Finite Group Databasebeta

← #29600342

Subgroups — #29600342

Order 15616.

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

normal subgroups only

This group has 408 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#21447503C212yesmaximal, complemented
3#21443987C212yesmaximal, complemented
4#21443986C211yesmaximal, characteristic, complemented
5C61×#1410C211yesmaximal, characteristic, complemented
6C2×#14519798—22no—
7#16840313C2211yescharacteristic
8#16840310C411yescharacteristic
9#16840143—22no—
10#16840141C2211yescharacteristic
11#16840138C411yescharacteristic
12#16840135C411yescharacteristic
13#16839745—22no—
14C4×C8×D61C411yescharacteristic
15#16839561C2212yes—
16#16839558—22no—
17C61×#492C2211yescharacteristic
18C61×#356C2212yes—
19C2×#11758226D412yes—
20C2×#11758224—21no—
21#14519798—44no—
22#14519798—44no—
23C2×#11758202—22no—
24C2×#11758190—22no—
25C2×#11758199—22no—
26M16×D61—42no—
27C2×C8×D61—21no—
28#14519777C2×C411yescharacteristic
29#14519774C2×C411yescharacteristic
30#14519766D412yes—
31#14519764—21no—
32#14519752—41no—
33#14519748—41no—
34D4×Dic61—41no—
35#14519739—41no—
36C42×D61C2×C411yescharacteristic
37#14519696—22no—
38#14519693—42no—
39C4×#92418C2×C411yescharacteristic
40C8×Dic61—21no—
41C2×SD16×C61—22no—
42C4⋊Q8×C61C2311yescharacteristic
43#127×C61C2×C411yescharacteristic
44#101×C61—22no—
45C4×C8×C61C2×C411yescharacteristic
46C2×Q8×C61C2×D412yes—
47C2×D4×C61—21no—
48C2×D4×C61—41no—
49C2×C22⋊D61—41no—
50C22×Dic61—41no—
51#11758226—42no—
52#11758226—42no—
53#11758224—42no—
54#11758224—42no—
55C2×D244—22no—
56C2×D244—41no—
57C2×D244C22⋊C412yes—
58C2×C4×D61—21no—
59C2×C4×D61—22no—
60C2×C4×D61C22⋊C411yescharacteristic
61C2×Dic122—21no—
62SD16×C61—44no—
63M16×C61—42no—
64C2×C8×C61—21no—
65C4⋊C4×C61—41no—
66C42×C61C22×C411yescharacteristic
67C22⋊Dic61—42no—
68#11758202—44no—
69#11758199—44no—
70#11758198—42no—
71C4⋊Dic61—41no—
72C4×Dic61—21no—
73C4×Dic61C22⋊C411yescharacteristic
74#11758194—42no—
75C2×#92418—21no—
76#11758190—44no—
77#11758189—82no—
78C8×D61—22no—
79C8×D61—22no—
80C8×D61—82no—
81C23×C61—41no—
82C22×D61—22no—
83C22×D61#9911yescharacteristic
84Q8×C61—42no—
85D4×C61—42no—
86D4×C61—82no—
87C2×C4×C61—21no—
88C2×C4×C61—22no—
89C2×C4×C61C2×C22⋊C411yescharacteristic
90C22⋊D61—82no—
91C2×Dic61—21no—
92C2×Dic61—41no—
93C2×Dic61—81no—
94C2×Dic61—81no—
95C2×Dic61#9911yescharacteristic
96D244—42no—
97D244—82no—
98D244—82no—
99C4×D61—21no—
100C4×D61—21no—
101C4×D61—42no—
102C4×D61—42no—
103C4×D61—42no—
104Dic122—42no—
105C8×C61—22no—
106C8×C61—82no—
107#92418—22no—
108#92418—82no—
109#13589—611nomaximal, Sylow
110C22×C61—82no—
111C22×C61—81no—
112C22×C61C2×#9911yescharacteristic
113C2×D61—21no—
114C2×D61—41no—
115C2×D61—42no—
116C2×D61—82no—
117C4×C61—22no—
118C4×C61—21no—
119C4×C61—42no—
120Dic61—41no—
121Dic61—81no—
122Dic61#35311yescharacteristic
123Dic61#35311yescharacteristic
124#3147—611no—
125#2132—612no—
126#1411—612no—
