Finite Group Databasebeta

← #29593790

Subgroups — #29593790

Order 15616.

424 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 69 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#21457494C211yesmaximal, characteristic
2#21447953C211yesmaximal, characteristic, complemented
3#21445205C211yesmaximal, characteristic, complemented
4#21443865C211yesmaximal, characteristic, complemented
5#21443781C211yesmaximal, characteristic, complemented
6#21443775C211yesmaximal, characteristic, complemented
7C61×#2240C211yesmaximal, characteristic, complemented
8C2×D8×D61—21no—
9#16840397—21no—
10#16839995—21no—
11#16839964—21no—
12#16839954—21no—
13#16839894—22no—
14C4×D488C2211yescharacteristic
15C8×D244C2211yescharacteristic
16#16839587—21no—
17C4⋊#11758185C2211yescharacteristic, complemented
18#16839562—21no—
19#16839515C2211yescharacteristic, complemented
20#16839508—21no—
21C61×#500C2211yescharacteristic, complemented
22C61×#493C2211yescharacteristic
23C61×#357—22no—
24C4×C16×C61C2211yescharacteristic, complemented
25C2×D4×D61—22no—
26C2×D4×D61—41no—
27C2×#11758201—41no—
28C2×D488D411yescharacteristic
29C2×#11758199—22no—
30D8×D61—42no—
31D8×D61—42no—
32C2×C8×D61—21no—
33#14519748—22no—
34#14519748—41no—
35#14519745—42no—
36C4×D244—21no—
37C4×D244D411yescharacteristic
38C2×#11758185—21no—
39#14519711D411yescharacteristic
40C4⋊#92418—21no—
41C4⋊#92418D411yescharacteristic
42#14519701—41no—
43#14519699—21no—
44#14519698—22no—
45#14519689—21no—
46C2×D8×C61—21no—
47C2×D8×C61D412yes—
48C4⋊Q8×C61—21no—
49#127×C61—21no—
50C2×C16×C61—21no—
51#107×C61—21no—
52#107×C61—21no—
53C4×C8×C61C2311yescharacteristic
54C23×D61—42no—
55C2×Q8×C61—41no—
56C2×D4×C61—22no—
57C2×D4×C61—41no—
58C2×D4×C61—41no—
59C2×C22⋊D61—42no—
60C2×C22⋊D61—42no—
61D4×D61—44no—
62D4×D61—82no—
63D4×D61—82no—
64D4×D61—82no—
65C2×D244—21no—
66C2×D244D811yescharacteristic
67C2×C4×D61—21no—
68C2×C4×D61—41no—
69D8×C61—42no—
70D8×C61—41no—
71D8×C61—41no—
72C2×C8×C61C2×D411yescharacteristic
73C2×C8×C61C2×D411yescharacteristic
74C4⋊C4×C61—41no—
75C4⋊C4×C61—41no—
76C42×C61C2×D411yescharacteristic
77C22⋊Dic61—42no—
78#11758201—82no—
79#11758199—44no—
80#11758198—21no—
81#11758198—41no—
82C4⋊Dic61—21no—
83C4⋊Dic61D811yescharacteristic
84C2×#92418—21no—
85C2×#92418—41no—
86D488—21no—
87D488SD1612yescomplemented
88C8×D61—41no—
89C8×D61—41no—
90C16×C61—41no—
91#11758185—41no—
92C23×C61—42no—
93C22×D61—21no—
94C22×D61—41no—
95C22×D61—84no—
96C22×D61—84no—
97C22×D61—162no—
98Q8×C61—81no—
99Q8×C61—81no—
100D4×C61—42no—
101D4×C61—42no—
102D4×C61—81no—
103D4×C61—81no—
104D4×C61—161no—
105C2×C4×C61—81no—
106C2×C4×C61#12011yescharacteristic
107C2×C4×C61C2×D811yescharacteristic
108C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
109C22⋊D61—84no—
110C22⋊D61—84no—
111C2×Dic61—21no—
112C2×Dic61—41no—
113D244—22no—
114D244—21no—
115D244—41no—
116D244D1612yes—
117C4×D61—41no—
118C4×D61—41no—
119C4×D61—81no—
120C4×D61—81no—
121C8×C61—21no—
122C8×C61C2×SD1611yescharacteristic
123C8×C61AGL(1,ℤ/8)11yescharacteristic
124#92418—41no—
125#92418—81no—
126#12368—611nomaximal, Sylow
127C22×C61—82no—
128C22×C61—82no—
129C22×C61—82no—
130C22×C61#44711yescharacteristic
131C2×D61—22no—
132C2×D61—41no—
133C2×D61—41no—
134C2×D61—42no—
135C2×D61—81no—
136C2×D61—81no—
137C2×D61—168no—
138C4×C61—21no—
139C4×C61—161no—
140C4×C61#45011yescharacteristic
141C4×C61#45511yescharacteristic
142C4×C61C2×D1611yescharacteristic
143C4×C61#50911yescharacteristic
144Dic61—41no—
145Dic61—81no—
146D4×D8—611no—
147#2240D6111yescharacteristic, complemented
148#2224—611no—
149#1673—611no—
150#1361—611no—
151#1331—611no—
152D4⋊C16—611no—
153C2×C61—162no—
154C2×C61#163911yescharacteristic
