Finite Group Databasebeta

← #28590943

Subgroups — #28590943

Order 15104.

798 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837429C211yesmaximal, characteristic
2C2×#16837358C211yesmaximal, characteristic
3#21424697C218yesmaximal, complemented
4#21423823C211yesmaximal, characteristic
5#21423805C211yesmaximal, characteristic
6#21423805C211yesmaximal, characteristic
7#21423794C211yesmaximal, characteristic
8C59×#2911C211yesmaximal, characteristic
9C22×#11745664C2211yescharacteristic
10C22×#11745664C2211yescharacteristic
11C23×C59⋊C8C2211yescharacteristic
12C2×#14273686C2211yescharacteristic
13#16837429—21no—
14#16837429—21no—
15#16837429C2214yes—
16C2×C4⋊(C59⋊C8)—22no—
17#16837358—22no—
18#16837358C2214yes—
19C2×#14273638—21no—
20C2×C4×C59⋊C8—21no—
21#16837167C2214yes—
22#16837161C2214yes—
23#16837161C2214yes—
24#16837156C2214yes—
25C22×M16×C59C2211yescharacteristic
26C2×#117×C59C2211yescharacteristic
27C2×#117×C59C2211yescharacteristic
28C59×#428C414yescomplemented
29C59×#428C414yescomplemented
30C59×#427C2214yes—
31C2×#11745664—21no—
32C2×#11745664—22no—
33C2×#11745664—21no—
34C2×#11745664—22no—
35C2×#11745664—22no—
36C2×#11745664Q814yes—
37C2×#11745664C2312yes—
38C2×#11745664C2312yes—
39C22×C59⋊C8—21no—
40C22×C59⋊C8—22no—
41C22×C59⋊C8—22no—
42C22×C59⋊C8—21no—
43C22×C59⋊C8D412yes—
44C22×C59⋊C8D411yescharacteristic
45C22×C59⋊C8D411yescharacteristic
46C22×C59⋊C8C2312yes—
47#14273686—22no—
48#14273686—22no—
49C2×C4⋊Dic59—21no—
50C2×C22⋊Dic59—21no—
51C2×C4×Dic59C2312yes—
52C4⋊(C59⋊C8)—24no—
53C4⋊(C59⋊C8)—24no—
54#14273638—24no—
55C4×C59⋊C8—24no—
56C23×C4×C59C2311yescharacteristic
57C2×M16×C59—22no—
58C2×M16×C59—22no—
59C2×M16×C59C2×C412yes—
60C2×M16×C59C2×C412yes—
61C22×C8×C59C2312yes—
62#117×C59—21no—
63#117×C59—21no—
64#117×C59—21no—
65#117×C59—21no—
66#117×C59C2×C412yes—
67#117×C59C2×C412yes—
68#117×C59C2312yes—
69#117×C59C2312yes—
70C2×C4⋊C4×C59C2×C412yes—
71C2×C4⋊C4×C59C2×C412yes—
72C2×C22⋊C4×C59—21no—
73C2×C22⋊C4×C59—21no—
74C2×C42×C59—21no—
75C2×C42×C59—21no—
76#107×C59C2×C418yes—
77#106×C59C2×C418yes—
78C24×C59#4111yescharacteristic
79C22×C4×C59—22no—
80C22×C4×C59—22no—
81C22×C4×C59C22×C411yescharacteristic
82C22×C4×C59C22×C411yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59C2×D411yescharacteristic
85C22×C4×C59C2×Q812yes—
86C22×C4×C59#4112yes—
87C22×C4×C59#4112yes—
88C22×C4×C59#4111yescharacteristic
89C22×C4×C59#4112yes—
90C22×C4×C59C2411yescharacteristic
91C22×Dic59—22no—
92M16×C59—24no—
93M16×C59—24no—
94C2×C8×C59—22no—
95C2×C8×C59—22no—
96C2×C8×C59C22×C414yes—
97C4⋊C4×C59—22no—
98C4⋊C4×C59—22no—
99C4⋊C4×C59C22×C414yes—
100C4⋊C4×C59C22×C414yes—
101C22⋊C4×C59—22no—
102C22⋊C4×C59—22no—
103C22⋊C4×C59—22no—
104C22⋊C4×C59—22no—
105C42×C59—22no—
106C42×C59—22no—
107C42×C59—22no—
108C42×C59—22no—
109C22⋊Dic59—42no—
110C4⋊Dic59—42no—
111C4×Dic59—22no—
112C4×Dic59—22no—
113#11745664—24no—
114#11745664—22no—
115#11745664—22no—
116#11745664—22no—
117#11745664—42no—
118#11745664C2×Q814yes—
119C2×C59⋊C8—24no—
120C2×C59⋊C8—24no—
121C2×C59⋊C8—22no—
122C2×C59⋊C8—22no—
123C2×C59⋊C8—22no—
124C2×C59⋊C8—22no—
125C2×C59⋊C8—22no—
126C2×C59⋊C8—22no—
