Finite Group Databasebeta

← #28489806

Subgroups — #28489806

Order 15104.

702 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×#16837776C211yesmaximal, characteristic
2#21429095C211yesmaximal, characteristic
3#21425045C211yesmaximal, characteristic, complemented
4C2×#16837145C211yesmaximal, characteristic
5C2×Q8⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
6#21421935C211yesmaximal, characteristic, complemented
7C2×C59×#477C211yesmaximal, characteristic, complemented
8C22×#11745671—22no—
9C2×#14273724C2211yescharacteristic
10C2×#14273716—21no—
11#16837776—22no—
12#16837776—22no—
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×C4×C59⋊C8C2211yescharacteristic
15#16837319—22no—
16#16837304—22no—
17#16837303—21no—
18C4×C4⋊Dic59C2211yescharacteristic
19#16837164—22no—
20#16837145—22no—
21Q8⋊(C59⋊C8)—22no—
22C2×C4⋊Q8×C59C2211yescharacteristic
23C2×C4×Q8×C59C2211yescharacteristic
24C59×#477—22no—
25C59×#477—22no—
26C2×#107×C59—21no—
27C2×C4⋊C8×C59C2211yescharacteristic
28C2×#103×C59—21no—
29C22×D236—21no—
30C22×D236—21no—
31C22×D236D412yes—
32C2×#11745671—28no—
33C2×#11745671—48no—
34C22×C59⋊C8—21no—
35C22×C59⋊C8—21no—
36#14273724—21no—
37#14273724—21no—
38#14273724—21no—
39#14273724—21no—
40#14273716—42no—
41#14273716—42no—
42C2×C4⋊Dic59D412yes—
43C2×(C2×D59)⋊C4—24no—
44C2×(C2×D59)⋊C4—24no—
45C2×C4×Dic59—21no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)—21no—
48C4×C59⋊C8—21no—
49C4×C59⋊C8D412yes—
50C59⋊#95—21no—
51C22×Q8×C59—21no—
52C22×Q8×C59D411yescharacteristic
53C22×C8×C59—21no—
54C4⋊Q8×C59—21no—
55C4⋊Q8×C59—21no—
56C4⋊Q8×C59—21no—
57C4⋊Q8×C59D412yes—
58C4×Q8×C59—21no—
59C4×Q8×C59—21no—
60C4×Q8×C59—21no—
61C4×Q8×C59—21no—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59—21no—
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59D411yescharacteristic
66C2×C42×C59—21no—
67C2×C42×C59C2311yescharacteristic
68#107×C59—24no—
69C4⋊C8×C59—21no—
70C4⋊C8×C59D412yes—
71#103×C59—24no—
72C23×D59—22no—
73C2×Q8×C59—21no—
74C2×Q8×C59—21no—
75C2×Q8×C59—22no—
76C2×Q8×C59—21no—
77C2×Q8×C59—21no—
78C2×Q8×C59—21no—
79C2×Q8×C59—44no—
80C2×Q8×C59—41no—
81C2×Q8×C59SD1612yescomplemented
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—41no—
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×Dic59—22no—
89C2×D236—22no—
90C2×D236—22no—
91C2×D236—44no—
92C2×D236—41no—
93C2×D236—41no—
94C2×D236—41no—
95C2×D236—41no—
96C2×D236—41no—
97C2×D236—41no—
98C2×D236—42no—
99C2×D236—41no—
100C2×D236—41no—
101C2×D236—42no—
102C2×C8×C59—22no—
103C2×C8×C59—41no—
104C2×C8×C59—41no—
105C4⋊C4×C59—22no—
106C4⋊C4×C59—21no—
107C4⋊C4×C59—22no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59—41no—
110C4⋊C4×C59—41no—
111C4⋊C4×C59—41no—
112C4⋊C4×C59—41no—
113C42×C59—41no—
114C42×C59—41no—
115C42×C59C2×D411yescharacteristic
116C42×C59C2×D411yescharacteristic
117C42×C59C2×D411yescharacteristic
118C42×C59#4111yescharacteristic
119#11745671—416no—
120(C2×D59)⋊C4—48no—
121(C2×D59)⋊C4—84no—
122C4⋊Dic59—22no—
123C4⋊Dic59—22no—
124C4×Dic59—41no—
125C4×Dic59—41no—
126C2×C59⋊C8—24no—
127C2×C59⋊C8—22no—
128C2×C59⋊C8—41no—
129C2×C59⋊C8—42no—
