Finite Group Databasebeta

← #29568273

Subgroups — #29568273

Order 15616.

520 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 107 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#21448362C211yesmaximal, characteristic
2#21445671C212yesmaximal, complemented
3#21443645C211yesmaximal, characteristic, complemented
4#21443639C212yesmaximal
5C61×#1272C211yesmaximal, characteristic, complemented
6C2×#14519766C411yescharacteristic
7C2×Q8×Dic61C411yescharacteristic
8C2×#14519739C411yescharacteristic
9C2×#14519706—22no—
10C2×#14519687—22no—
11#16839664C2211yescharacteristic
12#16839654—22no—
13#16839627C411yescharacteristic
14#16839609—22no—
15C42×Dic61C2211yescharacteristic
16#16839517C2212yes—
17C61×#391C2211yescharacteristic
18C61×#336C2212yes—
19C22×D244C2×C411yescharacteristic
20C22×C4×D61—21no—
21C22×#92418—22no—
22#14519766—22no—
23#14519766—22no—
24Q8×Dic61—42no—
25#14519739—42no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×#11758198—22no—
29C2×#11758198—22no—
30C2×#11758198—21no—
31C2×C4×Dic61—21no—
32C2×C4×Dic61C2×C411yescharacteristic
33C2×C4×Dic61C2×C411yescharacteristic
34C2×C4×Dic61D412yes—
35C2×C4×Dic61D412yes—
36#14519706—28no—
37#14519687—44no—
38#14519684—22no—
39C22×Q8×C61C2×C411yescharacteristic
40C22×C8×C61—22no—
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C42×C61C2311yescharacteristic
43#95×C61—22no—
44C23×D61—21no—
45C2×Q8×C61—21no—
46C2×Q8×C61—41no—
47C2×Q8×C61—41no—
48C2×Q8×C61M1612yes—
49C22×C4×C61—21no—
50C22×C4×C61—21no—
51C22×C4×C61C22×C411yescharacteristic
52C22×C4×C61C2×D412yes—
53C22×Dic61—21no—
54C22×Dic61C22⋊C412yes—
55C22×Dic61C22⋊C411yescharacteristic
56C22×Dic61C22⋊C411yescharacteristic
57C2×D244—21no—
58C2×D244—41no—
59C2×D244—41no—
60C2×D244M1612yes—
61C2×C4×D61—42no—
62C2×C4×D61—44no—
63C2×C8×C61—24no—
64C2×C8×C61—42no—
65C2×C8×C61—42no—
66C4⋊C4×C61—41no—
67C42×C61—41no—
68#11758198—28no—
69#11758198—42no—
70#11758198—42no—
71#11758198—42no—
72C4⋊Dic61—42no—
73C4⋊Dic61—41no—
74C4×Dic61—21no—
75C4×Dic61—21no—
76C4×Dic61—22no—
77C4×Dic61—22no—
78C4×Dic61—22no—
79C4×Dic61—24no—
80C4×Dic61—42no—
81C2×#92418—24no—
82C2×#92418—42no—
83C2×#92418—42no—
84C23×C61C2×C22⋊C411yescharacteristic
85C22×D61—22no—
86C22×D61—41no—
87C22×D61—41no—
88C22×D61—42no—
89C22×D61—42no—
90Q8×C61—42no—
91Q8×C61—42no—
92C2×C4×C61—22no—
93C2×C4×C61—21no—
94C2×C4×C61—21no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—41no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—42no—
101C2×C4×C61C2×M1611yescharacteristic
102C2×C4×C61C2×M1611yescharacteristic
103C2×C4×C61AGL(1,ℤ/8)12yes—
104C2×C4×C61#13712yes—
105C2×Dic61—22no—
106C2×Dic61—21no—
107C2×Dic61—22no—
108C2×Dic61—22no—
109C2×Dic61—22no—
110C2×Dic61—41no—
111C2×Dic61—42no—
112C2×Dic61#10012yes—
113C2×Dic61#10414yes—
114D244—42no—
115D244—42no—
116C4×D61—84no—
117C8×C61—44no—
118#92418—44no—
119#7635—611nomaximal, Sylow
120C22×C61#40711yescharacteristic
121C22×C61#41011yescharacteristic
122C22×C61C2×#10011yescharacteristic
123C22×C61#41712yes—
124C22×C61C2×#10412yes—
125C2×D61—42no—
126C2×D61—44no—
127C2×D61—81no—
128C2×D61—82no—
129C2×D61—81no—
130C4×C61—21no—
131C4×C61—21no—
132C4×C61—22no—
133C4×C61—22no—
134C4×C61—42no—
135C4×C61—81no—
136Dic61—41no—
137Dic61—42no—
138Dic61—81no—
139Dic61#32914yes—
140C4×#124—611no—
