Finite Group Databasebeta

← #29595164

Subgroups — #29595164

Order 15616.

470 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 109 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#21454516C211yesmaximal, characteristic
2#21447458C212yesmaximal
3#21445621C212yesmaximal, complemented
4#21443942C211yesmaximal, characteristic, complemented
5C61×#1391C211yesmaximal, characteristic, complemented
6C2×#14519784C2211yescharacteristic
7C22×C8×D61C411yescharacteristic, complemented
8#16840269C2211yescharacteristic
9#16839836—22no—
10#16839825—22no—
11#16839788C411yescharacteristic, complemented
12#16839746C2212yes—
13#16839725C411yescharacteristic, complemented
14#16839720C411yescharacteristic, complemented
15#16839718—22no—
16#16839551C414yescomplemented
17#16839548C414yescomplemented
18#16839532—22no—
19C61×#410C2212yes—
20C61×#408C2211yescharacteristic
21C22×C4×D61C2×C411yescharacteristic
22#14519784—41no—
23#14519784—41no—
24C2×#11758194—22no—
25C2×#11758194C2×C411yescharacteristic
26C2×#11758194D412yes—
27C22×#92418C2×C411yescharacteristic
28C22×#92418D412yes—
29C2×C8×D61—21no—
30C2×C8×D61—21no—
31C2×C8×D61—41no—
32C2×C8×D61—41no—
33#14519758C2×C412yes—
34#14519753—21no—
35#14519748—21no—
36#14519745C2×C412yes—
37C4×C22⋊D61—22no—
38#14519736C2×C411yescharacteristic
39#14519706—22no—
40#14519696—22no—
41#14519694C2×C414yes—
42#14519689—22no—
43#14519688—42no—
44#14519687—21no—
45C4×#92418—42no—
46C8×Dic61—21no—
47C2×#41×C61C2311yescharacteristic
48C2×M16×C61C2×C411yescharacteristic
49C22×C8×C61C2×C411yescharacteristic
50#117×C61—22no—
51#99×C61—22no—
52#99×C61C2×C414yes—
53C22⋊C8×C61—22no—
54C23×D61C4211yescharacteristic
55#41×C61—41no—
56#41×C61—41no—
57C2×Q8×C61#4111yescharacteristic
58C2×D4×C61C22×C412yes—
59C2×D4×C61#4111yescharacteristic
60C22×C4×C61C22×C411yescharacteristic
61C22×C4×C61C2×D412yes—
62C2×C22⋊D61—22no—
63C22×Dic61C4211yescharacteristic
64C2×C4×D61—21no—
65C2×C4×D61—21no—
66C2×C4×D61—41no—
67C2×C4×D61—41no—
68M16×C61—41no—
69C2×C8×C61—21no—
70C2×C8×C61—21no—
71C2×C8×C61—41no—
72C4⋊C4×C61—42no—
73C22⋊C4×C61—24no—
74C42×C61—42no—
75C22⋊Dic61—22no—
76C22⋊Dic61C4212yes—
77#11758198—22no—
78C4⋊Dic61—21no—
79#11758196—22no—
80C4×Dic61—21no—
81#11758194—24no—
82#11758194—41no—
83#11758194—42no—
84#11758194—42no—
85C2×#92418—22no—
86C2×#92418—22no—
87C2×#92418—21no—
88C2×#92418—21no—
89C2×#92418—42no—
90C2×#92418C22⋊C411yescharacteristic
91C2×#92418C22⋊C412yes—
92C2×#92418C22⋊C411yescharacteristic
93C8×D61—42no—
94C8×D61—41no—
95C8×D61—41no—
96C23×C61C2×C4211yescharacteristic
97C23×C61C4×D412yes—
98C22×D61—21no—
99C22×D61—42no—
100C22×D61—41no—
101Q8×C61—41no—
102D4×C61—42no—
103D4×C61—41no—
104C2×C4×C61—22no—
105C2×C4×C61—21no—
106C2×C4×C61—22no—
107C2×C4×C61—21no—
108C2×C4×C61—44no—
109C2×C4×C61C2×C22⋊C411yescharacteristic
110C2×C4×C61#11711yescharacteristic
111C2×C4×C61C4×D412yes—
112C22⋊D61—44no—
113C2×Dic61—22no—
114C2×Dic61—21no—
115C2×Dic61—41no—
116C4×D61—42no—
117C4×D61—41no—
118C4×D61—41no—
119C8×C61—41no—
120C8×C61—41no—
121#92418—21no—
122#92418—22no—
123#92418—44no—
124#92418#9911yescharacteristic
125#92418#9911yescharacteristic
126#12623—611nomaximal, Sylow
127C22×C61—21no—
128C22×C61—22no—
129C22×C61—42no—
130C22×C61—41no—
131C22×C61C4×C22⋊C411yescharacteristic
132C2×D61—22no—
133C2×D61—41no—
134C2×D61—42no—
135C2×D61—41no—
136C2×D61—42no—
137C4×C61—22no—
138C4×C61—21no—
139C4×C61—84no—
140C4×C61C2×#9911yescharacteristic
141C4×C61#41011yescharacteristic
142Dic61—42no—
143Dic61—41no—
144#2929—611no—
