Finite Group Databasebeta

← #29572187

Subgroups — #29572187

Order 15616.

1014 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 115 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#21459463C211yesmaximal, characteristic
2#21445737C211yesmaximal, characteristic, complemented
3#21444376C214yesmaximal, complemented
4C61×#1780C211yesmaximal, characteristic, complemented
5C2×#14519822C411yescharacteristic, complemented
6#16840411—24no—
7#16840400—24no—
8#16840336C412yescomplemented
9#16840321—22no—
10#16840273C2212yes—
11C2×#14519760—22no—
12#16840237C411yescharacteristic, complemented
13C2×#14519692—22no—
14#16839625C2212yescomplemented
15#16839500—24no—
16#16839497—24no—
17C22×#41×C61C2211yescharacteristic
18C2×C22⋊C8×C61—22no—
19C2×C22⋊C8×C61C2212yescomplemented
20#14519822—44no—
21C2×#11758222C2×C412yes—
22C2×#11758224—22no—
23C22×C22⋊D61—22no—
24C2×D4×D61—22no—
25#14519775—22no—
26#14519766—22no—
27#14519760—28no—
28#14519760—24no—
29#14519760—22no—
30#14519758—24no—
31#14519756D414yes—
32#14519753—24no—
33#14519751D414yes—
34#14519749—28no—
35#14519746—22no—
36#14519745—24no—
37D4×Dic61—24no—
38C4×C22⋊D61—24no—
39#14519736C2×C412yes—
40C2×#11758196—24no—
41C2×C22⋊Dic61—22no—
42C2×C22⋊Dic61—22no—
43C2×#11758198—24no—
44#14519692—44no—
45#14519684—24no—
46C2×#41×C61—24no—
47C2×#41×C61—44no—
48C22×Q8×C61C2×C411yescharacteristic
49C22×D4×C61C2×C411yescharacteristic
50C22×D4×C61D412yes—
51C23×C4×C61D412yes—
52C23×C4×C61C2311yescharacteristic
53C22×C8×C61—22no—
54C22⋊C8×C61—24no—
55C22⋊C8×C61—44no—
56C24×C61C22⋊C412yes—
57C24×C61C2×D411yescharacteristic
58C23×D61—22no—
59#41×C61—416no—
60C2×Q8×C61—22no—
61C2×Q8×C61—42no—
62C2×D4×C61—24no—
63C2×D4×C61—24no—
64C2×D4×C61—24no—
65C2×D4×C61—22no—
66C2×D4×C61—42no—
67C22×C4×C61—24no—
68C22×C4×C61—24no—
69C22×C4×C61—42no—
70C22×C4×C61—42no—
71C22×C4×C61C22⋊C412yes—
72C22×C4×C61C22×C411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D414yes—
75C2×C22⋊D61—24no—
76C2×C22⋊D61—22no—
77C2×C22⋊D61—24no—
78C2×C22⋊D61—44no—
79C2×C22⋊D61—44no—
80C22×Dic61—22no—
81C22×Dic61—24no—
82C22×Dic61—42no—
83#11758224—48no—
84D4×D61—48no—
85#11758222—44no—
86C2×D244C22⋊C412yes—
87C2×C4×D61—22no—
88C2×Dic122C22⋊C412yes—
89C2×C8×C61—44no—
90C2×C8×C61—42no—
91C22⋊Dic61—24no—
92C22⋊Dic61—22no—
93C22⋊Dic61—24no—
94C22⋊Dic61—44no—
95#11758198—28no—
96#11758198—24no—
97C4⋊Dic61C22⋊C414yes—
98#11758196—28no—
99#11758196—24no—
100C4×Dic61—22no—
101C23×C61—22no—
102C23×C61—22no—
103C23×C61—22no—
104C23×C61—42no—
105C23×C61—44no—
106C23×C61#9914yes—
107C23×C61C2×C22⋊C411yescharacteristic
108C23×C61#12012yes—
109C22×D61—22no—
110C22×D61—44no—
111C22×D61—84no—
112Q8×C61—44no—
113D4×C61—28no—
114D4×C61—48no—
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×C61C2×C22⋊C412yes—
123C2×C4×C61#12012yes—
124C22⋊D61—48no—
125C22⋊D61—44no—
126C22⋊D61—84no—
127C2×Dic61—24no—
128C2×Dic61—22no—
129C2×Dic61—44no—
130C2×Dic61—84no—
131C2×Dic61—82no—
132C2×Dic61—84no—
133D244—42no—
134C4×D61—44no—
135Dic122—42no—
136C8×C61—82no—
137#8462—611nomaximal, Sylow
138C22×C61—22no—
139C22×C61—24no—
140C22×C61—24no—
141C22×C61—22no—
142C22×C61—44no—
143C22×C61—44no—
144C22×C61—82no—
145C22×C61—82no—
146C22×C61#37911yescharacteristic
147C22×C61C2×#9912yes—
148C2×D61—44no—
149C2×D61—42no—
150C2×D61—164no—
151C4×C61—22no—
152C4×C61—22no—
153C4×C61—42no—
154Dic61—44no—
155Dic61—42no—
156Dic61—162no—
157#3485—611no—
158#1782—611no—
159#1780D6111yescharacteristic, complemented
160#1516—614no—
161C2×C61—22no—
162C2×C61—42no—
163C2×C61—44no—
164C2×C61#178111yescharacteristic
