Finite Group Databasebeta

← #29707416

Subgroups — #29707416

Order 15616.

1378 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1378 conjugacy classes of subgroups — too many to draw. Only its 275 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#21448198C213yesmaximal
2#21446985C214yesmaximal, complemented
3#21446982C214yesmaximal
4#21445552C213yesmaximal, complemented
5C61×#3458C211yesmaximal, characteristic, complemented
6C23×C4×D61C2211yescharacteristic
7C2×#14519736C2213yes—
8#16839623C2216yes—
9#16839619—26no—
10#16839617C2216yes—
11C4×#11758198—26no—
12#16839529C22112yes—
13C61×#544—26no—
14C61×#543C2214yes—
15C61×#533—26no—
16C2×#117×C61C2213yes—
17C24×D61C2311yescharacteristic
18C23×Dic61C2311yescharacteristic
19C22×C4×D61—23no—
20C22×C4×D61—23no—
21C22×C4×D61—26no—
22C22×C4×D61Q814yescomplemented
23#14519736—212no—
24C2×C4⋊Dic61—23no—
25C2×C22⋊Dic61C2316yes—
26C2×#11758198—212no—
27C2×C4×Dic61—23no—
28#14519684—212no—
29C23×C4×C61C2311yescharacteristic
30C4⋊Q8×C61—212no—
31#125×C61—212no—
32C22⋊Q8×C61—224no—
33C4×Q8×C61—28no—
34#117×C61—26no—
35#117×C61—26no—
36C2×C4⋊C4×C61D416yes—
37C2×C22⋊C4×C61C2316yes—
38C2×C42×C61D416yes—
39C24×C61C2411yescharacteristic
40C23×D61—23no—
41C23×D61—23no—
42C23×D61—44no—
43C23×D61C2×Q812yes—
44C2×Q8×C61—28no—
45C22×C4×C61C2×D4112yes—
46C22×C4×C61C2×D413yes—
47C22×C4×C61C2×D413yes—
48C22×C4×C61C2×Q812yes—
49C22×C4×C61#4116yes—
50C22×Dic61—23no—
51C22×Dic61—23no—
52C22×Dic61—26no—
53C22×Dic61C2×Q812yes—
54C2×C4×D61—212no—
55C2×C4×D61—212no—
56C2×C4×D61—43no—
57C2×C4×D61—41no—
58C2×C4×D61—46no—
59C2×C4×D61—46no—
60C4⋊C4×C61—224no—
61C4⋊C4×C61—212no—
62C22⋊C4×C61—212no—
63C42×C61—26no—
64C42×C61—26no—
65C22⋊Dic61—212no—
66#11758198—424no—
67C4⋊Dic61—46no—
68C4×Dic61—46no—
69C23×C61—42no—
70C23×C61C22×D413yes—
71C23×C61C22×Q811yescharacteristic
72C23×C61C2×#4113yes—
73C22×D61—212no—
74C22×D61—26no—
75C22×D61—44no—
76C22×D61—412no—
77C22×D61—46no—
78C22×D61—41no—
79C22×D61—43no—
80Q8×C61—88no—
81C2×C4×C61—212no—
82C2×C4×C61—412no—
83C2×C4×C61#12013yes—
84C2×C4×C61#12011yescharacteristic
85C2×C4×C61C22⋊Q816yes—
86C2×C4×C61C22⋊Q816yes—
87C2×C4×C61#12416yes—
88C2×C4×C61#12416yes—
89C2×Dic61—26no—
90C2×Dic61—41no—
91C2×Dic61—43no—
92C2×Dic61—412no—
93C2×Dic61—46no—
94C4×D61—412no—
95C4×D61—44no—
96#23467—611nomaximal, Sylow
97C22×C61—26no—
98C22×C61—41no—
99C22×C61—43no—
100C22×C61C2×#12011yescharacteristic
101C22×C61C2×C22⋊Q813yes—
102C22×C61C2×#12413yes—
103C2×D61—212no—
104C2×D61—412no—
105C2×D61—46no—
106C2×D61—42no—
107C2×D61—44no—
108C4×C61—412no—
109C4×C61#39316yes—
110C4×C61#39312yes—
111Dic61—42no—
112Dic61—86no—
113#3458D6111yescharacteristic, complemented
114#2280—613no—
115#2030—614no—
116#2029—614no—
117#1740—613no—
118C2×C61—23no—
119C2×C61—23no—
120C2×C61C2×#39311yescharacteristic
