Finite Group Databasebeta

← C31×#8100 #21464859

Subgroups — C31×#8100 #21464859

Order 7936.

994 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2280C212yesmaximal
2C31×#2280C211yesmaximal, characteristic
3C31×#1304C214yesmaximal
4C24×C4×C31C2211yescharacteristic
5C2×C31×#117—22no—
6C2×C31×#117C412yescomplemented
7C31×#381—22no—
8C31×#381C412yescomplemented
9C31×#380—22no—
10C31×#380—22no—
11C31×#380C414yescomplemented
12C31×#379C2214yes—
13C31×#379C2212yes—
14C4×C22⋊C4×C31—22no—
15C4×C22⋊C4×C31—22no—
16C4×C22⋊C4×C31C414yescomplemented
17C25×C31C2311yescharacteristic
18C23×C4×C31—44no—
19C23×C4×C31C2×C411yescharacteristic
20C23×C4×C31C2×C412yes—
21C23×C4×C31C2×C412yes—
22C23×C4×C31C2×C411yescharacteristic
23C23×C4×C31D412yes—
24C23×C4×C31D411yescharacteristic
25C23×C4×C31D411yescharacteristic
26C23×C4×C31D412yes—
27C23×C4×C31Q811yescharacteristic
28C23×C4×C31Q811yescharacteristic
29C31×#117—48no—
30C31×#117—44no—
31C2×C4⋊C4×C31—24no—
32C2×C4⋊C4×C31—22no—
33C2×C22⋊C4×C31—24no—
34C2×C22⋊C4×C31—24no—
35C2×C22⋊C4×C31—24no—
36C2×C22⋊C4×C31C2×C414yes—
37C2×C22⋊C4×C31C2×C418yes—
38C2×C42×C31—24no—
39C2×C42×C31—22no—
40C31×#95—24no—
41C31×#95—24no—
42C31×#95—24no—
43C24×C31—42no—
44C24×C31—44no—
45C24×C31C22×C411yescharacteristic
46C24×C31C22×C412yes—
47C24×C31C2×D412yes—
48C24×C31C2×D411yescharacteristic
49C24×C31C2×Q811yescharacteristic
50C22×C4×C31—24no—
51C22×C4×C31—28no—
52C22×C4×C31—28no—
53C22×C4×C31—24no—
54C22×C4×C31—44no—
55C22×C4×C31—44no—
56C22×C4×C31—42no—
57C22×C4×C31—42no—
58C22×C4×C31—42no—
59C22×C4×C31—42no—
60C22×C4×C31—42no—
61C22×C4×C31—44no—
62C22×C4×C31C4211yescharacteristic
63C22×C4×C31C4211yescharacteristic
64C22×C4×C31C4212yes—
65C22×C4×C31C22⋊C412yes—
66C22×C4×C31C22⋊C411yescharacteristic
67C22×C4×C31C22⋊C412yes—
68C22×C4×C31C22⋊C412yes—
69C22×C4×C31C22⋊C412yes—
70C22×C4×C31C22⋊C412yes—
71C22×C4×C31C22⋊C411yescharacteristic
72C22×C4×C31C4⋊C411yescharacteristic
73C22×C4×C31C4⋊C412yes—
74C22×C4×C31C4⋊C411yescharacteristic
75C22×C4×C31C4⋊C412yes—
76C22×C4×C31C4⋊C412yes—
77C22×C4×C31C4⋊C412yes—
78C22×C4×C31C4⋊C411yescharacteristic
79C22×C4×C31C4⋊C411yescharacteristic
80C4⋊C4×C31—44no—
81C4⋊C4×C31—48no—
82C22⋊C4×C31—48no—
83C22⋊C4×C31—48no—
84C22⋊C4×C31—48no—
85C22⋊C4×C31—48no—
86C22⋊C4×C31—44no—
87C42×C31—48no—
88C42×C31—44no—
89#8100C3111yesmaximal, characteristic, direct product factor, Sylow
90C23×C31—22no—
91C23×C31—24no—
92C23×C31—24no—
93C23×C31—24no—
94C23×C31—24no—
95C23×C31—42no—
96C23×C31—41no—
97C23×C31—41no—
98C23×C31—42no—
99C23×C31—42no—
