Finite Group Databasebeta

← #29596814

Subgroups — #29596814

Order 15616.

536 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 103 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#21448931C211yesmaximal, characteristic
2#21446208C211yesmaximal, characteristic
3#21446102C212yesmaximal
4#21443936C212yesmaximal
5C2×C61×#354C211yesmaximal, characteristic
6#16840237C2211yescharacteristic
7#16839771—21no—
8C2×#14519694C2211yescharacteristic
9#16839669—21no—
10#16839655—22no—
11#16839642C2212yes—
12#16839637—22no—
13C22⋊C4×Dic61—22no—
14#16839608—21no—
15#16839528—22no—
16#16839522—21no—
17C2×#124×C61C2211yescharacteristic
18C2×#99×C61C2212yes—
19C61×#354C414yescomplemented
20C23×Dic61D412yes—
21C2×#11758194—21no—
22D4×Dic61—42no—
23#14519736—21no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—42no—
26C2×C22⋊Dic61—22no—
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61D412yes—
29C2×C4×Dic61—22no—
30C2×C4×Dic61—21no—
31#14519701—42no—
32#14519694—22no—
33#14519684—21no—
34#14519684—21no—
35#14519684—22no—
36#14519684—22no—
37#14519684—42no—
38C22×Q8×C61D411yescharacteristic
39C22×D4×C61C2311yescharacteristic
40#124×C61—22no—
41#124×C61C2×C412yes—
42#124×C61Q812yes—
43C2×C22⋊C4×C61—22no—
44C2×C22⋊C4×C61—22no—
45C2×C42×C61D411yescharacteristic
46#99×C61—22no—
47#99×C61C2×C414yes—
48C24×C61C2×D412yes—
49C2×Q8×C61—21no—
50C2×Q8×C61—41no—
51C2×Q8×C61—41no—
52C2×Q8×C61C4⋊C412yescomplemented
53C2×D4×C61—41no—
54C2×D4×C61—41no—
55C2×D4×C61C22×C411yescharacteristic
56C2×D4×C61C2×Q811yescharacteristic
57C2×D4×C61#4112yes—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—42no—
61C22×C4×C61C2×D411yescharacteristic
62C22×Dic61—22no—
63C22×Dic61—22no—
64C22×Dic61—22no—
65C22×Dic61—22no—
66C22×Dic61—44no—
67C22×Dic61—42no—
68C22⋊C4×C61—24no—
69C22⋊C4×C61—24no—
70C22⋊C4×C61—24no—
71C22⋊C4×C61—42no—
72C42×C61—21no—
73C42×C61C4⋊C412yescomplemented
74C22⋊Dic61—44no—
75C22⋊Dic61—42no—
76C22⋊Dic61—41no—
77C22⋊Dic61—42no—
78C4⋊Dic61—22no—
79C4⋊Dic61—81no—
80C4⋊Dic61—84no—
81C4×Dic61—44no—
82C4×Dic61—41no—
83C4×Dic61—81no—
84#11758194—41no—
85#11758194—41no—
86C2×#92418—41no—
87C23×C61—22no—
88C23×C61—22no—
89C23×C61—42no—
90C23×C61—42no—
91C23×C61C4×D412yes—
92C23×C61#12011yescharacteristic
93C23×C61C22⋊Q812yes—
94Q8×C61—42no—
95Q8×C61—81no—
96D4×C61—41no—
97D4×C61—41no—
98D4×C61—42no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—42no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61—44no—
107C2×C4×C61—82no—
108C2×C4×C61—82no—
109C2×C4×C61C2×C4⋊C411yescharacteristic
110C2×C4×C61#12311yescharacteristic
111C2×Dic61—24no—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—44no—
115C2×Dic61—42no—
116C2×Dic61—42no—
117C2×Dic61—42no—
118C2×Dic61—42no—
119C2×Dic61—82no—
120C2×Dic61—82no—
121C2×Dic61—82no—
122#92418—81no—
123#12965—611nomaximal, Sylow
124C22×C61—22no—
125C22×C61—22no—
126C22×C61—22no—
127C22×C61—42no—
128C22×C61—42no—
129C22×C61—42no—
130C22×C61—42no—
131C22×C61—82no—
132C22×C61#38511yescharacteristic
133C22×C61B(4,𝔽2)11yescharacteristic
134C22×C61#45811yescharacteristic
135C4×C61—21no—
136C4×C61—21no—
137C4×C61—42no—
138C4×C61—42no—
139C4×C61—84no—
140Dic61—44no—
141Dic61—82no—
142Dic61—82no—
143#2372—611no—
144C2×#354D6111yescharacteristic
145#1885—611no—
