Finite Group Databasebeta

← C22×C31×#355 #21490122

Subgroups — C22×C31×#355 #21490122

Order 7936.

1382 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

normal subgroups only

This group has 1382 conjugacy classes of subgroups — too many to draw. Only its 382 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
1C22×C31×#124C211yesmaximal, characteristic
2C22×C31×#101C212yesmaximal, complemented
3C2×C31×#355C2112yesmaximal, direct product factor
4C23×Q8×C31C2211yescharacteristic
5C23×D4×C31C411yescharacteristic
6C22×M16×C31—22no—
7C2×C31×#124—26no—
8C2×C31×#124C416yes—
9C2×C31×#124C2216yes—
10C22×C22⋊C4×C31—22no—
11C22×C42×C31C411yescharacteristic
12C2×C31×#101—26no—
13C2×C31×#101C22112yescomplemented
14C31×#355C22116yesdirect product factor
15C25×C31—21no—
16C22×Q8×C31—43no—
17C22×Q8×C31—41no—
18C22×Q8×C31C2×C413yes—
19C22×Q8×C31D416yes—
20C22×Q8×C31C2313yes—
21C22×D4×C31—23no—
22C22×D4×C31—43no—
23C22×D4×C31—41no—
24C22×D4×C31C2×C416yes—
25C23×C4×C31—21no—
26C23×C4×C31C2×C411yescharacteristic
27C23×C4×C31D412yes—
28C2×M16×C31—212no—
29C2×M16×C31—46no—
30C22×C8×C31—42no—
31C31×#124—212no—
32C31×#124—212no—
33C31×#124C2×C4112yes—
34C31×#124C2314yes—
35C2×C22⋊C4×C31—212no—
36C2×C22⋊C4×C31—212no—
37C2×C42×C31—23no—
38C2×C42×C31C2×C416yes—
39C31×#101—212no—
40C31×#101C2318yescomplemented
41C24×C31—26no—
42C24×C31—48no—
43C24×C31—43no—
44C24×C31—41no—
45C24×C31C22⋊C411yescharacteristic
46C2×Q8×C31—43no—
47C2×Q8×C31—43no—
48C2×Q8×C31—46no—
49C2×Q8×C31—46no—
50C2×Q8×C31—43no—
51C2×Q8×C31—43no—
52C2×Q8×C31C22⋊C416yes—
53C2×Q8×C31C22×C413yes—
54C2×Q8×C31C2×D416yes—
55C2×Q8×C31C2411yescharacteristic
56C2×D4×C31—26no—
57C2×D4×C31—46no—
58C2×D4×C31—46no—
59C2×D4×C31—46no—
60C2×D4×C31—46no—
61C2×D4×C31C22×C414yes—
62C22×C4×C31—26no—
63C22×C4×C31—23no—
64C22×C4×C31—22no—
65C22×C4×C31—21no—
66C22×C4×C31—26no—
67C22×C4×C31—44no—
68C22×C4×C31C22⋊C413yes—
69C22×C4×C31C22⋊C416yes—
70C22×C4×C31C22×C413yes—
71C22×C4×C31C2×D416yes—
72M16×C31—28no—
73M16×C31—412no—
74C2×C8×C31—412no—
75C22⋊C4×C31—224no—
76C22⋊C4×C31—28no—
77C42×C31—26no—
78C42×C31C22×C414yes—
79C22×#355C3111yesmaximal, characteristic, direct product factor, Sylow
80C23×C31—24no—
81C23×C31—24no—
82C23×C31—424no—
83C23×C31—46no—
84C23×C31—46no—
85C23×C31—812no—
86C23×C31—84no—
87C23×C31#9913yes—
88C23×C31#9911yescharacteristic
89C23×C31C2×C22⋊C413yes—
90Q8×C31—46no—
91Q8×C31—43no—
92Q8×C31—46no—
93Q8×C31—41no—
94D4×C31—44no—
95D4×C31—412no—
96C2×C4×C31—26no—
97C2×C4×C31—26no—
98C2×C4×C31—23no—
99C2×C4×C31—23no—
100C2×C4×C31—26no—
101C2×C4×C31—26no—
102C2×C4×C31—23no—
103C2×C4×C31—23no—
104C2×C4×C31—24no—
105C2×C4×C31—412no—
106C2×C4×C31C2×C22⋊C413yes—
107C2×C4×C31C2×C22⋊C416yes—
108C2×C4×C31C23×C411yescharacteristic
109C2×C4×C31C22×D412yes—
110C8×C31—48no—
111C22×#124C2×C3111yescharacteristic
112C22×#101C2×C3112yescomplemented
113C2×#355C2×C31112yesdirect product factor
114C22×C31—26no—
115C22×C31—26no—
116C22×C31—48no—
117C22×C31—44no—
118C22×C31—812no—
119C22×C31—812no—
120C22×C31#35514yesdirect product factor
121C22×C31C2×#9913yes—
122C22×C31C2×#9913yes—
123C22×C31C22×C22⋊C411yescharacteristic
124C4×C31—22no—
125C4×C31—23no—
126C4×C31—26no—
127C4×C31—21no—
128C4×C31—44no—
129C23×Q8C22×C3111yescharacteristic
130C23×D4C4×C3111yescharacteristic
131C22×M16—22no—
132C2×#124—26no—
