Finite Group Databasebeta

← C22×C31×#324 #21474103

Subgroups — C22×C31×#324 #21474103

Order 7936.

1806 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊C4×C31C211yesmaximal, characteristic
2C22×C31×C22⋊C8C212yesmaximal
3C2×C31×#324C2112yesmaximal, direct product factor
4C24×C4×C31C411yescharacteristic
5C24×C4×C31C411yescharacteristic
6C24×C4×C31C2211yescharacteristic
7C23×C8×C31—22no—
8C22×C4⋊C4×C31—26no—
9C22×C4⋊C4×C31—22no—
10C22×C4⋊C4×C31C416yes—
11C22×C4⋊C4×C31C2216yes—
12C2×C31×C22⋊C8—26no—
13C2×C31×C22⋊C8C22112yes—
14C31×#324C22116yesdirect product factor
15C25×C31C2×C411yescharacteristic
16C23×C4×C31—23no—
17C23×C4×C31—21no—
18C23×C4×C31—24no—
19C23×C4×C31—23no—
20C23×C4×C31—21no—
21C23×C4×C31—44no—
22C23×C4×C31—44no—
23C23×C4×C31C818yescomplemented
24C23×C4×C31C2×C416yes—
25C23×C4×C31C2×C416yes—
26C23×C4×C31C2×C413yes—
27C23×C4×C31D412yes—
28C23×C4×C31D416yes—
29C23×C4×C31C2313yes—
30C22×C8×C31—212no—
31C22×C8×C31—48no—
32C2×C4⋊C4×C31—212no—
33C2×C4⋊C4×C31—212no—
34C2×C4⋊C4×C31—212no—
35C2×C4⋊C4×C31—212no—
36C2×C4⋊C4×C31C2×C4112yes—
37C2×C4⋊C4×C31C2314yes—
38C31×C22⋊C8—212no—
39C31×C22⋊C8C2318yes—
40C24×C31—24no—
41C24×C31—24no—
42C24×C31C22⋊C413yes—
43C24×C31C22⋊C411yescharacteristic
44C24×C31C2×C814yes—
45C24×C31M1614yes—
46C24×C31C22×C413yes—
47C22×C4×C31—224no—
48C22×C4×C31—28no—
49C22×C4×C31—28no—
50C22×C4×C31—26no—
51C22×C4×C31—26no—
52C22×C4×C31—212no—
53C22×C4×C31—26no—
54C22×C4×C31—26no—
55C22×C4×C31—46no—
56C22×C4×C31—412no—
57C22×C4×C31—46no—
58C22×C4×C31—412no—
59C22×C4×C31—44no—
60C22×C4×C31—412no—
61C22×C4×C31C22⋊C416yes—
62C22×C4×C31C22⋊C416yes—
63C22×C4×C31C2×C8124yescomplemented
64C22×C4×C31C22×C414yes—
65C22×C4×C31C22×C414yes—
66C22×C4×C31C22×C413yes—
67C22×C4×C31C2×D416yes—
68C22×C4×C31C2×D416yes—
69C22×C4×C31C2411yescharacteristic
70C2×C8×C31—28no—
71C2×C8×C31—424no—
72C4⋊C4×C31—224no—
73C4⋊C4×C31—28no—
74C22×#324C3111yesmaximal, characteristic, direct product factor, Sylow
75C23×C31—216no—
76C23×C31—216no—
77C23×C31—26no—
78C23×C31—26no—
79C23×C31—26no—
80C23×C31—26no—
81C23×C31C22⋊C8112yes—
82C23×C31C22⋊C814yes—
83C23×C31#9914yes—
84C23×C31#10114yes—
85C23×C31C2×C22⋊C413yes—
86C23×C31C2×C22⋊C413yes—
87C23×C31C22×C816yes—
88C23×C31C2×M1616yes—
89C23×C31C23×C411yescharacteristic
90C2×C4×C31—24no—
91C2×C4×C31—24no—
92C2×C4×C31—224no—
93C2×C4×C31—224no—
94C2×C4×C31—212no—
95C2×C4×C31—212no—
96C2×C4×C31—24no—
97C2×C4×C31—46no—
98C2×C4×C31—43no—
99C2×C4×C31—46no—
100C2×C4×C31—41no—
101C2×C4×C31—412no—
102C2×C4×C31—412no—
103C2×C4×C31—44no—
104C2×C4×C31C2×C22⋊C416yes—
105C2×C4×C31C22×C818yescomplemented
106C2×C4×C31C22×D412yes—
107C8×C31—48no—
108C23×C4⋊C4C2×C3111yescharacteristic
109C22×C22⋊C8C2×C3112yes—
110C2×#324C2×C31112yesdirect product factor
111C22×C31—23no—
112C22×C31—21no—
113C22×C31—23no—
114C22×C31—21no—
115C22×C31—224no—
116C22×C31—212no—
117C22×C31—212no—
118C22×C31#324116yesdirect product factor
119C22×C31C2×C22⋊C816yes—
120C22×C31C2×C22⋊C816yes—
121C22×C31C2×#9916yes—
122C22×C31C2×#10116yes—
123C22×C31C22×C22⋊C411yescharacteristic
124C22×C31C23×C811yescharacteristic
125C22×C31C22×M1611yescharacteristic
126C4×C31—28no—
127C4×C31—22no—
128C4×C31—26no—
129C4×C31—44no—
130C24×C4C4×C3111yescharacteristic
131C24×C4C4×C3111yescharacteristic
132C24×C4C22×C3111yescharacteristic
133C23×C8—22no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4—26no—
136C22×C4⋊C4C4×C3116yes—
137C22×C4⋊C4C22×C3116yes—
138C2×C22⋊C8—26no—
139C2×C22⋊C8C22×C31112yes—
140#324C22×C31116yesdirect product factor
141C2×C31—21no—
