Finite Group Databasebeta

← #29591146

Subgroups — #29591146

Order 15616.

452 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 99 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#21450992C211yesmaximal, characteristic
2#21445807C211yesmaximal, characteristic
3#21443841C214yesmaximal, complemented
4C61×#1459C211yesmaximal, characteristic
5C2×#14519777C411yescharacteristic, complemented
6C2×#14519692—21no—
7#16839674C412yescomplemented
8#16839674C2212yes—
9#16839665—21no—
10#16839627—21no—
11#16839622—21no—
12#16839622—21no—
13#16839622—21no—
14C4×C4⋊Dic61C411yescharacteristic, complemented
15#16839529C2212yes—
16C2×C4×D4×C61C2211yescharacteristic
17C2×C22⋊C8×C61—21no—
18C61×#324—21no—
19C61×#323C2212yes—
20C22×Dic122C2×C412yes—
21#14519777—41no—
22#14519777—41no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—41no—
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61—22no—
29C2×C22⋊Dic61—22no—
30C2×C4×Dic61—21no—
31#14519692—42no—
32#14519684—22no—
33#14519684—22no—
34#14519684—41no—
35#14519684C2×C412yes—
36C22×D4×C61D411yescharacteristic
37C23×C4×C61D411yescharacteristic
38C23×C4×C61D411yescharacteristic
39C22×C8×C61—41no—
40C4×D4×C61—41no—
41C4×D4×C61—41no—
42C2×C4⋊C4×C61D411yescharacteristic
43C2×C22⋊C4×C61C2×C411yescharacteristic
44C2×C22⋊C4×C61C2311yescharacteristic
45C2×C42×C61C2×C411yescharacteristic
46C22⋊C8×C61—22no—
47C22⋊C8×C61—41no—
48C24×C61C22⋊C411yescharacteristic
49C24×C61C2×D411yescharacteristic
50C2×D4×C61—41no—
51C2×D4×C61—41no—
52C2×D4×C61SD1612yes—
53C2×D4×C61Q1612yes—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—42no—
59C22×C4×C61—42no—
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22×C411yescharacteristic
64C22×C4×C61C2×D411yescharacteristic
65C22×Dic61—22no—
66C22×Dic61—22no—
67C22×Dic61—42no—
68C2×Dic122—42no—
69C2×Dic122—82no—
70C2×C8×C61—42no—
71C2×C8×C61—82no—
72C4⋊C4×C61—41no—
73C22⋊C4×C61—41no—
74C22⋊C4×C61—41no—
75C42×C61—21no—
76C42×C61—21no—
77C22⋊Dic61—42no—
78C22⋊Dic61—42no—
79C22⋊Dic61—82no—
80C22⋊Dic61—82no—
81C4⋊Dic61—42no—
82C4⋊Dic61—42no—
83C4⋊Dic61—41no—
84C4⋊Dic61—41no—
85C4⋊Dic61—82no—
86C4×Dic61—42no—
87C23×C61—21no—
88C23×C61—21no—
89C23×C61—21no—
90C23×C61—21no—
91C23×C61—41no—
92C23×C61—41no—
93C23×C61—41no—
94C23×C61—41no—
95C23×C61#10312yes—
96C23×C61C2×C22⋊C411yescharacteristic
97C23×C61C2×SD1611yescharacteristic
98C23×C61C2×Q1611yescharacteristic
99D4×C61—42no—
100D4×C61—42no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—22no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—81no—
113C2×C4×C61—81no—
114C2×C4×C61#9911yescharacteristic
115C2×C4×C61#9911yescharacteristic
116C2×C4×C61#10312yes—
117C2×Dic61—42no—
118C2×Dic61—42no—
119C2×Dic61—42no—
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123C2×Dic61—82no—
124C2×Dic61—82no—
125C2×Dic61—82no—
126Dic122—84no—
127C8×C61—82no—
128#11900—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—21no—
132C22×C61—22no—
133C22×C61—21no—
134C22×C61—42no—
135C22×C61—41no—
136C22×C61—41no—
137C22×C61—41no—
138C22×C61—41no—
139C22×C61—41no—
140C22×C61—41no—
141C22×C61—81no—
142C22×C61—81no—
143C22×C61C2×#9911yescharacteristic
144C22×C61C2×#10311yescharacteristic
145C22×C61#42111yescharacteristic
146C4×C61—21no—
147C4×C61—22no—
148C4×C61—41no—
149C4×C61—41no—
150C4×C61—41no—
151C4×C61#35412yes—
152Dic61—82no—
153Dic61—82no—
154Dic61—162no—
155#2563—611no—
156#1799—611no—
157#1459D6111yescharacteristic
