Finite Group Databasebeta

← #29577827

Subgroups — #29577827

Order 15616.

610 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 111 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#21450106C211yesmaximal, characteristic
2#21445919C212yesmaximal
3#21444149C211yesmaximal, characteristic
4#21443672C212yesmaximal
5C61×#2274C211yesmaximal, characteristic
6C2×#14519736C2211yescharacteristic
7#16839726C2212yes—
8#16839701C412yescomplemented
9#16839692C412yescomplemented
10#16839643—22no—
11#16839635—21no—
12#16839625—21no—
13#16839625C2211yescharacteristic
14#16839522—24no—
15#16839488—22no—
16C2×#117×C61—21no—
17C22×C4⋊C4×C61C2211yescharacteristic
18C4×C4⋊C4×C61—24no—
19C4×C22⋊C4×C61—21no—
20C4×C22⋊C4×C61C2212yes—
21C2×#11758194—22no—
22#14519736—22no—
23#14519736—22no—
24C2×C4⋊Dic61C2×C411yescharacteristic
25C2×C4⋊Dic61C2×C411yescharacteristic
26C2×#11758196—22no—
27C2×C22⋊Dic61—21no—
28C2×C22⋊Dic61—21no—
29C2×C4×Dic61—21no—
30C22⋊#92418—22no—
31C22⋊#92418—22no—
32#14519684—22no—
33#14519684—22no—
34#14519684—22no—
35#14519684—21no—
36#14519684C2×C412yes—
37C23×C4×C61D411yescharacteristic
38C23×C4×C61D411yescharacteristic
39C23×C4×C61C2311yescharacteristic
40#117×C61—28no—
41C2×C4⋊C4×C61—22no—
42C2×C4⋊C4×C61—22no—
43C2×C4⋊C4×C61—22no—
44C2×C4⋊C4×C61C2×C412yes—
45C2×C4⋊C4×C61D412yes—
46C2×C22⋊C4×C61—22no—
47C2×C42×C61—22no—
48C2×C42×C61—22no—
49C2×C42×C61D414yes—
50#95×C61—22no—
51#95×C61—22no—
52C24×C61C2×D411yescharacteristic
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—24no—
56C22×C4×C61—24no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—42no—
60C22×C4×C61—42no—
61C22×C4×C61C22⋊C412yes—
62C22×C4×C61C22⋊C412yes—
63C22×C4×C61C22×C411yescharacteristic
64C22×C4×C61C2×D412yes—
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61#4112yes—
67C22×Dic61—22no—
68C22×Dic61—21no—
69C22×Dic61—21no—
70C4⋊C4×C61—28no—
71C4⋊C4×C61—24no—
72C4⋊C4×C61—24no—
73C22⋊C4×C61—28no—
74C42×C61—28no—
75C42×C61—44no—
76C42×C61—82no—
77C22⋊Dic61—42no—
78C22⋊Dic61—41no—
79C22⋊Dic61—41no—
80C4⋊Dic61—22no—
81C4⋊Dic61—22no—
82#11758196—44no—
83C4×Dic61—42no—
84#11758194—44no—
85C2×#92418—42no—
86C23×C61—22no—
87C23×C61—22no—
88C23×C61—41no—
89C23×C61C2×C22⋊C411yescharacteristic
90C23×C61#12011yescharacteristic
91C23×C61#12311yescharacteristic
92C2×C4×C61—22no—
93C2×C4×C61—22no—
94C2×C4×C61—24no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—24no—
98C2×C4×C61—28no—
99C2×C4×C61—22no—
100C2×C4×C61—41no—
101C2×C4×C61—41no—
102C2×C4×C61—42no—
103C2×C4×C61—44no—
104C2×C4×C61—44no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—81no—
108C2×C4×C61#10414yes—
109C2×C4×C61C4×D412yes—
110C2×C4×C61#12112yes—
111C2×Dic61—42no—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—42no—
117C2×Dic61—42no—
118#92418—82no—
119#9478—611nomaximal, Sylow
120C22×C61—21no—
121C22×C61—21no—
122C22×C61—21no—
123C22×C61—21no—
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—21no—
127C22×C61—44no—
128C22×C61#38611yescharacteristic
129C22×C61C2×#10412yes—
130C22×C61#45812yes—
131C4×C61—22no—
132C4×C61—22no—
133C4×C61—24no—
134C4×C61—48no—
135C4×C61—42no—
