Finite Group Databasebeta

← #29569333

Subgroups — #29569333

Order 15616.

438 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 83 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#21453765C211yesmaximal, characteristic
2#21447356C212yesmaximal, complemented
3#21447352C211yesmaximal, characteristic
4#21443644C212yesmaximal, complemented
5C61×#1270C211yesmaximal, characteristic
6C2×#14519711—22no—
7C2×#14519708—21no—
8C2×#14519702—22no—
9C2×#14519698—21no—
10#16839701—21no—
11#16839674—22no—
12#16839668C2212yes—
13#16839642—21no—
14#16839637—22no—
15#16839632C2211yescharacteristic
16#16839517C2211yescharacteristic
17C61×#390C2211yescharacteristic
18C61×#336C2212yes—
19C22×Dic122D412yes—
20C23×Dic61—21no—
21C22×#92418—21no—
22C2×C4⋊Dic61—21no—
23C2×C4⋊Dic61—22no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61D412yes—
26C2×C22⋊Dic61—22no—
27C2×C22⋊Dic61—21no—
28#14519711—44no—
29#14519708—42no—
30#14519702—44no—
31#14519698—42no—
32#14519684—22no—
33#14519684—21no—
34#14519684—22no—
35#14519684—22no—
36C22×D4×C61D411yescharacteristic
37C22×C8×C61—22no—
38C2×C4⋊C4×C61D411yescharacteristic
39C2×C22⋊C4×C61—22no—
40C2×C42×C61C2311yescharacteristic
41C24×C61—21no—
42C2×D4×C61—21no—
43C2×D4×C61—41no—
44C2×D4×C61—41no—
45C2×D4×C61Q1612yes—
46C22×C4×C61—21no—
47C22×C4×C61C2×D411yescharacteristic
48C22×C4×C61C2×D412yes—
49C22×Dic61—22no—
50C22×Dic61—22no—
51C22×Dic61—41no—
52C22×Dic61—82no—
53C22×Dic61—81no—
54C2×Dic122—22no—
55C2×Dic122—22no—
56C2×Dic122—42no—
57C2×Dic122—42no—
58C2×C8×C61—24no—
59C2×C8×C61—42no—
60C2×C8×C61—42no—
61C4⋊C4×C61—41no—
62C22⋊C4×C61—42no—
63C22⋊C4×C61—42no—
64C42×C61—41no—
65C22⋊Dic61—82no—
66C22⋊Dic61—81no—
67C4⋊Dic61—42no—
68C4⋊Dic61—41no—
69C4⋊Dic61—42no—
70C4⋊Dic61—41no—
71C4⋊Dic61—42no—
72C4⋊Dic61—81no—
73C2×#92418—22no—
74C2×#92418—41no—
75C2×#92418—41no—
76C23×C61—22no—
77C23×C61—42no—
78C23×C61—42no—
79C23×C61—41no—
80C23×C61—41no—
81C23×C61#12011yescharacteristic
82D4×C61—42no—
83D4×C61—81no—
84C2×C4×C61—21no—
85C2×C4×C61—22no—
86C2×C4×C61—21no—
87C2×C4×C61—22no—
88C2×C4×C61—42no—
89C2×C4×C61—41no—
90C2×C4×C61C2×Q1611yescharacteristic
91C2×C4×C61#13512yes—
92C2×C4×C61#13711yescharacteristic
93C2×C4×C61#13712yes—
94C2×Dic61—42no—
95C2×Dic61—42no—
96C2×Dic61—42no—
97C2×Dic61—42no—
98C2×Dic61—42no—
99C2×Dic61—42no—
100C2×Dic61—84no—
101C2×Dic61—82no—
102C2×Dic61—81no—
103Dic122—44no—
104Dic122—82no—
105C8×C61—44no—
106#92418—42no—
107#7833—611nomaximal, Sylow
108C22×C61—42no—
109C22×C61—44no—
110C22×C61—82no—
111C22×C61—81no—
112C22×C61—81no—
113C22×C61#45111yescharacteristic
114C22×C61#45212yes—
115C22×C61#45312yes—
116C22×C61#45311yescharacteristic
117C22×C61#45811yescharacteristic
118C4×C61—21no—
119C4×C61—21no—
120C4×C61—22no—
121C4×C61—22no—
122C4×C61—81no—
123Dic61—82no—
124Dic61—82no—
125Dic61—161no—
126#2804—611no—
127#2095—612no—
128#2094—611no—
129#1272—612no—
130#1270D6111yescharacteristic
131C2×C61—82no—
