Finite Group Databasebeta

← #30617883

Subgroups — #30617883

Order 15616.

1050 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 220 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#21462266C212yesmaximal
2#21462010C218yesmaximal
3#21461717C214yesmaximal
4C2×#16840318C211yesmaximal, characteristic
5#16840707—22no—
6#16840706—22no—
7#16840705C2214yes—
8#16840704C2214yes—
9C4×C4⋊#6923—22no—
10C4×C22⋊#6923C2214yes—
11#16840698—22no—
12#16840640C22116yes—
13#16840318—22no—
14#16840318—22no—
15#16840318—22no—
16#16840318—22no—
17C2×#14519776C412yescomplemented
18C2×#14519769C411yescharacteristic, complemented
19C2×#14519755C2214yes—
20C2×#14519742C414yescomplemented
21C2×#14519735C2212yes—
22C2×C42×D61C2211yescharacteristic
23C2×#125×C61C411yescharacteristic, complemented
24C2×C4⋊#6923—24no—
25C2×C22⋊#6923C2318yes—
26C2×C4×#6923—24no—
27(C2×#6923)⋊C4—24no—
28(C2×#6923)⋊C4—24no—
29(C2×#6923)⋊C4—24no—
30(C2×#6923)⋊C4—24no—
31C22×Dic122C2×C414yes—
32C22×C4×D61C2311yescharacteristic
33C22×C4×D61C2312yes—
34#14519776—22no—
35#14519776—22no—
36#14519776—22no—
37#14519776—22no—
38#14519769—21no—
39#14519769—22no—
40#14519769—21no—
41#14519755—24no—
42#14519755—24no—
43#14519755—24no—
44#14519755—24no—
45#14519742—24no—
46#14519742—24no—
47#14519742—24no—
48#14519742—24no—
49#14519735—24no—
50#14519735—24no—
51C2×C4⋊Dic61C2×C412yes—
52C2×#11758196C2×C414yes—
53C2×#11758196C2×C414yes—
54C2×#11758198C2314yes—
55C2×C4×Dic61C2×C414yes—
56C2×C4×Dic61C2×C412yes—
57C2×C4×Dic61C2×C411yescharacteristic
58C42×D61D412yes—
59C42×D61D412yes—
60C42×D61D412yes—
61C42×D61D412yes—
62#125×C61—21no—
63#125×C61—22no—
64#125×C61—21no—
65C2×C4⋊C4×C61C2×C412yes—
66C2×C4⋊C4×C61C2×C414yes—
67C2×C42×C61C2×C411yescharacteristic
68C23×D61C2411yescharacteristic
69C22×#6923—28no—
70C22×C4×C61C22×C411yescharacteristic
71C22×C4×C61C22×C414yes—
72C22×C4×C61C22×C412yes—
73C22×Dic61C22×C412yes—
74C22×Dic61C22×C411yescharacteristic
75C22×Dic61C22×C414yes—
76C2×C4×D61—24no—
77C2×C4×D61—44no—
78C2×C4×D61C2×D412yes—
79C2×C4×D61C2×D412yes—
80C2×C4×D61C2×D412yes—
81C2×C4×D61C2×D412yes—
82C2×C4×D61C2×D412yes—
83C2×C4×D61C2×D412yes—
84C2×Dic122—24no—
85C2×Dic122—24no—
86C2×Dic122—48no—
87C22⋊#6923—216no—
88C4⋊#6923—48no—
89C4×#6923—48no—
90C4⋊C4×C61—22no—
91C4⋊C4×C61—22no—
92C4⋊C4×C61—24no—
93C4⋊C4×C61—24no—
94C42×C61C22⋊C411yescharacteristic
95C42×C61C22⋊C412yes—
96C42×C61C22⋊C411yescharacteristic
97#11758198—28no—
98C4⋊Dic61—22no—
99C4⋊Dic61—22no—
100#11758196—24no—
101#11758196—24no—
102#11758196—24no—
103#11758196—24no—
104C4×Dic61—28no—
105C4×Dic61C22⋊C412yes—
106C4×Dic61C22⋊C412yes—
107C4×Dic61C22⋊C412yes—
108C4×Dic61C22⋊C412yes—
109C4×Dic61C22⋊C411yescharacteristic
110C4×Dic61C22⋊C412yes—
111C4×Dic61C22⋊C411yescharacteristic
112C23×C61C23×C411yescharacteristic
113C22×D61—42no—
114C22×D61C22×D412yes—
115C22×D612−1+414yes—
116C2×#6923—48no—
117C2×#6923—48no—
118C2×#6923—48no—
119C2×C4×C61—22no—
120C2×C4×C61—24no—
121C2×C4×C61—42no—
122C2×C4×C61—44no—
123C2×C4×C61C2×C22⋊C412yes—
124C2×C4×C61C2×C22⋊C412yes—
125C2×C4×C61C2×C22⋊C411yescharacteristic
126C2×C4×C61C2×C22⋊C411yescharacteristic
127C2×Dic61—22no—
128C2×Dic61—24no—
129C2×Dic61—24no—
130C2×Dic61—44no—
131C2×Dic61C2×C22⋊C412yes—
132C2×Dic61C2×C22⋊C412yes—
133C2×Dic61C2×C22⋊C412yes—
134C4×D61—24no—
135C4×D61—24no—
136C4×D61—24no—
137C4×D61—24no—
138C4×D61—42no—
139C4×D61—42no—
140Dic122—48no—
141Dic122—48no—
142#21826—611nomaximal, Sylow
143C22×C61—22no—
144C22×C61C22×C22⋊C411yescharacteristic
145C22×C61#51912yes—
146C2×D61—22no—
147C2×D61—22no—
148C2×D61—44no—
149C2×D61#53614yes—
150#6923—88no—
151C4×C61—21no—
152C4×C61—21no—
153C4×C61—22no—
154C4×C61—22no—
