Finite Group Databasebeta

← #29600423

Subgroups — #29600423

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#21456892C211yesmaximal, characteristic
2#21447489C211yesmaximal, characteristic
3#21447485C211yesmaximal, characteristic
4#21443990C211yesmaximal, characteristic, complemented
5#21443983C211yesmaximal, characteristic, complemented
6#21443983C211yesmaximal, characteristic, complemented
7C61×#1412C211yesmaximal, characteristic, complemented
8#16840375C411yescharacteristic, complemented
9#16840325C2211yescharacteristic
10#16840156C411yescharacteristic, complemented
11#16840134C2211yescharacteristic
12#16840119C411yescharacteristic, complemented
13#16839983—21no—
14#16839961—21no—
15#16839925—21no—
16#16839914—21no—
17#16839824C411yescharacteristic, complemented
18#16839734—21no—
19#99×D61—21no—
20#16839556—21no—
21#16839555C2211yescharacteristic
22#16839555C2211yescharacteristic
23#16839554—21no—
24C61×#490C2211yescharacteristic
25C61×#353C2211yescharacteristic
26C61×#353C2211yescharacteristic
27C2×#11758224D411yescharacteristic
28C2×D4×D61D411yescharacteristic
29C2×Dic244—21no—
30C2×#11758190—21no—
31C2×#11758200—21no—
32C2×#11758200—21no—
33#14519777C2×C411yescharacteristic, complemented
34#14519773D411yescharacteristic
35#14519772—41no—
36#14519768—41no—
37#14519765C2×C411yescharacteristic
38#14519764D411yescharacteristic
39#14519757—21no—
40#14519749—41no—
41#14519745—41no—
42#14519742—21no—
43#14519734C2×C411yescharacteristic
44#14519733—41no—
45C22⋊C4×D61—41no—
46#14519706—21no—
47#14519700—21no—
48#14519697—21no—
49#14519692—21no—
50#14519692—21no—
51#14519691—41no—
52#14519691—41no—
53#14519689—21no—
54C4×#92418C2×C411yescharacteristic, complemented
55#127×C61C2311yescharacteristic
56#125×C61C2×C411yescharacteristic, complemented
57#102×C61—21no—
58#102×C61—21no—
59#99×C61—21no—
60#99×C61—21no—
61#97×C61C2×C411yescharacteristic, complemented
62C23×D61—21no—
63C2×D4×C61—41no—
64C2×D4×C61C2×D411yescharacteristic
65C2×D4×C61C2×D411yescharacteristic
66C2×C22⋊D61—21no—
67C2×C22⋊D61—21no—
68C2×C22⋊D61—41no—
69C22×Dic61—21no—
70#11758224—81no—
71D4×D61—81no—
72C2×D244C22⋊C411yescharacteristic
73C2×C4×D61—81no—
74C2×C4×D61C22⋊C411yescharacteristic
75C2×Dic122—21no—
76C2×Dic122C22⋊C411yescharacteristic
77C2×C8×C61—21no—
78C4⋊C4×C61—21no—
79C4⋊C4×C61—41no—
80C22⋊C4×C61—41no—
81C22⋊C4×C61—41no—
82C42×C61C22×C411yescharacteristic
83C22⋊Dic61—41no—
84C22⋊Dic61—41no—
85C22⋊Dic61—41no—
86#11758200—81no—
87#11758200—81no—
88#11758198—21no—
89#11758198—41no—
90#11758198—41no—
91#11758198—81no—
92C4⋊Dic61—41no—
93C4⋊Dic61—41no—
94#11758196—21no—
95#11758196—41no—
96#11758196—81no—
97C4×Dic61—41no—
98C4×Dic61—81no—
99C4×Dic61C22⋊C411yescharacteristic
100C2×#92418—21no—
101Dic244—81no—
102#11758190—81no—
103C23×C61—21no—
104C23×C61—21no—
105C22×D61—21no—
106C22×D61—41no—
107C22×D61—41no—
108C22×D61—161no—
109C22×D61#9911yescharacteristic
110D4×C61—41no—
111D4×C61—41no—
112D4×C61—161no—
113C2×C4×C61—21no—
114C2×C4×C61—41no—
115C2×C4×C61—81no—
116C2×C4×C61—81no—
117C2×C4×C61C2×C22⋊C411yescharacteristic
118C22⋊D61—81no—
119C22⋊D61—81no—
120C22⋊D61—161no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—81no—
127C2×Dic61—81no—
128C2×Dic61#9911yescharacteristic
129D244—41no—
130C4×D61—41no—
131C4×D61—162no—
132Dic122—41no—
133Dic122—41no—
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#1412D6111yescharacteristic, complemented
165#1409—611no—
166#1409—611no—
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#492—611no—
177#490C2×D6111yescharacteristic
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—1221no—
197C2×SD16—1221no—
198C2×SD16—1221no—
199C4⋊Q8—611no—
200#127—611no—
201#127C22×D6111yescharacteristic
202#126—2441no—
203#125—2441no—
204#125C4×D6111yescharacteristic, complemented
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—21no—
217#102—21no—
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#97C4×D6111yescharacteristic, complemented
227C4×C8—611no—
228C24—1221no—
229#41—4881no—
230C2×Q8—611no—
231C2×Q8—1221no—
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—4881no—
245SD16—4881no—
246SD16—4881no—
247C2×C8—21no—
248C2×C8—1221no—
249C4⋊C4—21no—
250C4⋊C4—41no—
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—2441no—
277Q8—2441no—
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#2144748711yescharacteristic
subgroupG/H
centerC2#21447487
commutatorC42×C61C22×C4
Frattini#127C22×D61
FittingC61×#1412C2
socleC2×C61#2127
prime radical#16840325C22
cosocle#127×C61C23
hypercenter#1412D61
hypocenterC61#13610
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #1412

Complements

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

NH ≅ G/Ntype
4. #21443990C2semidirect
5. #21443983C2semidirect
6. #21443983C2semidirect
7. C61×#1412C2semidirect
8. #16840375C4semidirect
10. #16840156C4semidirect
12. #16840119C4semidirect
17. #16839824C4semidirect
33. #14519777C2×C4semidirect
54. C4×#92418C2×C4semidirect
56. #125×C61C2×C4semidirect
61. #97×C61C2×C4semidirect
164. #1412D61semidirect
192. C61#13610semidirect
204. #125C4×D61semidirect
226. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214568922
#214474892
#214474852
#214439902
#214439832
C61×#14122
#1361061