136Dic61—82no—
137Dic61—81no—
138Dic61—81no—
139#2484—611no—
140#2274D6111yescharacteristic
141#1826—612no—
142#1463—611no—
143#1284—612no—
144C2×C61—22no—
145C2×C61—22no—
146C2×C61—41no—
147C2×C61#189011yescharacteristic
148C2×C61#189611yescharacteristic
149C2×C61#191211yescharacteristic
150C2×#117—21no—
151C2×#117—611no—
152C22×C4⋊C4C2×D6111yescharacteristic
153#407—612no—
154#400—612no—
155#398—612no—
156#387—1222no—
157#385—1221no—
158#381—611no—
159#381—1221no—
160C4×C4⋊C4—24no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4C2×D6112yes—
163#339—1224no—
164#324—1222no—
165C61#947811yescharacteristic, complemented, Sylow
166C23×C4C22⋊D6111yescharacteristic
167C23×C4C22⋊D6111yescharacteristic
168C23×C4C22×D6111yescharacteristic
169C2×M16—1222no—
170#117—28no—
171#117—1222no—
172#117—1222no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—611no—
177C2×C4⋊C4—611no—
178C2×C4⋊C4—1222no—
179C2×C4⋊C4C4×D6112yes—
180C2×C4⋊C4D24412yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—1221no—
183C2×C22⋊C4—1221no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—1221no—
187C2×C42C22⋊D6114yes—
188C22⋊C8—1222no—
189C22⋊C8—1222no—
190#95—22no—
191#95—22no—
192#95—612no—
193#95—1222no—
194#95—1222no—
195#95—1222no—
196#95—1221no—
197C24C2×C22⋊D6111yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—1222no—
207C22×C4—1221no—
208C22×C4—1221no—
209C22×C4#1175819812yes—
210C22×C4#1175819812yes—
211C22×C4C2×C4×D6111yescharacteristic
212C22×C4C2×D24411yescharacteristic
213C22×C4#1175822212yes—
214C22×C4C2×C22⋊D6112yes—
215M16—2444no—
216C2×C8—2442no—
217C4⋊C4—28no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—1222no—
221C4⋊C4—1222no—
222C4⋊C4—2444no—
223C22⋊C4—28no—
224C22⋊C4—2442no—
225C22⋊C4—2441no—
226C22⋊C4—2441no—
227C42—28no—
228C42—44no—
229C42—82no—
230C42—2442no—
231C23—22no—
232C23—22no—
233C23—41no—
234C23C2×#1175819811yescharacteristic
235C23#1451974611yescharacteristic
236C23#1451976011yescharacteristic
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—81no—
253C2×C4—2442no—
254C2×C4—2442no—
255C2×C4—2442no—
256C2×C4—2442no—
257C2×C4—2442no—
258C2×C4—2441no—
259C2×C4—2441no—
260C2×C4#1451970514yes—
261C2×C4C4×C22⋊D6112yes—
262C2×C4#1451975112yes—
263C8—4882no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22#1683964111yescharacteristic
273C22C2×#1451970512yes—
274C22#1683995212yes—
275C4—22no—
276C4—22no—
277C4—24no—
278C4—48no—
279C4—42no—
280C4—4882no—
281C4—4881no—
282C4—4881no—
283C2—22no—
284C2—22no—
285C2—41no—
286C2#2144623211yescharacteristic
287C2#2144626211yescharacteristic
288C2#2144635611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2274C2
socleC22×C61#386
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#2274D61
hypocenterC61#9478
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2274

Complements

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

NH ≅ G/Ntype
8. #16839701C4semidirect
9. #16839692C4semidirect
165. C61#9478semidirect

Maximal subgroups

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

subgroupindex
#214501062
#214459192
#214441492
#214436722
C61×#22742
#947861