Finite Group Databasebeta

← #30594561

Subgroups — #30594561

Order 15616.

340 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 64 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⋊#349C211yesmaximal, characteristic
2#21461697C211yesmaximal, characteristic
3H(3,ℤ/4)×D61C211yesmaximal, characteristic
4D61⋊M32—21no—
5C2×#14519846C411yescharacteristic, complemented
6C2×#14519845C411yescharacteristic, complemented
7#16840704—21no—
8#16840659C412yescomplemented
9#16840658—21no—
10#16840636—21no—
11C2×M16×D61C2211yescharacteristic
12#117×D61—21no—
13#16839519C411yescharacteristic, complemented
14#16839518—21no—
15C61×H(3,ℤ/4)C411yescharacteristic, complemented
16C2×C4⋊#6923—21no—
17C2×C4⋊#6923—21no—
18C2×C22⋊#6923—41no—
19#14519852—41no—
20#14519851—22no—
21D61⋊C16—41no—
22#14519846—41no—
23#14519845—41no—
24(C2×#6923)⋊C4—41no—
25C22×C4×D61D411yescharacteristic
26C2×#11758194D411yescharacteristic
27C2×#11758189C2×C411yescharacteristic
28C2×#11758189Q811yescharacteristic
29M16×D61—41no—
30M16×D61—41no—
31C2×C8×D61C2×C411yescharacteristic
32C2×C8×D61D411yescharacteristic
33#14519736—21no—
34#14519735—41no—
35#14519734—22no—
36#14519733—41no—
37C4⋊C4×D61—41no—
38C22⋊C4×D61—41no—
39C42×D61—22no—
40C2×M16×C61C2×C411yescharacteristic
41#117×C61—21no—
42C23×D61D811yescharacteristic
43C22×#6923—41no—
44C22×#6923—41no—
45C22×C4×C61C22⋊C411yescharacteristic
46C22×Dic61SD1611yescharacteristic
47C2×C4×D61—21no—
48C2×C4×D61—21no—
49C2×C4×D61—21no—
50C2×C4×D61—21no—
51C2×C4×D61—42no—
52C2×C4×D61C22⋊C411yescharacteristic
53C2×C4×D61C4⋊C411yescharacteristic
54C2×C4×D61SD1611yescharacteristic
55C2×C4×D61Q1611yescharacteristic
56C22⋊#6923—82no—
57C4⋊#6923—41no—
58C4⋊#6923—41no—
59C4⋊#6923—41no—
60C4⋊#6923—41no—
61M16×C61—41no—
62C2×C8×C61C4211yescharacteristic
63C2×C8×C61C4⋊C411yescharacteristic
64C4⋊C4×C61—41no—
65C22⋊C4×C61—41no—
66C42×C61—22no—
67C22⋊Dic61—41no—
68#11758198—42no—
69C4⋊Dic61—41no—
70#11758196—42no—
71C4×Dic61—22no—
72C4×Dic61—42no—
73#11758194—41no—
74C2×#92418C22⋊C411yescharacteristic
75C2×#92418C4⋊C411yescharacteristic
76#11758189—41no—
77#11758189—41no—
78C8×D61—21no—
79C8×D61—21no—
80C8×D61—21no—
81C8×D61—21no—
82#11758187—42no—
83C23×C61#10211yescharacteristic
84C22×D61—21no—
85C22×D61—21no—
86C22×D61—41no—
87C22×D61—41no—
88C22×D61#10211yescharacteristic
89C22×D61#10711yescharacteristic
90C2×#6923—81no—
91C2×#6923—81no—
92C2×#6923—81no—
93C2×#6923—81no—
94C2×#6923—81no—
95C2×#6923—81no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—42no—
99C2×C4×C61#9511yescharacteristic
100C2×C4×C61#10311yescharacteristic
101C2×Dic61—21no—
102C2×Dic61—21no—
103C2×Dic61—42no—
104C2×Dic61#10311yescharacteristic
105C2×Dic61#10611yescharacteristic
106C4×D61—21no—
107C4×D61—21no—
108C4×D61—21no—
109C4×D61—21no—
110C4×D61—21no—
111C4×D61—21no—
112C4×D61—82no—
113C4×D61—82no—
114C4×D61#9911yescharacteristic
115C4×D61#9911yescharacteristic
116C4×D61#10411yescharacteristic
117C4×D61#10411yescharacteristic
118C8×C61—21no—
119C8×C61—21no—
120#92418—21no—
121#92418—21no—
122#7254—611nomaximal, Sylow
123C22×C61—21no—
124C22×C61—41no—
125C22×C61#34011yescharacteristic
126C2×D61—21no—
127C2×D61—21no—
128C2×D61—21no—
129C2×D61—41no—
130C2×D61—41no—
131C2×D61—41no—
132C2×D61—81no—
133C2×D61#32711yescharacteristic
134C2×D61#32711yescharacteristic
135#6923—161no—
136#6923—161no—
137C4×C61—21no—
138C4×C61—82no—
139C4×C61#33911yescharacteristic
140C4×C61#34211yescharacteristic
141Dic61—21no—
142Dic61—82no—
143Dic61#32811yescharacteristic