127C2×C59⋊C8—42no—
128C2×C59⋊C8C2×D412yes—
129C2×C59⋊C8C2×D412yes—
130C23×C59—42no—
131C23×C59C4×D411yescharacteristic
132C23×C59C4×D411yescharacteristic
133C23×C59#12512yes—
134C23×C59#12612yes—
135C23×C59C2×#4111yescharacteristic
136C2×C4×C59—24no—
137C2×C4×C59—24no—
138C2×C4×C59—44no—
139C2×C4×C59—44no—
140C2×C4×C59#11712yes—
141C2×C4×C59#11712yes—
142C2×C4×C59C4×D411yescharacteristic
143C2×C4×C59C4×D411yescharacteristic
144C2×C4×C59C4×Q812yes—
145C2×C4×C59C4×Q812yes—
146C2×C4×C59C22⋊Q812yes—
147C2×C4×C59C22⋊Q812yes—
148C2×C4×C59#12312yes—
149C2×C4×C59#12312yes—
150C2×C4×C59#12512yes—
151C2×C4×C59#12612yes—
152C2×C4×C59C23×C411yescharacteristic
153C2×C4×C59C22×D411yescharacteristic
154C2×C4×C59C22×Q811yescharacteristic
155C2×C4×C59C2×#4111yescharacteristic
156C2×C4×C59C2×#4111yescharacteristic
157C2×C4×C59C2×#4111yescharacteristic
158C2×Dic59—44no—
159C2×Dic59—42no—
160C8×C59—24no—
161C59⋊C8—24no—
162C59⋊C8—22no—
163C59⋊C8—22no—
164C59⋊C8—42no—
165#21041—591nomaximal, Sylow
166C22×C59—22no—
167C22×C59—22no—
168C22×C59—22no—
169C22×C59—42no—
170C22×C59—42no—
171C22×C59#38712yes—
172C22×C59#38712yes—
173C22×C59C2×C4×D411yescharacteristic
174C22×C59C2×#12511yescharacteristic
175C22×C59C2×#12611yescharacteristic
176C4×C59—44no—
177C4×C59—44no—
178C4×C59#38712yes—
179C4×C59#38712yes—
180C4×C59C2×#11711yescharacteristic
181C4×C59C2×C4×Q811yescharacteristic
182C4×C59C2×C22⋊Q811yescharacteristic
183C4×C59C2×#12311yescharacteristic
184Dic59—82no—
185#2911D5911yescharacteristic
186C2×#424—591no—
187C2×#405—591no—
188#1922—598no—
189#1738—591no—
190#1733—591no—
191#1733—591no—
192#1729—591no—
193C2×C59—21no—
194C2×C59—21no—
195C2×C59—22no—
196C2×C59—22no—
197C2×C59C2×#38711yescharacteristic
198C2×C59#295512yes—
199C22×M16—591no—
200C22×M16—591no—
201C22×M16C2×D5911yescharacteristic
202C23×C8—591no—
203C2×#117—591no—
204C2×#117C2×D5911yescharacteristic
205C2×#117C2×D5911yescharacteristic
206#428Dic5914yescomplemented
207#428Dic5914yescomplemented
208#427C2×D5914yes—
209#424—594no—
210#424—1181no—
211#424—1181no—
212C2×C4⋊C8—1182no—
213#405—594no—
214#405—1182no—
215C2×#97—1181no—
216C2×C4×C8—1181no—
217#341—594no—
218#339—594no—
219#339—594no—
220#336—594no—
221C59#2104111yescharacteristic, complemented, Sylow
222C23×C4C22×D5911yescharacteristic
223C2×M16—22no—
224C2×M16—22no—
225C2×M16—594no—
226C2×M16—592no—
227C2×M16—592no—
228C2×M16—1182no—
229C2×M16—1181no—
230C2×M16—1181no—
231C2×M16—1182no—
232C2×M16—1182no—
233C2×M16C2×Dic5912yes—
234C2×M16C2×Dic5912yes—
235C22×C8—591no—
236C22×C8—591no—
237C22×C8—592no—
238C22×C8—592no—
239C22×C8—1182no—
240C22×C8—1181no—
241C22×C8—1182no—
242C22×C8—1181no—
243C22×C8C22×D5912yes—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—1182no—
249#117—1182no—
250#117C2×Dic5912yes—
251#117C2×Dic5912yes—
252#117C22×D5912yes—
253#117C22×D5912yes—
254C2×C4⋊C4—1181no—
255C2×C4⋊C4C2×Dic5912yes—
256C2×C4⋊C4C2×Dic5912yes—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—1181no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—592no—
263#107C2×Dic5918yes—
264#106C2×Dic5918yes—
265C4⋊C8—1184no—
266C4⋊C8—1184no—
