Finite Group Databasebeta

← D61⋊#1319 #30603834

Subgroups — D61⋊#1319 #30603834

Order 15616.

618 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 98 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
1D61⋊#324C212yesmaximal
2#21460750C212yesmaximal, complemented
3#21460749C212yesmaximal, complemented
4#391×D61C211yesmaximal, characteristic
5D61⋊(C22⋊C8)—22no—
6#16840608—22no—
7#16840607C2214yes—
8C22×Q8×D61C411yescharacteristic
9C2×C4⋊C4×D61C2211yescharacteristic
10C2×C42×D61C411yescharacteristic
11#16839669C2211yescharacteristic
12#16839668C2211yescharacteristic
13#16839667C411yescharacteristic
14#16839666—22no—
15#95×D61—22no—
16C61×#391C411yescharacteristic
17C2×D61⋊C8—42no—
18#14519834—42no—
19C22⋊#92420—24no—
20C22×Dic122—21no—
21C22×Dic122C2×C411yescharacteristic
22C2×Q8×D61—22no—
23C2×Q8×D61—41no—
24C2×Q8×D61—41no—
25C2×Q8×D61—42no—
26C2×Q8×D61C814yescomplemented
27C22×C4×D61—21no—
28C22×C4×D61—21no—
29C22×C4×D61C2×C411yescharacteristic
30C22×C4×D61D412yes—
31C2×C4⋊Dic61C2311yescharacteristic
32C2×#11758196D412yes—
33C4⋊C4×D61—41no—
34C4⋊C4×D61—41no—
35C2×C4×Dic61—21no—
36C2×C4×Dic61C2×C411yescharacteristic
37C42×D61—42no—
38#14519684—22no—
39#14519684—22no—
40#14519684—22no—
41C22×Q8×C61C2×C411yescharacteristic
42C2×C4⋊C4×C61C2×C411yescharacteristic
43C2×C42×C61C2×C411yescharacteristic
44#95×C61—22no—
45C23×D61C22⋊C411yescharacteristic
46C2×Q8×C61—21no—
47C2×Q8×C61—41no—
48C2×Q8×C61—41no—
49C2×Q8×C61C2×C812yescomplemented
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61C22⋊C412yes—
53C22×C4×C61C22×C411yescharacteristic
54C22×Dic61—21no—
55C22×Dic61—21no—
56C22×Dic61C22⋊C411yescharacteristic
57C22×Dic61C2×D412yes—
58Q8×D61—44no—
59Q8×D61—44no—
60Q8×D61—44no—
61Q8×D61—44no—
62C2×C4×D61—21no—
63C2×C4×D61—21no—
64C2×C4×D61—22no—
65C2×C4×D61—24no—
66C2×C4×D61—24no—
67C2×C4×D61—22no—
68C2×C4×D61—22no—
69C2×C4×D61—42no—
70C2×C4×D61—41no—
71C2×C4×D61—41no—
72C2×C4×D61—42no—
73C2×C4×D61—42no—
74C2×C4×D61—44no—
75C2×C4×D61C2×C812yes—
76C2×C4×D61M1612yes—
77C2×Dic122—21no—
78C2×Dic122—24no—
79C2×Dic122—42no—
80C2×Dic122—42no—
81C2×Dic122—41no—
82C2×Dic122—41no—
83C2×Dic122C2×C812yescomplemented
84C2×#92420—44no—
85D61⋊C8—84no—
86C4⋊C4×C61—41no—
87C42×C61—41no—
88C4⋊Dic61—41no—
89#11758196—42no—
90C4×Dic61—42no—
91C4×Dic61—41no—
92C23×C61C2×C22⋊C411yescharacteristic
93C22×D61—22no—
94C22×D61—22no—
95C22×D61C22⋊C812yes—
96C22×D61#9911yescharacteristic
97C22×D61#9911yescharacteristic
98C22×D61#10012yes—
99Q8×C61—42no—
100Q8×C61—42no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—22no—
104C2×C4×C61—22no—
105C2×C4×C61—22no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—42no—
111C2×C4×C61C22×C811yescharacteristic
112C2×C4×C61C2×M1611yescharacteristic
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—22no—
116C2×Dic61—22no—
117C2×Dic61—22no—
118C2×Dic61—41no—
119C2×Dic61—41no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—42no—
123C2×Dic61C22⋊C812yes—
124C4×D61—22no—
125C4×D61—22no—
126C4×D61—22no—
127C4×D61—22no—
128C4×D61—48no—
129C4×D61—44no—
130C4×D61—44no—
131C4×D61—44no—
132C4×D61—82no—
133Dic122—48no—
134Dic122—42no—
135Dic122—42no—
136#92420—84no—
137C2×#1319—611nomaximal, Sylow
138C22×C61—21no—
139C22×C61—21no—
140C22×C61C2×C22⋊C811yescharacteristic
141C22×C61C2×#9911yescharacteristic
142C22×C61C2×#10011yescharacteristic
143C2×D61—21no—
144C2×D61—21no—
145C2×D61—24no—
146C2×D61—24no—
147C2×D61#32312yes—
148C2×D61#32312yes—
149C2×D61#35412yes—
150C2×D61#35612yes—
151C4×C61—21no—
152C4×C61—21no—
153C4×C61—42no—
154C4×C61—42no—
155C4×C61—42no—
156Dic61—22no—
157Dic61—42no—
158Dic61—42no—
159Dic61—42no—
160C2×#391—611no—
