Finite Group Databasebeta

← #29835246

Subgroups — #29835246

Order 15616.

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

normal subgroups only

This group has 564 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#21456557C211yesmaximal, characteristic, complemented
2#21454730C211yesmaximal, characteristic, complemented
3#21448034C212yesmaximal, complemented
4#21447991C212yesmaximal, complemented
5#21447896C212yesmaximal, complemented
6#21447767C212yesmaximal, complemented
7#21447719C214yesmaximal, complemented
8C61×#2165C211yesmaximal, characteristic, complemented
9#16840562C2211yescharacteristic
10#16840528—21no—
11#16840312—21no—
12#16840200C2214yes—
13#16840160C2211yescharacteristic, complemented
14#16840139C2211yescharacteristic, complemented
15#16840109—21no—
16#16840015—21no—
17#16840015C2212yes—
18#16839885—21no—
19#16839885C2212yes—
20#16839880—21no—
21#16839880C2212yes—
22#16839875—21no—
23#16839875C2212yescomplemented
24#16839811C2211yescharacteristic
25C2×#14519705—21no—
26#16839585C2214yescomplemented
27#16839576C2214yescomplemented
28#16839573C2214yes—
29C4×M16×C61C2211yescharacteristic
30C61×#364C2212yescomplemented
31C61×#349C2212yes—
32C61×#347C2212yescomplemented
33C2×#11758222—21no—
34C2×#11758222—41no—
35#14519810—22no—
36#14519805—22no—
37#14519799—21no—
38#14519799C2312yes—
39C2×D488—21no—
40C2×D488C2312yes—
41#14519785—42no—
42#14519783—22no—
43#14519783—22no—
44C2×#11758194—21no—
45M16×D61—42no—
46#14519777—21no—
47#14519774—21no—
48#14519765—41no—
49#14519751—22no—
50#14519751—41no—
51C4×D244—22no—
52C4×D244—41no—
53#14519722—24no—
54D976—24no—
55#14519713—21no—
56#14519713C2312yes—
57#14519709—21no—
58#14519709C2312yes—
59#14519705—22no—
60#14519705—22no—
61#14519699—22no—
62#14519699—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—
74C2×C22⋊D61—42no—
75#11758222—22no—
76#11758222—22no—
77#11758222—41no—
78#11758222—41no—
79#11758222—84no—
80C2×D244—21no—
81C2×D244—22no—
82C2×D244—41no—
83C2×D244—41no—
84C2×C4×D61—41no—
85C2×C4×D61—42no—
86C2×Dic122—21no—
87C2×Dic122—41no—
88M16×C61—22no—
89M16×C61—22no—
90C2×C8×C61C2×D411yescharacteristic
91C2×C8×C61C2×D411yescharacteristic
92C2×C8×C61C2×D411yescharacteristic
93C2×C8×C61C2411yescharacteristic
94C42×C61—21no—
95C42×C61C2×D411yescharacteristic
96C42×C61C2×D411yescharacteristic
97#11758198—42no—
98#11758198—42no—
99C4⋊Dic61—22no—
100C4⋊Dic61—41no—
101#11758194—22no—
102#11758194—41no—
103C2×#92418—41no—
104Dic244—21no—
105Dic244#4112yescomplemented
106D488—21no—
107D488—24no—
108D488—42no—
109D488#4112yescomplemented
110#11758190—22no—
111#11758190—22no—
112#11758189—42no—
113#11758189—42no—
114C8×D61—42no—
115C8×D61—42no—
116C16×C61—24no—
117C23×C61AGL(1,ℤ/8)11yescharacteristic
118C22×D61—41no—
119C22×D61—42no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—42no—
125C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C61C22×D411yescharacteristic
127C22⋊D61—42no—
128C22⋊D61—42no—
129C22⋊D61—84no—
130C2×Dic61—41no—
131C2×Dic61—42no—
132D244—22no—
133D244—41no—
134D244—42no—
135D244—42no—
136D244—82no—
137D244—82no—
138C4×D61—42no—
139C4×D61—42no—
140C4×D61—84no—
141Dic122—22no—
142Dic122—41no—
143Dic122—82no—
144C8×C61—21no—
145C8×C61—21no—
146C8×C61AGL(1,ℤ/8)12yes—
147C8×C61C2×#4112yes—
148#92418—42no—
149#33688—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—41no—
152C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
153C2×D61—42no—
154C2×D61—82no—
155C2×D61—82no—
156C2×D61—82no—
157C2×D61—161no—
158C4×C61—22no—
159C4×C61—21no—
160C4×C61—41no—
161C4×C61#52511yescharacteristic
162C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
163Dic61—42no—
164Dic61—82no—
165#3184—611no—
166#2956—611no—
167#2238—612no—
168#2230—612no—
169#2213—612no—
170#2190—612no—
171AGL(1,ℤ/16)—614no—
172#2165D6111yescharacteristic, complemented
173C2×C61—21no—
174C2×C61—41no—
175C2×C61#310911yescharacteristic
176D61—82no—