144Dic61#32811yescharacteristic
145C2×#349—611no—
146#1809—611no—
147C2×H(3,ℤ/4)—611no—
148C2×C61—21no—
149C2×C61—41no—
150C2×C61#129411yescharacteristic
151D61—21no—
152D61—41no—
153D61#133911yescharacteristic
154D61#134011yescharacteristic
155C22×M16—611no—
156C2×#117—1221no—
157C2×M32—1221no—
158C2×#107—611no—
159C2×#106—611no—
160#380—1221no—
161#349—612no—
162#349—1221no—
163#340—1221no—
164H(3,ℤ/4)—611no—
165H(3,ℤ/4)—1221no—
166H(3,ℤ/4)#692311yescharacteristic, complemented
167C61#725411yescharacteristic, complemented, Sylow
168C23×C4—611no—
169C2×M16—611no—
170C2×M16—611no—
171C2×M16—611no—
172C2×M16—2441no—
173C2×M16—2441no—
174C2×M16C2×#692311yescharacteristic
175C22×C8—611no—
176C22×C8—611no—
177#117—21no—
178#117—1222no—
179#117—1221no—
180#117—2441no—
181#117—2441no—
182C2×C4⋊C4—1221no—
183C2×C4⋊C4—1221no—
184C2×C4⋊C4—2441no—
185C2×C22⋊C4—2441no—
186C2×C22⋊C4—2441no—
187C2×C42—1222no—
188M32—1222no—
189M32—2441no—
190C2×C16—2441no—
191#107—2441no—
192#106—2441no—
193#95—2441no—
194C24—611no—
195C22×C4—611no—
196C22×C4—611no—
197C22×C4—611no—
198C22×C4—611no—
199C22×C4—611no—
200C22×C4—1221no—
201C22×C4—1221no—
202C22×C4—1221no—
203C22×C4—1221no—
204C22×C4—2442no—
205C22×C4—2441no—
206C22×C4—2441no—
207C22×C4C22⋊#692311yescharacteristic
208M16—41no—
209M16—2441no—
210M16—2441no—
211M16—2441no—
212C2×C8—611no—
213C2×C8—611no—
214C2×C8—1221no—
215C2×C8—1221no—
216C2×C8—1221no—
217C2×C8—1221no—
218C2×C8C4×#692311yescharacteristic
219C2×C8C4⋊#692311yescharacteristic
220C4⋊C4—41no—
221C4⋊C4—2441no—
222C4⋊C4—2442no—
223C4⋊C4—2441no—
224C4⋊C4—2441no—
225C4⋊C4—2441no—
226C4⋊C4—2441no—
227C22⋊C4—41no—
228C22⋊C4—4882no—
229C22⋊C4—2442no—
230C22⋊C4—2441no—
231C42—22no—
232C42—1222no—
233C42—2442no—
234C16—2442no—
235C23—611no—
236C23—611no—
237C23—1221no—
238C23—1221no—
239C23—2441no—
240C23—2441no—
241C23#1451983911yescharacteristic
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—42no—
245C2×C4—611no—
246C2×C4—611no—
247C2×C4—611no—
248C2×C4—611no—
249C2×C4—611no—
250C2×C4—611no—
251C2×C4—1221no—
252C2×C4—1221no—
253C2×C4—1221no—
254C2×C4—1221no—
255C2×C4—1221no—
256C2×C4—1221no—
257C2×C4—1221no—
258C2×C4—1221no—
259C2×C4—4882no—
260C2×C4—4882no—
261C2×C4—4881no—
262C2×C4—4881no—
263C2×C4—4881no—
264C2×C4—4881no—
265C2×C4—4881no—
266C2×C4—4881no—
267C2×C4—2442no—
268C2×C4(C2×#6923)⋊C411yescharacteristic
269C2×C4#1451984011yescharacteristic
270C8—21no—
271C8—21no—
272C8—1221no—
273C8—1221no—
274C22—21no—
275C22—41no—
276C22—611no—
277C22—611no—
278C22—1221no—
279C22—1221no—
280C22—1221no—
281C22—4881no—
282C22—2441no—
283C22—2441no—
284C22—2441no—
285C22#1684063511yescharacteristic
286C4—21no—
287C4—82no—
288C4—611no—
289C4—611no—
290C4—1221no—
291C4—9761no—
292C4—9761no—
293C4—4882no—
294C4#1684063311yescharacteristic
295C4#1684063911yescharacteristic
296C2—21no—
297C2—41no—
298C2—611no—
299C2—611no—
300C2—1221no—
301C2—2441no—
302C2#2146069311yescharacteristic
subgroupG/H
centerC2#21460693
commutatorC2×C8×C61C42
FrattiniC2×M16C2×#6923
FittingC61×H(3,ℤ/4)C4
socleC2×C61#1294
prime radicalC23×D61D8
cosocleC2×M16×D61C22
hypercenterH(3,ℤ/4)#6923
hypocenterC61#7254
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
5. C2×#14519846C4semidirect
6. C2×#14519845C4semidirect
8. #16840659C4semidirect
13. #16839519C4semidirect
15. C61×H(3,ℤ/4)C4semidirect
166. H(3,ℤ/4)#6923semidirect
167. C61#7254semidirect

Maximal subgroups

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

subgroupindex
D61⋊#3492
#214616972
H(3,ℤ/4)×D612
#725461