Finite Group Databasebeta

← C31×#7563 #21464322

Subgroups — C31×#7563 #21464322

Order 7936.

690 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2297C211yesmaximal, characteristic
2C4×C31×C22⋊C8C212yesmaximal, complemented
3C31×#1272C212yesmaximal, complemented
4C31×#1270C212yesmaximal, complemented
5C22×#41×C31C411yescharacteristic
6C2×C31×#117C411yescharacteristic
7C22×C42×C31C2211yescharacteristic
8C2×C31×C22⋊C8—22no—
9C2×C4×C8×C31—22no—
10C31×#391C411yescharacteristic
11C31×#391C2211yescharacteristic
12C31×#390C411yescharacteristic
13C31×#390C2211yescharacteristic
14C31×#386—22no—
15C4×C22⋊C4×C31—22no—
16C31×#336C2214yes—
17C2×#41×C31—22no—
18C2×#41×C31—42no—
19C2×#41×C31—42no—
20C2×#41×C31C814yescomplemented
21C22×Q8×C31C2×C411yescharacteristic
22C22×D4×C31—21no—
23C22×D4×C31C2×C411yescharacteristic
24C23×C4×C31—21no—
25C23×C4×C31C2×C411yescharacteristic
26C23×C4×C31D412yes—
27C22×C8×C31—24no—
28C31×#117—42no—
29C2×C4⋊C4×C31C2×C411yescharacteristic
30C2×C4⋊C4×C31C2×C411yescharacteristic
31C2×C22⋊C4×C31—21no—
32C2×C22⋊C4×C31—22no—
33C2×C22⋊C4×C31—22no—
34C2×C42×C31—21no—
35C2×C42×C31—42no—
36C2×C42×C31C2×C411yescharacteristic
37C2×C42×C31D412yes—
38C2×C42×C31C2311yescharacteristic
39C31×C22⋊C8—28no—
40C4×C8×C31—44no—
41C31×#95—22no—
42C31×#95—22no—
43C24×C31—21no—
44C24×C31C22⋊C411yescharacteristic
45#41×C31—48no—
46#41×C31—48no—
47C2×Q8×C31—21no—
48C2×Q8×C31—41no—
49C2×Q8×C31—41no—
50C2×Q8×C31C2×C812yes—
51C2×D4×C31—24no—
52C2×D4×C31—21no—
53C2×D4×C31—41no—
54C2×D4×C31—41no—
55C2×D4×C31—42no—
56C2×D4×C31—42no—
57C2×D4×C31C2×C812yes—
58C22×C4×C31—22no—
59C22×C4×C31—22no—
60C22×C4×C31—21no—
61C22×C4×C31—24no—
62C22×C4×C31—24no—
63C22×C4×C31—24no—
64C22×C4×C31—21no—
65C22×C4×C31—21no—
66C22×C4×C31—42no—
67C22×C4×C31—42no—
68C22×C4×C31—42no—
69C22×C4×C31C22⋊C412yes—
70C22×C4×C31C22⋊C411yescharacteristic
71C22×C4×C31C2×C812yes—
72C22×C4×C31M1612yes—
73C22×C4×C31C22×C411yescharacteristic
74C22×C4×C31C2×D412yes—
75C2×C8×C31—28no—
76C2×C8×C31—44no—
77C2×C8×C31—44no—
78C4⋊C4×C31—41no—
79C4⋊C4×C31—41no—
80C22⋊C4×C31—42no—
81C22⋊C4×C31—44no—
82C22⋊C4×C31—44no—
83C42×C31—24no—
84C42×C31—41no—
85C42×C31—41no—
86C42×C31—42no—
87#7563C3111yesmaximal, characteristic, direct product factor, Sylow
88C23×C31—21no—
89C23×C31—21no—
90C23×C31—22no—
91C23×C31—22no—
92C23×C31—22no—
93C23×C31—41no—
94C23×C31—41no—
95C23×C31—42no—
96C23×C31—42no—
97C23×C31C22⋊C812yes—
98C23×C31C2×C22⋊C411yescharacteristic
99Q8×C31—42no—
100Q8×C31—42no—
101D4×C31—42no—
102D4×C31—42no—
103D4×C31—48no—
104C2×C4×C31—22no—
105C2×C4×C31—22no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—22no—
109C2×C4×C31—22no—
110C2×C4×C31—24no—
111C2×C4×C31—24no—
112C2×C4×C31—22no—
113C2×C4×C31—48no—
114C2×C4×C31—41no—
115C2×C4×C31—41no—
116C2×C4×C31—48no—
117C2×C4×C31—41no—
118C2×C4×C31—41no—
119C2×C4×C31—42no—
120C2×C4×C31—42no—
121C2×C4×C31C22⋊C812yes—
122C2×C4×C31#9912yes—
123C2×C4×C31#10012yes—
124C2×C4×C31#10414yes—
125C2×C4×C31C22×C811yescharacteristic
126C2×C4×C31C2×M1611yescharacteristic
127C2×C4×C31#13514yes—
128C8×C31—48no—
129#2297C2×C3111yescharacteristic
130C4×C22⋊C8C2×C3112yescomplemented
131#1272C2×C3112yescomplemented
132#1270C2×C3112yescomplemented
133C22×C31—22no—
134C22×C31—24no—
135C22×C31—41no—
136C22×C31—42no—
137C22×C31—41no—
138C22×C31—42no—
139C22×C31—44no—
140C22×C31—81no—
141C22×C31—82no—
142C22×C31—81no—
143C22×C31C2×C22⋊C811yescharacteristic
144C22×C31C2×#9911yescharacteristic
145C22×C31C2×#10011yescharacteristic
146C22×C31#41612yes—
147C22×C31C2×#10412yes—
148C4×C31—21no—
149C4×C31—21no—
150C4×C31—24no—
151C4×C31—24no—