100C23×C31—42no—
101C23×C31—42no—
102C23×C31—416no—
103C23×C31C2×C4211yescharacteristic
104C23×C31C2×C22⋊C411yescharacteristic
105C23×C31C2×C22⋊C412yes—
106C23×C31C2×C4⋊C411yescharacteristic
107C23×C31C2×C4⋊C412yes—
108C2×C4×C31—22no—
109C2×C4×C31—24no—
110C2×C4×C31—24no—
111C2×C4×C31—24no—
112C2×C4×C31—22no—
113C2×C4×C31—22no—
114C2×C4×C31—22no—
115C2×C4×C31—24no—
116C2×C4×C31—22no—
117C2×C4×C31—24no—
118C2×C4×C31—42no—
119C2×C4×C31—42no—
120C2×C4×C31—42no—
121C2×C4×C31—42no—
122C2×C4×C31—42no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—48no—
126C2×C4×C31—48no—
127C2×C4×C31—48no—
128C2×C4×C31—44no—
129C2×C4×C31—48no—
130C2×C4×C31#9511yescharacteristic
131C2×C4×C31#9511yescharacteristic
132C2×C4×C31#9512yes—
133C2×C4×C31#9512yes—
134C2×C4×C31#9511yescharacteristic
135C2×C4×C31#9511yescharacteristic
136C2×C4×C31#9914yes—
137#2280C2×C3112yes—
138#2280C2×C3111yescharacteristic
139#1304C2×C3114yes—
140C22×C31—22no—
141C22×C31—22no—
142C22×C31—22no—
143C22×C31—21no—
144C22×C31—21no—
145C22×C31—22no—
146C22×C31—22no—
147C22×C31—22no—
148C22×C31—22no—
149C22×C31—22no—
150C22×C31—48no—
151C22×C31—44no—
152C22×C31—44no—
153C22×C31—48no—
154C22×C31—44no—
155C22×C31C2×#9511yescharacteristic
156C22×C31C2×#9912yes—
157C22×C31#41014yes—
158C4×C31—21no—
159C4×C31—21no—
160C4×C31—22no—
161C4×C31—22no—
162C4×C31—84no—
163C4×C31—88no—
164C4×C31#34212yes—
165C4×C31#34212yes—
166C24×C4C22×C3111yescharacteristic
167C2×#117—22no—
168C2×#117C4×C3112yescomplemented
169#381—22no—
170#381C4×C3112yescomplemented
171#380—22no—
172#380—22no—
173#380C4×C3114yescomplemented
174#379C22×C3114yes—
175#379C22×C3112yes—
176C4×C22⋊C4—22no—
177C4×C22⋊C4—22no—
178C4×C22⋊C4C4×C3114yescomplemented
179C2×C31—22no—
180C2×C31—24no—
181C2×C31—41no—
182C2×C31—41no—
183C2×C31—42no—
184C2×C31C2×#34211yescharacteristic
185C2×C31#174012yes—
186C25C23×C3111yescharacteristic
187C23×C4—44no—
188C23×C4C2×C4×C3111yescharacteristic
189C23×C4C2×C4×C3112yes—
190C23×C4C2×C4×C3112yes—
191C23×C4C2×C4×C3111yescharacteristic
192C23×C4D4×C3112yes—
193C23×C4D4×C3111yescharacteristic
194C23×C4D4×C3111yescharacteristic
195C23×C4D4×C3112yes—
196C23×C4Q8×C3111yescharacteristic
197C23×C4Q8×C3111yescharacteristic
198#117—48no—
199#117—44no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4C2×C4×C3114yes—
206C2×C22⋊C4C2×C4×C3118yes—
207C2×C42—24no—
208C2×C42—22no—
209#95—24no—
210#95—24no—
211#95—24no—
212C31#810011yescharacteristic, direct product factor, Sylow
213C24—42no—
214C24—44no—
215C24C22×C4×C3111yescharacteristic
