Finite Group Databasebeta

← #29600421

Subgroups — #29600421

Order 15616.

336 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 57 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#21456884C211yesmaximal, characteristic
2#21447489C211yesmaximal, characteristic
3#21447485C211yesmaximal, characteristic
4#21443989C211yesmaximal, characteristic, complemented
5#21443989C211yesmaximal, characteristic, complemented
6#21443984C211yesmaximal, characteristic, complemented
7C61×#1409C211yesmaximal, characteristic, complemented
8#16840373C411yescharacteristic, complemented
9#16840325C2211yescharacteristic
10#16840154C411yescharacteristic, complemented
11#16840126C2211yescharacteristic
12#16840117C411yescharacteristic, complemented
13#16839982—21no—
14#16839960—21no—
15#16839921—21no—
16#16839911—21no—
17#16839821C411yescharacteristic, complemented
18#16839734—21no—
19#99×D61—21no—
20#16839556—21no—
21#16839555C2211yescharacteristic
22#16839555C2211yescharacteristic
23#16839554—21no—
24C61×#492C2211yescharacteristic
25C61×#353C2211yescharacteristic
26C61×#353C2211yescharacteristic
27C2×#11758224D411yescharacteristic
28C2×D4×D61D411yescharacteristic
29C2×#11758202—21no—
30C2×#11758201—21no—
31#14519773D411yescharacteristic
32#14519772—41no—
33#14519769C2×C411yescharacteristic, complemented
34#14519767—41no—
35#14519765C2×C411yescharacteristic
36#14519764D411yescharacteristic
37#14519753—21no—
38#14519749—41no—
39#14519745—41no—
40Q8×Dic61—21no—
41#14519734C2×C411yescharacteristic
42#14519733—41no—
43C22⋊C4×D61—41no—
44#14519706—21no—
45#14519702—21no—
46#14519699—21no—
47#14519698—21no—
48#14519698—21no—
49#14519692—21no—
50#14519692—21no—
51#14519691—41no—
52#14519691—41no—
53#14519689—21no—
54#14519688C2×C411yescharacteristic, complemented
55C2×SD16×C61—21no—
56C2×SD16×C61—21no—
57C4⋊Q8×C61C2×C411yescharacteristic, complemented
58#127×C61C2311yescharacteristic
59#99×C61—21no—
60#99×C61—21no—
61C4×C8×C61C2×C411yescharacteristic, complemented
62C23×D61—21no—
63C2×Q8×C61—21no—
64C2×D4×C61—41no—
65C2×D4×C61C2×D411yescharacteristic
66C2×D4×C61C2×D411yescharacteristic
67C2×C22⋊D61—21no—
68C2×C22⋊D61—21no—
69C2×C22⋊D61—41no—
70C22×Dic61—21no—
71#11758224—81no—
72D4×D61—81no—
73C2×D244C22⋊C411yescharacteristic
74C2×C4×D61—81no—
75C2×C4×D61C22⋊C411yescharacteristic
76C2×Dic122C22⋊C411yescharacteristic
77SD16×C61—81no—
78SD16×C61—81no—
79C2×C8×C61—21no—
80C4⋊C4×C61—41no—
81C22⋊C4×C61—41no—
82C22⋊C4×C61—41no—
83C42×C61C22×C411yescharacteristic
84C22⋊Dic61—41no—
85C22⋊Dic61—41no—
86C22⋊Dic61—41no—
87#11758202—81no—
88#11758201—81no—
89#11758198—21no—
90#11758198—41no—
91#11758198—41no—
92#11758198—81no—
93C4⋊Dic61—21no—
94C4⋊Dic61—41no—
95#11758196—21no—
96#11758196—41no—
97#11758196—41no—
98#11758196—81no—
99C4×Dic61—41no—
100C4×Dic61—81no—
101C4×Dic61C22⋊C411yescharacteristic
102C2×#92418—21no—
103C23×C61—21no—
104C23×C61—21no—
105C22×D61—21no—
106C22×D61—41no—
107C22×D61—41no—
108C22×D61—161no—
109C22×D61#9911yescharacteristic
110Q8×C61—41no—
111Q8×C61—41no—
112D4×C61—41no—
113D4×C61—41no—
114D4×C61—161no—
115C2×C4×C61—21no—
116C2×C4×C61—41no—
117C2×C4×C61—81no—
118C2×C4×C61—81no—
119C2×C4×C61C2×C22⋊C411yescharacteristic
120C22⋊D61—81no—
121C22⋊D61—81no—
122C22⋊D61—161no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—81no—
129C2×Dic61—81no—
130C2×Dic61#9911yescharacteristic
131D244—41no—
132C4×D61—41no—
133C4×D61—162no—
134Dic122—41no—
135C8×C61—41no—
136#92418—41no—
137#13610—611nomaximal, Sylow
138C22×C61—41no—
139C22×C61—41no—
140C22×C61—81no—
141C22×C61—81no—
142C22×C61C2×#9911yescharacteristic
143C2×D61—21no—
144C2×D61—41no—
145C2×D61—41no—
146C2×D61—82no—
147C2×D61—81no—
148C2×D61—161no—
149C4×C61—21no—
150C4×C61—41no—
151C4×C61—81no—
152C4×C61—161no—
153C4×C61—161no—
154Dic61—21no—
155Dic61—41no—
156Dic61—41no—
157Dic61—81no—
158Dic61—161no—