127#1410—611no—
128#1410D6111yescharacteristic, complemented
129C2×C61—21no—
130C2×C61—161no—
131C2×C61C2×#35311yescharacteristic
132D61—41no—
133D61—82no—
134C2×#135—1222no—
135#532—611no—
136#532—611no—
137#495—1222no—
138#494—611no—
139#493—611no—
140#492—611no—
141#492C2×D6111yescharacteristic
142#413—1222no—
143C2×C4×C8—611no—
144#356—612no—
145#356C2×D6112yes—
146#355—1222no—
147C61#1358911yescharacteristic, complemented, Sylow
148C2×#41—612no—
149C2×#41—1221no—
150#135—2444no—
151#135—2444no—
152C2×Q16—1222no—
153C2×SD16—22no—
154C2×SD16—1222no—
155C2×D8—1222no—
156C2×M16—2442no—
157C22×C8—1221no—
158C4⋊Q8—611no—
159C4⋊Q8C22×D6111yescharacteristic
160#127—611no—
161#127C4×D6111yescharacteristic
162#124—612no—
163#124—1221no—
164#121—2441no—
165#121—2441no—
166C4×D4—2441no—
167C4×D4—2441no—
168C2×C42—611no—
169#101—22no—
170#101—1222no—
171#100—2442no—
172C4×C8—611no—
173C4×C8—1221no—
174C4×C8C4×D6111yescharacteristic
175#41—2442no—
176#41—2442no—
177#41—2442no—
178#41—2442no—
179C2×Q8—1221no—
180C2×Q8D4×D6112yes—
181C2×D4—21no—
182C2×D4—41no—
183C2×D4—612no—
184C2×D4—1222no—
185C2×D4—2441no—
186C2×D4—2441no—
187C22×C4—611no—
188C22×C4—1222no—
189C22×C4—1221no—
190C22×C4—2441no—
191Q16—2444no—
192SD16—44no—
193SD16—2444no—
194D8—2444no—
195M16—42no—
196M16—4882no—
197M16—2442no—
198C2×C8—21no—
199C2×C8—1222no—
200C2×C8—1222no—
201C2×C8—1221no—
202C2×C8—4882no—
203C4⋊C4—41no—
204C4⋊C4—2441no—
205C22⋊C4—2442no—
206C22⋊C4—2442no—
207C42—611no—
208C42—1221no—
209C42C2×C4×D6111yescharacteristic
210C23—41no—
211C23—611no—
212C23—1222no—
213Q8—42no—
214Q8—2442no—
215D4—42no—
216D4—82no—
217D4—4882no—
218D4—4882no—
219D4—4882no—
220D4—2442no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—611no—
224C2×C4—1221no—
225C2×C4—1221no—
226C2×C4—1221no—
227C2×C4—4881no—
228C2×C4—4881no—
229C2×C4—2442no—
230C2×C4—2442no—
231C2×C4—2442no—
232C2×C4—2441no—
233C2×C4C22⋊C4×D6111yescharacteristic
234C8—22no—
235C8—82no—
236C8—1222no—
237C8—4882no—
238C22—82no—
239C22—81no—
240C22—1221no—
241C22—4882no—
242C22—2441no—
243C22—2442no—
244C22#99×D6111yescharacteristic
245C4—22no—
246C4—21no—
247C4—42no—
248C4—611no—
249C4—611no—
250C4—4881no—
251C4—2441no—
252C2—21no—
253C2—161no—
254C2—4882no—
255C2—2441no—
256C2#353×D6111yescharacteristic
subgroupG/H
centerC2#353×D61
commutatorC42×C61C22×C4
FrattiniC4⋊Q8C22×D61
FittingC61×#1410C2
socleC2×C61C2×#353
prime radical#21456242C2
cosocleC4⋊Q8×C61C23
hypercenter#1410D61
hypocenterC61#13589
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #1410

Complements

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

NH ≅ G/Ntype
2. #21447503C2semidirect
3. #21443987C2semidirect
4. #21443986C2semidirect
5. C61×#1410C2semidirect
128. #1410D61semidirect
147. C61#13589semidirect

Maximal subgroups

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

subgroupindex
#214562422
#214475032
#214439872
#214439862
C61×#14102
#1358961