Finite Group Databasebeta

← #30603854

Subgroups — #30603854

Order 15616.

504 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 86 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
1D61⋊#349C212yesmaximal, complemented
2#21460752C212yesmaximal, complemented
3#21460751C212yesmaximal, complemented
4#21454262C211yesmaximal, characteristic
5D61⋊M32—22no—
6#16840659—22no—
7#16840658C2214yes—
8C22×#11758222C411yescharacteristic
9C2×M16×D61C411yescharacteristic
10C2×M16×D61C2211yescharacteristic
11#16839727—22no—
12#16839727—22no—
13#16839726C2211yescharacteristic
14#16839725C411yescharacteristic
15#16839725C2211yescharacteristic
16C61×#407C411yescharacteristic
17#14519852—42no—
18#14519851—24no—
19D61⋊C16—42no—
20C2×#11758222—21no—
21C2×#11758222—21no—
22C2×#11758222—21no—
23C2×#11758222—21no—
24C2×#11758222—42no—
25C2×#11758222—42no—
26C22×Dic122C811yescharacteristic
27C22×D244C811yescharacteristic
28C22×C22⋊D61C811yescharacteristic
29C22×C22⋊D61C811yescharacteristic
30C22×C4×D61C2×C411yescharacteristic
31C22×C4×D61C2×C411yescharacteristic
32C2×#11758194C2×C411yescharacteristic
33C2×#11758194C2311yescharacteristic
34C2×#11758189—21no—
35C2×#11758189D412yes—
36M16×D61—42no—
37M16×D61—42no—
38C2×C8×D61—21no—
39C2×C8×D61D412yes—
40C22⋊#92418—22no—
41#14519689—22no—
42#14519689—22no—
43C23×C4×C61C2×C411yescharacteristic
44C2×M16×C61C2×C411yescharacteristic
45C2×M16×C61C2×C411yescharacteristic
46C22⋊C8×C61—22no—
47C24×C61C2×C811yescharacteristic
48C23×D61C2×C811yescharacteristic
49C23×D61C2×C811yescharacteristic
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61—41no—
53C22×C4×C61—41no—
54C22×C4×C61C2×C811yescharacteristic
55C22×C4×C61C22×C411yescharacteristic
56C2×C22⋊D61—21no—
57C2×C22⋊D61—21no—
58C2×C22⋊D61—21no—
59C2×C22⋊D61—21no—
60C2×C22⋊D61—42no—
61C2×C22⋊D61—42no—
62C22×Dic61C2×C811yescharacteristic
63C22×Dic61C2×C811yescharacteristic
64#11758222—44no—
65#11758222—42no—
66#11758222—42no—
67#11758222—44no—
68#11758222—42no—
69#11758222—42no—
70C2×D244—21no—
71C2×D244—21no—
72C2×D244—42no—
73C2×C4×D61—22no—
74C2×C4×D61—22no—
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61—41no—
78C2×C4×D61—41no—
79C2×C4×D61C22⋊C411yescharacteristic
80C2×C4×D61C22⋊C411yescharacteristic
81C2×C4×D61M1611yescharacteristic
82C2×C4×D61M1611yescharacteristic
83C2×Dic122—21no—
84C2×Dic122—21no—
85C2×Dic122—42no—
86M16×C61—41no—
87M16×C61—41no—
88C2×C8×C61—21no—
89C2×C8×C61C22⋊C412yes—
90#11758194—41no—
91#11758194—41no—
92C2×#92418—21no—
93C2×#92418C2×D412yes—
94#11758189—42no—
95#11758189—42no—
96C8×D61—22no—
97C8×D61—22no—
98C8×D61—42no—
99#11758187—44no—
100C23×C61—21no—
101C23×C61—41no—
102C23×C61—81no—
103C23×C61C22×C811yescharacteristic
104C22×D61—22no—
105C22×D61—21no—
106C22×D61—41no—
107C22×D61—81no—
108C22×D61—81no—
109C22×D61C22⋊C811yescharacteristic
110C22×D61C22⋊C811yescharacteristic
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—42no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61C2×C22⋊C411yescharacteristic
119C2×C4×C61C2×M1611yescharacteristic
120C22⋊D61—42no—
121C22⋊D61—42no—
122C22⋊D61—42no—
123C22⋊D61—42no—
124C2×Dic61—22no—
125C2×Dic61—21no—
126C2×Dic61—41no—
127C2×Dic61C22⋊C811yescharacteristic
128C2×Dic61C22⋊C811yescharacteristic
129D244—42no—
130D244—42no—
131C4×D61—22no—
132C4×D61—21no—
133C4×D61—21no—
134C4×D61—22no—
135C4×D61—42no—
136C4×D61—41no—
137C4×D61—41no—
138C4×D61#9912yes—
139C4×D61#10012yes—
140Dic122—42no—
141Dic122—42no—
142C8×C61—22no—
143C8×C61—41no—
144#92418—22no—
145#92418—41no—
146#11436—611nomaximal, Sylow
147C22×C61—21no—
148C22×C61—41no—
149C22×C61—42no—
150C22×C61—41no—
151C22×C61—82no—
152C22×C61C2×C22⋊C811yescharacteristic
153C2×D61—21no—
154C2×D61—41no—
155C2×D61—41no—
156C2×D61—42no—
157C2×D61—41no—
158C2×D61—82no—
159C2×D61—82no—
160C2×D61#32312yes—
161C4×C61—21no—
162C4×C61—41no—
