Finite Group Databasebeta

← #28480402

Subgroups — #28480402

Order 15104.

1050 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21424088C211yesmaximal, characteristic, complemented
2#21422764C212yesmaximal, complemented
3#21422525C212yesmaximal, complemented
4#21422417C211yesmaximal, characteristic, complemented
5C2×C59×#550C211yesmaximal, characteristic
6#16837455C2212yes—
7#16837426C2212yes—
8C2×#14273644—21no—
9#16837171—21no—
10#16837146—24no—
11Q8⋊(C59⋊C8)—24no—
12#16837126—22no—
13C22×#41×C59C412yes—
14C59×#550—24no—
15C59×#550—24no—
16C2×#127×C59—21no—
17C2×#127×C59C2211yescharacteristic
18C2×#121×C59—22no—
19C2×#121×C59C412yes—
20C2×C4×Q8×C59C2211yescharacteristic
21C2×C4×D4×C59—21no—
22C2×C4×D4×C59C2211yescharacteristic
23C2×#11745664—22no—
24C4⋊(C59⋊C8)—22no—
25C4⋊(C59⋊C8)—22no—
26C4⋊(C59⋊C8)—22no—
27#14273644—42no—
28C22⋊(C59⋊C8)—22no—
29C4×C59⋊C8—22no—
30C2×#41×C59—24no—
31C2×#41×C59—24no—
32C2×#41×C59—48no—
33C22×Q8×C59C2×C411yescharacteristic
34C22×D4×C59—22no—
35C22×D4×C59—22no—
36C22×D4×C59—21no—
37C22×D4×C59—21no—
38C22×D4×C59C2×C412yes—
39C22×D4×C59D411yescharacteristic
40C23×C4×C59—22no—
41C23×C4×C59C2×C412yes—
42#127×C59—28no—
43#127×C59—22no—
44#127×C59D414yes—
45#121×C59—216no—
46#121×C59—24no—
47#121×C59—24no—
48C4×Q8×C59—22no—
49C4×Q8×C59D414yes—
50C4×D4×C59—28no—
51C4×D4×C59—22no—
52C4×D4×C59—22no—
53C2×C4⋊C4×C59—21no—
54C2×C4⋊C4×C59C2×C411yescharacteristic
55C2×C4⋊C4×C59D411yescharacteristic
56C2×C22⋊C4×C59—22no—
57C2×C22⋊C4×C59D412yes—
58C2×C42×C59—21no—
59C2×C42×C59C2311yescharacteristic
60C24×C59—22no—
61C24×C59C22⋊C412yes—
62#41×C59—416no—
63#41×C59—48no—
64#41×C59—48no—
65C2×Q8×C59—24no—
66C2×Q8×C59—21no—
67C2×Q8×C59C22⋊C412yes—
68C2×D4×C59—28no—
69C2×D4×C59—28no—
70C2×D4×C59—22no—
71C2×D4×C59—22no—
72C2×D4×C59—24no—
73C2×D4×C59—24no—
74C2×D4×C59—22no—
75C2×D4×C59—21no—
76C2×D4×C59—21no—
77C2×D4×C59—48no—
78C2×D4×C59—44no—
79C2×D4×C59—42no—
80C2×D4×C59—41no—
81C2×D4×C59—41no—
82C2×D4×C59—82no—
83C2×D4×C59—84no—
84C2×D4×C59C22⋊C414yes—
85C22×C4×C59—21no—
86C22×C4×C59—28no—
87C22×C4×C59—21no—
88C22×C4×C59—22no—
89C22×C4×C59—22no—
90C22×C4×C59—48no—
91C22×C4×C59—44no—
92C22×C4×C59C22⋊C412yes—
93C22×C4×C59C22×C411yescharacteristic
94C22×C4×C59C2×D411yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C4⋊C4×C59—21no—
97C4⋊C4×C59—24no—
98C4⋊C4×C59—22no—
99C4⋊C4×C59C22⋊C412yes—
100C22⋊C4×C59—28no—
101C22⋊C4×C59—24no—
102C42×C59—24no—
103C42×C59C2×D412yes—
104C42×C59C2×D412yes—
105#11745664—42no—
106#11745664—42no—
107C2×C59⋊C8—42no—
108C2×C59⋊C8—42no—
109C23×C59—22no—
110C23×C59—24no—
111C23×C59—22no—
112C23×C59—22no—
113C23×C59—44no—
114C23×C59—44no—
115C23×C59—42no—
116C23×C59—42no—
