Finite Group Databasebeta

← #30604740

Subgroups — #30604740

Order 15616.

568 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 88 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#21461518C211yesmaximal, characteristic, complemented
2#21461052C211yesmaximal, characteristic, complemented
3#21460797C214yesmaximal, complemented
4#21451570C211yesmaximal, characteristic
5C2×#14519856—21no—
6#16840778—21no—
7#16840640—21no—
8#16840640C2212yes—
9#16840608—21no—
10#16840606C2212yes—
11C2×#14519748—21no—
12C2×#14519734C2211yescharacteristic
13#16839676—21no—
14#16839676C412yescomplemented
15#16839670—21no—
16#16839670C2212yes—
17#16839665—21no—
18#16839628C411yescharacteristic, complemented
19C2×#127×C61C411yescharacteristic, complemented
20#14519856—24no—
21#14519856—42no—
22C2×C4⋊#6923—41no—
23C2×C22⋊#6923—22no—
24C2×C4×#6923—41no—
25C4⋊#92420—41no—
26C22⋊#92420—22no—
27C4×#92420—41no—
28C22×C22⋊D61—22no—
29C22×C4×D61D411yescharacteristic
30#14519748—42no—
31#14519748—42no—
32#14519734—41no—
33#14519734—41no—
34C2×C4⋊Dic61—21no—
35C2×#11758196C2×C411yescharacteristic
36C2×#11758196D411yescharacteristic
37C2×C22⋊Dic61—22no—
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—22no—
40C2×#11758198D411yescharacteristic
41C2×#11758198C2311yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43#14519684—21no—
44#14519684C2×C412yes—
45C22×D4×C61—21no—
46C22×D4×C61—21no—
47C22×D4×C61C2×C412yes—
48#127×C61—41no—
49#127×C61—41no—
50C2×C42×C61C2×C411yescharacteristic
51C24×C61—22no—
52C23×D61C2×D411yescharacteristic
53C22×#6923—42no—
54C2×D4×C61—42no—
55C2×D4×C61—42no—
56C2×D4×C61—41no—
57C2×D4×C61—41no—
58C2×D4×C61—42no—
59C2×D4×C61—42no—
60C2×D4×C61—82no—
61C2×D4×C61—82no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C2×C22⋊D61—42no—
66C2×C22⋊D61—42no—
67C2×C22⋊D61—82no—
68C2×C22⋊D61—82no—
69C22×Dic61—22no—
70C22×Dic61C22⋊C411yescharacteristic
71C22×Dic61C22⋊C411yescharacteristic
72C22×Dic61C2×D411yescharacteristic
73C2×C4×D61—21no—
74C2×C4×D61—21no—
75C2×C4×D61—22no—
76C2×C4×D61D812yes—
77C2×C4×D61SD1612yes—
78C22⋊#6923—24no—
79C22⋊#6923—42no—
80C2×#92420—42no—
81C4⋊#6923—44no—
82C4×#6923—44no—
83C42×C61—21no—
84C42×C61—21no—
85C22⋊Dic61—44no—
86C22⋊Dic61—82no—
87C22⋊Dic61—82no—
88#11758198—41no—
89#11758198—41no—
90C4⋊Dic61—42no—
91#11758196—41no—
92#11758196—41no—
93C4×Dic61—21no—
94C4×Dic61—21no—
95C23×C61—42no—
96C23×C61—42no—
97C23×C61—42no—
98C23×C61—42no—
99C23×C61—42no—
100C23×C61—42no—
101C23×C61—42no—
102C23×C61C2×C22⋊C411yescharacteristic
103C22×D61—21no—
104C22×D61—21no—
105C22×D61—41no—
106C22×D61—41no—
107C22×D61C2×D811yescharacteristic
108C22×D61C2×SD1611yescharacteristic
109C2×#6923—44no—
110C2×#6923—82no—
111C2×#6923—82no—
112D4×C61—84no—
113D4×C61—84no—
114D4×C61—82no—
115D4×C61—82no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61#9911yescharacteristic
121C2×C4×C61#9911yescharacteristic
122C2×C4×C61#10212yes—
123C2×C4×C61#10412yes—
124C22⋊D61—84no—
125C22⋊D61—84no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61#10212yes—
136C4×D61—22no—
137C4×D61—22no—
138C4×D61—22no—
139C4×D61—22no—
140#92420—82no—
141#11693—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—82no—
145C22×C61—82no—
146C22×C61—82no—
147C22×C61—82no—
148C22×C61—82no—
149C22×C61—82no—
150C22×C61—82no—
151C22×C61C2×#9911yescharacteristic
152C22×C61C2×#10211yescharacteristic
153C22×C61C2×#10411yescharacteristic
154C2×D61—21no—
155C2×D61—21no—
156C2×D61—41no—
157C2×D61—41no—
158C2×D61—41no—
159C2×D61—41no—
160C2×D61—81no—
161#6923—84no—
162C4×C61—41no—
163C4×C61—41no—
164C4×C61#32712yes—
165C4×C61#32712yes—
166Dic61—22no—
167Dic61—41no—
168Dic61—41no—
169Dic61—82no—
170#2619—611no—
171#1740—611no—
172#1458—611no—
173#1343—614no—