155C4×C61—44no—
156Dic61—22no—
157Dic61—22no—
158Dic61—44no—
159Dic61#41014yes—
160C2×#533—611no—
161#2283—612no—
162#1996—618no—
163#1820—614no—
164C2×C61—22no—
165C2×C61#231011yescharacteristic
166C2×C61C2×#41012yes—
167D61—24no—
168#533—1222no—
169#533—1222no—
170#533—1222no—
171#533—1222no—
172C2×C4⋊Q8—612no—
173C2×#125—611no—
174C2×#125#692311yescharacteristic, complemented
175C2×C22⋊Q8—614no—
176C2×C4×Q8—614no—
177C2×#117—612no—
178C22×C42—611no—
179#383—1222no—
180#382—1222no—
181#381—614no—
182#380—614no—
183C4×C4⋊C4—1222no—
184C4×C22⋊C4—614no—
185#376—1222no—
186#342—6116no—
187C61#2182611yescharacteristic, complemented, Sylow
188C22×Q8—614no—
189C23×C4—611no—
190C23×C4—612no—
191C4⋊Q8—1222no—
192C4⋊Q8—1222no—
193C4⋊Q8—1222no—
194C4⋊Q8—1222no—
195#125—21no—
196#125—22no—
197#125—21no—
198#125—1221no—
199#125—1222no—
200#125—1221no—
201C22⋊Q8—1224no—
202C22⋊Q8—1224no—
203C22⋊Q8—1224no—
204C22⋊Q8—1224no—
205C4×Q8—1224no—
206C4×Q8—1224no—
207C4×Q8—1224no—
208C4×Q8—1224no—
209#117—1224no—
210#117—1224no—
211C2×C4⋊C4—612no—
212C2×C4⋊C4—614no—
213C2×C4⋊C4—614no—
214C2×C4⋊C4—1224no—
215C2×C4⋊C4C2×#692312yes—
216C2×C4⋊C4C2×#692314yes—
217C2×C22⋊C4—614no—
218C2×C22⋊C4—618no—
219C2×C42—612no—
220C2×C42—611no—
221C2×C42—612no—
222C2×C42—612no—
223C2×C42—612no—
224C2×C42—612no—
225C2×C42—614no—
226C2×C42—1224no—
227C2×C42C2×#692311yescharacteristic
228#95—1224no—
229#95—1224no—
230#95—1224no—
231#95—1224no—
232C24—611no—
233C2×Q8—1224no—
234C2×Q8—1224no—
235C2×Q8—2448no—
236C22×C4—611no—
237C22×C4—612no—
238C22×C4—612no—
239C22×C4—612no—
240C22×C4—612no—
241C22×C4—612no—
242C22×C4—612no—
243C22×C4—612no—
244C22×C4—614no—
245C22×C4—1224no—
246C22×C4—1228no—
247C22×C4—2444no—
248C22×C4C22×#692311yescharacteristic
249C22×C4C22×#692314yes—
250C22×C4C22×#692312yes—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—1224no—
256C4⋊C4—1224no—
257C4⋊C4—1224no—
258C4⋊C4—1224no—
259C4⋊C4—1222no—
260C4⋊C4—1222no—
261C4⋊C4—2448no—
262C22⋊C4—1228no—
263C22⋊C4—12216no—
264C42—612no—
265C42—612no—
266C42—612no—
267C42—612no—
268C42—611no—
269C42—612no—
270C42—611no—
271C42—1228no—
272C42—2448no—
273C42C22⋊#692311yescharacteristic
274C42C22⋊#692312yes—
275C42C22⋊#692311yescharacteristic
276C23—612no—
277C23—614no—
278C23—2442no—
279C23C23×#692311yescharacteristic
280Q8—2448no—
281Q8—2448no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—612no—
287C2×C4—612no—
288C2×C4—612no—
289C2×C4—1224no—
290C2×C4—1224no—
291C2×C4—1224no—
292C2×C4—1224no—
293C2×C4—1224no—
294C2×C4—1222no—
295C2×C4—1224no—
296C2×C4—2442no—
297C2×C4—2442no—
298C2×C4—2448no—
299C2×C4—2448no—
300C2×C4—2448no—
301C2×C4—2444no—
302C2×C4C2×C22⋊#692311yescharacteristic
303C2×C4C2×C22⋊#692311yescharacteristic
304C2×C4C2×C22⋊#692312yes—
305C2×C4C2×C22⋊#692312yes—
306C22—22no—
307C22—614no—
308C22—1222no—
309C22—1222no—
310C22—2444no—
311C22C22×C22⋊#692311yescharacteristic
312C22#1684083012yes—
313C4—21no—
314C4—21no—
315C4—22no—
316C4—22no—
317C4—44no—
318C4—614no—
319C4—1222no—
320C4—1222no—
321C4—4888no—
322C4—2444no—
323C2—22no—
324C2—1224no—
325C2#2146229311yescharacteristic
326C2D61⋊#41012yes—
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#125×C61C4
socleC22×C61C22×C22⋊C4
prime radical, cosocleC23×D61C24
hypercenterC2×#125#6923
hypocenterC61#21826
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
17. C2×#14519776C4semidirect
18. C2×#14519769C4semidirect
20. C2×#14519742C4semidirect
23. C2×#125×C61C4semidirect
174. C2×#125#6923semidirect
187. C61#21826semidirect

Maximal subgroups

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

subgroupindex
#214622662
#214620102
#214617172
C2×#168403182
#2182661