155C2×C61#218411yescharacteristic
156C2×C61#219411yescharacteristic
157D61—42no—
158D61—82no—
159C22×D8—1221no—
160D42—1221no—
161#500C2×D6111yescharacteristic, complemented
162#493C2×D6111yescharacteristic
163#466—1221no—
164#461—1221no—
165#459—1221no—
166#447—1222no—
167C4×D8—611no—
168C8×D4—611no—
169#366—1221no—
170C4⋊C16—611no—
171#357—22no—
172#357—1221no—
173C4×C16C2×D6111yescharacteristic, complemented
174#335—611no—
175#332—1221no—
176C61#1236811yescharacteristic, complemented, Sylow
177C22×D4—1222no—
178C22×D4—2441no—
179C2×SD16—2441no—
180C2×D8—21no—
181C2×D8—611no—
182C2×D8—1222no—
183C2×D8—2442no—
184C2×D8—2442no—
185C2×D8C22⋊D6112yes—
186C22×C8—1221no—
187C4⋊Q8—21no—
188#127—21no—
189#121—1222no—
190#121—2441no—
191#120—2442no—
192C4×D4—611no—
193C4×D4—1221no—
194C2×C16—21no—
195C2×C16—1221no—
196#107—21no—
197#107—21no—
198#107—611no—
199C4⋊C8—611no—
200C4⋊C8—1221no—
201#103—2441no—
202#102—1222no—
203#102—1221no—
204C22⋊C8—1221no—
205C4×C8C22×D6111yescharacteristic
206C24—2442no—
207C2×Q8—41no—
208C2×D4—22no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—611no—
212C2×D4—1221no—
213C2×D4—4882no—
214C2×D4—4882no—
215C2×D4—4882no—
216C2×D4—2442no—
217C2×D4—2444no—
218C2×D4—2442no—
219C22×C4—1221no—
220C22×C4—2441no—
221SD16—4882no—
222D8—42no—
223D8—41no—
224D8—41no—
225D8—612no—
226D8—1221no—
227D8—2444no—
228C2×C8—1221no—
229C2×C8—2441no—
230C2×C8—2441no—
231C2×C8—2441no—
232C2×C8D4×D6111yescharacteristic
233C2×C8C2×C22⋊D6111yescharacteristic
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—611no—
237C4⋊C4—1221no—
238C22⋊C4—1221no—
239C22⋊C4—2442no—
240C22⋊C4—2441no—
241C42D4×D6111yescharacteristic
242C16—41no—
243C16—2441no—
244C23—42no—
245C23—1221no—
246C23—9762no—
247C23—4884no—
248C23—4884no—
249C23—2441no—
250Q8—81no—
251Q8—81no—
252D4—42no—
253D4—42no—
254D4—81no—
255D4—81no—
256D4—161no—
257D4—612no—
258D4—1221no—
259D4—1222no—
260D4—4884no—
261D4—4884no—
262D4—2441no—
263C2×C4—81no—
264C2×C4—1221no—
265C2×C4—4881no—
266C2×C4—4881no—
267C2×C4—2441no—
268C2×C4—2441no—
269C2×C4—2441no—
270C2×C4#1451974511yescharacteristic
271C2×C4D8×D6111yescharacteristic
272C2×C4#1451980111yescharacteristic
273C8—21no—
274C8—4881no—
275C8—2441no—
276C8SD16×D6111yescharacteristic
277C8#1451980411yescharacteristic
278C22—82no—
279C22—82no—
280C22—82no—
281C22—1222no—
282C22—9768no—
283C22—4881no—
284C22—4881no—
285C22—2442no—
286C22—2441no—
287C22—2441no—
288C22#1683989411yescharacteristic
289C4—21no—
290C4—161no—
291C4—4881no—
292C4—2441no—
293C4#1683991111yescharacteristic
294C4#1683993911yescharacteristic
295C4D16×D6111yescharacteristic
296C4#1684019611yescharacteristic
297C2—162no—
298C2—4882no—
299C2—2442no—
300C2#2144499511yescharacteristic
301C2#2144772911yescharacteristic
302C2#2144778911yescharacteristic
subgroupG/H
centerC22#16839894
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#2240C2
socleC22×C61#447
prime radical#21457494C2
hypercenter#2240D61
hypocenterC61#12368
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2240

Complements

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

NH ≅ G/Ntype
2. #21447953C2semidirect
3. #21445205C2semidirect
4. #21443865C2semidirect
5. #21443781C2semidirect
6. #21443775C2semidirect
7. C61×#2240C2semidirect
17. C4⋊#11758185C22semidirect
19. #16839515C22semidirect
21. C61×#500C22semidirect
24. C4×C16×C61C22semidirect
87. D488SD16semidirect
147. #2240D61semidirect
161. #500C2×D61semidirect
173. C4×C16C2×D61semidirect
176. C61#12368semidirect

Maximal subgroups

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

subgroupindex
#214574942
#214479532
#214452052
#214438652
#214437812
#214437752
C61×#22402
#1236861