Finite Group Databasebeta

← #30604760

Subgroups — #30604760

Order 15616.

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

normal subgroups only

This group has 552 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#21461524C211yesmaximal, characteristic, complemented
2#21461052C211yesmaximal, characteristic, complemented
3#21460801C214yesmaximal, complemented
4#21451570C211yesmaximal, characteristic
5C2×D61⋊M16—21no—
6#16840778—21no—
7#16840642—21no—
8#16840642C2212yes—
9#16840608C2212yes—
10#16840606—21no—
11C2×#14519748—21no—
12C2×#14519734C2211yescharacteristic
13#16839676—21no—
14#16839676C2212yes—
15#16839670—21no—
16#16839670C412yes—
17#16839665—21no—
18#16839628C411yescharacteristic
19C2×#127×C61C411yescharacteristic
20C2×#11758213—41no—
21D61⋊M16—24no—
22D61⋊M16—42no—
23C2×#11758217—22no—
24C2×D61⋊C8—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×#11758196D411yescharacteristic
36C2×#11758196C2311yescharacteristic
37C2×C22⋊Dic61—22no—
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—22no—
40C2×#11758198C2×C411yescharacteristic
41C2×#11758198D411yescharacteristic
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×D61C22⋊C411yescharacteristic
53C2×D4×C61—42no—
54C2×D4×C61—42no—
55C2×D4×C61—41no—
56C2×D4×C61—41no—
57C2×D4×C61—42no—
58C2×D4×C61—42no—
59C2×D4×C61—82no—
60C2×D4×C61—82no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22×C411yescharacteristic
64C2×C22⋊D61—44no—
65C2×C22⋊D61—82no—
66C2×C22⋊D61—82no—
67C22×Dic61—22no—
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C2×D411yescharacteristic
70C22×Dic61C2×D411yescharacteristic
71C2×C4×D61—21no—
72C2×C4×D61—21no—
73C2×C4×D61—22no—
74C2×C4×D61D812yes—
75C2×C4×D61SD1612yes—
76#11758217—24no—
77#11758217—42no—
78C2×#92420—42no—
79C2×#92420—42no—
80#11758213—44no—
81D61⋊C8—44no—
82C42×C61—21no—
83C42×C61—21no—
84C22⋊Dic61—42no—
85C22⋊Dic61—42no—
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×D61#10212yes—
108D4×C61—84no—
109D4×C61—84no—
110D4×C61—82no—
111D4×C61—82no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61#9911yescharacteristic
117C2×C4×C61#9911yescharacteristic
118C2×C4×C61#10212yes—
119C2×C4×C61#10412yes—
120C22⋊D61—88no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61C2×D811yescharacteristic
131C2×Dic61C2×SD1611yescharacteristic
132C4×D61—22no—
133C4×D61—22no—
134C4×D61—22no—
135C4×D61—22no—
136#92420—44no—
137#92420—82no—
138#11699—611nomaximal, Sylow
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—82no—
142C22×C61—82no—
143C22×C61—82no—
144C22×C61—82no—
145C22×C61—82no—
146C22×C61—82no—
147C22×C61—82no—
148C22×C61C2×#9911yescharacteristic
149C22×C61C2×#10211yescharacteristic
150C22×C61C2×#10411yescharacteristic
151C2×D61—22no—
152C2×D61—41no—
153C2×D61—41no—
154C2×D61—42no—
155C2×D61—81no—
156C4×C61—41no—
157C4×C61—41no—
158C4×C61#32712yes—
159C4×C61#32712yes—
160Dic61—21no—
161Dic61—21no—
162Dic61—41no—
163Dic61—41no—
164Dic61—82no—
165#2619—611no—
166#1742—611no—
167#1458—611no—
168#1345—614no—
169C2×C61—21no—
170C2×C61—21no—
171C2×C61—162no—
172C2×C61C2×#32711yescharacteristic
173C2×C61#212211yescharacteristic
174C2×C61#213411yescharacteristic
175D61—42no—
176C22×M16—1221no—
177C2×#127#692311yescharacteristic
178C2×#121—1221no—
179C2×#117—611no—
180#431—1221no—
