Finite Group Databasebeta

← #29600402

Subgroups — #29600402

Order 15616.

320 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 67 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#21456880C211yesmaximal, characteristic
2#21447508C211yesmaximal, characteristic, complemented
3#21443993C212yesmaximal, complemented
4#21443975C212yesmaximal, complemented
5C61×#2129C211yesmaximal, characteristic, complemented
6#16840367C411yescharacteristic, complemented
7#16840152C2211yescharacteristic
8#16840136C411yescharacteristic, complemented
9#16840130C411yescharacteristic, complemented
10#16840116C2211yescharacteristic
11#16839974—21no—
12#16839964—21no—
13#16839907—21no—
14#16839901—21no—
15#16839823C411yescharacteristic, complemented
16C2×#14519696—21no—
17#16839561—21no—
18#16839559C2212yes—
19C61×#536C2211yescharacteristic
20C61×#410—21no—
21C61×#354—21no—
22C61×#354C2212yes—
23C2×#11758201—21no—
24C2×#11758201—21no—
25C2×#11758201—21no—
26C2×#11758199—21no—
27C2×#11758194—41no—
28#14519774C2×C411yescharacteristic
29#14519769C2×C411yescharacteristic
30#14519762—41no—
31#14519751—21no—
32#14519750—41no—
33C4×D244—21no—
34C4⋊#92418—21no—
35#14519701—21no—
36#14519701—21no—
37#14519701—21no—
38#14519698—21no—
39#14519696—22no—
40#14519696—41no—
41C22⋊#92418—21no—
42#14519688C2×C411yescharacteristic
43C4×#92418C2×C411yescharacteristic
44C2×#41×C61D411yescharacteristic
45C22×Q8×C61D411yescharacteristic
46C4⋊Q8×C61C2×C411yescharacteristic
47C4⋊Q8×C61D411yescharacteristic
48#124×C61D411yescharacteristic
49#124×C61C2311yescharacteristic
50#123×C61—41no—
51C22⋊Q8×C61—41no—
52#117×C61—41no—
53#117×C61C2×C411yescharacteristic
54#99×C61—22no—
55#99×C61—41no—
56#41×C61—81no—
57C2×Q8×C61—21no—
58C2×Q8×C61—81no—
59C2×Q8×C61C22⋊C411yescharacteristic
60C2×Q8×C61C22⋊C411yescharacteristic
61C2×Q8×C61C2×D411yescharacteristic
62C2×D4×C61—21no—
63C2×D4×C61C2×D411yescharacteristic
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61C22⋊C411yescharacteristic
67C2×C22⋊D61—41no—
68C2×D244—21no—
69C2×D244—41no—
70C2×C4×D61—41no—
71C4⋊C4×C61—21no—
72C4⋊C4×C61—41no—
73C4⋊C4×C61—41no—
74C4⋊C4×C61—81no—
75C22⋊C4×C61—21no—
76C22⋊C4×C61—41no—
77C22⋊C4×C61—41no—
78C22⋊C4×C61—42no—
79C42×C61—81no—
80C42×C61C22⋊C411yescharacteristic
81C42×C61C22×C411yescharacteristic
82#11758201—81no—
83#11758201—81no—
84#11758201—81no—
85#11758199—81no—
86#11758198—41no—
87#11758198—41no—
88C4⋊Dic61—21no—
89#11758196—41no—
90#11758194—42no—
91#11758194—81no—
92C2×#92418—21no—
93C2×#92418—21no—
94C2×#92418—81no—
95C23×C61—21no—
96C23×C61#9911yescharacteristic
97C22×D61—41no—
98Q8×C61—41no—
99Q8×C61—41no—
100Q8×C61—41no—
101Q8×C61—81no—
102D4×C61—41no—
103D4×C61—81no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—82no—
115C2×C4×C61#9911yescharacteristic
116C2×C4×C61C2×C22⋊C411yescharacteristic
117C22⋊D61—161no—
118C2×Dic61—41no—
119D244—41no—
120D244—41no—
121D244—161no—
122C4×D61—161no—
123#92418—41no—
124#92418—41no—
125#92418—82no—
126#13606—611nomaximal, Sylow
127C22×C61—21no—
128C22×C61—41no—
129C22×C61—41no—
130C22×C61—81no—
131C22×C61C2×#9911yescharacteristic
132C2×D61—81no—
133C2×D61—81no—
134C2×D61—81no—
135C4×C61—21no—
136C4×C61—21no—
137C4×C61—41no—
138C4×C61—41no—
139C4×C61—41no—
140C4×C61—41no—
141C4×C61—41no—
142C4×C61—162no—
143Dic61—81no—
144#3221—611no—
145#2133—611no—
146#2129D6111yescharacteristic, complemented