130C23×C59#12011yescharacteristic
131C22×D59—42no—
132C22×D59—42no—
133C22×D59—42no—
134C22×D59—44no—
135C22×D59—44no—
136Q8×C59—22no—
137Q8×C59—22no—
138Q8×C59—44no—
139Q8×C59—41no—
140Q8×C59—41no—
141Q8×C59—82no—
142C2×C4×C59—22no—
143C2×C4×C59—22no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—42no—
148C2×C4×C59—42no—
149C2×C4×C59—82no—
150C2×C4×C59#12011yescharacteristic
151C2×C4×C59#12011yescharacteristic
152C2×C4×C59#12111yescharacteristic
153C2×C4×C59#12111yescharacteristic
154C2×C4×C59#12111yescharacteristic
155C2×C4×C59#12711yescharacteristic
156C2×C4×C59C2×SD1611yescharacteristic
157C2×C4×C59#13511yescharacteristic
158C2×C4×C59#13512yes—
159C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C59#13711yescharacteristic
161C2×C4×C59#13712yes—
162C2×Dic59—42no—
163C2×Dic59—42no—
164C2×Dic59—42no—
165D236—44no—
166D236—82no—
167D236—81no—
168D236—81no—
169D236—82no—
170D236—81no—
171D236—81no—
172D236—81no—
173D236—81no—
174C8×C59—42no—
175C59⋊C8—24no—
176C59⋊C8—81no—
177#10055—591nomaximal, Sylow
178C22×C59#39211yescharacteristic
179C22×C59#45011yescharacteristic
180C22×C59#45211yescharacteristic
181C22×C59#45212yes—
182C22×C59#45312yes—
183C2×D59—84no—
184C2×D59—84no—
185C2×D59—82no—
186C2×D59—82no—
187C2×D59—82no—
188C4×C59—21no—
189C4×C59—21no—
190C4×C59—42no—
191C4×C59—81no—
192C4×C59—82no—
193C4×C59#44811yescharacteristic
194C4×C59#44911yescharacteristic
195C4×C59#46412yes—
196C4×C59#46511yescharacteristic
197C4×C59#46811yescharacteristic
198C4×C59#49511yescharacteristic
199C4×C59#49711yescharacteristic
200Dic59—82no—
201C2×#492—591no—
202C2×#477D5911yescharacteristic, complemented
203#2548—591no—
204#1987—591no—
205C2×#331—591no—
206C2×Q8⋊C8—591no—
207#1279—591no—
208C2×C59#163012yes—
209C2×C59#163112yes—
210C2×C59#200011yescharacteristic
211C2×C59#200211yescharacteristic
212C2×C59#201511yescharacteristic
213D59—162no—
214C22×SD16—1182no—
215C2×C4⋊Q8C2×D5911yescharacteristic
216C2×#127—591no—
217C2×#124—1181no—
218C2×C4×Q8C2×D5911yescharacteristic
219#492—1182no—
220#492—1182no—
221#477—22no—
222#477—22no—
223C2×#107—21no—
224C2×C4⋊C8—591no—
225C2×C4⋊C8C2×D5911yescharacteristic
226C2×#103—21no—
227C2×C4×C8—591no—
228#394—1182no—
229#390—1182no—
230#390—1181no—
231C4×C4⋊C4—591no—
232#340—1182no—
233#331—1182no—
234Q8⋊C8—1182no—
235C59#1005511yescharacteristic, complemented, Sylow
236C22×Q8—21no—
237C22×Q8C22⋊D5911yescharacteristic
238C22×D4—592no—
239C22×D4—1181no—
240C22×D4—1181no—
241C2×SD16—1188no—
242C2×SD16—2368no—
243C22×C8—21no—
244C22×C8—1181no—
245C22×C8—1181no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8C22⋊D5912yes—
250#127—1181no—
251#127—1181no—
252#127—1181no—
253#127—1181no—
254#124—2362no—
255#124—2362no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—592no—
264C2×C4⋊C4C22⋊D5911yescharacteristic
265C2×C22⋊C4—1184no—
266C2×C22⋊C4—1184no—
267C2×C42—21no—
268C2×C42—1181no—
269C2×C42C22×D5911yescharacteristic
270#107—24no—
271C4⋊C8—21no—
272C4⋊C8—1181no—
273C4⋊C8—1181no—
274C4⋊C8C22⋊D5912yes—
275#103—24no—
276C4×C8—592no—
277C4×C8—1181no—
278#95—1181no—
279C24—1182no—
280C2×Q8—21no—