158#1351—614no—
159C2×C61—21no—
160C2×C61—21no—
161C2×C61—41no—
162C2×C61—42no—
163C2×C61—41no—
164C2×C61#152011yescharacteristic
165C2×C61#212211yescharacteristic
166C2×C61C2×#35411yescharacteristic
167C2×C4⋊Q8—611no—
168C2×C4×D4C2×D6111yescharacteristic
169C2×#99—1221no—
170C2×C22⋊C8—21no—
171#393—612no—
172#393—612no—
173#390—1221no—
174#382—1221no—
175#380—1221no—
176#380—1221no—
177#380—1221no—
178C4×C4⋊C4—611no—
179#342—612no—
180#324—21no—
181#323C2×D6112yes—
182C61#1190011yescharacteristic, complemented, Sylow
183C22×Q8—612no—
184C22×D4D24411yescharacteristic
185C23×C4D24411yescharacteristic
186C23×C4C22⋊D6111yescharacteristic
187C22×C8—41no—
188C4⋊Q8—2441no—
189C4⋊Q8—2441no—
190C4×D4—41no—
191C4×D4—41no—
192C2×C4⋊C4—1221no—
193C2×C4⋊C4—1221no—
194C2×C4⋊C4—1221no—
195C2×C4⋊C4—2441no—
196C2×C4⋊C4C22⋊D6111yescharacteristic
197C2×C22⋊C4—1222no—
198C2×C22⋊C4—1222no—
199C2×C22⋊C4—1222no—
200C2×C22⋊C4C4×D6111yescharacteristic
201C2×C22⋊C4C22×D6111yescharacteristic
202C2×C42—1221no—
203C2×C42C4×D6111yescharacteristic
204#99—2442no—
205C22⋊C8—22no—
206C22⋊C8—41no—
207#95—612no—
208#95—1222no—
209#95—1222no—
210#95—2441no—
211C24#1175819811yescharacteristic
212C24C2×D24411yescharacteristic
213C2×Q8—4882no—
214C2×Q8—2442no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4#1175819012yes—
218C2×D4Dic24412yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—1222no—
226C22×C4—1222no—
227C22×C4—2442no—
228C22×C4#1175819811yescharacteristic
229C22×C4#1175819811yescharacteristic
230C22×C4#1175819811yescharacteristic
231C22×C4C2×C4×D6111yescharacteristic
232C22×C4C2×C22⋊D6111yescharacteristic
233C2×C8—42no—
234C2×C8—82no—
235C4⋊C4—41no—
236C4⋊C4—4882no—
237C4⋊C4—2442no—
238C4⋊C4—2442no—
239C4⋊C4—2441no—
240C4⋊C4—2441no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—4882no—
244C22⋊C4—4882no—
245C22⋊C4—2442no—
246C22⋊C4—2442no—
247C42—21no—
248C42—21no—
249C42—2442no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23#1451970212yes—
259C23C2×#1175819811yescharacteristic
260C23C2×#1175819011yescharacteristic
261C23C2×Dic24411yescharacteristic
262Q8—4884no—
263D4—42no—
264D4—42no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—4882no—
279C2×C4—4882no—
280C2×C4—4882no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4—2442no—
284C2×C4—2442no—
285C2×C4—2442no—
286C2×C4—2442no—
287C2×C4#1451969111yescharacteristic
288C2×C4#1451969111yescharacteristic
289C2×C4#1451970212yes—
290C8—82no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—21no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22—81no—
305C22C2×#1451969111yescharacteristic
306C22C2×#1451970211yescharacteristic
307C22#1683977811yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—9762no—
314C4—4882no—
315C4—4882no—
316C4#1683955612yes—
317C2—21no—
318C2—21no—
319C2—41no—
320C2—42no—
321C2—41no—
322C2#2144439211yescharacteristic
323C2#2144747011yescharacteristic
324C2C2×#1683955611yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1459C2
socleC22×C61C2×#99
prime radicalC22×D4×C61D4
cosocleC2×C22⋊C4×C61C23
hypercenter#1459D61
hypocenterC61#11900
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #21443841C2semidirect
5. C2×#14519777C4semidirect
7. #16839674C4semidirect
14. C4×C4⋊Dic61C4semidirect
182. C61#11900semidirect

Maximal subgroups

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

subgroupindex
#214509922
#214458072
#214438412
C61×#14592
#1190061