Finite Group Databasebeta

← #30604715

Subgroups — #30604715

Order 15616.

448 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 76 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, complemented
2#21461491C211yesmaximal, characteristic, complemented
3#21460791C212yesmaximal, complemented
4#21460790C212yesmaximal, complemented
5#21455430C211yesmaximal, characteristic
6C2×#14519856—21no—
7D61⋊M32—21no—
8#16840659C2212yes—
9#16840658—21no—
10#16840630—21no—
11#16840630C2212yes—
12C2×#14519800C411yescharacteristic, complemented
13C2×M16×D61C2211yescharacteristic
14#16840275—21no—
15#16840017C2211yescharacteristic
16#16840016C2211yescharacteristic
17#16840015C411yescharacteristic, complemented
18#16840011—21no—
19#16840010—21no—
20#16839797C411yescharacteristic, complemented
21#16839757—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×#11758226—21no—
34C22×C4×D61D411yescharacteristic
35#14519800—41no—
36#14519800—41no—
37C2×#11758201—21no—
38C2×D488C2×C411yescharacteristic, complemented
39C2×#11758194D411yescharacteristic
40C2×#11758189D411yescharacteristic
41C2×#11758189C2311yescharacteristic
42M16×D61—41no—
43M16×D61—41no—
44C2×C8×D61C2×C411yescharacteristic
45C2×C8×D61D411yescharacteristic
46#14519762—41no—
47#14519753—21no—
48#14519752—21no—
49#14519751—21no—
50#14519747—41no—
51#14519739—41no—
52#14519737—41no—
53#14519735—21no—
54#14519710—21no—
55#14519709C2×C411yescharacteristic, complemented
56#14519701—21no—
57#14519699—21no—
58#14519697—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
68C2×C22⋊D61—41no—
69C22×Dic61SD1611yescharacteristic
70#11758226—41no—
71#11758226—41no—
72#11758226—41no—
73#11758226—41no—
74C2×D244—21no—
75C2×D244—41no—
76C2×D244—41no—
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—21no—
81C2×C4×D61—41no—
82C2×C4×D61C22⋊C411yescharacteristic
83C2×C4×D61D811yescharacteristic
84C2×C4×D61SD1611yescharacteristic
85C2×C4×D61C2×D411yescharacteristic
86C22⋊#6923—42no—
87C22⋊#6923—42no—
88C4⋊#6923—42no—
89C4⋊#6923—42no—
90C4×#6923—24no—
91C4×#6923—42no—
92Q16×C61—41no—
93M16×C61—41no—
94C2×C8×C61C22⋊C411yescharacteristic
95C2×C8×C61C22×C411yescharacteristic
96C4⋊C4×C61—21no—
97C4⋊C4×C61—41no—
98C22⋊C4×C61—41no—
99#11758201—42no—
100#11758198—41no—
101#11758198—41no—
102#11758198—41no—
103#11758198—41no—
104C4⋊Dic61—21no—
105#11758196—41no—
106C4×Dic61—41no—
107#11758194—41no—
108C2×#92418C22⋊C411yescharacteristic
109C2×#92418C2×D411yescharacteristic
110D488—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—41no—
122C22×D61—81no—
123C22×D61#10211yescharacteristic
124C22×D61C2×D811yescharacteristic
125C2×#6923—44no—
126C2×#6923—84no—
127Q8×C61—41no—
128Q8×C61—41no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61#10211yescharacteristic
134C2×C4×C61C2×C22⋊C411yescharacteristic
135C22⋊D61—82no—
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—41no—
139C2×Dic61#10211yescharacteristic
140C2×Dic61C2×SD1611yescharacteristic
141D244—41no—
142D244—41no—
143D244—82no—
144D244—81no—
145D244—81no—
146C4×D61—21no—
147C4×D61—21no—
148C4×D61—21no—
149C4×D61—21no—
150C4×D61—21no—
151C4×D61—21no—
152C4×D61—81no—
153C4×D61—81no—
154C4×D61#9911yescharacteristic
155C4×D61#9911yescharacteristic
156C4×D61#10411yescharacteristic
157C4×D61#10411yescharacteristic
158C8×C61—21no—
159C8×C61—21no—
160#92418—21no—
161#92418—21no—
162#11685—611nomaximal, Sylow
163C22×C61—21no—
164C22×C61—41no—
165C22×C61C2×#10211yescharacteristic
166C2×D61—21no—
167C2×D61—41no—
168C2×D61—42no—
169C2×D61—41no—
170C2×D61—81no—
171C2×D61—81no—
172C2×D61—81no—
173C2×D61—81no—
174C2×D61#32711yescharacteristic
175C2×D61#32711yescharacteristic
176#6923—84no—
177C4×C61—21no—
178C4×C61—81no—
179C4×C61—81no—
180C4×C61C2×#9911yescharacteristic
181C4×C61C2×#10411yescharacteristic
182Dic61—21no—
183Dic61—81no—
184Dic61#32711yescharacteristic
185Dic61#32711yescharacteristic
186#3054—611no—
187C2×#349—611no—
188C2×H(3,ℤ/4)—611no—
189#1340—612no—
190#1339—612no—
191C2×C61—21no—
192C2×C61—41no—
193C2×C61C2×#32711yescharacteristic
194D61—21no—
195D61—21no—
196D61—41no—
197D61—161no—