121C2×C61#302512yes—
122D61—44no—
123C24×C4—611no—
124#544—26no—
125#543C2×D6114yes—
126#533—26no—
127C2×#117—613no—
128C2×#117C2×D6113yes—
129#381—616no—
130#380—1226no—
131#379—616no—
132C4×C22⋊C4—1226no—
133#342—6112no—
134C61#2346711yescharacteristic, complemented, Sylow
135C25—611no—
136C23×C4—611no—
137C23×C4—614no—
138C23×C4—1226no—
139C23×C4—1223no—
140C23×C4—1223no—
141C23×C4C22×D6111yescharacteristic
142C4⋊Q8—212no—
143#125—212no—
144C22⋊Q8—224no—
145C4×Q8—28no—
146#117—26no—
147#117—26no—
148#117—12212no—
149C2×C4⋊C4—1223no—
150C2×C4⋊C4C22⋊D6116yes—
151C2×C22⋊C4—616no—
152C2×C22⋊C4—12212no—
153C2×C22⋊C4C22×D6116yes—
154C2×C42—1223no—
155C2×C42C22⋊D6116yes—
156#95—12212no—
157C24—612no—
158C24—1223no—
159C24—1223no—
160C24—2444no—
161C24C23×D6111yescharacteristic
162C2×Q8—28no—
163C22×C4—612no—
164C22×C4—1223no—
165C22×C4—1223no—
166C22×C4—12212no—
167C22×C4—12212no—
168C22×C4—1226no—
169C22×C4—2443no—
170C22×C4—2441no—
171C22×C4—2446no—
172C22×C4—2446no—
173C22×C4Q8×D6112yes—
174C22×C4#1175822616yes—
175C22×C4C2×C22⋊D61112yes—
176C22×C4C2×C22⋊D6113yes—
177C22×C4C2×C22⋊D6113yes—
178C4⋊C4—224no—
179C4⋊C4—212no—
180C4⋊C4—2446no—
181C22⋊C4—212no—
182C22⋊C4—12212no—
183C22⋊C4—24424no—
184C42—26no—
185C42—26no—
186C42—2446no—
187C23—42no—
188C23—12212no—
189C23—1226no—
190C23—2446no—
191C23—2444no—
192C23—24412no—
193C23—2441no—
194C23—2443no—
195C23C22×C22⋊D6113yes—
196C23C2×Q8×D6111yescharacteristic
197C23C2×#1175822613yes—
198Q8—88no—
199C2×C4—212no—
200C2×C4—412no—
201C2×C4—1226no—
202C2×C4—24412no—
203C2×C4—2446no—
204C2×C4—2441no—
205C2×C4—2443no—
206C2×C4—24412no—
207C2×C4—2444no—
208C2×C4#1451974613yes—
209C2×C4#1451974611yescharacteristic
210C2×C4#1451975316yes—
211C2×C4#1451975316yes—
212C2×C4#1451976616yes—
213C2×C4#1451976616yes—
214C22—26no—
215C22—41no—
216C22—43no—
217C22—12212no—
218C22—24412no—
219C22—2446no—
220C22—2442no—
221C22—2444no—
222C22C2×#1451974611yescharacteristic
223C22C2×#1451975313yes—
224C22C2×#1451976613yes—
225C4—412no—
226C4—4886no—
227C4—2442no—
228C4#1683967316yes—
229C4#1683967312yes—
230C2—23no—
231C2—23no—
232C2—2444no—
233C2C2×#1683967311yescharacteristic
234C2#2145521112yes—
subgroupG/H
centerC2×C4#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#3458C2
socleC22×C61C2×#120
prime radicalC24×D61C23
hypercenter#3458D61
hypocenterC61#23467
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #3458

Complements

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

NH ≅ G/Ntype
2. #21446985C2semidirect
4. #21445552C2semidirect
5. C61×#3458C2semidirect
22. C22×C4×D61Q8semidirect
113. #3458D61semidirect
134. C61#23467semidirect

Maximal subgroups

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

subgroupindex
#214481982
#214469852
#214469822
#214455522
C61×#34582
#2346761