281C2×Q8—22no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—44no—
287C2×Q8—41no—
288C2×Q8#1174567112yescomplemented
289C2×D4—1182no—
290C2×D4—1182no—
291C2×D4—2364no—
292C2×D4—2361no—
293C2×D4—2361no—
294C2×D4—2361no—
295C2×D4—2361no—
296C2×D4—2361no—
297C2×D4—2361no—
298C2×D4—2362no—
299C2×D4—2361no—
300C2×D4—2361no—
301C2×D4—2362no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—1182no—
306C22×C4D4×D5911yescharacteristic
307C22×C4D4×D5911yescharacteristic
308C22×C4C2×C22⋊D5911yescharacteristic
309SD16—23616no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—1184no—
314C2×C8—1182no—
315C2×C8—2361no—
316C2×C8—2362no—
317C4⋊C4—22no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—1182no—
326C4⋊C4—1182no—
327C22⋊C4—4724no—
328C22⋊C4—2368no—
329C42—41no—
330C42—41no—
331C42—2361no—
332C42—2361no—
333C42D4×D5911yescharacteristic
334C42#1174568711yescharacteristic
335C42C2×C22⋊D5911yescharacteristic
336C42C2×C22⋊D5911yescharacteristic
337C23—2362no—
338C23—2362no—
339C23—2364no—
340C23—2362no—
341C23—2364no—
342C23#1427369511yescharacteristic
343Q8—22no—
344Q8—22no—
345Q8—44no—
346Q8—41no—
347Q8—41no—
348Q8—82no—
349D4—4722no—
350D4—4721no—
351D4—4721no—
352D4—4721no—
353D4—4721no—
354D4—4721no—
355D4—4721no—
356D4—4722no—
357D4—2364no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—2362no—
367C2×C4—2362no—
368C2×C4—2362no—
369C2×C4#1427369511yescharacteristic
370C2×C4#1427369611yescharacteristic
371C2×C4#1427369811yescharacteristic
372C2×C4#1427369911yescharacteristic
373C2×C4#1427369911yescharacteristic
374C2×C4#1427372311yescharacteristic
375C2×C4C2×#1174567111yescharacteristic
376C2×C4#1427374711yescharacteristic
377C2×C4#1427375012yes—
378C2×C4#1427375211yescharacteristic
379C2×C4#1427375811yescharacteristic
380C2×C4#1427375912yes—
381C8—42no—
382C8—1184no—
383C8—4721no—
384C22—4722no—
385C22—4724no—
386C22—4724no—
387C22—4722no—
388C22—4722no—
389C22#1683731011yescharacteristic
390C22#1683755211yescharacteristic
391C22#1683756211yescharacteristic
392C22#1683756312yes—
393C22#1683756812yes—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—81no—
398C4—82no—
399C4—4722no—
400C4#1683754111yescharacteristic
401C4#1683754711yescharacteristic
402C4#1683762612yes—
403C4#1683763311yescharacteristic
404C4#1683765011yescharacteristic
405C4#1683778711yescharacteristic
406C4#1683779511yescharacteristic
407C2—9442no—
408C2#2142321612yes—
409C2#2142322112yes—
410C2#2142509711yescharacteristic
411C2#2142510811yescharacteristic
412C2#2142518711yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC2×C59×#477C2
socleC23×C59#120
prime radicalC2×#14273724C22
hypercenterC2×#477D59
hypocenterC59#10055
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #21425045C2semidirect
5. C2×Q8⋊(C59⋊C8)C2semidirect
6. #21421935C2semidirect
7. C2×C59×#477C2semidirect
81. C2×Q8×C59SD16semidirect
202. C2×#477D59semidirect
235. C59#10055semidirect
288. C2×Q8#11745671semidirect

Maximal subgroups

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

subgroupindex
C2×#168377762
#214290952
#214250452
C2×#168371452
C2×Q8⋊(C59⋊C8)2
#214219352
C2×C59×#4772
#1005559