Finite Group Databasebeta

← #28480835

Subgroups — #28480835

Order 15104.

746 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 143 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#21424895C211yesmaximal, characteristic
2#21422796C212yesmaximal, complemented
3#21422711C212yesmaximal, complemented
4#21422437C211yesmaximal, characteristic
5C59×#3501C211yesmaximal, characteristic
6#16837459C2212yes—
7C2×C4⋊(C59⋊C8)C2211yescharacteristic
8C2×C22⋊(C59⋊C8)—22no—
9C2×#14273638C2211yescharacteristic
10#16837148—22no—
11#16837148—22no—
12#16837144—24no—
13#16837126—22no—
14C59×#554—22no—
15C59×#554—24no—
16D4×Q8×C59—22no—
17C59×#544C412yes—
18C59×#543—22no—
19C59×#533C2212yes—
20C2×C22⋊Q8×C59—22no—
21C59×#518C412yes—
22C2×C4×D4×C59C2211yescharacteristic
23C22×C59⋊C8—21no—
24C22×C59⋊C8—21no—
25C4⋊(C59⋊C8)—22no—
26C4⋊(C59⋊C8)—22no—
27C22⋊(C59⋊C8)—22no—
28C22⋊(C59⋊C8)—44no—
29#14273638—22no—
30#14273638—22no—
31C22×Q8×C59—22no—
32C22×D4×C59C2×C411yescharacteristic
33C23×C4×C59D412yes—
34C4⋊Q8×C59—24no—
35C4⋊Q8×C59D412yes—
36C4⋊Q8×C59D412yes—
37#125×C59—24no—
38#125×C59D414yes—
39C22⋊Q8×C59—24no—
40C22⋊Q8×C59—24no—
41C22⋊Q8×C59—28no—
42C22⋊Q8×C59—28no—
43C22⋊Q8×C59—24no—
44C22⋊Q8×C59—22no—
45C22⋊Q8×C59—22no—
46C4×Q8×C59—22no—
47C4×Q8×C59—22no—
48C4×D4×C59—24no—
49C4×D4×C59—24no—
50C4×D4×C59—22no—
51C4×D4×C59—22no—
52#117×C59C2×C414yes—
53C2×C4⋊C4×C59—22no—
54C2×C4⋊C4×C59—24no—
55C2×C4⋊C4×C59C2×C411yescharacteristic
56C2×C22⋊C4×C59—24no—
57C2×C22⋊C4×C59D412yes—
58C2×C42×C59C2311yescharacteristic
59C24×C59C22⋊C412yes—
60C2×Q8×C59—24no—
61C2×Q8×C59—22no—
62C2×Q8×C59—22no—
63C2×Q8×C59—82no—
64C2×Q8×C59—82no—
65C2×D4×C59—24no—
66C2×D4×C59—22no—
67C22×C4×C59—24no—
68C22×C4×C59—24no—
69C22×C4×C59—22no—
70C22×C4×C59—22no—
71C22×C4×C59—42no—
72C22×C4×C59—42no—
73C22×C4×C59C22⋊C412yes—
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C4⋊C4×C59—24no—
78C4⋊C4×C59—28no—
79C4⋊C4×C59—22no—
80C4⋊C4×C59—24no—
81C4⋊C4×C59—22no—
82C4⋊C4×C59—24no—
83C4⋊C4×C59—22no—
84C4⋊C4×C59C22⋊C414yes—
85C4⋊C4×C59C22⋊C414yes—
86C22⋊C4×C59—28no—
87C22⋊C4×C59—24no—
88C22⋊C4×C59—24no—
89C42×C59—24no—
90C42×C59C2×D412yes—
91C42×C59C2×D412yes—
92C2×C59⋊C8—22no—
93C2×C59⋊C8—41no—
94C2×C59⋊C8—41no—
95C2×C59⋊C8—42no—
96C2×C59⋊C8—42no—
97C2×C59⋊C8—41no—
98C23×C59—22no—
99C23×C59—22no—
100C23×C59—42no—
101C23×C59—42no—
102C23×C59#10114yes—
103C23×C59C2×C22⋊C411yescharacteristic
104Q8×C59—42no—
105Q8×C59—42no—
106Q8×C59—82no—
107Q8×C59—84no—
108D4×C59—28no—
109C2×C4×C59—22no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—24no—
113C2×C4×C59—24no—
114C2×C4×C59—22no—
115C2×C4×C59—41no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59—82no—
120C2×C4×C59—84no—
121C2×C4×C59—84no—
122C2×C4×C59C2×C22⋊C412yes—
123C2×C4×C59#12012yes—
124C2×C4×C59#12011yescharacteristic
125C2×C4×C59#12011yescharacteristic
126C59⋊C8—42no—
127C59⋊C8—82no—
128C59⋊C8—81no—
129#8457—591nomaximal, Sylow
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—24no—
133C22×C59—44no—
134C22×C59—42no—
135C22×C59—42no—
136C22×C59—82no—
137C22×C59#37911yescharacteristic
138C22×C59C2×#10112yes—
139C4×C59—21no—
140C4×C59—22no—
141C4×C59—22no—
142C4×C59—42no—
143C4×C59—44no—
144C4×C59—44no—
145C4×C59#45311yescharacteristic
146C4×C59#45511yescharacteristic