152C4×C31—42no—
153C4×C31—81no—
154C4×C31—81no—
155C4×C31#32314yes—
156C22×#41C4×C3111yescharacteristic
157C2×#117C4×C3111yescharacteristic
158C22×C42C22×C3111yescharacteristic
159C2×C22⋊C8—22no—
160C2×C4×C8—22no—
161#391C4×C3111yescharacteristic
162#391C22×C3111yescharacteristic
163#390C4×C3111yescharacteristic
164#390C22×C3111yescharacteristic
165#386—22no—
166C4×C22⋊C4—22no—
167#336C22×C3114yes—
168C2×C31—42no—
169C2×C31—82no—
170C2×C31C2×#32311yescharacteristic
171C2×C31#147612yes—
172C2×C31#150112yes—
173C2×C31#152511yescharacteristic
174C2×C31#154311yescharacteristic
175C2×#41—22no—
176C2×#41—42no—
177C2×#41—42no—
178C2×#41C8×C3114yescomplemented
179C22×Q8C2×C4×C3111yescharacteristic
180C22×D4—21no—
181C22×D4C2×C4×C3111yescharacteristic
182C23×C4—21no—
183C23×C4C2×C4×C3111yescharacteristic
184C23×C4D4×C3112yes—
185C22×C8—24no—
186#117—42no—
187C2×C4⋊C4C2×C4×C3111yescharacteristic
188C2×C4⋊C4C2×C4×C3111yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C42—21no—
193C2×C42—42no—
194C2×C42C2×C4×C3111yescharacteristic
195C2×C42D4×C3112yes—
196C2×C42C23×C3111yescharacteristic
197C22⋊C8—28no—
198C4×C8—44no—
199#95—22no—
200#95—22no—
201C31#756311yescharacteristic, direct product factor, Sylow
202C24—21no—
203C24C22⋊C4×C3111yescharacteristic
204#41—48no—
205#41—48no—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8C2×C8×C3112yes—
210C2×D4—24no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4C2×C8×C3112yes—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4C22⋊C4×C3112yes—
229C22×C4C22⋊C4×C3111yescharacteristic
230C22×C4C2×C8×C3112yes—
231C22×C4M16×C3112yes—
232C22×C4C22×C4×C3111yescharacteristic
233C22×C4C2×D4×C3112yes—
234C2×C8—28no—
235C2×C8—44no—
236C2×C8—44no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—44no—
241C22⋊C4—44no—
242C42—24no—
243C42—41no—
244C42—41no—
245C42—42no—
246C23—21no—
247C23—21no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—41no—
252C23—41no—
253C23—42no—
254C23—42no—
255C23C31×C22⋊C812yes—
256C23C2×C22⋊C4×C3111yescharacteristic
257Q8—42no—
258Q8—42no—
259D4—42no—
260D4—42no—
261D4—48no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—48no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—48no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4C31×C22⋊C812yes—
280C2×C4C31×#9912yes—
281C2×C4C31×#10012yes—
282C2×C4C31×#10414yes—
283C2×C4C22×C8×C3111yescharacteristic
284C2×C4C2×M16×C3111yescharacteristic
285C2×C4C31×#13514yes—
286C8—48no—
287C22—22no—
288C22—24no—
289C22—41no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—44no—
294C22—81no—
295C22—82no—
296C22—81no—
297C22C2×C31×C22⋊C811yescharacteristic
298C22C2×C31×#9911yescharacteristic
299C22C2×C31×#10011yescharacteristic
300C22C31×#41612yes—
301C22C2×C31×#10412yes—
302C4—21no—
303C4—21no—
304C4—24no—
305C4—24no—
306C4—42no—
307C4—81no—
308C4—81no—
309C4C31×#32314yes—
310C2—42no—
311C2—82no—
312C2C2×C31×#32311yescharacteristic
313C2C31×#147612yes—
314C2C31×#150112yes—
315C2C31×#152511yescharacteristic
316C2C31×#154311yescharacteristic
subgroupG/H
centerC22×C4×C31C22⋊C4
commutatorC2×C4C22×C8×C31
Frattini, cosocleC2×C42C23×C31
socleC23×C31C2×C22⋊C4
prime radicalC22×#41×C31C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×C31×C22⋊C8C2semidirect
3. C31×#1272C2semidirect
4. C31×#1270C2semidirect
20. C2×#41×C31C8semidirect
87. #7563C31direct (N × H)
130. C4×C22⋊C8C2×C31semidirect
131. #1272C2×C31semidirect
132. #1270C2×C31semidirect
178. C2×#41C8×C31semidirect
201. C31#7563direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#22972
C4×C31×C22⋊C82
C31×#12722
C31×#12702
#756331