Finite Group Databasebeta

← D61⋊#1340 #30604710

Subgroups — D61⋊#1340 #30604710

Order 15616.

444 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 78 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⋊#349C211yesmaximal, characteristic
2#21461491C211yesmaximal, characteristic
3#21460792C212yesmaximal, complemented
4#21460791C212yesmaximal, complemented
5#470×D61C211yesmaximal, characteristic
6C2×#14519856—21no—
7D61⋊M32—21no—
8#16840659C2212yes—
9#16840658—21no—
10#16840630—21no—
11#16840630C2212yes—
12C2×Q16×D61C411yescharacteristic, complemented
13C2×M16×D61C2211yescharacteristic
14C22⋊Q8×D61—21no—
15#16840021C2211yescharacteristic
16#16840020C411yescharacteristic, complemented
17#16840019—21no—
18#16840018—21no—
19#16840017C2211yescharacteristic
20#106×D61C411yescharacteristic, complemented
21#103×D61—21no—
22C61×#470C411yescharacteristic, complemented
23#14519856—22no—
24#14519856—22no—
25#14519856—41no—
26#14519856—41no—
27C2×C4⋊#6923—41no—
28C2×C22⋊#6923—41no—
29C2×C4×#6923—22no—
30#14519852—41no—
31#14519851—22no—
32D61⋊C16—41no—
33C2×Q8×D61—21no—
34C22×C4×D61D411yescharacteristic
35C2×Dic244C2×C411yescharacteristic, complemented
36C2×#11758202—21no—
37Q16×D61—41no—
38Q16×D61—41no—
39C2×#11758194D411yescharacteristic
40C2×#11758189D411yescharacteristic
41C2×#11758189C2311yescharacteristic
42M16×D61—41no—
43M16×D61—41no—
44C2×C8×D61C2×C411yescharacteristic
45C2×C8×D61D411yescharacteristic
46#14519757—21no—
47#14519756—21no—
48#14519755—41no—
49#14519754—41no—
50#14519753—21no—
51C4⋊C4×D61—21no—
52C4⋊C4×D61—41no—
53C22⋊C4×D61—41no—
54#14519709C2×C411yescharacteristic, complemented
55#14519708—21no—
56#14519702—21no—
57#14519701—21no—
58#14519700—21no—
59C2×Q16×C61C2×C411yescharacteristic, complemented
60C2×M16×C61C2×C411yescharacteristic
61C22⋊Q8×C61—21no—
62#106×C61C2×C411yescharacteristic, complemented
63#103×C61—21no—
64C23×D61D811yescharacteristic
65C22×#6923—42no—
66C2×Q8×C61—21no—
67C22×C4×C61C22⋊C411yescharacteristic
68C22×Dic61SD1611yescharacteristic
69Q8×D61—41no—
70Q8×D61—41no—
71Q8×D61—41no—
72Q8×D61—41no—
73C2×C4×D61—21no—
74C2×C4×D61—21no—
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61—41no—
78C2×C4×D61—41no—
79C2×C4×D61C22⋊C411yescharacteristic
80C2×C4×D61D811yescharacteristic
81C2×C4×D61SD1611yescharacteristic
82C2×C4×D61C2×D411yescharacteristic
83C2×Dic122—21no—
84C2×Dic122—41no—
85C22⋊#6923—42no—
86C22⋊#6923—42no—
87C4⋊#6923—42no—
88C4⋊#6923—42no—
89C4×#6923—24no—
90C4×#6923—42no—
91Q16×C61—41no—
92M16×C61—41no—
93C2×C8×C61C22⋊C411yescharacteristic
94C2×C8×C61C22×C411yescharacteristic
95C4⋊C4×C61—21no—
96C4⋊C4×C61—41no—
97C22⋊C4×C61—41no—
98C22⋊Dic61—41no—
99#11758202—42no—
100#11758198—41no—
101#11758198—41no—
102C4⋊Dic61—21no—
103C4⋊Dic61—41no—
104#11758196—41no—
105#11758196—41no—
106#11758196—41no—
107#11758194—41no—
108C2×#92418C22⋊C411yescharacteristic
109C2×#92418C2×D411yescharacteristic
110Dic244—41no—
111#11758189—41no—