117C23×C59—42no—
118C23×C59—42no—
119C23×C59C2×C22⋊C411yescharacteristic
120Q8×C59—24no—
121Q8×C59—22no—
122Q8×C59#10014yescomplemented
123D4×C59—44no—
124D4×C59—416no—
125D4×C59—44no—
126D4×C59—42no—
127D4×C59—44no—
128D4×C59—42no—
129D4×C59—44no—
130D4×C59—84no—
131D4×C59—88no—
132C2×C4×C59—24no—
133C2×C4×C59—22no—
134C2×C4×C59—22no—
135C2×C4×C59—22no—
136C2×C4×C59—22no—
137C2×C4×C59—41no—
138C2×C4×C59—416no—
139C2×C4×C59—44no—
140C2×C4×C59—42no—
141C2×C4×C59—44no—
142C2×C4×C59—42no—
143C2×C4×C59C2×C22⋊C411yescharacteristic
144C2×C4×C59C2×C22⋊C411yescharacteristic
145C2×C4×C59#12012yes—
146C2×C4×C59#12012yes—
147C2×C4×C59#13512yes—
148C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C59#13711yescharacteristic
150C59⋊C8—82no—
151C59⋊C8—82no—
152#8385—591nomaximal, Sylow
153C22×C59—21no—
154C22×C59—21no—
155C22×C59—42no—
156C22×C59—44no—
157C22×C59—42no—
158C22×C59—44no—
159C22×C59—42no—
160C22×C59—44no—
161C22×C59—82no—
162C22×C59—82no—
163C22×C59—84no—
164C22×C59—84no—
165C22×C59#37911yescharacteristic
166C22×C59#41611yescharacteristic
167C22×C59#41711yescharacteristic
168C4×C59—22no—
169C4×C59—24no—
170C4×C59—21no—
171C4×C59—42no—
172C4×C59—42no—
173C4×C59C2×#10012yes—
174C4×C59#44912yes—
175C4×C59#45212yes—
176C2×#550D5911yescharacteristic
177#1800—591no—
178#1545—592no—
179#1487—592no—
180#1460—591no—
181C2×C59—21no—
182C2×C59—21no—
183C2×C59—82no—
184C2×C59—84no—
185C2×C59#178311yescharacteristic
186C2×C59#178811yescharacteristic
187C2×C59#179511yescharacteristic
188C22×#41Dic5912yes—
189#550—24no—
190#550—24no—
191C2×#127—21no—
192C2×#127C2×D5911yescharacteristic
193C2×#121—22no—
194C2×#121Dic5912yes—
195C2×C4×Q8C2×D5911yescharacteristic
196C2×C4×D4—21no—
197C2×C4×D4C2×D5911yescharacteristic
198#431—592no—
199#423—592no—
200C2×#100—1181no—
201#343—1181no—
202#331—1184no—
203Q8⋊C8—1184no—
204#323—1182no—
205C59#838511yescharacteristic, complemented, Sylow
206C2×#41—24no—
207C2×#41—24no—
208C2×#41—48no—
209C22×Q8C2×Dic5911yescharacteristic
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4C2×Dic5912yes—
215C22×D4C22⋊D5911yescharacteristic
216C23×C4—22no—
217C23×C4C2×Dic5912yes—
218C2×M16—1182no—
219#127—28no—
220#127—22no—
221#127C22⋊D5914yes—
222#121—216no—
223#121—24no—
224#121—24no—
225C4×Q8—22no—
226C4×Q8C22⋊D5914yes—
227C4×D4—28no—
228C4×D4—22no—
229C4×D4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C2×Dic5911yescharacteristic
232C2×C4⋊C4C22⋊D5911yescharacteristic
233C2×C22⋊C4—22no—
234C2×C22⋊C4C22⋊D5912yes—
235C2×C42—21no—
236C2×C42C22×D5911yescharacteristic
237C4⋊C8—1182no—
238C4⋊C8—1182no—
239C4⋊C8—1182no—
240#100—2362no—
241C22⋊C8—1182no—
242C4×C8—1182no—
243C24—22no—