146#1867—612no—
147#1390—612no—
148C2×C61—21no—
149C2×C61—21no—
150C2×C61—42no—
151C2×C61—42no—
152C2×C61#186711yescharacteristic
153C2×C61#219712yes—
154C2×#124C2×D6111yescharacteristic
155#518—611no—
156#419—1221no—
157C2×#100—611no—
158C2×#99C2×D6112yes—
159#391—1221no—
160#388—1222no—
161#386—612no—
162#385—1222no—
163C4×C22⋊C4—1222no—
164#376—1221no—
165#354Dic6114yescomplemented
166#342—1222no—
167#339—1221no—
168C61#1296511yescharacteristic, complemented, Sylow
169C22×Q8D24411yescharacteristic
170C22×D4C22×D6111yescharacteristic
171C23×C4—612no—
172C2×M16—1221no—
173#124—22no—
174#124Dic12212yes—
175#124C2×Dic6112yes—
176C4×D4—2442no—
177#117—1221no—
178C2×C4⋊C4—1221no—
179C2×C4⋊C4—2442no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—612no—
183C2×C22⋊C4—1222no—
184C2×C22⋊C4—1222no—
185C2×C42—1221no—
186C2×C42—1222no—
187C2×C42D24411yescharacteristic
188#103—2442no—
189#100—1222no—
190#99—22no—
191#99C2×Dic6114yes—
192#95—1221no—
193#95—1221no—
194#95—1222no—
195#95—1222no—
196#95—2442no—
197C24D4×D6112yes—
198C2×Q8—21no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8C4⋊Dic6112yescomplemented
202C2×D4—41no—
203C2×D4—41no—
204C2×D4C2×Dic12211yescharacteristic
205C2×D4#1175822412yes—
206C2×D4C22×Dic6111yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—1222no—
211C22×C4—1222no—
212C22×C4—1222no—
213C22×C4—1222no—
214C22×C4—2444no—
215C22×C4—2442no—
216C22×C4C2×D24411yescharacteristic
217M16—2441no—
218M16—2441no—
219C2×C8—2441no—
220C4⋊C4—1222no—
221C4⋊C4—4881no—
222C4⋊C4—4884no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—42no—
227C22⋊C4—2444no—
228C22⋊C4—2442no—
229C22⋊C4—2441no—
230C22⋊C4—2442no—
231C42—21no—
232C42—4881no—
233C42—2441no—
234C42—2444no—
235C42C4⋊Dic6112yescomplemented
236C23—22no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23D4×Dic6112yes—
241C23#1451974711yescharacteristic
242C23#1451975412yes—
243Q8—42no—
244Q8—81no—
245D4—41no—
246D4—42no—
247D4—41no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—44no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—1224no—
259C2×C4—4882no—
260C2×C4—4882no—
261C2×C4—4882no—
262C2×C4—2442no—
263C2×C4—2442no—
264C2×C4—2442no—
265C2×C4—2442no—
266C2×C4—2442no—
267C2×C4—2442no—
268C2×C4—2444no—
269C2×C4C2×C4⋊Dic6111yescharacteristic
270C2×C4#1451976111yescharacteristic
271C8—4881no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—82no—
280C22#1683963711yescharacteristic
281C22#1683995011yescharacteristic
282C22#1683995311yescharacteristic
283C4—21no—
284C4—21no—
285C4—42no—
286C4—42no—
287C4—84no—
288C4—4882no—
289C4—4882no—
290C4—2444no—
291C2—21no—
292C2—21no—
293C2—42no—
294C2—42no—
295C2#2144610211yescharacteristic
296C2#2144781212yes—
subgroupG/H
centerC22#16839950
commutatorC2×D4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC2×C61×#354C2
socleC22×C61B(4,𝔽2)
prime radical, cosocleC22×D4×C61C23
hypercenterC2×#354D61
hypocenterC61#12965
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
19. C61×#354C4semidirect
52. C2×Q8×C61C4⋊C4semidirect
73. C42×C61C4⋊C4semidirect
165. #354Dic61semidirect
168. C61#12965semidirect
201. C2×Q8C4⋊Dic61semidirect
235. C42C4⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
#214489312
#214462082
#214461022
#214439362
C2×C61×#3542
#1296561