Finite Group Databasebeta

← #29600324

Subgroups — #29600324

Order 15616.

416 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 59 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#21456242C211yesmaximal, characteristic
2#21447481C212yesmaximal, complemented
3#21443984C211yesmaximal, characteristic, complemented
4#21443981C212yesmaximal, complemented
5C61×#1409C211yesmaximal, characteristic, complemented
6C2×#14519798—22no—
7#16840313C411yescharacteristic, complemented
8#16840310C2211yescharacteristic
9#16840143—22no—
10#16840141C411yescharacteristic, complemented
11#16840138C2211yescharacteristic
12#16840135C411yescharacteristic, complemented
13#16839734—22no—
14C4×C8×D61C411yescharacteristic, complemented
15#16839556—22no—
16#16839555C2212yes—
17C61×#492C2211yescharacteristic
18C61×#353C2212yes—
19C2×#11758226—21no—
20C2×#11758224D412yes—
21#14519798—44no—
22#14519798—44no—
23C2×#11758202—22no—
24C2×#11758190—22no—
25C2×#11758199—22no—
26C2×C8×D61—21no—
27#14519777C2×C411yescharacteristic
28#14519774C2×C411yescharacteristic
29#14519766—21no—
30#14519764D412yes—
31#14519752—41no—
32#14519748—41no—
33D4×Dic61—41no—
34#14519739—41no—
35#14519733—42no—
36C42×D61C2×C411yescharacteristic
37#14519692—22no—
38#14519691—42no—
39C4×#92418C2×C411yescharacteristic
40C8×Dic61—21no—
41C2×SD16×C61—22no—
42C4⋊Q8×C61C2×C411yescharacteristic
43#127×C61C2311yescharacteristic
44#99×C61—22no—
45C4×C8×C61C2×C411yescharacteristic
46C2×Q8×C61—21no—
47C2×D4×C61—41no—
48C2×D4×C61C2×D412yes—
49C2×C22⋊D61—22no—
50C22×Dic61—22no—
51#11758226—42no—
52#11758226—42no—
53#11758224—42no—
54#11758224—42no—
55C2×D244—21no—
56C2×D244—41no—
57C2×D244—41no—
58C2×C4×D61—21no—
59C2×C4×D61—41no—
60C2×C4×D61C22⋊C411yescharacteristic
61C2×Dic122C22⋊C412yes—
62SD16×C61—44no—
63C2×C8×C61—21no—
64C4⋊C4×C61—41no—
65C22⋊C4×C61—42no—
66C42×C61C22×C411yescharacteristic
67C22⋊Dic61—42no—
68C22⋊Dic61—42no—
69#11758202—44no—
70#11758199—44no—
71#11758198—42no—
72C4⋊Dic61—41no—
73#11758196—82no—
74C4×Dic61—21no—
75C4×Dic61—82no—
76C4×Dic61C22⋊C411yescharacteristic
77C2×#92418—21no—
78#11758190—44no—
79C8×D61—22no—
80C8×D61—22no—
81C23×C61—22no—
82C22×D61—41no—
83C22×D61#9911yescharacteristic
84Q8×C61—42no—
85D4×C61—42no—
86D4×C61—82no—
87C2×C4×C61—21no—
88C2×C4×C61—41no—
89C2×C4×C61—82no—
90C2×C4×C61C2×C22⋊C411yescharacteristic
91C22⋊D61—82no—
92C2×Dic61—21no—
93C2×Dic61—22no—
94C2×Dic61—42no—
95C2×Dic61—42no—
96C2×Dic61—82no—
97C2×Dic61#9911yescharacteristic
98D244—42no—
99D244—82no—
100D244—82no—
101C4×D61—21no—
102C4×D61—21no—
103C4×D61—42no—
104C4×D61—81no—
105C4×D61—81no—
106Dic122—42no—
107C8×C61—22no—
108#92418—22no—
109#13583—611nomaximal, Sylow
110C22×C61—42no—
111C22×C61—82no—
112C22×C61C2×#9911yescharacteristic
113C2×D61—21no—
114C2×D61—41no—
115C2×D61—82no—
116C2×D61—81no—
117C4×C61—22no—
118C4×C61—21no—
119C4×C61—81no—
120C4×C61—162no—
121Dic61—41no—
122Dic61—42no—
123Dic61—162no—
124Dic61#35311yescharacteristic
125Dic61#35311yescharacteristic
126#3147—611no—
127#2126—612no—
128#1409—611no—
