Finite Group Databasebeta

← #30595207

Subgroups — #30595207

Order 15616.

286 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 50 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
1C2×#16840671C211yesmaximal, characteristic
2C2×#16840669C211yesmaximal, characteristic
3#21446628C211yesmaximal, characteristic
4C2×#14519838—21no—
5C2×#14519836—21no—
6#16840671—21no—
7#16840671C412yescomplemented
8#16840669—21no—
9#16840669C412yescomplemented
10C2×#14519765C2211yescharacteristic
11C2×#14519700—21no—
12C2×#14519697—21no—
13C2×C4×#92418C411yescharacteristic, complemented
14C61×#382C411yescharacteristic, complemented
15C2×#11758217—41no—
16C2×C22⋊#6923—41no—
17#14519838—22no—
18#14519838—41no—
19#14519836—22no—
20#14519836—41no—
21C22×Dic122D411yescharacteristic
22C22×D244D411yescharacteristic
23C22×#92418—21no—
24#14519765—41no—
25#14519765C2×C411yescharacteristic
26#14519765C2×C411yescharacteristic
27#14519765D411yescharacteristic
28#14519765Q811yescharacteristic
29C2×#11758198—41no—
30#14519700—42no—
31#14519697—42no—
32C4×#92418—41no—
33C2×C4⋊C4×C61—21no—
34C2×C42×C61C2×C411yescharacteristic
35#95×C61—41no—
36C23×D61—21no—
37C22×#6923—81no—
38C22×C4×C61—21no—
39C22×C4×C61—41no—
40C22×C4×C61C22⋊C411yescharacteristic
41C22×Dic61—21no—
42C2×D244—21no—
43C2×D244—41no—
44C2×D244—41no—
45C2×D244C22⋊C411yescharacteristic
46C2×D244C4⋊C411yescharacteristic
47C2×Dic122—21no—
48C2×Dic122—41no—
49C2×Dic122—41no—
50C2×Dic122C22⋊C411yescharacteristic
51C2×Dic122C4⋊C411yescharacteristic
52C22⋊#6923—42no—
53C22⋊#6923—81no—
54#11758217—42no—
55#11758217—81no—
56C2×#92420—81no—
57C4⋊C4×C61—41no—
58C4⋊C4×C61—41no—
59C42×C61—21no—
60C42×C61C4211yescharacteristic
61C42×C61C4⋊C411yescharacteristic
62#11758198—41no—
63#11758198—41no—
64#11758198—41no—
65#11758198—41no—
66C2×#92418—22no—
67C2×#92418—41no—
68C2×#92418—41no—
69C23×C61#9911yescharacteristic
70C22×D61—21no—
71C22×D61—21no—
72C22×D61—41no—
73C22×D61—41no—
74C22×D61—81no—
75C22×D61—81no—
76C2×#6923—82no—
77C2×#6923—161no—
78C2×#6923—161no—
79C2×C4×C61—21no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—21no—
83C2×C4×C61—41no—
84C2×C4×C61—41no—
85C2×C4×C61—81no—
86C2×C4×C61—81no—
87C2×C4×C61—81no—
88C2×C4×C61#9511yescharacteristic
89C2×C4×C61#9911yescharacteristic
90C2×Dic61—21no—
91C2×Dic61—21no—
92C2×Dic61—41no—
93C2×Dic61—41no—
94D244—41no—
95D244—41no—
96D244—81no—
97Dic122—41no—
98Dic122—41no—
99Dic122—81no—
100#92420—82no—
101#92418—42no—
102#7502—611nomaximal, Sylow
103C22×C61—21no—
104C22×C61—21no—
105C22×C61#34211yescharacteristic
106C22×C61#35311yescharacteristic
107C22×C61#35411yescharacteristic
108C2×D61—41no—
109C2×D61—41no—
110C2×D61—81no—
111C2×D61—81no—
112C2×D61—81no—
113C2×D61—81no—
114C2×D61—161no—
115#6923—162no—
116C4×C61—21no—
117C4×C61—21no—
118C4×C61—41no—
119C4×C61—41no—
120C4×C61—161no—
121Dic61—41no—
122Dic61—41no—
123C2×#355—611no—
124C2×#354—611no—
125#1958—611no—
126C2×C61—21no—