112#11758189—41no—
113C8×D61—21no—
114C8×D61—21no—
115C8×D61—21no—
116C8×D61—21no—
117#11758187—42no—
118C23×C61#10211yescharacteristic
119C22×D61—21no—
120C22×D61—21no—
121C22×D61—42no—
122C22×D61#10211yescharacteristic
123C22×D61C2×D811yescharacteristic
124C2×#6923—44no—
125C2×#6923—84no—
126Q8×C61—41no—
127Q8×C61—41no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61#10211yescharacteristic
133C2×C4×C61C2×C22⋊C411yescharacteristic
134C2×Dic61—21no—
135C2×Dic61—21no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61#10211yescharacteristic
139C2×Dic61C2×SD1611yescharacteristic
140C4×D61—21no—
141C4×D61—21no—
142C4×D61—21no—
143C4×D61—21no—
144C4×D61—21no—
145C4×D61—21no—
146C4×D61—82no—
147C4×D61—81no—
148C4×D61—81no—
149C4×D61#9911yescharacteristic
150C4×D61#9911yescharacteristic
151C4×D61#10411yescharacteristic
152C4×D61#10411yescharacteristic
153Dic122—41no—
154Dic122—41no—
155Dic122—81no—
156Dic122—81no—
157C8×C61—21no—
158C8×C61—21no—
159#92418—21no—
160#92418—21no—
161C2×#1340—611nomaximal, Sylow
162C22×C61—21no—
163C22×C61—41no—
164C22×C61C2×#10211yescharacteristic
165C2×D61—22no—
166C2×D61—21no—
167C2×D61—42no—
168C2×D61—41no—
169C2×D61—81no—
170C2×D61#32711yescharacteristic
171C2×D61#32711yescharacteristic
172#6923—84no—
173C4×C61—21no—
174C4×C61—81no—
175C4×C61—81no—
176C4×C61C2×#9911yescharacteristic
177C4×C61C2×#10411yescharacteristic
178Dic61—21no—
179Dic61—81no—
180Dic61—81no—
181Dic61#32711yescharacteristic
182Dic61#32711yescharacteristic
183C2×#470—611no—
184C2×#349—611no—
185C2×H(3,ℤ/4)—611no—
186#1340—612no—
187#1340—612no—
188C2×C61—21no—
189C2×C61—41no—
190C2×C61C2×#32711yescharacteristic
191D61—21no—
192D61—41no—
193D61#134012yescomplemented
194C22×Q16—611no—
195C22×M16—611no—
196C2×C22⋊Q8—1221no—
197C2×#117—1221no—
198C2×M32—1221no—
199#470—611no—
200#470—611no—
201#470—611no—
202#470—1221no—
203#470—1221no—
204#470#692311yescharacteristic, complemented
205C2×#106—611no—
206C2×#103—1221no—
207#349—612no—
208#349—1221no—
209H(3,ℤ/4)—612no—
210H(3,ℤ/4)—1221no—
211C61C2×#134011yescharacteristic, complemented, Sylow
212C22×Q8—1221no—
213C23×C4—611no—
214C2×Q16—611no—
215C2×Q16—1221no—
216C2×Q16—2441no—
217C2×Q16—2441no—
218C2×Q16C2×#692311yescharacteristic, complemented
219C2×M16—611no—
220C2×M16—611no—
221C2×M16—611no—
222C2×M16—2441no—
223C2×M16—2441no—
224C2×M16C2×#692311yescharacteristic
225C22×C8—611no—
226C22×C8—611no—
227C22⋊Q8—21no—
228C22⋊Q8—1221no—
229C22⋊Q8—1221no—
230C22⋊Q8—1221no—
231C22⋊Q8—2441no—
232C22⋊Q8—2441no—
233#117—1222no—
234#117—1222no—
235#117—2441no—
236#117—2441no—
237C2×C4⋊C4—1221no—
238C2×C4⋊C4—2441no—
239C2×C4⋊C4—2441no—
240C2×C22⋊C4—2441no—
241C2×C22⋊C4—2441no—
242C2×C42—1222no—
243M32—1222no—
244M32—2441no—
245C2×C16—2441no—
246#106—611no—
247#106—1221no—
248#106C2×#692311yescharacteristic, complemented
249#103—21no—
250#103—1221no—
251#103—1221no—
252#103—1221no—
253C24—611no—
254C2×Q8—21no—
255C2×Q8—1221no—
