Finite Group Databasebeta

← #28480103

Subgroups — #28480103

Order 15104.

710 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 151 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#21425033C211yesmaximal, characteristic, complemented
2#21422483C212yesmaximal, complemented
3#21422478C212yesmaximal, complemented
4#21422420C211yesmaximal, characteristic, complemented
5C59×#3461C211yesmaximal, characteristic
6#16837455C2212yes—
7C2×C4⋊(C59⋊C8)C2211yescharacteristic
8C2×C4×C59⋊C8C2211yescharacteristic
9#16837157—22no—
10Q8⋊(C59⋊C8)—22no—
11Q8⋊(C59⋊C8)—22no—
12D4⋊(C59⋊C8)—22no—
13D4⋊(C59⋊C8)—22no—
14#16837128—22no—
15C59×#544C411yescharacteristic
16C59×#542—22no—
17C59×#541—22no—
18C59×#539—22no—
19C59×#537C411yescharacteristic
20C59×#536C412yes—
21C59×#533—22no—
22C59×#529—22no—
23C59×#529—22no—
24C4×#41×C59C2212yes—
25C2×C4×Q8×C59C2211yescharacteristic
26C22×C59⋊C8—21no—
27C22×C59⋊C8—21no—
28C4⋊(C59⋊C8)—22no—
29C4⋊(C59⋊C8)—22no—
30C22⋊(C59⋊C8)—22no—
31C4×C59⋊C8—22no—
32C4×C59⋊C8—22no—
33C2×#41×C59C2×C412yes—
34C22×Q8×C59C2×C411yescharacteristic
35C4⋊Q8×C59—22no—
36C4⋊Q8×C59—22no—
37#126×C59—24no—
38#126×C59—24no—
39#125×C59—22no—
40#125×C59—24no—
41#125×C59—22no—
42#124×C59—22no—
43#124×C59—22no—
44#123×C59—24no—
45#123×C59—24no—
46C22⋊Q8×C59—24no—
47C22⋊Q8×C59—22no—
48C22⋊Q8×C59—22no—
49C22⋊Q8×C59—22no—
50C22⋊Q8×C59—22no—
51#121×C59—22no—
52#121×C59—22no—
53C4×Q8×C59—22no—
54C4×Q8×C59—22no—
55C4×Q8×C59—22no—
56C4×Q8×C59D412yes—
57C4×Q8×C59D412yes—
58C4×D4×C59—22no—
59C4×D4×C59—22no—
60C4×D4×C59—22no—
61C4×D4×C59D412yes—
62C4×D4×C59D412yes—
63#117×C59—24no—
64#117×C59—22no—
65#117×C59C2×C412yes—
66C2×C4⋊C4×C59C2×C411yescharacteristic
67C2×C4⋊C4×C59D412yes—
68C2×C42×C59—22no—
69C2×C42×C59D412yes—
70C2×C42×C59C2311yescharacteristic
71#41×C59—28no—
72C2×Q8×C59—22no—
73C2×Q8×C59—22no—
74C2×Q8×C59—24no—
75C2×Q8×C59C22⋊C412yes—
76C2×D4×C59—22no—
77C2×D4×C59—22no—
78C2×D4×C59C22⋊C412yes—
79C22×C4×C59—22no—
80C22×C4×C59—22no—
81C22×C4×C59—22no—
82C22×C4×C59C22⋊C412yes—
83C22×C4×C59C22⋊C412yes—
84C22×C4×C59C22×C411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C4⋊C4×C59—24no—
88C4⋊C4×C59—24no—
89C4⋊C4×C59—22no—
90C4⋊C4×C59—22no—
91C4⋊C4×C59—24no—
92C4⋊C4×C59—22no—
93C4⋊C4×C59—24no—
94C4⋊C4×C59—22no—
95C4⋊C4×C59—22no—
96C4⋊C4×C59C22⋊C412yes—
97C4⋊C4×C59C22⋊C412yes—
98C22⋊C4×C59—24no—
99C22⋊C4×C59—24no—
100C22⋊C4×C59—24no—
101C22⋊C4×C59—22no—
102C22⋊C4×C59—22no—
103C22⋊C4×C59—22no—
104C42×C59—22no—
105C42×C59—22no—
106C42×C59—22no—
107C42×C59—22no—
108C42×C59—22no—
109C42×C59C2×D412yes—
110C42×C59C2×D412yes—
111C2×C59⋊C8—22no—
112C2×C59⋊C8—41no—
113C2×C59⋊C8—41no—
114C2×C59⋊C8—42no—
115C2×C59⋊C8—42no—
116C2×C59⋊C8—41no—
117C23×C59—22no—
118C23×C59C2×C22⋊C411yescharacteristic
119Q8×C59—24no—
120Q8×C59—44no—
121Q8×C59—82no—
122D4×C59—24no—
123D4×C59—44no—
124D4×C59—82no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—22no—
128C2×C4×C59—22no—
129C2×C4×C59—22no—
130C2×C4×C59—22no—
131C2×C4×C59—22no—
132C2×C4×C59—24no—
133C2×C4×C59—41no—
134C2×C4×C59—44no—
135C2×C4×C59—42no—
136C2×C4×C59—42no—
137C2×C4×C59—42no—
138C2×C4×C59—82no—
139C2×C4×C59C2×C22⋊C412yes—
140C2×C4×C59#12011yescharacteristic
141C2×C4×C59#12012yes—
142C2×C4×C59#12011yescharacteristic
143C2×C4×C59#13114yes—
144C59⋊C8—42no—
145C59⋊C8—82no—
146C59⋊C8—81no—
147#8321—591nomaximal, Sylow
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—42no—
151C22×C59—42no—
152C22×C59—42no—
153C22×C59#37911yescharacteristic
154C22×C59#40812yes—
155C4×C59—21no—
156C4×C59—22no—
157C4×C59—42no—
158C4×C59—42no—
159C4×C59—42no—
160C4×C59—42no—
161C4×C59—42no—