244C24C22⋊Dic5912yes—
245#41—416no—
246#41—48no—
247#41—48no—
248C2×Q8—24no—
249C2×Q8—21no—
250C2×Q8C22⋊Dic5912yes—
251C2×D4—28no—
252C2×D4—28no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—22no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—48no—
261C2×D4—44no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—82no—
266C2×D4—84no—
267C2×D4C22⋊Dic5914yes—
268C22×C4—21no—
269C22×C4—28no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—48no—
274C22×C4—44no—
275C22×C4C22⋊Dic5912yes—
276C22×C4C22×Dic5911yescharacteristic
277C22×C4C2×C22⋊D5911yescharacteristic
278C22×C4C2×C22⋊D5911yescharacteristic
279M16—2362no—
280M16—2362no—
281C2×C8—2362no—
282C2×C8—2362no—
283C4⋊C4—21no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4C22⋊Dic5912yes—
287C22⋊C4—28no—
288C22⋊C4—24no—
289C42—24no—
290C42C2×C22⋊D5912yes—
291C42C2×C22⋊D5912yes—
292C23—22no—
293C23—24no—
294C23—22no—
295C23—22no—
296C23—44no—
297C23—44no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23C2×C22⋊Dic5911yescharacteristic
303Q8—24no—
304Q8—22no—
305Q8#1427364414yescomplemented
306D4—44no—
307D4—416no—
308D4—44no—
309D4—42no—
310D4—44no—
311D4—42no—
312D4—44no—
313D4—84no—
314D4—88no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—416no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4C2×C22⋊Dic5911yescharacteristic
327C2×C4C2×C22⋊Dic5911yescharacteristic
328C2×C4#1427369612yes—
329C2×C4#1427369612yes—
330C2×C4#1427374712yes—
331C2×C4#1427375211yescharacteristic
332C2×C4#1427375711yescharacteristic
333C8—4722no—
334C8—4722no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—44no—
339C22—42no—
340C22—44no—
341C22—44no—
342C22—42no—
343C22—82no—
344C22—82no—
345C22—84no—
346C22—84no—
347C22#1683725811yescharacteristic
348C22#1683739811yescharacteristic
349C22#1683740311yescharacteristic
350C4—24no—
351C4—21no—
352C4—22no—
353C4—42no—
354C4—42no—
355C4C2×#1427364412yes—
356C4#1683754712yes—
357C4#1683756412yes—
358C2—21no—
359C2—21no—
360C2—84no—
361C2—82no—
362C2#2142401711yescharacteristic
363C2#2142403611yescharacteristic
364C2#2142406611yescharacteristic
subgroupG/H
centerC22#16837258
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
Fitting, prime radicalC2×C59×#550C2
socleC22×C59#379
cosocleC2×C42×C59C23
hypercenterC2×#550D59
hypocenterC59#8385
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #21424088C2semidirect
2. #21422764C2semidirect
3. #21422525C2semidirect
4. #21422417C2semidirect
122. Q8×C59#100semidirect
205. C59#8385semidirect
305. Q8#14273644semidirect

Maximal subgroups

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

subgroupindex
#214240882
#214227642
#214225252
#214224172
C2×C59×#5502
#838559