Finite Group Databasebeta

← #30603903

Subgroups — #30603903

Order 15616.

462 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 112 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#21461050C212yesmaximal
2#21460763C214yesmaximal
3#21448303C211yesmaximal, characteristic
4#16840778—22no—
5#16840607—22no—
6#16840605C2214yes—
7C2×#14519736C2211yescharacteristic
8C22×C4×Dic61C411yescharacteristic
9#16839661—22no—
10#16839637C412yes—
11#16839637C2212yes—
12C4×C4⋊Dic61—22no—
13C2×#117×C61C411yescharacteristic
14C4⋊#92420—42no—
15C22⋊#92420—24no—
16C4×#92420—42no—
17C23×Dic61C2×C412yes—
18#14519736—41no—
19#14519736—41no—
20C2×C4⋊Dic61—22no—
21C2×C4⋊Dic61—22no—
22C2×C4⋊Dic61D412yes—
23C2×C22⋊Dic61C2×C411yescharacteristic
24C2×C22⋊Dic61C2311yescharacteristic
25C2×C4×Dic61—21no—
26C2×C4×Dic61—21no—
27C2×C4×Dic61—41no—
28C2×C4×Dic61—41no—
29C2×C4×Dic61D412yes—
30#14519684—22no—
31#14519684—22no—
32C23×C4×C61C2×C411yescharacteristic
33#117×C61—22no—
34#117×C61C814yescomplemented
35C2×C4⋊C4×C61—21no—
36C2×C22⋊C4×C61C2×C412yes—
37C2×C42×C61—21no—
38C24×C61C22×C411yescharacteristic
39C22×C4×C61—21no—
40C22×C4×C61—21no—
41C22×C4×C61—22no—
42C22×C4×C61—22no—
43C22×C4×C61C22⋊C411yescharacteristic
44C22×C4×C61C22⋊C411yescharacteristic
45C22×C4×C61C2×C812yes—
46C22×C4×C61M1612yes—
47C22×Dic61—22no—
48C22×Dic61—44no—
49C22×Dic61—42no—
50C22×Dic61C22⋊C412yes—
51C22×Dic61C2×D412yes—
52C2×#92420—44no—
53C4⋊C4×C61—24no—
54C22⋊C4×C61—22no—
55C22⋊C4×C61C2×C814yes—
56C42×C61—24no—
57C22⋊Dic61—41no—
58C22⋊Dic61—41no—
59C4⋊Dic61—42no—
60C4⋊Dic61—44no—
61C4⋊Dic61—44no—
62C4×Dic61—22no—
63C4×Dic61—22no—
64C4×Dic61—42no—
65C4×Dic61—41no—
66C4×Dic61—41no—
67C23×C61—21no—
68C23×C61—21no—
69C23×C61—42no—
70C23×C61C2×C22⋊C411yescharacteristic
71C23×C61C22×C811yescharacteristic
72C23×C61C2×M1611yescharacteristic
73C2×C4×C61—21no—
74C2×C4×C61—21no—
75C2×C4×C61—22no—
76C2×C4×C61—22no—
77C2×C4×C61—22no—
78C2×C4×C61—22no—
79C2×C4×C61—24no—
80C2×C4×C61—42no—
81C2×C4×C61—42no—
82C2×C4×C61C22⋊C812yes—
83C2×C4×C61C22⋊C812yes—
84C2×C4×C61#9911yescharacteristic
85C2×C4×C61#9911yescharacteristic
86C2×C4×C61#10112yes—
87C2×Dic61—24no—
88C2×Dic61—22no—
89C2×Dic61—22no—
90C2×Dic61—42no—
91C2×Dic61—44no—
92C2×Dic61—42no—
93C2×Dic61—42no—
94C2×Dic61—42no—
95C2×Dic61—44no—
96#92420—84no—
97#11452—611nomaximal, Sylow
98C22×C61—22no—
99C22×C61—21no—
100C22×C61—21no—
101C22×C61—41no—
102C22×C61—41no—
103C22×C61—42no—
104C22×C61—42no—
105C22×C61C2×C22⋊C811yescharacteristic
106C22×C61C2×#9911yescharacteristic
107C22×C61C2×#10111yescharacteristic
108C4×C61—22no—
109C4×C61—44no—
110C4×C61#32412yes—
111C4×C61#32412yes—
112Dic61—42no—
113Dic61—42no—
114Dic61—44no—
115#2296—611no—
116#1458—612no—
117#1326—614no—
118C2×C61—21no—