129#1409D6111yescharacteristic, complemented
130#1408—612no—
131C2×C61—21no—
132C2×C61—82no—
133C2×C61C2×#35311yescharacteristic
134D61—41no—
135D61—161no—
136C2×#135—1222no—
137#532—611no—
138#532—611no—
139#495—1222no—
140#494—611no—
141#493—611no—
142#492—611no—
143#492C2×D6111yescharacteristic
144#410—1222no—
145C2×C4×C8—611no—
146#354—1222no—
147#353—612no—
148#353C2×D6112yes—
149C61#1358311yescharacteristic, complemented, Sylow
150C2×#41—612no—
151C2×#41—1221no—
152#135—2444no—
153#135—2444no—
154C2×Q16—1222no—
155C2×SD16—22no—
156C2×SD16—1222no—
157C2×D8—1222no—
158C22×C8—1221no—
159C4⋊Q8—611no—
160C4⋊Q8C4×D6111yescharacteristic
161#127—611no—
162#127C22×D6111yescharacteristic
163#124—612no—
164#124—1221no—
165#121—2441no—
166#121—2441no—
167C4×D4—2441no—
168C4×D4—2441no—
169#117—2442no—
170C2×C42—611no—
171#99—22no—
172#99—1222no—
173#99—2442no—
174C4×C8—611no—
175C4×C8—1221no—
176C4×C8C4×D6111yescharacteristic
177#41—2442no—
178#41—2442no—
179#41—2442no—
180#41—2442no—
181C2×Q8—21no—
182C2×Q8—612no—
183C2×D4—41no—
184C2×D4—1222no—
185C2×D4—1221no—
186C2×D4—2441no—
187C2×D4—2441no—
188C2×D4D4×D6112yes—
189C22×C4—611no—
190C22×C4—1221no—
191C22×C4—1222no—
192C22×C4—2441no—
193Q16—2444no—
194SD16—44no—
195SD16—2444no—
196D8—2444no—
197C2×C8—21no—
198C2×C8—1222no—
199C2×C8—1222no—
200C2×C8—1221no—
201C4⋊C4—41no—
202C4⋊C4—4882no—
203C4⋊C4—2441no—
204C22⋊C4—42no—
205C22⋊C4—2442no—
206C22⋊C4—2442no—
207C22⋊C4—2442no—
208C42—611no—
209C42—1221no—
210C42—4882no—
211C42C2×C4×D6111yescharacteristic
212C23—22no—
213C23—611no—
214C23—2441no—
215Q8—42no—
216Q8—2442no—
217D4—42no—
218D4—82no—
219D4—4882no—
220D4—4882no—
221D4—4882no—
222D4—2442no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—82no—
226C2×C4—611no—
227C2×C4—1221no—
228C2×C4—1221no—
229C2×C4—1221no—
230C2×C4—1222no—
231C2×C4—4881no—
232C2×C4—4881no—
233C2×C4—4882no—
234C2×C4—2442no—
235C2×C4—2442no—
236C2×C4—2442no—
237C2×C4C22⋊C4×D6111yescharacteristic
238C8—22no—
239C8—1222no—
240C22—42no—
241C22—82no—
242C22—1221no—
243C22—4882no—
244C22—4881no—
245C22—2441no—
246C22#99×D6111yescharacteristic
247C4—22no—
248C4—21no—
249C4—81no—
250C4—162no—
251C4—611no—
252C4—611no—
253C4—9762no—
254C4—2441no—
255C4—2442no—
256C2—21no—
257C2—82no—
258C2—9761no—
259C2—2441no—
260C2#353×D6111yescharacteristic
subgroupG/H
centerC2#353×D61
commutatorC42×C61C22×C4
Frattini#127C22×D61
FittingC61×#1409C2
socleC2×C61C2×#353
prime radical#21456242C2
cosocle#127×C61C23
hypercenter#1409D61
hypocenterC61#13583
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #1409

Complements

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

NH ≅ G/Ntype
2. #21447481C2semidirect
3. #21443984C2semidirect
4. #21443981C2semidirect
5. C61×#1409C2semidirect
7. #16840313C4semidirect
10. #16840141C4semidirect
12. #16840135C4semidirect
14. C4×C8×D61C4semidirect
129. #1409D61semidirect
149. C61#13583semidirect

Maximal subgroups

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

subgroupindex
#214562422
#214474812
#214439842
#214439812
C61×#14092
#1358361