145#2120—612no—
146#1757—612no—
147#1392—611no—
148#1391D6111yescharacteristic, complemented
149C2×C61—21no—
150C2×C61—42no—
151C2×C61—41no—
152C2×C61C4×#9911yescharacteristic
153D61—42no—
154C2×#131—611no—
155C23×C8—611no—
156#525—611no—
157#435—1222no—
158C8×D4—1222no—
159#424—611no—
160#413—612no—
161#410C2×D6112yes—
162#408C2×D6111yescharacteristic
163#407—611no—
164#405—611no—
165#405—1222no—
166#352—614no—
167AΣL(1,𝔽16)—614no—
168#344—1222no—
169C61#1262311yescharacteristic, complemented, Sylow
170C2×#41C22×D6111yescharacteristic
171C23×C4—611no—
172#131—2441no—
173#131—2441no—
174C2×M16—611no—
175C2×M16—612no—
176C2×M16—1222no—
177C2×M16C4×D6111yescharacteristic
178C22×C8—611no—
179C22×C8—612no—
180C22×C8—1221no—
181C22×C8—1221no—
182C22×C8—2441no—
183C22×C8—2441no—
184C22×C8C4×D6111yescharacteristic
185#123—612no—
186C22⋊Q8—1221no—
187#121—1221no—
188#120—612no—
189C4×D4—1222no—
190#117—22no—
191#117—611no—
192C4⋊C8—1222no—
193#101—1222no—
194#100—614no—
195#99—22no—
196#99C4×D6114yes—
197C22⋊C8—22no—
198C22⋊C8—1222no—
199#97—1221no—
200#97—2442no—
201C4×C8—1221no—
202C4×C8—2442no—
203C24—611no—
204#41—41no—
205#41—41no—
206C2×Q8#1175822611yescharacteristic
207C2×D4—1222no—
208C2×D4C2×C4×D6112yes—
209C2×D4#1175822411yescharacteristic
210C22×C4—611no—
211C22×C4—1221no—
212C22×C4—1221no—
213C22×C4—2441no—
214C22×C4—2441no—
215C22×C4C2×C4×D6111yescharacteristic
216C22×C4D4×D6112yes—
217M16—41no—
218M16—1224no—
219M16—2442no—
220M16—2441no—
221M16—2442no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—611no—
226C2×C8—611no—
227C2×C8—612no—
228C2×C8—1222no—
229C2×C8—1222no—
230C2×C8—1221no—
231C2×C8—1221no—
232C2×C8—2441no—
233C2×C8—2441no—
234C2×C8—2442no—
235C2×C8—2442no—
236C4⋊C4—42no—
237C4⋊C4—1221no—
238C4⋊C4—1222no—
239C22⋊C4—24no—
240C22⋊C4—612no—
241C22⋊C4—1222no—
242C22⋊C4—1222no—
243C42—42no—
244C42—1221no—
245C23—1221no—
246C23—2441no—
247C23—2442no—
248C23C42×D6111yescharacteristic
249C23#1451973712yes—
250Q8—41no—
251D4—42no—
252D4—41no—
253D4—2444no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—44no—
259C2×C4—1222no—
260C2×C4—1221no—
261C2×C4—2442no—
262C2×C4—2441no—
263C2×C4—2441no—
264C2×C4—2441no—
265C2×C4C22⋊C4×D6111yescharacteristic
266C2×C4#1451973511yescharacteristic
267C2×C4#1451973912yes—
268C8—41no—
269C8—41no—
270C8—611no—
271C8—611no—
272C8—1221no—
273C8—1222no—
274C8—2444no—
275C22—22no—
276C22—21no—
277C22—42no—
278C22—41no—
279C22—1222no—
280C22—2441no—
281C22—2442no—
282C22—2442no—
283C22—2441no—
284C22#1683960911yescharacteristic
285C4—22no—
286C4—21no—
287C4—84no—
288C4—2442no—
289C4—2441no—
290C4#99×D6111yescharacteristic
291C4#1683973411yescharacteristic
292C2—21no—
293C2—42no—
294C2—41no—
295C2—2442no—
296C2#2144560611yescharacteristic
subgroupG/H
centerC4#99×D61
commutatorC23×C61C2×C42
FrattiniC2×#41C22×D61
FittingC61×#1391C2
socleC2×C61C4×#99
prime radical#16840269C22
cosocleC2×#41×C61C23
hypercenter#1391D61
hypocenterC61#12623
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1391

Complements

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

NH ≅ G/Ntype
3. #21445621C2semidirect
4. #21443942C2semidirect
5. C61×#1391C2semidirect
7. C22×C8×D61C4semidirect
11. #16839788C4semidirect
13. #16839725C4semidirect
14. #16839720C4semidirect
16. #16839551C4semidirect
17. #16839548C4semidirect
148. #1391D61semidirect
169. C61#12623semidirect

Maximal subgroups

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

subgroupindex
#214545162
#214474582
#214456212
#214439422
C61×#13912
#1262361