198C2×#135—611no—
199C22×M16—611no—
200#525—1221no—
201C2×#117—1221no—
202C2×M32—1221no—
203#470—611no—
204#470#692311yescharacteristic, complemented
205#469—611no—
206#469—611no—
207#468—1221no—
208#468—1221no—
209#427—611no—
210#416—1221no—
211#349—612no—
212#349—1221no—
213H(3,ℤ/4)—612no—
214H(3,ℤ/4)—1221no—
215C61#1168511yescharacteristic, complemented, Sylow
216C2×#41—1221no—
217C23×C4—611no—
218#135—2441no—
219#135—2441no—
220C2×Q16C2×#692311yescharacteristic, complemented
221C2×SD16—1221no—
222C2×D8—611no—
223C2×M16—611no—
224C2×M16—611no—
225C2×M16—611no—
226C2×M16—2441no—
227C2×M16—2441no—
228C2×M16C2×#692311yescharacteristic
229C22×C8—611no—
230C22×C8—611no—
231#123—2441no—
232C22⋊Q8—21no—
233C22⋊Q8—1221no—
234#121—1221no—
235#121—1221no—
236#120—2441no—
237C4×D4—2441no—
238C4×D4—2441no—
239#117—1222no—
240#117—1222no—
241#117—1221no—
242#117—2441no—
243#117—2441no—
244C2×C4⋊C4—2441no—
245C2×C22⋊C4—2441no—
246C2×C42—1222no—
247M32—1222no—
248M32—2441no—
249C2×C16—2441no—
250#107—1221no—
251#106—611no—
252#106C2×#692311yescharacteristic, complemented
253#103—21no—
254#103—1221no—
255#102—1221no—
256#102—1221no—
257C24—611no—
258#41—2441no—
259#41—2441no—
260#41—2441no—
261#41—2441no—
262C2×Q8—21no—
263C2×D4—1221no—
264C2×D4—2441no—
265C2×D4—2441no—
266C2×D4—2441no—
267C22×C4—611no—
268C22×C4—611no—
269C22×C4—611no—
270C22×C4—611no—
271C22×C4—611no—
272C22×C4—1221no—
273C22×C4—1221no—
274C22×C4—1221no—
275C22×C4—1221no—
276C22×C4—2441no—
277C22×C4—2442no—
278C22×C4C22⋊#692311yescharacteristic
279Q16—41no—
280SD16—2442no—
281D8—2441no—
282M16—41no—
283M16—2441no—
284M16—2441no—
285M16—2441no—
286C2×C8—611no—
287C2×C8—611no—
288C2×C8—1221no—
289C2×C8—1221no—
290C2×C8—1221no—
291C2×C8—1221no—
292C2×C8C22⋊#692311yescharacteristic
293C2×C8C22×#692311yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—1221no—
297C4⋊C4—2441no—
298C4⋊C4—2442no—
299C4⋊C4—2442no—
300C22⋊C4—41no—
301C22⋊C4—2441no—
302C22⋊C4—2441no—
303C22⋊C4—2441no—
304C22⋊C4—2442no—
305C22⋊C4—2442no—
306C22⋊C4—2441no—
307C42—1224no—
308C42—2441no—
309C42—2442no—
310C16—2442no—
311C23—611no—
312C23—611no—
313C23—1221no—
314C23—1221no—
315C23—4881no—
316C23—2441no—
317C23#1451983911yescharacteristic
318Q8—41no—
319Q8—41no—
320D4—4881no—
321D4—4881no—
322D4—4882no—
323D4—4882no—
324D4—2441no—
325D4—2441no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—611no—
331C2×C4—611no—
332C2×C4—611no—
333C2×C4—611no—
334C2×C4—611no—
335C2×C4—611no—
336C2×C4—1221no—
337C2×C4—1221no—
338C2×C4—1221no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—1221no—
342C2×C4—1221no—
343C2×C4—1221no—
344C2×C4—4881no—
345C2×C4—4881no—
346C2×C4—4884no—
347C2×C4—2444no—
348C2×C4—2441no—
349C2×C4#1451983911yescharacteristic
350C2×C4C2×C22⋊#692311yescharacteristic
351C8—21no—
352C8—21no—
353C8—1221no—
354C8—1221no—
355C22—21no—
356C22—41no—
357C22—611no—
358C22—611no—
359C22—1221no—
360C22—4881no—
361C22—4881no—
362C22—4881no—
363C22—4881no—
364C22—2441no—
365C22—2441no—
366C22—2442no—
367C22C2×#1451983911yescharacteristic
368C4—21no—
369C4—81no—
370C4—81no—
371C4—611no—
372C4—611no—
373C4—1221no—
374C4—4881no—
375C4—4884no—
376C4D61⋊#9911yescharacteristic
377C4D61⋊#10411yescharacteristic
378C2—21no—
379C2—41no—
380C2—1221no—
381C2—1221no—
382C2—9761no—
383C2—2441no—
384C2D61⋊#32711yescharacteristic
subgroupG/H
centerC2D61⋊#327
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#470C4
socleC2×C61C2×#327
prime radical#21455430C2
cosocleC2×#11758189C23
hypercenter#470#6923
hypocenterC61#11685
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
1. D61⋊#349C2semidirect
2. #21461491C2semidirect
3. #21460791C2semidirect
4. #21460790C2semidirect
12. C2×#14519800C4semidirect
17. #16840015C4semidirect
20. #16839797C4semidirect
22. C61×#470C4semidirect
38. C2×D488C2×C4semidirect
55. #14519709C2×C4semidirect
59. C2×Q16×C61C2×C4semidirect
62. #106×C61C2×C4semidirect
204. #470#6923semidirect
215. C61#11685semidirect
220. C2×Q16C2×#6923semidirect
252. #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
#214607912
#214607902
#214554302
#1168561