159Dic61—161no—
160#3222—611no—
161#2128—611no—
162#2127—611no—
163#1412—611no—
164#1412—611no—
165#1409—611no—
166#1409D6111yescharacteristic, complemented
167C2×C61—21no—
168C2×C61—81no—
169C2×C61—81no—
170C2×C61#212711yescharacteristic
171D61—41no—
172D61—81no—
173#541—611no—
174#535—611no—
175#497—611no—
176#492C2×D6111yescharacteristic
177#490—611no—
178#488—611no—
179#464—1221no—
180#460—1221no—
181#452—1221no—
182#450—1221no—
183#433—611no—
184#410—1221no—
185C2×#99—1221no—
186#354—1221no—
187#353—611no—
188#353—611no—
189#353—1221no—
190#353C2×D6111yescharacteristic
191#353C2×D6111yescharacteristic
192C61#1361011yescharacteristic, complemented, Sylow
193C2×#41—611no—
194C22×D4—611no—
195C2×Q16—1221no—
196C2×SD16—21no—
197C2×SD16—21no—
198C2×SD16—1221no—
199C4⋊Q8C4×D6111yescharacteristic, complemented
200#127—611no—
201#127C22×D6111yescharacteristic
202#126—2441no—
203#125—611no—
204#125—2441no—
205#124—611no—
206#124—611no—
207C22⋊Q8—1221no—
208#121—2441no—
209#120—2441no—
210C4×Q8—1221no—
211#117—611no—
212#117—2441no—
213C2×C22⋊C4—2441no—
214C4⋊C8—1221no—
215#103—1221no—
216#102—1221no—
217#102—1221no—
218#102—1221no—
219#99—21no—
220#99—21no—
221#99—1221no—
222#99—1221no—
223#99—2441no—
224#99—2441no—
225C22⋊C8—1221no—
226#97—611no—
227C4×C8C4×D6111yescharacteristic, complemented
228C24—1221no—
229#41—4881no—
230C2×Q8—21no—
231C2×Q8—611no—
232C2×D4—41no—
233C2×D4—611no—
234C2×D4—1221no—
235C2×D4—1221no—
236C2×D4—4881no—
237C2×D4—2441no—
238C2×D4D4×D6111yescharacteristic
239C2×D4D4×D6111yescharacteristic
240C22×C4—611no—
241C22×C4—1221no—
242C22×C4—4881no—
243Q16—4881no—
244SD16—81no—
245SD16—81no—
246SD16—4881no—
247C2×C8—21no—
248C2×C8—1221no—
249C4⋊C4—41no—
250C4⋊C4—1221no—
251C4⋊C4—1221no—
252C4⋊C4—4881no—
253C4⋊C4—2441no—
254C4⋊C4—2441no—
255C4⋊C4—2441no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—1221no—
259C22⋊C4—4881no—
260C22⋊C4—2441no—
261C22⋊C4—2441no—
262C22⋊C4—2441no—
263C22⋊C4—2441no—
264C22⋊C4—2441no—
265C42—611no—
266C42—4881no—
267C42—2441no—
268C42C2×C4×D6111yescharacteristic
269C23—21no—
270C23—21no—
271C23—611no—
272C23—1221no—
273C23—9761no—
274C23—2441no—
275C23—2441no—
276Q8—41no—
277Q8—41no—
278Q8—2441no—
279D4—41no—
280D4—41no—
281D4—161no—
282D4—9761no—
283D4—4881no—
284D4—4881no—
285D4—2441no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—611no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—9762no—
294C2×C4—4881no—
295C2×C4—4881no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4C22⋊C4×D6111yescharacteristic
301C8—41no—
302C8—2441no—
303C22—41no—
304C22—41no—
305C22—81no—
306C22—81no—
307C22—1221no—
308C22—9761no—
309C22—4881no—
310C22—4882no—
311C22—2441no—
312C22—2441no—
313C22#99×D6111yescharacteristic
314C4—21no—
315C4—41no—
316C4—81no—
317C4—161no—
318C4—161no—
319C4—1221no—
320C4—9761no—
321C4—9761no—
322C4—4881no—
323C4—2441no—
324C4—2441no—
325C2—21no—
326C2—81no—
327C2—81no—
328C2—4881no—
329C2—2441no—
330C2#2144748511yescharacteristic
subgroupG/H
centerC2#21447485
commutatorC42×C61C22×C4
Frattini#127C22×D61
FittingC61×#1409C2
socleC2×C61#2127
prime radical#16840325C22
cosocle#127×C61C23
hypercenter#1409D61
hypocenterC61#13610
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #1409

Complements

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

NH ≅ G/Ntype
4. #21443989C2semidirect
5. #21443989C2semidirect
6. #21443984C2semidirect
7. C61×#1409C2semidirect
8. #16840373C4semidirect
10. #16840154C4semidirect
12. #16840117C4semidirect
17. #16839821C4semidirect
33. #14519769C2×C4semidirect
54. #14519688C2×C4semidirect
57. C4⋊Q8×C61C2×C4semidirect
61. C4×C8×C61C2×C4semidirect
166. #1409D61semidirect
192. C61#13610semidirect
199. C4⋊Q8C4×D61semidirect
227. C4×C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214568842
#214474892
#214474852
#214439892
#214439842
C61×#14092
#1361061