177D61—162no—
178#575—611no—
179#568—1221no—
180#532—1221no—
181#509—614no—
182#498—611no—
183#493—611no—
184#486—1221no—
185#469—612no—
186#469—1221no—
187#444—612no—
188#444—1221no—
189#443—612no—
190#443—1221no—
191#442—612no—
192#442—1221no—
193#430—611no—
194C2×#104—1221no—
195C4×M16C2×D6111yescharacteristic
196#365—614no—
197#364C2×D6112yescomplemented
198#361—614no—
199#360—614no—
200#349C2×D6112yes—
201#347C2×D6112yescomplemented
202C61#3368811yescharacteristic, complemented, Sylow
203C2×#41—1221no—
204C2×#41—2441no—
205#137—1222no—
206AGL(1,ℤ/8)—1222no—
207#135—612no—
208#135—1221no—
209C2×D8—612no—
210C2×D8—1221no—
211#131—1222no—
212#131—1222no—
213#131—2442no—
214C2×M16—1221no—
215C2×M16—2442no—
216C2×M16D24411yescharacteristic
217C2×M16C22×D6111yescharacteristic
218C4⋊Q8—1221no—
219#127—1221no—
220#124—2441no—
221#121—1222no—
222#121—2441no—
223C4×D4—1222no—
224C4×D4—2441no—
225C2×C42D24411yescharacteristic
226SD32—1224no—
227D16—1224no—
228M32C22×D6114yes—
229#108—612no—
230#108—1221no—
231#106—612no—
232#106—1221no—
233#104—1222no—
234#104—1222no—
235#102—1222no—
236#102—1222no—
237#97D24411yescharacteristic
238#97C22×D6111yescharacteristic
239C4×C8D24411yescharacteristic
240C4×C8C22×D6111yescharacteristic
241#41—1222no—
242#41—1222no—
243#41—4884no—
244#41—2441no—
245#41—2441no—
246C2×Q8—1221no—
247C2×Q8—2441no—
248C2×D4—1221no—
249C2×D4—1222no—
250C2×D4—2441no—
251C2×D4—2441no—
252C2×D4—2442no—
253C2×D4—2441no—
254C22×C4—21no—
255C22×C4—2442no—
256C22×C4—2441no—
257C22×C4C2×D24411yescharacteristic
258Q16—612no—
259Q16—1221no—
260SD16—1222no—
261SD16—1222no—
262D8—612no—
263D8—1221no—
264D8—1224no—
265D8—2442no—
266M16—22no—
267M16—22no—
268M16—1222no—
269M16—2442no—
270M16—2441no—
271M16—2442no—
272C2×C8—2442no—
273C2×C8—2441no—
274C2×C8—2442no—
275C2×C8C2×D24411yescharacteristic
276C2×C8C2×D24411yescharacteristic
277C2×C8C2×D24411yescharacteristic
278C2×C8C23×D6111yescharacteristic
279C4⋊C4—1222no—
280C4⋊C4—2441no—
281C22⋊C4—2442no—
282C22⋊C4—2442no—
283C42—21no—
284C42C2×D24411yescharacteristic
285C42C2×D24411yescharacteristic
286C16—24no—
287C23—2441no—
288C23—2442no—
289C23#1451980511yescharacteristic
290Q8—1222no—
291Q8—4882no—
292Q8—2441no—
293D4—1222no—
294D4—4882no—
295D4—4882no—
296D4—4884no—
297D4—2441no—
298D4—2442no—
299D4—2442no—
300D4—2442no—
301D4—2442no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—4884no—
308C2×C4—2442no—
309C2×C4—2442no—
310C2×C4—2441no—
311C2×C4—2442no—
312C2×C4#1451980511yescharacteristic
313C2×C4C22×D24411yescharacteristic
314C8—21no—
315C8—21no—
316C8—2442no—
317C8#1451980512yes—
318C8#41×D6112yes—
319C22—21no—
320C22—41no—
321C22—9761no—
322C22—4882no—
323C22—4882no—
324C22—4882no—
325C22—2442no—
326C22C2×#1451980511yescharacteristic
327C4—22no—
328C4—21no—
329C4—41no—
330C4—4882no—
331C4—2442no—
332C4#1684027411yescharacteristic
333C4C2×#1451980511yescharacteristic
334C2—21no—
335C2—41no—
336C2—9762no—
337C2—4882no—
338C2#2145591111yescharacteristic
subgroupG/H
centerC2#21455911
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2165C2
socleC2×C61#3109
hypercenter#2165D61
hypocenterC61#33688
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. #21456557C2semidirect
2. #21454730C2semidirect
3. #21448034C2semidirect
4. #21447991C2semidirect
5. #21447896C2semidirect
6. #21447767C2semidirect
7. #21447719C2semidirect
8. C61×#2165C2semidirect
13. #16840160C22semidirect
14. #16840139C22semidirect
23. #16839875C22semidirect
26. #16839585C22semidirect
27. #16839576C22semidirect
30. C61×#364C22semidirect
32. C61×#347C22semidirect
105. Dic244#41semidirect
109. D488#41semidirect
172. #2165D61semidirect
197. #364C2×D61semidirect
201. #347C2×D61semidirect
202. C61#33688semidirect

Maximal subgroups

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

subgroupindex
#214565572
#214547302
#214480342
#214479912
#214478962
#214477672
#214477192
C61×#21652
#3368861