147#1414—612no—
148#1406—612no—
149C2×C61—21no—
150C2×C61—41no—
151C2×C61—81no—
152C2×C61#212711yescharacteristic
153D61—161no—
154#540—611no—
155#536C2×D6111yescharacteristic
156#496—611no—
157#492—611no—
158#491—611no—
159#488—611no—
160#463—1221no—
161#461—1221no—
162#449—1221no—
163#448—1221no—
164#433—611no—
165C2×#101—1221no—
166#410—21no—
167#356—1221no—
168#355—612no—
169#354—21no—
170#354C2×D6112yes—
171C61#1360611yescharacteristic, complemented, Sylow
172C2×#41C22⋊D6111yescharacteristic
173C22×Q8D24411yescharacteristic
174C2×SD16—1221no—
175C2×SD16—1221no—
176C2×SD16—1221no—
177C2×D8—1221no—
178C2×M16—2441no—
179C4⋊Q8C4×D6111yescharacteristic
180C4⋊Q8D24411yescharacteristic
181#127—611no—
182#125—611no—
183#124C22⋊D6111yescharacteristic
184#124C22×D6111yescharacteristic
185#123—41no—
186#123—2441no—
187C22⋊Q8—41no—
188#121—1221no—
189#121—2441no—
190C4×D4—1221no—
191#117—41no—
192#117C4×D6111yescharacteristic
193C4⋊C8—1221no—
194#103—1221no—
195#103—1221no—
196#103—1221no—
197#102—1221no—
198#101—1222no—
199#101—2441no—
200#99—22no—
201#99—41no—
202C22⋊C8—1221no—
203#97—611no—
204C4×C8—611no—
205#41—81no—
206C2×Q8—21no—
207C2×Q8—81no—
208C2×Q8#1175819811yescharacteristic
209C2×Q8#1175819811yescharacteristic
210C2×Q8C2×D24411yescharacteristic
211C2×D4—21no—
212C2×D4—1221no—
213C2×D4—2441no—
214C2×D4—2441no—
215C2×D4C2×C22⋊D6111yescharacteristic
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—2441no—
219C22×C4#1175819811yescharacteristic
220SD16—4881no—
221SD16—4881no—
222SD16—4881no—
223D8—4881no—
224M16—4881no—
225M16—2442no—
226C2×C8—1221no—
227C2×C8—1221no—
228C2×C8—4881no—
229C4⋊C4—21no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—81no—
233C4⋊C4—1221no—
234C4⋊C4—2441no—
235C22⋊C4—21no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—2441no—
240C22⋊C4—2441no—
241C42—81no—
242C42#1175819811yescharacteristic
243C42C2×C4×D6111yescharacteristic
244C23—21no—
245C23—2441no—
246C23#1451969111yescharacteristic
247Q8—41no—
248Q8—41no—
249Q8—41no—
250Q8—81no—
251D4—41no—
252D4—81no—
253D4—9761no—
254D4—9761no—
255D4—2441no—
256D4—2441no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—82no—
268C2×C4—9761no—
269C2×C4—2441no—
270C2×C4#1451969111yescharacteristic
271C2×C4C2×#1175819811yescharacteristic
272C8—4882no—
273C8—2441no—
274C8—2441no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—81no—
279C22—4881no—
280C22—4881no—
281C22—4881no—
282C22C2×#1451969111yescharacteristic
283C4—21no—
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—162no—
291C4—4881no—
292C2—21no—
293C2—41no—
294C2—81no—
295C2—9761no—
296C2#2144748611yescharacteristic
subgroupG/H
centerC2#21447486
commutatorC42×C61C22×C4
Frattini#124C22×D61
FittingC61×#2129C2
socleC2×C61#2127
prime radical#21456880C2
cosocle#124×C61C23
hypercenter#2129D61
hypocenterC61#13606
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ #2129

Complements

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

NH ≅ G/Ntype
2. #21447508C2semidirect
3. #21443993C2semidirect
4. #21443975C2semidirect
5. C61×#2129C2semidirect
6. #16840367C4semidirect
8. #16840136C4semidirect
9. #16840130C4semidirect
15. #16839823C4semidirect
146. #2129D61semidirect
171. C61#13606semidirect

Maximal subgroups

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

subgroupindex
#214568802
#214475082
#214439932
#214439752
C61×#21292
#1360661