147#3501D5911yescharacteristic
148#1955—591no—
149#1552—592no—
150#1531—592no—
151#1466—591no—
152C2×C59—22no—
153C2×C59—44no—
154C2×C59#178411yescharacteristic
155C2×C59#179211yescharacteristic
156C2×C59#179411yescharacteristic
157#554—22no—
158#554—24no—
159D4×Q8—22no—
160#544Dic5912yes—
161#543—22no—
162#533C2×D5912yes—
163C2×C22⋊Q8—22no—
164#518Dic5912yes—
165C2×C4×D4C2×D5911yescharacteristic
166#432—592no—
167C2×C4⋊C8—591no—
168C2×C22⋊C8—1182no—
169C2×#97—591no—
170#332—1182no—
171#332—1182no—
172#330—1184no—
173#323—1182no—
174C59#845711yescharacteristic, complemented, Sylow
175C22×Q8—22no—
176C22×D4C2×Dic5911yescharacteristic
177C23×C4C22⋊D5912yes—
178C22×C8—1181no—
179C22×C8—1181no—
180C4⋊Q8—24no—
181C4⋊Q8C22⋊D5912yes—
182C4⋊Q8C22⋊D5912yes—
183#125—24no—
184#125C22⋊D5914yes—
185C22⋊Q8—24no—
186C22⋊Q8—24no—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8—24no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—22no—
197C4×D4—22no—
198#117C2×Dic5914yes—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4C2×Dic5911yescharacteristic
202C2×C22⋊C4—24no—
203C2×C22⋊C4C22⋊D5912yes—
204C2×C42C22×D5911yescharacteristic
205C4⋊C8—1182no—
206C4⋊C8—1182no—
207C22⋊C8—1182no—
208C22⋊C8—2364no—
209#97—1182no—
210#97—1182no—
211C24C22⋊Dic5912yes—
212C2×Q8—24no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—82no—
216C2×Q8—82no—
217C2×D4—24no—
218C2×D4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4C22⋊Dic5912yes—
226C22×C4C22×Dic5911yescharacteristic
227C22×C4C2×C22⋊D5911yescharacteristic
228C22×C4C2×C22⋊D5911yescharacteristic
229C2×C8—1182no—
230C2×C8—2362no—
231C2×C8—2362no—
232C2×C8—2361no—
233C2×C8—2361no—
234C2×C8—2361no—
235C4⋊C4—24no—
236C4⋊C4—28no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4C22⋊Dic5914yes—
243C4⋊C4C22⋊Dic5914yes—
244C22⋊C4—28no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C42—24no—
248C42C2×C22⋊D5912yes—
249C42C2×C22⋊D5912yes—
250C23—22no—
251C23—22no—
252C23—42no—
253C23—42no—
254C23#1427364614yes—
255C23C2×C22⋊Dic5911yescharacteristic
256Q8—42no—
257Q8—42no—
258Q8—82no—
259Q8—84no—
260D4—28no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—82no—
272C2×C4—84no—
273C2×C4—84no—
274C2×C4C2×C22⋊Dic5912yes—
275C2×C4#1427369612yes—
276C2×C4#1427369611yescharacteristic
277C2×C4#1427369611yescharacteristic
278C8—4722no—
279C8—4721no—
280C8—2362no—
281C22—21no—
282C22—21no—
283C22—24no—
284C22—44no—
285C22—42no—
286C22—42no—
287C22—82no—
288C22#1683725811yescharacteristic
289C22C2×#1427364612yes—
290C4—22no—
291C4—21no—
292C4—22no—
293C4—42no—
294C4—44no—
295C4—44no—
296C4#1683757011yescharacteristic
297C4#1683758011yescharacteristic
298C2—22no—
299C2—44no—
300C2#2142402011yescharacteristic
301C2#2142405011yescharacteristic
302C2#2142406011yescharacteristic
subgroupG/H
centerC22#16837258
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3501C2
socleC22×C59#379
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenter#3501D59
hypocenterC59#8457
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
2. #21422796C2semidirect
3. #21422711C2semidirect
174. C59#8457semidirect

Maximal subgroups

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

subgroupindex
#214248952
#214227962
#214227112
#214224372
C59×#35012
#845759