Finite Group Databasebeta

← #29572151

Subgroups — #29572151

Order 15616.

1038 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1038 conjugacy classes of subgroups — too many to draw. Only its 123 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#21459470C211yesmaximal, characteristic
2#21444324C214yesmaximal, complemented
3#21444090C211yesmaximal, characteristic, complemented
4C61×#1784C211yesmaximal, characteristic, complemented
5C2×#14519825C411yescharacteristic, complemented
6Q8×C22⋊D61—24no—
7#16840423—24no—
8#16840336—22no—
9#16840336C412yescomplemented
10#16840273—22no—
11#16840273C2212yes—
12#16840240C411yescharacteristic, complemented
13C2×#14519684—22no—
14C2×#14519684C2212yes—
15#16839500—24no—
16#16839497—24no—
17C23×Q8×C61C2211yescharacteristic
18C2×#101×C61—22no—
19C61×#407C2212yes—
20#14519825—44no—
21C2×#11758226—22no—
22C2×#11758222—22no—
23C2×#11758222C2×C412yes—
24C2×Q8×D61—22no—
25C23×Dic61—22no—
26#14519775—24no—
27#14519775—22no—
28#14519766—24no—
29#14519766—22no—
30#14519760—24no—
31#14519760—22no—
32#14519756—24no—
33#14519756D414yes—
34#14519753—28no—
35#14519753—24no—
36#14519751—24no—
37#14519751D414yes—
38#14519746—22no—
39Q8×Dic61—24no—
40C4×C22⋊D61—28no—
41C4×C22⋊D61—24no—
42#14519736—22no—
43#14519736C2×C412yes—
44#14519684—24no—
45#14519684—44no—
46C22×Q8×C61—24no—
47C22×Q8×C61—44no—
48C22×Q8×C61C2×C411yescharacteristic
49C22×Q8×C61C2×C411yescharacteristic
50C22×Q8×C61D412yes—
51C23×C4×C61D412yes—
52C23×C4×C61C2311yescharacteristic
53C2×M16×C61—22no—
54#101×C61—44no—
55C22⋊C8×C61—24no—
56C24×C61C2×D411yescharacteristic
57C2×Q8×C61—24no—
58C2×Q8×C61—24no—
59C2×Q8×C61—22no—
60C2×Q8×C61—22no—
61C2×Q8×C61—24no—
62C2×Q8×C61—42no—
63C2×Q8×C61—42no—
64C2×Q8×C61—416no—
65C22×C4×C61—24no—
66C22×C4×C61—24no—
67C22×C4×C61—42no—
68C22×C4×C61—42no—
69C22×C4×C61C22⋊C412yes—
70C22×C4×C61C22⋊C412yes—
71C22×C4×C61C22×C411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D414yes—
74C2×C22⋊D61—22no—
75C2×C22⋊D61—24no—
76C22×Dic61—24no—
77C22×Dic61—42no—
78C22×Dic61—44no—
79C22×Dic61—44no—
80C22×Dic61—42no—
81#11758226—48no—
82Q8×D61—48no—
83#11758222—44no—
84#11758222—44no—
85#11758222—44no—
86C2×D244—22no—
87C2×D244C22⋊C412yes—
88C2×C4×D61—24no—
89C2×C4×D61—22no—
90C2×Dic122—22no—
91C2×Dic122C22⋊C412yes—
92M16×C61—44no—
93C2×C8×C61—42no—
94C22⋊Dic61—22no—
95C22⋊Dic61—24no—
96#11758198—28no—
97#11758198—24no—
98C4⋊Dic61—24no—
99C4⋊Dic61C22⋊C414yes—
100#11758196—28no—
101#11758196—24no—
102C4×Dic61—24no—
103C4×Dic61—22no—
104C23×C61—22no—
105C23×C61—22no—
106C23×C61—22no—
107C23×C61C2×C22⋊C411yescharacteristic
108C23×C61#12012yes—
109C22×D61—22no—
110Q8×C61—28no—
111Q8×C61—48no—
112Q8×C61—44no—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—22no—
117C2×C4×C61—22no—
118C2×C4×C61—22no—
119C2×C4×C61—22no—
120C2×C4×C61—44no—
121C2×C4×C61—48no—
122C2×C4×C61#9914yes—
123C2×C4×C61C2×C22⋊C412yes—
124C2×C4×C61#12012yes—
125C22⋊D61—44no—
126C22⋊D61—44no—
127C22⋊D61—44no—
128C2×Dic61—24no—
129C2×Dic61—22no—
130C2×Dic61—48no—
131C2×Dic61—44no—
132C2×Dic61—84no—
133C2×Dic61—82no—
134C2×Dic61—82no—
135D244—42no—
136D244—44no—
137C4×D61—48no—
138C4×D61—44no—
139Dic122—44no—
140Dic122—42no—
141C8×C61—82no—
142#8455—611nomaximal, Sylow
143C22×C61—24no—
144C22×C61—22no—
145C22×C61—22no—
146C22×C61—22no—
147C22×C61—22no—
148C22×C61—22no—
149C22×C61#37911yescharacteristic
150C22×C61C2×#9912yes—
151C22×C61#45412yes—
152C22×C61#45612yes—
153C2×D61—44no—
154C2×D61—42no—
155C4×C61—22no—
156C4×C61—24no—
157C4×C61—22no—
158C4×C61—42no—
159Dic61—44no—
160Dic61—42no—