127C2×C61—21no—
128C2×C61#139311yescharacteristic
129C2×C61#141411yescharacteristic
130C2×C61#141411yescharacteristic
131D61—81no—
132D61—81no—
133C2×#124—611no—
134C2×#103—1221no—
135C2×#102—1221no—
136C2×#101—1221no—
137C2×#99—1221no—
138C2×C4×C8—611no—
139#382#692311yescharacteristic, complemented
140#355—612no—
141#355—1221no—
142#354—612no—
143#354—1221no—
144C61#750211yescharacteristic, complemented, Sylow
145C22×Q8—611no—
146C22×D4—611no—
147C2×M16—2441no—
148C22×C8—1221no—
149#124—611no—
150#124—611no—
151#124—611no—
152#124—611no—
153#124—2441no—
154C2×C4⋊C4—21no—
155C2×C22⋊C4—2441no—
156C2×C22⋊C4—2441no—
157C2×C42C2×#692311yescharacteristic
158#103—2442no—
159#102—2442no—
160#101—1222no—
161#101—2441no—
162#99—1222no—
163#99—2441no—
164C4×C8—2441no—
165#95—41no—
166C24—1221no—
167C2×Q8—611no—
168C2×Q8—611no—
169C2×Q8—1221no—
170C2×Q8—2441no—
171C2×Q8—2441no—
172C2×D4—611no—
173C2×D4—611no—
174C2×D4—1221no—
175C2×D4—2441no—
176C2×D4—2441no—
177C22×C4—21no—
178C22×C4—41no—
179C22×C4—1221no—
180C22×C4—4881no—
181C22×C4C22⋊#692311yescharacteristic
182M16—4881no—
183M16—2442no—
184C2×C8—1222no—
185C2×C8—4881no—
186C2×C8—2441no—
187C2×C8—2441no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C22⋊C4—4881no—
191C22⋊C4—2441no—
192C22⋊C4—2441no—
193C22⋊C4—2441no—
194C22⋊C4—2441no—
195C22⋊C4—2442no—
196C42—21no—
197C42C4×#692311yescharacteristic
198C42C4⋊#692311yescharacteristic
199C23—1221no—
200C23—1221no—
201C23—4881no—
202C23—4881no—
203C23—2441no—
204C23—2441no—
205C23#1451983511yescharacteristic
206Q8—4881no—
207Q8—2441no—
208Q8—2441no—
209D4—4881no—
210D4—2441no—
211D4—2441no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—1221no—
222C2×C4—1221no—
223C2×C4—9761no—
224C2×C4—9761no—
225C2×C4—4882no—
226C2×C4—2441no—
227C2×C4—2441no—
228C2×C4(C2×#6923)⋊C411yescharacteristic
229C2×C4#1451983611yescharacteristic
230C8—4882no—
231C8—2442no—
232C22—21no—
233C22—21no—
234C22—9761no—
235C22—4881no—
236C22—4881no—
237C22—4881no—
238C22—4881no—
239C22—2441no—
240C22—2441no—
241C22#1684064011yescharacteristic
242C22#1684066611yescharacteristic
243C22#1684066811yescharacteristic
244C4—21no—
245C4—21no—
246C4—41no—
247C4—41no—
248C4—161no—
249C4—9762no—
250C4—2441no—
251C4—2441no—
252C2—21no—
253C2—21no—
254C2—4881no—
255C2—4881no—
256C2#2146091411yescharacteristic
257C2#2146096411yescharacteristic
258C2#2146096511yescharacteristic
subgroupG/H
centerC22#16840668
commutatorC42×C61C42
FrattiniC2×C42C2×#6923
FittingC61×#382C4
socleC22×C61#354
prime radicalC22×D244D4
cosocleC2×#14519765C22
hypercenter#382#6923
hypocenterC61#7502
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
7. #16840671C4semidirect
9. #16840669C4semidirect
13. C2×C4×#92418C4semidirect
14. C61×#382C4semidirect
139. #382#6923semidirect
144. C61#7502semidirect

Maximal subgroups

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

subgroupindex
C2×#168406712
C2×#168406692
#214466282
#750261