Finite Group Databasebeta

← #29593771

Subgroups — #29593771

Order 15616.

358 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 61 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#21457613C211yesmaximal, characteristic
2#21447987C211yesmaximal, characteristic, complemented
3#21445407C211yesmaximal, characteristic, complemented
4#21443859C211yesmaximal, characteristic, complemented
5#21443792C211yesmaximal, characteristic, complemented
6#21443791C211yesmaximal, characteristic, complemented
7C61×#2230C211yesmaximal, characteristic, complemented
8#16840574—21no—
9#16840569—21no—
10#16840158—21no—
11#16840136—21no—
12#16840135C2211yescharacteristic
13#16840026—21no—
14#16840025—21no—
15#16839946—21no—
16#16839931—21no—
17#16839880C2211yescharacteristic
18#16839584—21no—
19#16839583C2211yescharacteristic, complemented
20#16839571—21no—
21C8×#92418C2211yescharacteristic, complemented
22C61×#498C2211yescharacteristic, complemented
23C61×#443C2211yescharacteristic
24C61×#365—21no—
25C61×#364C2211yescharacteristic, complemented
26C61×#360—21no—
27#14519824—21no—
28#14519823—21no—
29#14519810—21no—
30#14519810—41no—
31#14519809—41no—
32#14519807—21no—
33#14519805—21no—
34#14519805—41no—
35#14519802—21no—
36#14519801—41no—
37#14519799—41no—
38#14519799D411yescharacteristic
39#14519798—41no—
40C2×#11758190—21no—
41C2×#11758190—21no—
42C2×#11758190—41no—
43C2×#11758201—42no—
44SD16×D61—41no—
45#14519783—21no—
46#14519777D411yescharacteristic
47#14519774D411yescharacteristic
48#14519717—21no—
49#14519713D411yescharacteristic
50#14519708—42no—
51#14519705—21no—
52#14519695—21no—
53#14519693—21no—
54C4×#92418—21no—
55#135×C61D411yescharacteristic
56#131×C61—21no—
57C4⋊Q8×C61—21no—
58M32×C61—21no—
59#108×C61D411yescharacteristic
60#106×C61—21no—
61#106×C61—21no—
62#104×C61—21no—
63C4×C8×C61C2311yescharacteristic
64#41×C61—21no—
65C2×Q8×C61—41no—
66#11758226—41no—
67Q8×D61—41no—
68#11758224—41no—
69#11758224—81no—
70D4×D61—41no—
71D4×D61—81no—
72#11758222—21no—
73#11758222—41no—
74#11758222—41no—
75C2×D244—21no—
76C2×D244—41no—
77C2×D244—41no—
78C2×Dic122—21no—
79C2×Dic122—41no—
80Q16×C61SD1611yescharacteristic, complemented
81SD16×C61—21no—
82D8×C61SD1611yescharacteristic, complemented
83M16×C61—21no—
84M16×C61—41no—
85C2×C8×C61—41no—
86C2×C8×C61C2×D411yescharacteristic
87C2×C8×C61C2×D411yescharacteristic
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C42×C61C2×D411yescharacteristic
91#11758202—41no—
92#11758201—41no—
93#11758201—84no—
94#11758200—41no—
95#11758199—41no—
96C4⋊Dic61—41no—
97#11758194—21no—
98C2×#92418—22no—
99Dic244—41no—
100Dic244SD1611yescharacteristic, complemented
101D488—41no—
102D488SD1611yescharacteristic, complemented
103#11758190—21no—
104#11758190—41no—
105#11758190—41no—
106#11758190—41no—
107#11758190—41no—
108#11758190—41no—
109#11758190—41no—
110#11758189—41no—
111C8×D61—41no—
112C16×C61—41no—
113#11758185—41no—
114C22×D61—41no—
115C22×D61—81no—
116Q8×C61—21no—
117Q8×C61—82no—
118D4×C61—21no—
119D4×C61—41no—
120C2×C4×C61—21no—
121C2×C4×C61—41no—
122C2×C4×C61—81no—
123C2×C4×C61#12011yescharacteristic
124C22⋊D61—41no—
125C22⋊D61—81no—
126C22⋊D61—81no—
127C2×Dic61—41no—
128C2×Dic61—81no—
129D244—21no—
130D244—41no—
131D244—41no—
132D244—82no—
133D244—81no—
134C4×D61—41no—
135C4×D61—81no—
136C4×D61—81no—
137Dic122—21no—
138Dic122—41no—
139Dic122—41no—
140Dic122—81no—
141C8×C61—41no—
142C8×C61C2×SD1611yescharacteristic
143C8×C61C2×SD1611yescharacteristic
144C8×C61AGL(1,ℤ/8)11yescharacteristic
145C8×C61AGL(1,ℤ/8)11yescharacteristic
146#92418—24no—
147#92418—41no—
148#12364—611nomaximal, Sylow
149C22×C61—41no—
150C22×C61#45311yescharacteristic
151C2×D61—41no—
152C2×D61—81no—
153C2×D61—161no—
154C2×D61—321no—
155C4×C61—22no—
156C4×C61—41no—
157C4×C61—161no—
158C4×C61#45011yescharacteristic
159C4×C61#45011yescharacteristic
160Dic61—41no—
161Dic61—81no—
162#3291—611no—
163#2230—611no—
164#2230D6111yescharacteristic, complemented
165#1705—611no—
166#1359—611no—
167#1335—611no—