256C2×Q8—2441no—
257C2×Q8—2441no—
258C2×Q8—2441no—
259C2×Q8—2441no—
260C2×Q8—2441no—
261C22×C4—611no—
262C22×C4—611no—
263C22×C4—611no—
264C22×C4—611no—
265C22×C4—611no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4—1221no—
269C22×C4—1221no—
270C22×C4—2441no—
271C22×C4—2441no—
272C22×C4—2442no—
273C22×C4C22⋊#692311yescharacteristic
274Q16—41no—
275Q16—2441no—
276Q16—2442no—
277M16—41no—
278M16—2441no—
279M16—2441no—
280M16—2441no—
281C2×C8—611no—
282C2×C8—611no—
283C2×C8—1221no—
284C2×C8—1221no—
285C2×C8—1221no—
286C2×C8—1221no—
287C2×C8C22⋊#692311yescharacteristic
288C2×C8C22×#692311yescharacteristic
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—1221no—
292C4⋊C4—2441no—
293C4⋊C4—2441no—
294C4⋊C4—2441no—
295C4⋊C4—2442no—
296C4⋊C4—2442no—
297C4⋊C4—2441no—
298C22⋊C4—41no—
299C22⋊C4—2441no—
300C22⋊C4—2441no—
301C22⋊C4—2442no—
302C22⋊C4—2442no—
303C22⋊C4—2441no—
304C42—1224no—
305C42—2442no—
306C16—2442no—
307C23—611no—
308C23—611no—
309C23—1221no—
310C23—1221no—
311C23—2442no—
312C23#1451983911yescharacteristic
313Q8—41no—
314Q8—41no—
315Q8—4881no—
316Q8—4881no—
317Q8—2441no—
318Q8—2441no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—611no—
324C2×C4—611no—
325C2×C4—611no—
326C2×C4—611no—
327C2×C4—611no—
328C2×C4—611no—
329C2×C4—1221no—
330C2×C4—1221no—
331C2×C4—1221no—
332C2×C4—1221no—
333C2×C4—1221no—
334C2×C4—1221no—
335C2×C4—1221no—
336C2×C4—1221no—
337C2×C4—4881no—
338C2×C4—4881no—
339C2×C4—4882no—
340C2×C4—4884no—
341C2×C4—2444no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4#1451983911yescharacteristic
345C2×C4C2×C22⋊#692311yescharacteristic
346C8—21no—
347C8—21no—
348C8—1221no—
349C8—1221no—
350C22—21no—
351C22—41no—
352C22—611no—
353C22—611no—
354C22—1222no—
355C22—1221no—
356C22—4881no—
357C22—2441no—
358C22—2442no—
359C22C2×#1451983911yescharacteristic
360C4—21no—
361C4—81no—
362C4—81no—
363C4—611no—
364C4—611no—
365C4—1221no—
366C4—4881no—
367C4—4884no—
368C4—4881no—
369C4D61⋊#9911yescharacteristic
370C4D61⋊#10411yescharacteristic
371C2—21no—
372C2—41no—
373C2—612no—
374C2—1221no—
375C2—2441no—
376C2D61⋊#32711yescharacteristic
subgroupG/H
centerC2D61⋊#327
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#470C4
socleC2×C61C2×#327
prime radicalC23×D61D8
cosocleC2×#11758189C23
hypercenter#470#6923
hypocenterC61C2×#1340
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #470

Complements

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

NH ≅ G/Ntype
3. #21460792C2semidirect
4. #21460791C2semidirect
12. C2×Q16×D61C4semidirect
16. #16840020C4semidirect
20. #106×D61C4semidirect
22. C61×#470C4semidirect
35. C2×Dic244C2×C4semidirect
54. #14519709C2×C4semidirect
59. C2×Q16×C61C2×C4semidirect
62. #106×C61C2×C4semidirect
193. D61#1340semidirect
204. #470#6923semidirect
211. C61C2×#1340semidirect
218. C2×Q16C2×#6923semidirect
248. #106C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#3492
#214614912
#214607922
#214607912
#470×D612
C2×#134061