174C2×C61—21no—
175C2×C61—21no—
176C2×C61—162no—
177C2×C61C2×#32711yescharacteristic
178C2×C61#212211yescharacteristic
179C2×C61#212811yescharacteristic
180D61—41no—
181D61—41no—
182C2×#127#692311yescharacteristic, complemented
183C2×#121—1221no—
184C2×#117—611no—
185C2×#117—1221no—
186#431—1221no—
187#394—612no—
188#394—1221no—
189#392—612no—
190#392—1221no—
191#390—1221no—
192#382—611no—
193#342—612no—
194#342—1221no—
195#324—1221no—
196#323—612no—
197C61#1169311yescharacteristic, complemented, Sylow
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—1222no—
201C22×D4C2×#692312yes—
202C23×C4—611no—
203#127—41no—
204#127—41no—
205#121—2442no—
206#121—2442no—
207#117—1224no—
208#117—2441no—
209#117—2441no—
210#117—2442no—
211C2×C4⋊C4—611no—
212C2×C4⋊C4—611no—
213C2×C4⋊C4—1221no—
214C2×C4⋊C4—2441no—
215C2×C22⋊C4—611no—
216C2×C22⋊C4—611no—
217C2×C22⋊C4—1222no—
218C2×C22⋊C4—1222no—
219C2×C22⋊C4—1222no—
220C2×C22⋊C4—1222no—
221C2×C42—611no—
222C2×C42—2441no—
223C2×C42C2×#692311yescharacteristic
224C4⋊C8—2441no—
225C22⋊C8—1222no—
226C4×C8—2441no—
227#95—612no—
228#95—1221no—
229C24—22no—
230C24—611no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4—4882no—
240C2×D4—4882no—
241C2×D4—2442no—
242C2×D4—2442no—
243C22×C4—611no—
244C22×C4—612no—
245C22×C4—612no—
246C22×C4—611no—
247C22×C4—611no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4—1222no—
251C22×C4—1222no—
252C22×C4—2442no—
253C22×C4C22⋊#692311yescharacteristic
254C22×C4C22⋊#692311yescharacteristic
255C22×C4C22×#692311yescharacteristic
256C2×C8—2442no—
257C4⋊C4—2441no—
258C4⋊C4—2441no—
259C4⋊C4—2444no—
260C4⋊C4—2442no—
261C22⋊C4—1224no—
262C22⋊C4—4882no—
263C22⋊C4—4882no—
264C22⋊C4—2441no—
265C22⋊C4—2441no—
266C22⋊C4—2444no—
267C22⋊C4—2442no—
268C42—21no—
269C42—21no—
270C42—1221no—
271C42—1221no—
272C42—2444no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—611no—
281C23—611no—
282C23—1221no—
283C23—1221no—
284C23—2441no—
285C23—2441no—
286C23C2×C22⋊#692311yescharacteristic
287D4—84no—
288D4—84no—
289D4—82no—
290D4—82no—
291D4—4884no—
292D4—4884no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—612no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1222no—
301C2×C4—1222no—
302C2×C4—1222no—
303C2×C4—1222no—
304C2×C4—1221no—
305C2×C4—1221no—
306C2×C4—4882no—
307C2×C4—4882no—
308C2×C4—2442no—
309C2×C4—2442no—
310C2×C4—2444no—
311C2×C4—2442no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4#1451983511yescharacteristic
315C2×C4#1451983511yescharacteristic
316C2×C4#1451983912yes—
317C2×C4#1451984112yes—
318C8—4882no—
319C22—21no—
320C22—21no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—1221no—
329C22—1221no—
330C22—4881no—
331C22—2441no—
332C22—2441no—
333C22—2441no—
334C22—2441no—
335C22C2×#1451983511yescharacteristic
336C22C2×#1451983911yescharacteristic
337C22C2×#1451984111yescharacteristic
338C4—41no—
339C4—41no—
340C4—1222no—
341C4—4882no—
342C4—4884no—
343C4—2441no—
344C4—2441no—
345C4#1684061312yes—
346C4#1684061312yes—
347C2—21no—
348C2—21no—
349C2—162no—
350C2—2441no—
351C2—2441no—
352C2C2×#1684061311yescharacteristic
353C2#2146207811yescharacteristic
354C2#2146209811yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#127×C61C4
socleC22×C61C2×#99
prime radical#21451570C2
cosocleC2×#11758198C23
hypercenterC2×#127#6923
hypocenterC61#11693
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #21461518C2semidirect
2. #21461052C2semidirect
3. #21460797C2semidirect
14. #16839676C4semidirect
18. #16839628C4semidirect
19. C2×#127×C61C4semidirect
182. C2×#127#6923semidirect
197. C61#11693semidirect

Maximal subgroups

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

subgroupindex
#214615182
#214610522
#214607972
#214515702
#1169361