165C2×C61#179412yes—
166D61—82no—
167C2×H(5,𝔽2)—611no—
168C22×#41C2×D6111yescharacteristic
169#547—1224no—
170#546—1224no—
171#536—612no—
172#534—1222no—
173#525—612no—
174C2×#123—1222no—
175#518—611no—
176C2×#99—1222no—
177C2×C22⋊C8—22no—
178C2×C22⋊C8C2×D6112yescomplemented
179#381—612no—
180#328—1224no—
181#327—1224no—
182C61#846211yescharacteristic, complemented, Sylow
183H(5,𝔽2)—2444no—
184C2×#41—24no—
185C2×#41—44no—
186C2×#41—612no—
187C2×#41—1222no—
188C22×Q8C4×D6111yescharacteristic
189C22×D4—1222no—
190C22×D4—1222no—
191C22×D4C4×D6111yescharacteristic
192C22×D4D24412yes—
193C23×C4C22⋊D6112yes—
194C23×C4C22×D6111yescharacteristic
195C22×C8—22no—
196C4⋊Q8—1222no—
197#124—1222no—
198#123—1228no—
199#123—1224no—
200#123—1224no—
201#123—1222no—
202C22⋊Q8—614no—
203C22⋊Q8—1224no—
204#121—614no—
205#121—1228no—
206#120—1224no—
207#120—1222no—
208C4×D4—1224no—
209C4×D4—1224no—
210#117—612no—
211C2×C4⋊C4—1224no—
212C2×C22⋊C4—1224no—
213C2×C22⋊C4—1222no—
214C2×C22⋊C4—1222no—
215#99—2444no—
216C22⋊C8—24no—
217C22⋊C8—44no—
218#95—1224no—
219C24—1222no—
220C24#1175819812yes—
221C24C2×D24411yescharacteristic
222#41—416no—
223#41—2444no—
224#41—2448no—
225C2×Q8—22no—
226C2×Q8—42no—
227C2×Q8—612no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—22no—
232C2×D4—42no—
233C2×D4—612no—
234C2×D4—1224no—
235C2×D4—1224no—
236C2×D4—1222no—
237C2×D4—2448no—
238C2×D4—2444no—
239C2×D4—2444no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—1224no—
245C22×C4—1222no—
246C22×C4—1222no—
247C22×C4—2442no—
248C22×C4#1175819812yes—
249C22×C4C2×C4×D6111yescharacteristic
250C22×C4D4×D6114yes—
251C22×C4C2×C22⋊D6111yescharacteristic
252C2×C8—44no—
253C2×C8—42no—
254C4⋊C4—614no—
255C4⋊C4—1228no—
256C4⋊C4—1224no—
257C22⋊C4—1228no—
258C22⋊C4—1224no—
259C22⋊C4—1222no—
260C22⋊C4—1224no—
261C22⋊C4—1224no—
262C22⋊C4—2444no—
263C42—1222no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—42no—
268C23—44no—
269C23—1222no—
270C23—4884no—
271C23—2444no—
272C23#1451969114yes—
273C23C2×#1175819811yescharacteristic
274C23#1451974712yes—
275Q8—44no—
276Q8—2442no—
277D4—28no—
278D4—48no—
279D4—4884no—
280D4—2442no—
281D4—2448no—
282D4—2444no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—44no—
289C2×C4—48no—
290C2×C4—1224no—
291C2×C4—1222no—
292C2×C4—4882no—
293C2×C4—4884no—
294C2×C4—4884no—
295C2×C4—2444no—
296C2×C4—2444no—
297C2×C4C22⋊C4×D6112yes—
298C2×C4#1451974512yes—
299C8—82no—
300C22—22no—
301C22—22no—
302C22—24no—
303C22—24no—
304C22—44no—
305C22—44no—
306C22—82no—
307C22—82no—
308C22—9764no—
309C22—2442no—
310C22—2444no—
311C22#1683961711yescharacteristic
312C22C2×#1451969112yes—
313C4—22no—
314C4—22no—
315C4—42no—
316C4—9762no—
317C4—2442no—
318C4—2444no—
319C2—22no—
320C2—42no—
321C2—44no—
322C2—4882no—
323C2#2144573311yescharacteristic
324C2#2144578212yes—
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1780C2
socleC22×C61#379
prime radical#21459463C2
cosocleC23×C4×C61C23
hypercenter#1780D61
hypocenterC61#8462
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1780

Complements

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

NH ≅ G/Ntype
2. #21445737C2semidirect
3. #21444376C2semidirect
4. C61×#1780C2semidirect
5. C2×#14519822C4semidirect
8. #16840336C4semidirect
12. #16840237C4semidirect
14. #16839625C22semidirect
19. C2×C22⋊C8×C61C22semidirect
159. #1780D61semidirect
178. C2×C22⋊C8C2×D61semidirect
182. C61#8462semidirect

Maximal subgroups

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

subgroupindex
#214594632
#214457372
#214443762
C61×#17802
#846261