141#1767—612no—
142#1272—611no—
143#1272D6111yescharacteristic, complemented
144#1270—612no—
145C2×C61#146611yescharacteristic
146C2×C61#148912yes—
147C2×C61#150912yes—
148C2×C61C2×#32911yescharacteristic
149C2×C61#156011yescharacteristic
150D61—82no—
151C2×#124—611no—
152C2×C4×Q8—611no—
153C2×C4×D4—611no—
154C2×C4⋊C8—1222no—
155C2×#97—1222no—
156#391C2×D6111yescharacteristic
157#390—611no—
158#388—1222no—
159#382—611no—
160C4×C22⋊C4—1222no—
161C43—611no—
162#336—612no—
163#336C2×D6112yes—
164C61#763511yescharacteristic, complemented, Sylow
165C22×Q8C4×D6111yescharacteristic
166C22×D4—611no—
167C23×C4—1221no—
168C22×C8—22no—
169C22×C8—1222no—
170#124—1222no—
171#124—1222no—
172C4×Q8—2442no—
173C4×D4—2442no—
174C2×C4⋊C4—611no—
175C2×C4⋊C4—1221no—
176C2×C4⋊C4C4×D6111yescharacteristic
177C2×C22⋊C4—1222no—
178C2×C22⋊C4—1221no—
179C2×C22⋊C4—1222no—
180C2×C42—611no—
181C2×C42—611no—
182C2×C42—612no—
183C2×C42—612no—
184C2×C42—1221no—
185C2×C42C22×D6111yescharacteristic
186C4⋊C8—1228no—
187#97—2444no—
188#95—22no—
189#95—1222no—
190C24—1221no—
191C2×Q8—21no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8#1175818912yes—
195C2×D4—612no—
196C2×D4—1221no—
197C2×D4—2441no—
198C2×D4—2441no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—611no—
202C22×C4—612no—
203C22×C4—611no—
204C22×C4—1221no—
205C22×C4—2442no—
206C22×C4—2444no—
207C22×C4C2×C4×D6111yescharacteristic
208C22×C4D4×D6112yes—
209C2×C8—24no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—1224no—
213C2×C8—2442no—
214C2×C8—2442no—
215C4⋊C4—41no—
216C4⋊C4—2441no—
217C4⋊C4—2442no—
218C22⋊C4—1228no—
219C22⋊C4—2442no—
220C22⋊C4—2442no—
221C22⋊C4—2442no—
222C42—41no—
223C42—1224no—
224C42—1222no—
225C42—1222no—
226C42—1221no—
227C42—1221no—
228C42—1222no—
229C42—2442no—
230C23—1222no—
231C23—2441no—
232C23—2441no—
233C23—2442no—
234C23—2442no—
235C23C22⋊C4×D6111yescharacteristic
236Q8—42no—
237Q8—42no—
238D4—2442no—
239D4—2442no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—612no—
250C2×C4—614no—
251C2×C4—1222no—
252C2×C4—1221no—
253C2×C4—1222no—
254C2×C4—1222no—
255C2×C4—1222no—
256C2×C4—4884no—
257C2×C4—2441no—
258C2×C4—2442no—
259C2×C4M16×D6111yescharacteristic
260C2×C4C2×#1175818911yescharacteristic
261C2×C4#1451980412yes—
262C2×C4#1451980912yes—
263C8—44no—
264C8—2444no—
265C22—4881no—
266C22—4881no—
267C22—4882no—
268C22—2442no—
269C22—2444no—
270C22#1683972411yescharacteristic
271C22#1683973411yescharacteristic
272C22#100×D6111yescharacteristic
273C22#1683976212yes—
274C22#104×D6112yes—
275C4—21no—
276C4—21no—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—81no—
281C4—614no—
282C4—4881no—
283C4—2442no—
284C4—2441no—
285C2—4882no—
286C2#2144415811yescharacteristic
287C2#2144425712yes—
288C2#2144434312yes—
289C2#329×D6111yescharacteristic
290C2#2144455311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1272C2
socleC23×C61C2×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1272D61
hypocenterC61#7635
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #21445671C2semidirect
3. #21443645C2semidirect
5. C61×#1272C2semidirect
143. #1272D61semidirect
164. C61#7635semidirect

Maximal subgroups

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

subgroupindex
#214483622
#214456712
#214436452
#214436392
C61×#12722
#763561