168#1335—611no—
169C2×C61—21no—
170C2×C61—81no—
171C2×C61#162911yescharacteristic
172D61—81no—
173D61—161no—
174#577—1221no—
175#577—1221no—
176#498—1221no—
177#498C2×D6111yescharacteristic, complemented
178#492—611no—
179#492—1221no—
180#471—1221no—
181#471—1221no—
182#456—1221no—
183#453—1221no—
184#443—611no—
185#443C2×D6111yescharacteristic
186#365—21no—
187#365—1221no—
188#364—611no—
189#364C2×D6111yescharacteristic, complemented
190#360—21no—
191#360—1221no—
192C82—611no—
193C61#1236411yescharacteristic, complemented, Sylow
1942−1+4—1221no—
195H(5,𝔽2)—1221no—
196#137—1221no—
197#137—1221no—
198#137—2441no—
199#137—2441no—
200AGL(1,ℤ/8)—1221no—
201AGL(1,ℤ/8)—1221no—
202AGL(1,ℤ/8)—2441no—
203AGL(1,ℤ/8)—2441no—
204#135—611no—
205#135—2441no—
206#135—2441no—
207#135C22⋊D6111yescharacteristic
208C2×SD16—1221no—
209C2×SD16—1221no—
210C2×SD16—2441no—
211C2×SD16—2442no—
212C2×SD16—2441no—
213#131—21no—
214#131—1221no—
215C4⋊Q8—21no—
216C4⋊Q8—611no—
217#127—611no—
218M32—21no—
219M32—1221no—
220#108—611no—
221#108C22⋊D6111yescharacteristic
222#106—21no—
223#106—21no—
224#106—2442no—
225#104—21no—
226#104—1221no—
227#101—1221no—
228#100—1221no—
229C4×C8—1221no—
230C4×C8C22×D6111yescharacteristic
231#41—21no—
232#41—1221no—
233#41—4881no—
234#41—2441no—
235#41—2441no—
236#41—2441no—
237#41—2441no—
238C2×Q8—41no—
239C2×Q8—1221no—
240C2×Q8—2441no—
241C2×Q8—2441no—
242C2×D4—1221no—
243C2×D4—4881no—
244C2×D4—2441no—
245C2×D4—2441no—
246C2×D4—2441no—
247Q16—611no—
248Q16—2441no—
249Q16—2441no—
250Q16#1175820111yescharacteristic, complemented
251SD16—21no—
252SD16—1221no—
253SD16—4884no—
254SD16—2441no—
255SD16—2441no—
256SD16—2441no—
257SD16—2441no—
258SD16—2441no—
259SD16—2441no—
260SD16—2441no—
261SD16—2441no—
262D8—611no—
263D8—2441no—
264D8—2441no—
265D8#1175820111yescharacteristic, complemented
266M16—21no—
267M16—41no—
268M16—1221no—
269M16—2441no—
270C2×C8—41no—
271C2×C8—1222no—
272C2×C8—2441no—
273C2×C8D4×D6111yescharacteristic
274C2×C8C2×C22⋊D6111yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—2441no—
278C42D4×D6111yescharacteristic
279C16—41no—
280C16—2441no—
281C23—4881no—
282C23—2441no—
283Q8—21no—
284Q8—82no—
285Q8—1221no—
286Q8—4881no—
287Q8—2441no—
288Q8—2441no—
289D4—21no—
290D4—41no—
291D4—1221no—
292D4—4882no—
293D4—4881no—
294D4—4881no—
295D4—4881no—
296D4—2441no—
297D4—2441no—
298D4—2441no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—4881no—
303C2×C4—4881no—
304C2×C4—4881no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4#1451974511yescharacteristic
308C8—41no—
309C8—1224no—
310C8—2441no—
311C8SD16×D6111yescharacteristic
312C8C2×#1175820111yescharacteristic
313C8#1451980211yescharacteristic
314C8#1451980411yescharacteristic
315C22—41no—
316C22—19521no—
317C22—9761no—
318C22—4881no—
319C22—2441no—
320C22#1683992811yescharacteristic
321C4—22no—
322C4—41no—
323C4—161no—
324C4—4881no—
325C4—2441no—
326C4#1683991111yescharacteristic
327C4#1683991211yescharacteristic
328C2—21no—
329C2—81no—
330C2—9761no—
331C2—4881no—
332C2#2144493411yescharacteristic
subgroupG/H
centerC2#21444934
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#2230C2
socleC2×C61#1629
prime radical#21457613C2
hypercenter#2230D61
hypocenterC61#12364
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2230

Complements

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

NH ≅ G/Ntype
2. #21447987C2semidirect
3. #21445407C2semidirect
4. #21443859C2semidirect
5. #21443792C2semidirect
6. #21443791C2semidirect
7. C61×#2230C2semidirect
19. #16839583C22semidirect
21. C8×#92418C22semidirect
22. C61×#498C22semidirect
25. C61×#364C22semidirect
80. Q16×C61SD16semidirect
82. D8×C61SD16semidirect
100. Dic244SD16semidirect
102. D488SD16semidirect
164. #2230D61semidirect
177. #498C2×D61semidirect
189. #364C2×D61semidirect
193. C61#12364semidirect
250. Q16#11758201semidirect
265. D8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214576132
#214479872
#214454072
#214438592
#214437922
#214437912
C61×#22302
#1236461