161C2×#324—612no—
162#1319—612no—
163#1319—612no—
164C2×C61—21no—
165C2×C61—21no—
166C2×C61C2×#32311yescharacteristic
167C2×C61C2×#35411yescharacteristic
168C2×C61C2×#35611yescharacteristic
169D61—22no—
170D61#131914yescomplemented
171C23×Q8—611no—
172C22×C4⋊C4—611no—
173C22×C42—611no—
174C2×C22⋊C8—1222no—
175#391—611no—
176#391—611no—
177#391—611no—
178#391—1222no—
179#391#692311yescharacteristic
180C2×#95—1222no—
181#324—614no—
182#324—1222no—
183C61C2×#131911yescharacteristic, complemented, Sylow
184C22×Q8—614no—
185C22×Q8—611no—
186C22×Q8—1221no—
187C22×Q8—1222no—
188C22×Q8—2442no—
189C22×Q8—2441no—
190C22×Q8—2441no—
191C22×Q8C2×#692311yescharacteristic
192C23×C4—611no—
193C23×C4—612no—
194C23×C4—1221no—
195C23×C4—1221no—
196C22×C8—2442no—
197C2×C4⋊C4—612no—
198C2×C4⋊C4—611no—
199C2×C4⋊C4—2441no—
200C2×C4⋊C4—2441no—
201C2×C4⋊C4C2×#692311yescharacteristic
202C2×C42—611no—
203C2×C42—1221no—
204C2×C42—2442no—
205C2×C42C2×#692311yescharacteristic
206C22⋊C8—1224no—
207C22⋊C8—2442no—
208#95—22no—
209#95—1222no—
210#95—1222no—
211#95—1222no—
212C24—611no—
213C2×Q8—21no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—612no—
217C2×Q8—1224no—
218C2×Q8—1221no—
219C2×Q8—2441no—
220C2×Q8—2441no—
221C2×Q8—2442no—
222C2×Q8—2442no—
223C2×Q8—2444no—
224C2×Q8—2444no—
225C2×Q8—2444no—
226C2×Q8—2444no—
227C2×Q8D61⋊C812yescomplemented
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—611no—
231C22×C4—612no—
232C22×C4—612no—
233C22×C4—612no—
234C22×C4—1222no—
235C22×C4—1221no—
236C22×C4—1221no—
237C22×C4—1222no—
238C22×C4—1222no—
239C22×C4—1224no—
240C22×C4—1224no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4—2444no—
244C22×C4—2442no—
245C22×C4—2441no—
246C22×C4—2441no—
247C22×C4—2442no—
248C22×C4—2442no—
249C22×C4C22⋊#692312yes—
250C22×C4C22×#692311yescharacteristic
251C2×C8—4884no—
252C2×C8—2444no—
253C4⋊C4—41no—
254C4⋊C4—2442no—
255C4⋊C4—2441no—
256C42—41no—
257C42—2441no—
258C42—2442no—
259C23—611no—
260C23—612no—
261C23—611no—
262C23—612no—
263C23—1222no—
264C23—1222no—
265C23C2×C22⋊#692311yescharacteristic
266Q8—42no—
267Q8—42no—
268Q8—2442no—
269Q8—2442no—
270Q8—2448no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—612no—
282C2×C4—1222no—
283C2×C4—1222no—
284C2×C4—1222no—
285C2×C4—1222no—
286C2×C4—1221no—
287C2×C4—1221no—
288C2×C4—1222no—
289C2×C4—1222no—
290C2×C4—1222no—
291C2×C4—4882no—
292C2×C4—2444no—
293C2×C4—2444no—
294C2×C4—2444no—
295C2×C4—2448no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2442no—
301C2×C4C2×D61⋊C811yescharacteristic
302C2×C4D61⋊M1611yescharacteristic
303C8—4884no—
304C22—21no—
305C22—21no—
306C22—612no—
307C22—612no—
308C22—612no—
309C22—612no—
310C22—1224no—
311C22—1224no—
312C22—1221no—
313C22—1221no—
314C22D61⋊(C22⋊C8)11yescharacteristic
315C22D61⋊#9911yescharacteristic
316C22D61⋊#10011yescharacteristic
317C4—21no—
318C4—21no—
319C4—42no—
320C4—42no—
321C4—42no—
322C4—1222no—
323C4—2442no—
324C4—2442no—
325C4—2442no—
326C2—21no—
327C2—21no—
328C2—614no—
329C2—1222no—
330C2D61⋊#32311yescharacteristic
331C2D61⋊#35411yescharacteristic
332C2D61⋊#35611yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC22×C61C2×#99
prime radicalC23×D61C22⋊C4
cosocleC2×C4⋊Dic61C23
hypercenter#391#6923
hypocenterC61C2×#1319
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #21460750C2semidirect
3. #21460749C2semidirect
26. C2×Q8×D61C8semidirect
49. C2×Q8×C61C2×C8semidirect
83. C2×Dic122C2×C8semidirect
170. D61#1319semidirect
183. C61C2×#1319semidirect
227. C2×Q8D61⋊C8semidirect

Maximal subgroups

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

subgroupindex
D61⋊#3242
#214607502
#214607492
#391×D612
C2×#131961