132C2×C61#161411yescharacteristic
133C2×C61#161812yes—
134C2×C61#163011yescharacteristic
135C2×C61#163212yes—
136C2×C61#165511yescharacteristic
137C2×#107—1222no—
138C2×#106—1221no—
139C2×#103—1222no—
140C2×#102—1221no—
141#400—1221no—
142#393—1222no—
143#391—612no—
144#390C2×D6111yescharacteristic
145#386—1221no—
146#385—1222no—
147#384—611no—
148#336—611no—
149#336C2×D6112yes—
150C61#783311yescharacteristic, complemented, Sylow
151C22×Q8—612no—
152C22×D4D24411yescharacteristic
153C23×C4—1221no—
154C22×C8—22no—
155C22×C8—1221no—
156C2×C4⋊C4—612no—
157C2×C4⋊C4—1221no—
158C2×C4⋊C4—1221no—
159C2×C4⋊C4—1222no—
160C2×C4⋊C4D24411yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—1222no—
163C2×C22⋊C4—1221no—
164C2×C42C22×D6111yescharacteristic
165#107—2444no—
166#106—2442no—
167#103—2444no—
168#102—2442no—
169#95—1222no—
170#95—1221no—
171#95—1222no—
172#95—1222no—
173C24—21no—
174C2×Q8—1222no—
175C2×Q8—1222no—
176C2×Q8—2442no—
177C2×Q8—2442no—
178C2×D4—21no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4Dic24412yes—
182C22×C4—21no—
183C22×C4—1222no—
184C22×C4—1222no—
185C22×C4—4882no—
186C22×C4—4881no—
187C22×C4—2441no—
188C22×C4C2×D24411yescharacteristic
189C22×C4D4×D6112yes—
190C2×C8—24no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—1222no—
194C2×C8—2441no—
195C2×C8—2441no—
196C4⋊C4—41no—
197C4⋊C4—4881no—
198C4⋊C4—2441no—
199C4⋊C4—2442no—
200C4⋊C4—2441no—
201C4⋊C4—2442no—
202C4⋊C4—2442no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—4881no—
206C22⋊C4—4882no—
207C42—41no—
208C23—22no—
209C23—41no—
210C23—42no—
211C23—42no—
212C23—41no—
213C23#1451974711yescharacteristic
214Q8—4882no—
215Q8—2444no—
216D4—42no—
217D4—81no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—4882no—
225C2×C4—4881no—
226C2×C4—4884no—
227C2×C4—2442no—
228C2×C4—2442no—
229C2×C4—2442no—
230C2×C4—2442no—
231C2×C4—2442no—
232C2×C4—2442no—
233C2×C4C2×Dic24411yescharacteristic
234C2×C4#1451979612yes—
235C2×C4#1451980612yes—
236C2×C4#1451981011yescharacteristic
237C8—44no—
238C8—2442no—
239C22—44no—
240C22—42no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22#1683992011yescharacteristic
245C22#1683992512yes—
246C22#1683993112yes—
247C22#1683993211yescharacteristic
248C22#1683995311yescharacteristic
249C4—22no—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—81no—
254C4—9761no—
255C4—4882no—
256C4—4882no—
257C2—82no—
258C2#2144484611yescharacteristic
259C2#2144486812yes—
260C2#2144494111yescharacteristic
261C2#2144495512yes—
262C2#2144510911yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1270C2
socleC23×C61#120
prime radicalC22×D4×C61D4
hypercenter#1270D61
hypocenterC61#7833
LCSG ▷ C2×C42×C61 ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #21447356C2semidirect
4. #21443644C2semidirect
150. C61#7833semidirect

Maximal subgroups

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

subgroupindex
#214537652
#214473562
#214473522
#214436442
C61×#12702
#783361