216C24C22×C4×C3112yes—
217C24C2×D4×C3112yes—
218C24C2×D4×C3111yescharacteristic
219C24C2×Q8×C3111yescharacteristic
220C22×C4—24no—
221C22×C4—28no—
222C22×C4—28no—
223C22×C4—24no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4C42×C3111yescharacteristic
233C22×C4C42×C3111yescharacteristic
234C22×C4C42×C3112yes—
235C22×C4C22⋊C4×C3112yes—
236C22×C4C22⋊C4×C3112yes—
237C22×C4C22⋊C4×C3111yescharacteristic
238C22×C4C22⋊C4×C3112yes—
239C22×C4C22⋊C4×C3112yes—
240C22×C4C22⋊C4×C3112yes—
241C22×C4C22⋊C4×C3111yescharacteristic
242C22×C4C4⋊C4×C3111yescharacteristic
243C22×C4C4⋊C4×C3112yes—
244C22×C4C4⋊C4×C3111yescharacteristic
245C22×C4C4⋊C4×C3112yes—
246C22×C4C4⋊C4×C3112yes—
247C22×C4C4⋊C4×C3112yes—
248C22×C4C4⋊C4×C3111yescharacteristic
249C22×C4C4⋊C4×C3111yescharacteristic
250C4⋊C4—44no—
251C4⋊C4—48no—
252C22⋊C4—48no—
253C22⋊C4—48no—
254C22⋊C4—48no—
255C22⋊C4—48no—
256C22⋊C4—44no—
257C42—48no—
258C42—44no—
259C23—22no—
260C23—24no—
261C23—24no—
262C23—24no—
263C23—24no—
264C23—42no—
265C23—41no—
266C23—41no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—416no—
272C23C2×C42×C3111yescharacteristic
273C23C2×C22⋊C4×C3111yescharacteristic
274C23C2×C22⋊C4×C3112yes—
275C23C2×C4⋊C4×C3111yescharacteristic
276C23C2×C4⋊C4×C3112yes—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—48no—
295C2×C4—48no—
296C2×C4—48no—
297C2×C4—44no—
298C2×C4—48no—
299C2×C4C31×#9511yescharacteristic
300C2×C4C31×#9511yescharacteristic
301C2×C4C31×#9512yes—
302C2×C4C31×#9512yes—
303C2×C4C31×#9511yescharacteristic
304C2×C4C31×#9511yescharacteristic
305C2×C4C31×#9914yes—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—48no—
317C22—44no—
318C22—44no—
319C22—48no—
320C22—44no—
321C22C2×C31×#9511yescharacteristic
322C22C2×C31×#9912yes—
323C22C31×#41014yes—
324C4—21no—
325C4—21no—
326C4—22no—
327C4—22no—
328C4—84no—
329C4—88no—
330C4C31×#34212yes—
331C4C31×#34212yes—
332C2—22no—
333C2—24no—
334C2—41no—
335C2—41no—
336C2—42no—
337C2C2×C31×#34211yescharacteristic
338C2C31×#174012yes—
subgroupG/H
centerC2×C4×C31#95
commutatorC23C2×C42×C31
Frattini, cosocleC25C23×C31
socleC22×C31C2×#95
prime radicalC25×C31C23
LCSG ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×C31×#117C4semidirect
8. C31×#381C4semidirect
11. C31×#380C4semidirect
16. C4×C22⋊C4×C31C4semidirect
89. #8100C31direct (N × H)
168. C2×#117C4×C31semidirect
170. #381C4×C31semidirect
173. #380C4×C31semidirect
178. C4×C22⋊C4C4×C31semidirect
212. C31#8100direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#22802
C31×#13042
#810031