133C2×#124C4×C3116yes—
134C2×#124C22×C3116yes—
135C22×C22⋊C4—22no—
136C22×C42C4×C3111yescharacteristic
137C2×#101—26no—
138C2×#101C22×C31112yescomplemented
139#355C22×C31116yesdirect product factor
140C2×C31—21no—
141C2×C31—23no—
142C2×C31—84no—
143C2×C31C2×#35516yesdirect product factor
144C2×C31C22×#9911yescharacteristic
145C25—21no—
146C22×Q8—41no—
147C22×Q8—43no—
148C22×Q8C2×C4×C3113yes—
149C22×Q8D4×C3116yes—
150C22×Q8C23×C3113yes—
151C22×D4—23no—
152C22×D4—43no—
153C22×D4—41no—
154C22×D4C2×C4×C3116yes—
155C23×C4—21no—
156C23×C4C2×C4×C3111yescharacteristic
157C23×C4D4×C3112yes—
158C2×M16—212no—
159C2×M16—46no—
160C22×C8—42no—
161#124—212no—
162#124—212no—
163#124C2×C4×C31112yes—
164#124C23×C3114yes—
165C2×C22⋊C4—212no—
166C2×C22⋊C4—212no—
167C2×C42—23no—
168C2×C42C2×C4×C3116yes—
169#101—212no—
170#101C23×C3118yescomplemented
171C31C22×#35511yescharacteristic, direct product factor, Sylow
172C24—26no—
173C24—48no—
174C24—43no—
175C24—41no—
176C24C22⋊C4×C3111yescharacteristic
177C2×Q8—43no—
178C2×Q8—43no—
179C2×Q8—46no—
180C2×Q8—46no—
181C2×Q8—43no—
182C2×Q8—43no—
183C2×Q8C22⋊C4×C3116yes—
184C2×Q8C22×C4×C3113yes—
185C2×Q8C2×D4×C3116yes—
186C2×Q8C24×C3111yescharacteristic
187C2×D4—26no—
188C2×D4—46no—
189C2×D4—46no—
190C2×D4—46no—
191C2×D4—46no—
192C2×D4C22×C4×C3114yes—
193C22×C4—22no—
194C22×C4—26no—
195C22×C4—23no—
196C22×C4—21no—
197C22×C4—26no—
198C22×C4—44no—
199C22×C4C22⋊C4×C3113yes—
200C22×C4C22⋊C4×C3116yes—
201C22×C4C22×C4×C3113yes—
202C22×C4C2×D4×C3116yes—
203M16—28no—
204M16—412no—
205C2×C8—412no—
206C22⋊C4—224no—
207C22⋊C4—28no—
208C42—26no—
209C42C22×C4×C3114yes—
210C23—24no—
211C23—24no—
212C23—424no—
213C23—46no—
214C23—46no—
215C23—812no—
216C23—84no—
217C23C31×#9913yes—
218C23C31×#9911yescharacteristic
219C23C2×C22⋊C4×C3113yes—
220Q8—46no—
221Q8—43no—
222Q8—46no—
223Q8—41no—
224D4—44no—
225D4—412no—
226C2×C4—26no—
227C2×C4—26no—
228C2×C4—23no—
229C2×C4—23no—
230C2×C4—26no—
231C2×C4—26no—
232C2×C4—23no—
233C2×C4—23no—
234C2×C4—24no—
235C2×C4—412no—
236C2×C4C2×C22⋊C4×C3113yes—
237C2×C4C2×C22⋊C4×C3116yes—
238C2×C4C23×C4×C3111yescharacteristic
239C2×C4C22×D4×C3112yes—
240C8—48no—
241C22—26no—
242C22—26no—
243C22—48no—
244C22—44no—
245C22—812no—
246C22—812no—
247C22C31×#35514yesdirect product factor
248C22C2×C31×#9913yes—
249C22C2×C31×#9913yes—
250C22C22×C22⋊C4×C3111yescharacteristic
251C4—22no—
252C4—23no—
253C4—26no—
254C4—21no—
255C4—44no—
256C2—21no—
257C2—23no—
258C2—84no—
259C2C2×C31×#35516yesdirect product factor
260C2C22×C31×#9911yescharacteristic
subgroupG/H
center, socleC23×C31#99
commutatorC2×C4C23×C4×C31
Frattini, cosocleC2×Q8C24×C31
prime radicalC23×D4×C31C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C24×C31 ◁ C23×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C31×#101C2semidirect
3. C2×C31×#355C2direct (N × H)
13. C2×C31×#101C22semidirect
14. C31×#355C22direct (N × H)
40. C31×#101C23semidirect
79. C22×#355C31direct (N × H)
112. C22×#101C2×C31semidirect
113. C2×#355C2×C31direct (N × H)
120. C22×C31#355direct (N × H)
138. C2×#101C22×C31semidirect
139. #355C22×C31direct (N × H)
143. C2×C31C2×#355direct (N × H)
170. #101C23×C31semidirect
171. C31C22×#355direct (N × H)
247. C22C31×#355direct (N × H)
259. C2C2×C31×#355direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×#1242
C22×C31×#1012
C2×C31×#3552
C22×#35531