142C2×C31—21no—
143C2×C31—26no—
144C2×C31C2×#324112yesdirect product factor
145C2×C31C22×C22⋊C811yescharacteristic
146C2×C31C22×#9911yescharacteristic
147C2×C31C22×#10111yescharacteristic
148C25C2×C4×C3111yescharacteristic
149C23×C4—23no—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—23no—
153C23×C4—21no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4C8×C3118yescomplemented
157C23×C4C2×C4×C3116yes—
158C23×C4C2×C4×C3116yes—
159C23×C4C2×C4×C3113yes—
160C23×C4D4×C3112yes—
161C23×C4D4×C3116yes—
162C23×C4C23×C3113yes—
163C22×C8—212no—
164C22×C8—48no—
165C2×C4⋊C4—212no—
166C2×C4⋊C4—212no—
167C2×C4⋊C4—212no—
168C2×C4⋊C4—212no—
169C2×C4⋊C4C2×C4×C31112yes—
170C2×C4⋊C4C23×C3114yes—
171C22⋊C8—212no—
172C22⋊C8C23×C3118yes—
173C31C22×#32411yescharacteristic, direct product factor, Sylow
174C24—24no—
175C24—24no—
176C24C22⋊C4×C3113yes—
177C24C22⋊C4×C3111yescharacteristic
178C24C2×C8×C3114yes—
179C24M16×C3114yes—
180C24C22×C4×C3113yes—
181C22×C4—224no—
182C22×C4—28no—
183C22×C4—28no—
184C22×C4—26no—
185C22×C4—26no—
186C22×C4—212no—
187C22×C4—26no—
188C22×C4—26no—
189C22×C4—46no—
190C22×C4—412no—
191C22×C4—46no—
192C22×C4—412no—
193C22×C4—44no—
194C22×C4—412no—
195C22×C4C22⋊C4×C3116yes—
196C22×C4C22⋊C4×C3116yes—
197C22×C4C2×C8×C31124yescomplemented
198C22×C4C22×C4×C3114yes—
199C22×C4C22×C4×C3114yes—
200C22×C4C22×C4×C3113yes—
201C22×C4C2×D4×C3116yes—
202C22×C4C2×D4×C3116yes—
203C22×C4C24×C3111yescharacteristic
204C2×C8—28no—
205C2×C8—424no—
206C4⋊C4—224no—
207C4⋊C4—28no—
208C23—26no—
209C23—26no—
210C23—26no—
211C23—26no—
212C23—216no—
213C23—216no—
214C23C31×C22⋊C8112yes—
215C23C31×C22⋊C814yes—
216C23C31×#9914yes—
217C23C31×#10114yes—
218C23C2×C22⋊C4×C3113yes—
219C23C2×C22⋊C4×C3113yes—
220C23C22×C8×C3116yes—
221C23C2×M16×C3116yes—
222C23C23×C4×C3111yescharacteristic
223C2×C4—224no—
224C2×C4—224no—
225C2×C4—212no—
226C2×C4—212no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—46no—
231C2×C4—43no—
232C2×C4—46no—
233C2×C4—41no—
234C2×C4—412no—
235C2×C4—412no—
236C2×C4—44no—
237C2×C4C2×C22⋊C4×C3116yes—
238C2×C4C22×C8×C3118yescomplemented
239C2×C4C22×D4×C3112yes—
240C8—48no—
241C22—23no—
242C22—21no—
243C22—23no—
244C22—21no—
245C22—224no—
246C22—212no—
247C22—212no—
248C22C31×#324116yesdirect product factor
249C22C2×C31×C22⋊C816yes—
250C22C2×C31×C22⋊C816yes—
251C22C2×C31×#9916yes—
252C22C2×C31×#10116yes—
253C22C22×C22⋊C4×C3111yescharacteristic
254C22C23×C8×C3111yescharacteristic
255C22C22×M16×C3111yescharacteristic
256C4—28no—
257C4—22no—
258C4—26no—
259C4—44no—
260C2—26no—
261C2—21no—
262C2—21no—
263C2C2×C31×#324112yesdirect product factor
264C2C22×C31×C22⋊C811yescharacteristic
265C2C22×C31×#9911yescharacteristic
266C2C22×C31×#10111yescharacteristic
subgroupG/H
center, socleC24×C31C22⋊C4
commutatorC22C23×C8×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC25×C31C2×C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C24×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C31×#324C2direct (N × H)
14. C31×#324C22direct (N × H)
23. C23×C4×C31C8semidirect
63. C22×C4×C31C2×C8semidirect
74. C22×#324C31direct (N × H)
105. C2×C4×C31C22×C8semidirect
110. C2×#324C2×C31direct (N × H)
118. C22×C31#324direct (N × H)
140. #324C22×C31direct (N × H)
144. C2×C31C2×#324direct (N × H)
156. C23×C4C8×C31semidirect
173. C31C22×#324direct (N × H)
197. C22×C4C2×C8×C31semidirect
238. C2×C4C22×C8×C31semidirect
248. C22C31×#324direct (N × H)
263. C2C2×C31×#324direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4⋊C4×C312
C22×C31×C22⋊C82
C2×C31×#3242
C22×#32431