162C4×C59#40814yes—
163C4×C59#45411yescharacteristic
164C4×C59#45611yescharacteristic
165#3461D5911yescharacteristic
166#1984—591no—
167#1477—592no—
168#1476—592no—
169#1461—591no—
170C2×C59—22no—
171C2×C59—82no—
172C2×C59#178011yescharacteristic
173C2×C59#179311yescharacteristic
174C2×C59#179411yescharacteristic
175#544Dic5911yescharacteristic
176#542—22no—
177#541—22no—
178#539—22no—
179#537Dic5911yescharacteristic
180#536Dic5912yes—
181#533—22no—
182#529—22no—
183#529—22no—
184C4×#41C2×D5912yes—
185C2×C4×Q8C2×D5911yescharacteristic
186#431—592no—
187C2×C4⋊C8—591no—
188C2×C4×C8—591no—
189#336—1182no—
190Q8⋊C8—1182no—
191Q8⋊C8—1182no—
192D4⋊C8—1182no—
193D4⋊C8—1182no—
194#324—1182no—
195C59#832111yescharacteristic, complemented, Sylow
196C2×#41C2×Dic5912yes—
197C22×Q8C2×Dic5911yescharacteristic
198C22×C8—1181no—
199C22×C8—1181no—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202#126—24no—
203#126—24no—
204#125—22no—
205#125—24no—
206#125—22no—
207#124—22no—
208#124—22no—
209#123—24no—
210#123—24no—
211C22⋊Q8—24no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216#121—22no—
217#121—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8C22⋊D5912yes—
222C4×Q8C22⋊D5912yes—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4C22⋊D5912yes—
227C4×D4C22⋊D5912yes—
228#117—24no—
229#117—22no—
230#117C2×Dic5912yes—
231C2×C4⋊C4C2×Dic5911yescharacteristic
232C2×C4⋊C4C22⋊D5912yes—
233C2×C42—22no—
234C2×C42C22⋊D5912yes—
235C2×C42C22×D5911yescharacteristic
236C4⋊C8—1182no—
237C4⋊C8—1182no—
238C22⋊C8—1182no—
239C4×C8—1182no—
240C4×C8—1182no—
241#41—28no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8C22⋊Dic5912yes—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4C22⋊Dic5912yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4C22⋊Dic5912yes—
253C22×C4C22⋊Dic5912yes—
254C22×C4C22×Dic5911yescharacteristic
255C22×C4C2×C22⋊D5911yescharacteristic
256C22×C4C2×C22⋊D5911yescharacteristic
257C2×C8—1182no—
258C2×C8—2362no—
259C2×C8—2362no—
260C2×C8—2361no—
261C2×C8—2361no—
262C2×C8—2361no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4C22⋊Dic5912yes—
273C4⋊C4C22⋊Dic5912yes—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42C2×C22⋊D5912yes—
286C42C2×C22⋊D5912yes—
287C23—22no—
288C23C2×C22⋊Dic5911yescharacteristic
289Q8—24no—
290Q8—44no—
291Q8—82no—
292D4—24no—
293D4—44no—
294D4—82no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4C2×C22⋊Dic5912yes—
310C2×C4#1427369611yescharacteristic
311C2×C4#1427369612yes—
312C2×C4#1427369611yescharacteristic
313C2×C4#1427373414yes—
314C8—4722no—
315C8—4721no—
316C8—2362no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22#1683725811yescharacteristic
323C22#1683736912yes—
324C4—21no—
325C4—22no—
326C4—42no—
327C4—42no—
328C4—42no—
329C4—42no—
330C4—42no—
331C4#1683736914yes—
332C4#1683757511yescharacteristic
333C4#1683758511yescharacteristic
334C2—22no—
335C2—82no—
336C2#2142400811yescharacteristic
337C2#2142405511yescharacteristic
338C2#2142406011yescharacteristic
subgroupG/H
centerC2×C4#14273696
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3461C2
socleC22×C59#379
prime radicalC59×#537C4
cosocleC2×C42×C59C23
hypercenter#3461D59
hypocenterC59#8321
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3461

Complements

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

NH ≅ G/Ntype
1. #21425033C2semidirect
2. #21422483C2semidirect
3. #21422478C2semidirect
4. #21422420C2semidirect
195. C59#8321semidirect

Maximal subgroups

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

subgroupindex
#214250332
#214224832
#214224782
#214224202
C59×#34612
#832159