267#97—1184no—
268C4×C8—1184no—
269C24#1174568711yescharacteristic
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—1182no—
273C22×C4#1174568512yes—
274C22×C4#1174568512yes—
275C22×C4D4×D5911yescharacteristic
276C22×C4D4×D5911yescharacteristic
277C22×C4#1174568711yescharacteristic
278C22×C4Q8×D5912yes—
279C22×C4#1174568912yes—
280C22×C4C22×Dic5911yescharacteristic
281C22×C4C22×Dic5911yescharacteristic
282C22×C4C23×D5911yescharacteristic
283M16—24no—
284M16—24no—
285M16—594no—
286M16—1184no—
287M16—1182no—
288M16—1182no—
289M16—1182no—
290M16—2362no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—592no—
294C2×C8—592no—
295C2×C8—1184no—
296C2×C8—1184no—
297C2×C8—1182no—
298C2×C8—1182no—
299C2×C8—1182no—
300C2×C8—1182no—
301C2×C8—1182no—
302C2×C8—1182no—
303C2×C8—2362no—
304C2×C8C22×Dic5914yes—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—2362no—
308C4⋊C4C22×Dic5914yes—
309C4⋊C4C22×Dic5914yes—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—2362no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—1182no—
320C42—1182no—
321C23—42no—
322C23D4×Dic5911yescharacteristic
323C23D4×Dic5911yescharacteristic
324C23#1427371712yes—
325C23#1427372212yes—
326C23C2×#1174568711yescharacteristic
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—2364no—
332C2×C4—2362no—
333C2×C4#1427368612yes—
334C2×C4#1427368612yes—
335C2×C4D4×Dic5911yescharacteristic
336C2×C4D4×Dic5911yescharacteristic
337C2×C4Q8×Dic5912yes—
338C2×C4Q8×Dic5912yes—
339C2×C4#1427370512yes—
340C2×C4#1427370512yes—
341C2×C4#1427371212yes—
342C2×C4#1427371212yes—
343C2×C4#1427371712yes—
344C2×C4#1427372212yes—
345C2×C4C23×Dic5911yescharacteristic
346C2×C4C2×D4×D5911yescharacteristic
347C2×C4C2×Q8×D5911yescharacteristic
348C2×C4C2×#1174568511yescharacteristic
349C2×C4C2×#1174568511yescharacteristic
350C2×C4C2×#1174568911yescharacteristic
351C8—24no—
352C8—1184no—
353C8—1182no—
354C8—1182no—
355C8—2362no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—42no—
360C22—42no—
361C22#1683728812yes—
362C22#1683728812yes—
363C22C2×D4×Dic5911yescharacteristic
364C22C2×#1427371711yescharacteristic
365C22C2×#1427372211yescharacteristic
366C4—44no—
367C4—44no—
368C4—4722no—
369C4#1683728812yes—
370C4#1683728812yes—
371C4C2×#1427368611yescharacteristic
372C4C2×Q8×Dic5911yescharacteristic
373C4C2×#1427370511yescharacteristic
374C4C2×#1427371211yescharacteristic
375C2—21no—
376C2—21no—
377C2—22no—
378C2—22no—
379C2C2×#1683728811yescharacteristic
380C2#2143299912yes—
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2911C2
socleC22×C59C2×C4×D4
prime radicalC24×C59#41
cosocleC22×C4×C59C24
hypercenter#2911D59
hypocenterC59#21041
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2911

Complements

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

NH ≅ G/Ntype
3. #21424697C2semidirect
28. C59×#428C4semidirect
29. C59×#428C4semidirect
206. #428Dic59semidirect
207. #428Dic59semidirect
221. C59#21041semidirect

Maximal subgroups

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

subgroupindex
C2×#168374292
C2×#168373582
#214246972
#214238232
#214238052
#214237942
C59×#29112
#2104159