161Dic61—84no—
162#3486—611no—
163#1784D6111yescharacteristic, complemented
164#1504—614no—
165#1446—611no—
166C2×C61—22no—
167C2×C61—22no—
168C2×C61—22no—
169C2×C61#178111yescharacteristic
170C2×C61#179312yes—
171D61—82no—
172C2×2−1+4—611no—
173C23×Q8C2×D6111yescharacteristic
174D4×Q8—1224no—
175#548—1224no—
176#536—612no—
177#536—1222no—
178#525—612no—
179#525—1222no—
180#519—611no—
181C2×#101—22no—
182#407C2×D6112yes—
183C2×#95—612no—
184C2×#95—1222no—
185#328—1224no—
186#327—1224no—
187C61#845511yescharacteristic, complemented, Sylow
1882−1+4—2444no—
189C2×#41—612no—
190C2×#41—1222no—
191C2×#41—1222no—
192C22×Q8—24no—
193C22×Q8—44no—
194C22×Q8—1222no—
195C22×Q8C4×D6111yescharacteristic
196C22×Q8C4×D6111yescharacteristic
197C22×Q8C22⋊D6112yes—
198C23×C4—1222no—
199C23×C4D24412yes—
200C23×C4C22×D6111yescharacteristic
201C2×M16—22no—
202C4⋊Q8—1222no—
203C4⋊Q8—1224no—
204#124—1222no—
205#124—1224no—
206#123—1224no—
207#123—1222no—
208C22⋊Q8—614no—
209C22⋊Q8—1224no—
210C22⋊Q8—1224no—
211C22⋊Q8—1228no—
212#121—614no—
213#121—1224no—
214#120—1222no—
215C4×Q8—1224no—
216C4×D4—1228no—
217C4×D4—1224no—
218#117—612no—
219#117—1222no—
220#101—44no—
221C22⋊C8—24no—
222#95—1224no—
223#95—2444no—
224C24C2×D24411yescharacteristic
225#41—2444no—
226#41—2444no—
227#41—2448no—
228#41—2444no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—24no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—416no—
237C2×Q8—612no—
238C2×Q8—1222no—
239C2×Q8—2448no—
240C2×D4—612no—
241C2×D4—1222no—
242C2×D4—1222no—
243C2×D4—1224no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—1224no—
249C22×C4—1222no—
250C22×C4—1224no—
251C22×C4—2444no—
252C22×C4—2442no—
253C22×C4—2444no—
254C22×C4—2442no—
255C22×C4#1175819812yes—
256C22×C4#1175819812yes—
257C22×C4C2×C4×D6111yescharacteristic
258C22×C4D4×D6114yes—
259C22×C4C2×C22⋊D6111yescharacteristic
260M16—44no—
261C2×C8—42no—
262C4⋊C4—614no—
263C4⋊C4—1228no—
264C4⋊C4—1224no—
265C4⋊C4—1224no—
266C22⋊C4—1228no—
267C22⋊C4—1222no—
268C22⋊C4—1224no—
269C22⋊C4—1224no—
270C42—1224no—
271C42—1222no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—1222no—
276C23C2×#1175819811yescharacteristic
277C23#1451974712yes—
278Q8—28no—
279Q8—48no—
280Q8—44no—
281Q8—2442no—
282Q8—2444no—
283D4—2442no—
284D4—2444no—
285D4—2444no—
286D4—2444no—
287D4—2444no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—1224no—
298C2×C4—1222no—
299C2×C4—4884no—
300C2×C4—4882no—
301C2×C4—4882no—
302C2×C4—2448no—
303C2×C4—2444no—
304C2×C4—2448no—
305C2×C4—2444no—
306C2×C4#1451969114yes—
307C2×C4C22⋊C4×D6112yes—
308C2×C4#1451974512yes—
309C8—82no—
310C22—22no—
311C22—24no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—2442no—
317C22—2444no—
318C22#1683961711yescharacteristic
319C22C2×#1451969112yes—
320C22#1683993712yes—
321C22#1683994712yes—
322C4—22no—
323C4—24no—
324C4—22no—
325C4—42no—
326C4—4884no—
327C4—2442no—
328C4—2444no—
329C2—22no—
330C2—22no—
331C2—22no—
332C2—4882no—
333C2#2144573311yescharacteristic
334C2#2144578012yes—
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1784C2
socleC22×C61#379
prime radical#21459470C2
cosocleC23×C4×C61C23
hypercenter#1784D61
hypocenterC61#8455
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
2. #21444324C2semidirect
3. #21444090C2semidirect
4. C61×#1784C2semidirect
5. C2×#14519825C4semidirect
9. #16840336C4semidirect
12. #16840240C4semidirect
163. #1784D61semidirect
187. C61#8455semidirect

Maximal subgroups

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

subgroupindex
#214594702
#214443242
#214440902
C61×#17842
#845561