163C4×C61C2×#9911yescharacteristic
164C4×C61C2×#10011yescharacteristic
165Dic61—21no—
166Dic61—41no—
167Dic61#32312yes—
168#2879—611no—
169C2×#349—612no—
170#1320—612no—
171#1320—612no—
172C2×C61—21no—
173C2×C61—41no—
174C2×C61—81no—
175C2×C61C2×#32311yescharacteristic
176D61—22no—
177D61—41no—
178D61—81no—
179C22×#41—611no—
180C22×M16—611no—
181C22×M16—611no—
182C2×M32—1222no—
183#408—1222no—
184#408—1222no—
185#407—611no—
186#407—611no—
187#407—611no—
188#407#692311yescharacteristic
189#349—614no—
190#349—1222no—
191C61#1143611yescharacteristic, complemented, Sylow
192C2×#41—1221no—
193C2×#41—1221no—
194C2×#41—1221no—
195C2×#41—1221no—
196C2×#41—2442no—
197C2×#41—2442no—
198C22×Q8—611no—
199C22×D4—611no—
200C22×D4—611no—
201C22×D4—611no—
202C23×C4—611no—
203C23×C4—611no—
204C23×C4C2×#692311yescharacteristic
205C2×M16—611no—
206C2×M16—612no—
207C2×M16—611no—
208C2×M16—1221no—
209C2×M16—2442no—
210C2×M16—2442no—
211C2×M16C2×#692311yescharacteristic
212C2×M16C2×#692311yescharacteristic
213C22×C8—612no—
214C22×C8—1221no—
215M32—1224no—
216M32—2442no—
217C2×C16—2442no—
218C22⋊C8—22no—
219C22⋊C8—1222no—
220C22⋊C8—1222no—
221C22⋊C8—1222no—
222C24—611no—
223C24—611no—
224C24D61⋊C811yescharacteristic
225#41—2442no—
226#41—2442no—
227#41—2444no—
228#41—2444no—
229#41—2442no—
230#41—2442no—
231C2×Q8—1221no—
232C2×Q8—1221no—
233C2×Q8—2442no—
234C2×D4—1221no—
235C2×D4—1221no—
236C2×D4—1221no—
237C2×D4—1221no—
238C2×D4—1221no—
239C2×D4—1221no—
240C2×D4—2442no—
241C2×D4—2442no—
242C2×D4—2442no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—611no—
248C22×C4—611no—
249C22×C4—611no—
250C22×C4—611no—
251C22×C4—611no—
252C22×C4—611no—
253C22×C4—1222no—
254C22×C4—1222no—
255C22×C4—1221no—
256C22×C4—1221no—
257C22×C4—2441no—
258C22×C4—2441no—
259C22×C4D61⋊C811yescharacteristic
260C22×C4C22×#692311yescharacteristic
261M16—41no—
262M16—41no—
263M16—2442no—
264M16—2442no—
265M16—2441no—
266M16—2441no—
267C2×C8—21no—
268C2×C8—612no—
269C2×C8—1222no—
270C2×C8—1222no—
271C2×C8—1221no—
272C2×C8—2442no—
273C2×C8C22⋊#692312yes—
274C16—2444no—
275C23—21no—
276C23—41no—
277C23—81no—
278C23—611no—
279C23—611no—
280C23—1222no—
281C23—1221no—
282C23—4881no—
283C23—4881no—
284C23—2441no—
285C23C2×D61⋊C811yescharacteristic
286Q8—2442no—
287Q8—2442no—
288D4—2442no—
289D4—2442no—
290D4—2442no—
291D4—2442no—
292D4—2442no—
293D4—2442no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—611no—
302C2×C4—612no—
303C2×C4—612no—
304C2×C4—611no—
305C2×C4—1222no—
306C2×C4—1222no—
307C2×C4—1222no—
308C2×C4—1221no—
309C2×C4—1221no—
310C2×C4—1221no—
311C2×C4—2442no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4C2×C22⋊#692311yescharacteristic
316C2×C4D61⋊M1611yescharacteristic
317C8—22no—
318C8—41no—
319C8—1222no—
320C8—2441no—
321C22—21no—
322C22—41no—
323C22—42no—
324C22—41no—
325C22—82no—
326C22—612no—
327C22—1221no—
328C22—4882no—
329C22—4882no—
330C22—2441no—
331C22—2442no—
332C22—2441no—
333C22—2441no—
334C22D61⋊(C22⋊C8)11yescharacteristic
335C4—21no—
336C4—41no—
337C4—612no—
338C4—1221no—
339C4—2441no—
340C4D61⋊#9911yescharacteristic
341C4D61⋊#10011yescharacteristic
342C2—21no—
343C2—41no—
344C2—81no—
345C2—1222no—
346C2—4881no—
347C2—2441no—
348C2D61⋊#32311yescharacteristic
subgroupG/H
centerC4D61⋊#99
commutatorC23×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#407C4
socleC2×C61C2×#323
prime radicalC22×#11758222C4
cosocleC2×#11758194C23
hypercenter#407#6923
hypocenterC61#11436
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #407

Complements

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

NH ≅ G/Ntype
1. D61⋊#349C2semidirect
2. #21460752C2semidirect
3. #21460751C2semidirect
191. C61#11436semidirect

Maximal subgroups

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

subgroupindex
D61⋊#3492
#214607522
#214607512
#214542622
#1143661