Finite Group Databasebeta

← #29835254

Subgroups — #29835254

Order 15616.

520 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 131 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#21456564C211yesmaximal, characteristic, complemented
2#21454730C211yesmaximal, characteristic
3#21448039C212yesmaximal, complemented
4#21447991C212yesmaximal, complemented
5#21447901C212yesmaximal, complemented
6#21447772C212yesmaximal
7#21447712C214yesmaximal, complemented
8C61×#2165C211yesmaximal, characteristic, complemented
9#16840562C2211yescharacteristic
10#16840528—21no—
11#16840319—21no—
12#16840205C2214yes—
13#16840160C2211yescharacteristic, complemented
14#16840142C2211yescharacteristic, complemented
15#16840113—21no—
16#16840020—21no—
17#16840020C2212yes—
18#16839885—21no—
19#16839885C2212yes—
20#16839880—21no—
21#16839880C2212yes—
22#16839870—21no—
23#16839870C2212yescomplemented
24#16839811C2211yescharacteristic
25C2×#14519705—21no—
26#16839585C2214yescomplemented
27#16839579C2214yescomplemented
28#16839573C2214yes—
29C4×M16×C61C2211yescharacteristic
30C61×#364C2212yescomplemented
31C61×#349C2212yes—
32C61×#347C2212yescomplemented
33C2×#11758222—21no—
34#14519810—22no—
35#14519805—22no—
36#14519799—21no—
37#14519799C2312yes—
38C2×Dic244—21no—
39C2×Dic244C2312yes—
40#14519785—42no—
41#14519783—22no—
42#14519783—22no—
43C2×#11758194—21no—
44M16×D61—42no—
45#14519777—21no—
46#14519777—21no—
47#14519769—41no—
48#14519756—22no—
49#14519756—41no—
50C4×Dic122—22no—
51C4×Dic122—41no—
52#14519736—41no—
53Dic488—24no—
54#14519722—24no—
55#14519713—21no—
56#14519713C2312yes—
57#14519709—21no—
58#14519709C2312yes—
59#14519705—22no—
60#14519705—22no—
61#14519702—22no—
62#14519702—22no—
63C2×M16×C61D411yescharacteristic
64C2×M16×C61C2311yescharacteristic
65C2×C42×C61D411yescharacteristic
66M32×C61C2314yes—
67#97×C61D411yescharacteristic
68#97×C61C2311yescharacteristic
69C4×C8×C61D411yescharacteristic
70C4×C8×C61C2311yescharacteristic
71C22×C4×C61—21no—
72C22×C4×C61C2×D411yescharacteristic
73C2×C22⋊D61—41no—
74#11758222—22no—
75#11758222—22no—
76#11758222—41no—
77#11758222—41no—
78C2×D244—21no—
79C2×C4×D61—41no—
80C2×Dic122—21no—
81C2×Dic122—22no—
82C2×Dic122—41no—
83M16×C61—22no—
84M16×C61—22no—
85C2×C8×C61C2×D411yescharacteristic
86C2×C8×C61C2×D411yescharacteristic
87C2×C8×C61C2×D411yescharacteristic
88C2×C8×C61C2411yescharacteristic
89C42×C61—21no—
90C42×C61C2×D411yescharacteristic
91C42×C61C2×D411yescharacteristic
92C22⋊Dic61—42no—
93C4⋊Dic61—22no—
94C4⋊Dic61—41no—
95C4⋊Dic61—41no—
96C4⋊Dic61—41no—
97#11758196—42no—
98#11758196—42no—
99C4×Dic61—42no—
100#11758194—22no—
101#11758194—41no—
102C2×#92418—41no—
103Dic244—21no—
104Dic244—24no—
105Dic244—42no—
106Dic244#4112yes—
107D488—21no—
108D488#4112yes—
109#11758190—22no—
110#11758190—22no—
111#11758189—42no—
112#11758189—42no—
113C8×D61—42no—
114C8×D61—42no—
115C16×C61—24no—
116C23×C61AGL(1,ℤ/8)11yescharacteristic
117C22×D61—41no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—42no—
123C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C61C22×D411yescharacteristic
125C22⋊D61—42no—
126C22⋊D61—42no—
127C2×Dic61—41no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130D244—22no—
131D244—41no—
132C4×D61—42no—
133C4×D61—42no—
134Dic122—22no—
135Dic122—41no—
136Dic122—42no—
137Dic122—42no—
138Dic122—82no—
139C8×C61—21no—
140C8×C61—21no—
141C8×C61AGL(1,ℤ/8)12yes—
142C8×C61C2×#4112yes—
143#92418—42no—
144#33689—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—41no—
147C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
148C2×D61—42no—
149C2×D61—161no—
150C4×C61—22no—
151C4×C61—21no—
152C4×C61—41no—
153C4×C61#52511yescharacteristic
154C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
155Dic61—42no—
156Dic61—82no—
157Dic61—82no—
158#3185—611no—
159#2956—611no—
160#2239—612no—
161#2230—612no—
162#2214—612no—