181#394—612no—
182#394—1221no—
183#392—612no—
184#392—1221no—
185#390—1221no—
186#382—611no—
187#343—612no—
188#343—1221no—
189#324—612no—
190#323—1221no—
191C61#1169911yescharacteristic, complemented, Sylow
192C22×D4—21no—
193C22×D4—21no—
194C22×D4—1222no—
195C22×D4C2×#692312yes—
196C23×C4—611no—
197C2×M16—1222no—
198C2×M16—1224no—
199C2×M16—2442no—
200C2×M16—2441no—
201C22×C8—2441no—
202#127—41no—
203#127—41no—
204#121—2442no—
205#121—2442no—
206#117—2441no—
207#117—2441no—
208C2×C4⋊C4—611no—
209C2×C4⋊C4—611no—
210C2×C4⋊C4—1221no—
211C2×C22⋊C4—611no—
212C2×C22⋊C4—611no—
213C2×C22⋊C4—1222no—
214C2×C22⋊C4—1222no—
215C2×C22⋊C4—1222no—
216C2×C42—611no—
217C2×C42C2×#692311yescharacteristic
218C4⋊C8—2441no—
219C22⋊C8—1222no—
220C4×C8—2441no—
221#95—612no—
222#95—1221no—
223C24—22no—
224C24—611no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—82no—
232C2×D4—82no—
233C2×D4—4882no—
234C2×D4—4882no—
235C2×D4—2444no—
236C22×C4—611no—
237C22×C4—612no—
238C22×C4—612no—
239C22×C4—611no—
240C22×C4—611no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4—1222no—
244C22×C4—1222no—
245C22×C4C22⋊#692311yescharacteristic
246C22×C4C22⋊#692311yescharacteristic
247C22×C4C22×#692311yescharacteristic
248M16—1224no—
249M16—2442no—
250M16—2444no—
251C2×C8—2442no—
252C2×C8—2444no—
253C2×C8—2442no—
254C4⋊C4—2441no—
255C4⋊C4—2441no—
256C4⋊C4—2442no—
257C22⋊C4—4882no—
258C22⋊C4—4882no—
259C22⋊C4—2441no—
260C22⋊C4—2441no—
261C22⋊C4—2442no—
262C22⋊C4—2442no—
263C42—21no—
264C42—21no—
265C42—1221no—
266C42—1221no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—612no—
275C23—1221no—
276C23—1221no—
277C23—2441no—
278C23—2441no—
279C23C2×C22⋊#692311yescharacteristic
280D4—84no—
281D4—84no—
282D4—82no—
283D4—82no—
284D4—4888no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—611no—
290C2×C4—611no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—1222no—
294C2×C4—1222no—
295C2×C4—1222no—
296C2×C4—1222no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—2442no—
300C2×C4—2442no—
301C2×C4—2442no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4#1451983511yescharacteristic
305C2×C4#1451983511yescharacteristic
306C2×C4#1451983912yes—
307C2×C4#1451984112yes—
308C8—4882no—
309C8—2444no—
310C22—21no—
311C22—21no—
312C22—82no—
313C22—82no—
314C22—82no—
315C22—82no—
316C22—82no—
317C22—82no—
318C22—82no—
319C22—1222no—
320C22—4881no—
321C22—2442no—
322C22—2441no—
323C22—2441no—
324C22C2×#1451983511yescharacteristic
325C22C2×#1451983911yescharacteristic
326C22C2×#1451984111yescharacteristic
327C4—41no—
328C4—41no—
329C4—1221no—
330C4—1221no—
331C4—4882no—
332C4—2441no—
333C4—2441no—
334C4#1684061312yes—
335C4#1684061312yes—
336C2—21no—
337C2—21no—
338C2—162no—
339C2—2442no—
340C2C2×#1684061311yescharacteristic
341C2#2146207811yescharacteristic
342C2#2146211811yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#127×C61C4
socleC22×C61C2×#99
prime radical#21451570C2
cosocleC2×#11758196C23
hypercenterC2×#127#6923
hypocenterC61#11699
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. #21461524C2semidirect
2. #21461052C2semidirect
3. #21460801C2semidirect
191. C61#11699semidirect

Maximal subgroups

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

subgroupindex
#214615242
#214610522
#214608012
#214515702
#1169961