119C2×C61—21no—
120C2×C61—41no—
121C2×C61—41no—
122C2×C61C2×#32411yescharacteristic
123C2×C61#212012yes—
124C2×#117—611no—
125C2×#117#692311yescharacteristic
126C22×C42—611no—
127#431—1222no—
128#389—1222no—
129#385—612no—
130#385—612no—
131C4×C4⋊C4—1222no—
132#324—1222no—
133#323—614no—
134C61#1145211yescharacteristic, complemented, Sylow
135C23×C4—612no—
136C23×C4C2×#692311yescharacteristic
137#117—22no—
138#117—2441no—
139#117—2441no—
140#117#9242014yescomplemented
141C2×C4⋊C4—21no—
142C2×C4⋊C4—612no—
143C2×C4⋊C4—1222no—
144C2×C4⋊C4—1222no—
145C2×C22⋊C4—611no—
146C2×C22⋊C4—611no—
147C2×C22⋊C4C2×#692312yes—
148C2×C42—21no—
149C2×C42—612no—
150C2×C42—1221no—
151C2×C42—1221no—
152C2×C42—2441no—
153C2×C42—2441no—
154C4⋊C8—2442no—
155C22⋊C8—1224no—
156C4×C8—2442no—
157#95—1222no—
158#95—1222no—
159C24C22×#692311yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—612no—
165C22×C4—612no—
166C22×C4—1222no—
167C22×C4—2444no—
168C22×C4—2442no—
169C22×C4C2×#9242012yes—
170C22×C4#1175821712yes—
171C22×C4C22⋊#692311yescharacteristic
172C22×C4C22⋊#692311yescharacteristic
173C2×C8—2444no—
174C4⋊C4—24no—
175C4⋊C4—2442no—
176C4⋊C4—2444no—
177C4⋊C4—2444no—
178C22⋊C4—22no—
179C22⋊C4—2441no—
180C22⋊C4—2441no—
181C22⋊C4C2×#9242014yes—
182C42—24no—
183C42—1222no—
184C42—1222no—
185C42—2442no—
186C42—2441no—
187C42—2441no—
188C23—21no—
189C23—21no—
190C23—42no—
191C23C2×C22⋊#692311yescharacteristic
192C23C22×#9242011yescharacteristic
193C23C2×#1175821711yescharacteristic
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—1224no—
204C2×C4—1222no—
205C2×C4—1222no—
206C2×C4—2442no—
207C2×C4—2444no—
208C2×C4—2442no—
209C2×C4—2442no—
210C2×C4—2442no—
211C2×C4—2444no—
212C2×C4C22⋊#9242012yes—
213C2×C4C22⋊#9242012yes—
214C2×C4#1451983611yescharacteristic
215C2×C4#1451983611yescharacteristic
216C2×C4#1451983812yes—
217C8—4884no—
218C22—22no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—42no—
224C22—42no—
225C22C2×C22⋊#9242011yescharacteristic
226C22C2×#1451983611yescharacteristic
227C22C2×#1451983811yescharacteristic
228C4—22no—
229C4—44no—
230C4—2442no—
231C4—2442no—
232C4—2444no—
233C4#1684060712yes—
234C4#1684060712yes—
235C2—21no—
236C2—21no—
237C2—41no—
238C2—41no—
239C2C2×#1684060711yescharacteristic
240C2#2146207212yes—
subgroupG/H
centerC2×C4#14519836
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
FittingC2×#117×C61C4
socleC22×C61C2×#99
prime radicalC24×C61C22×C4
cosocleC2×C22⋊Dic61C23
hypercenterC2×#117#6923
hypocenterC61#11452
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
34. #117×C61C8semidirect
134. C61#11452semidirect
140. #117#92420semidirect

Maximal subgroups

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

subgroupindex
#214610502
#214607632
#214483032
#1145261