2362no—
358D4—2362no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4C2×(C2×D59)⋊C411yescharacteristic
369C2×C4C4×C22⋊D5911yescharacteristic
370C2×C4#1427369611yescharacteristic
371C2×C4#1427370111yescharacteristic
372C2×C4#1427370111yescharacteristic
373C2×C4#1427371011yescharacteristic
374C2×C4C2×#1174567111yescharacteristic
375C2×C4#1427374711yescharacteristic
376C2×C4#1427375211yescharacteristic
377C2×C4#1427375811yescharacteristic
378C8—22no—
379C8—41no—
380C8—42no—
381C8—4721no—
382C8—4721no—
383C8—2362no—
384C22—21no—
385C22—21no—
386C22—9442no—
387C22—9442no—
388C22—4722no—
389C22—4722no—
390C22—4722no—
391C22#1683728111yescharacteristic
392C22C4×#1174567111yescharacteristic
393C22#1683750011yescharacteristic
394C4—21no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4C2×#1427364311yescharacteristic
402C4#1683738411yescharacteristic
403C4#1683754111yescharacteristic
404C4#1683754711yescharacteristic
405C4#1683760211yescharacteristic
406C4#1683762411yescharacteristic
407C2—21no—
408C2—21no—
409C2—9442no—
410C2#2142445111yescharacteristic
411C2#2142455911yescharacteristic
412C2#2142457611yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingQ8×M16×C59C2
socleC22×C59#386
prime radicalC2×#14273724C22
cosocleC2×C42×C59C23
hypercenterQ8×M16D59
hypocenterC59#9254
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
4. #21422525C2semidirect
5. #21421970C2semidirect
6. #21421947C2semidirect
7. Q8×M16×C59C2semidirect
105. M16×C59SD16semidirect
142. Q8×C59#100semidirect
202. Q8×M16D59semidirect
237. C59#9254semidirect
316. M16#11745671semidirect
349. Q8#14273643semidirect

Maximal subgroups

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

subgroupindex
C2×#168376022
#214249362
#214227652
#214225252
#214219702
#214219472
Q8×M16×C592
#925459