163#2191—612no—
164#2180—614no—
165#2165D6111yescharacteristic, complemented
166C2×C61—21no—
167C2×C61—41no—
168C2×C61#310911yescharacteristic
169D61—82no—
170#575—611no—
171#568—1221no—
172#533—1221no—
173#510—614no—
174#498—611no—
175#494—611no—
176#487—1221no—
177#470—612no—
178#470—1221no—
179#444—612no—
180#444—1221no—
181#443—612no—
182#443—1221no—
183#441—612no—
184#441—1221no—
185#430—611no—
186C2×#104—1221no—
187C4×M16C2×D6111yescharacteristic
188#365—614no—
189#364C2×D6112yescomplemented
190#362—614no—
191#360—614no—
192#349C2×D6112yes—
193#347C2×D6112yescomplemented
194C61#3368911yescharacteristic, complemented, Sylow
195C2×#41—1221no—
196#137—1222no—
197AGL(1,ℤ/8)—1222no—
198#135—612no—
199#135—1221no—
200C2×Q16—612no—
201C2×Q16—1221no—
202#131—1222no—
203#131—1222no—
204#131—2442no—
205C2×M16—1221no—
206C2×M16—2442no—
207C2×M16D24411yescharacteristic
208C2×M16C22×D6111yescharacteristic
209C4⋊Q8—1221no—
210C4⋊Q8—1221no—
211#125—2441no—
212C22⋊Q8—1222no—
213C22⋊Q8—2441no—
214C4×Q8—1222no—
215C4×Q8—2441no—
216#117—2441no—
217C2×C42D24411yescharacteristic
218Q32—1224no—
219SD32—1224no—
220M32C22×D6114yes—
221#108—612no—
222#108—1221no—
223#106—612no—
224#106—1221no—
225#104—1222no—
226#104—1222no—
227#103—1222no—
228#103—1222no—
229#97D24411yescharacteristic
230#97C22×D6111yescharacteristic
231C4×C8D24411yescharacteristic
232C4×C8C22×D6111yescharacteristic
233#41—1222no—
234#41—1222no—
235#41—2441no—
236#41—2441no—
237C2×Q8—1221no—
238C2×Q8—1222no—
239C2×Q8—2441no—
240C2×D4—1221no—
241C2×D4—2441no—
242C22×C4—21no—
243C22×C4—2441no—
244C22×C4C2×D24411yescharacteristic
245Q16—612no—
246Q16—1221no—
247Q16—1224no—
248Q16—2442no—
249SD16—1222no—
250SD16—1222no—
251D8—612no—
252D8—1221no—
253M16—22no—
254M16—22no—
255M16—1222no—
256M16—2442no—
257M16—2441no—
258M16—2442no—
259C2×C8—2442no—
260C2×C8—2441no—
261C2×C8—2442no—
262C2×C8C2×D24411yescharacteristic
263C2×C8C2×D24411yescharacteristic
264C2×C8C2×D24411yescharacteristic
265C2×C8C23×D6111yescharacteristic
266C4⋊C4—1222no—
267C4⋊C4—2442no—
268C4⋊C4—2441no—
269C4⋊C4—2441no—
270C4⋊C4—2441no—
271C4⋊C4—2442no—
272C22⋊C4—2442no—
273C42—21no—
274C42—2442no—
275C42C2×D24411yescharacteristic
276C42C2×D24411yescharacteristic
277C16—24no—
278C23—2441no—
279C23#1451980511yescharacteristic
280Q8—1222no—
281Q8—4882no—
282Q8—2441no—
283Q8—2442no—
284Q8—2442no—
285D4—1222no—
286D4—2441no—
287D4—2442no—
288D4—2442no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—2442no—
295C2×C4—2442no—
296C2×C4—2442no—
297C2×C4—2441no—
298C2×C4—2442no—
299C2×C4#1451980511yescharacteristic
300C2×C4C22×D24411yescharacteristic
301C8—21no—
302C8—21no—
303C8—2442no—
304C8#1451980512yes—
305C8#41×D6112yes—
306C22—21no—
307C22—41no—
308C22—9761no—
309C22—2442no—
310C22C2×#1451980511yescharacteristic
311C4—22no—
312C4—21no—
313C4—41no—
314C4—4882no—
315C4—4882no—
316C4—2442no—
317C4#1684027411yescharacteristic
318C4C2×#1451980511yescharacteristic
319C2—21no—
320C2—41no—
321C2—4882no—
322C2#2145591111yescharacteristic
subgroupG/H
centerC2#21455911
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2165C2
socleC2×C61#3109
prime radical#16840562C22
hypercenter#2165D61
hypocenterC61#33689
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
1. #21456564C2semidirect
3. #21448039C2semidirect
4. #21447991C2semidirect
5. #21447901C2semidirect
7. #21447712C2semidirect
8. C61×#2165C2semidirect
13. #16840160C22semidirect
14. #16840142C22semidirect
23. #16839870C22semidirect
26. #16839585C22semidirect
27. #16839579C22semidirect
30. C61×#364C22semidirect
32. C61×#347C22semidirect
165. #2165D61semidirect
189. #364C2×D61semidirect
193. #347C2×D61semidirect
194. C61#33689semidirect

Maximal subgroups

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

subgroupindex
#214565642
#214547302
#214480392
#214479912
#214479012
#214477722
#214477122
C61×#21652
#3368961