Finite Group Databasebeta

← C31×#17632 #21474391

Subgroups — C31×#17632 #21474391

Order 7936.

874 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3502C211yesmaximal, characteristic
2C31×#2983C211yesmaximal, characteristic
3C31×#2356C211yesmaximal, characteristic
4C31×#1518C212yesmaximal
5C31×#1515C214yesmaximal
6C31×#1506C214yesmaximal
7C31×#1503C212yesmaximal
8Q82×C31C414yescomplemented
9C31×#557—24no—
10C31×#557C414yescomplemented
11C31×#544—21no—
12C31×#544—22no—
13C31×#533C2212yes—
14C31×#533C2214yes—
15C31×#519—21no—
16C2×C4×Q8×C31C2211yescharacteristic
17C31×#446—24no—
18C31×#432C2214yes—
19C31×#431C2212yes—
20C31×#391—21no—
21C31×#387—22no—
22C31×#383C2212yes—
23C31×#382C2211yescharacteristic
24C4×C4⋊C4×C31—21no—
25C4×C4⋊C4×C31C2211yescharacteristic
26C31×#376C2212yes—
27C31×#328C2214yes—
28C31×#328C2214yes—
29C31×#328C2218yes—
30C22×Q8×C31D411yescharacteristic
31C31×C4⋊Q8—28no—
32C31×C4⋊Q8—24no—
33C31×C4⋊Q8C2×C412yes—
34C31×C4⋊Q8C2×C414yes—
35C31×C4⋊Q8C2×C412yes—
36C31×C4⋊Q8C2×C414yes—
37C31×#125—28no—
38C31×#125—24no—
39C31×#125—22no—
40C31×#125—24no—
41C31×C22⋊Q8—24no—
42C31×C22⋊Q8—22no—
43C31×C22⋊Q8C2314yes—
44C4×Q8×C31—24no—
45C4×Q8×C31—24no—
46C4×Q8×C31—22no—
47C4×Q8×C31—24no—
48C4×Q8×C31C2×C414yes—
49C4×Q8×C31C2×C418yes—
50C4×Q8×C31C2×C414yes—
51C4×Q8×C31D414yes—
52C31×#117—24no—
53C31×#117—22no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C4⋊C4×C31D412yes—
58C2×C42×C31—22no—
59C2×C42×C31—21no—
60C2×C42×C31C2311yescharacteristic
61C2×C42×C31C2312yes—
62C31×C4⋊C8—22no—
63C31×C4⋊C8—24no—
64C31×C22⋊C8C2314yes—
65C31×#97—24no—
66C4×C8×C31—22no—
67C31×#95—22no—
68C31×#95—22no—
69C31×#95—21no—
70C31×#95—22no—
71C31×#95C2311yescharacteristic
72C31×#95C2312yes—
73C31×#95C2311yescharacteristic
74C2×Q8×C31—22no—
75C2×Q8×C31—24no—
76C2×Q8×C31—22no—
77C2×Q8×C31—22no—
78C2×Q8×C31C22⋊C414yes—
79C2×Q8×C31C22×C414yes—
80C22×C4×C31—21no—
81C22×C4×C31—22no—
82C22×C4×C31—21no—
83C22×C4×C31C2×D412yes—
84C22×C4×C31C2×D412yes—
85C22×C4×C31C2×D411yescharacteristic
86C22×C4×C31C2×D411yescharacteristic
87C22×C4×C31C2411yescharacteristic
88C2×C8×C31—24no—
89C4⋊C4×C31—24no—
90C4⋊C4×C31—22no—
91C4⋊C4×C31—24no—
92C4⋊C4×C31—22no—
93C4⋊C4×C31—22no—
94C4⋊C4×C31—24no—
95C4⋊C4×C31—24no—
96C4⋊C4×C31—28no—
97C4⋊C4×C31—24no—
98C4⋊C4×C31—42no—
99C4⋊C4×C31—42no—
100C4⋊C4×C31C22⋊C418yes—
101C4⋊C4×C31C22⋊C414yes—
102C4⋊C4×C31C22×C414yes—
103C22⋊C4×C31—24no—
104C42×C31—28no—
105C42×C31—24no—
106C42×C31—44no—
107C42×C31—42no—
108C42×C31C22×C414yes—
109C42×C31C22×C412yes—
110C42×C31C2×D412yes—
111C42×C31C2×D414yes—
112#17632C3111yesmaximal, characteristic, direct product factor, Sylow
113C23×C31C22×D411yescharacteristic
114Q8×C31—28no—
115Q8×C31—44no—
116Q8×C31—84no—
117Q8×C31—82no—
118C2×C4×C31—22no—
119C2×C4×C31—22no—
120C2×C4×C31—22no—
121C2×C4×C31—24no—
122C2×C4×C31—24no—
123C2×C4×C31—42no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
127C2×C4×C31—41no—
128C2×C4×C31—42no—
129C2×C4×C31—41no—
130C2×C4×C31—41no—
131C2×C4×C31—42no—
132C2×C4×C31—41no—
133C2×C4×C31—42no—
134C2×C4×C31C2×C22⋊C414yes—
135C2×C4×C31C2×C22⋊C412yes—
136C2×C4×C31C2×C22⋊C414yes—
137C2×C4×C31C2×C22⋊C412yes—
138C2×C4×C31#13712yes—
139C2×C4×C31#13712yes—
140C2×C4×C31C23×C411yescharacteristic
141C2×C4×C31C22×D411yescharacteristic
142C8×C31—44no—
143#3502C2×C3111yescharacteristic
144#2983C2×C3111yescharacteristic
145#2356C2×C3111yescharacteristic
146#1518C2×C3112yes—
147#1515C2×C3114yes—
148#1506C2×C3114yes—
149#1503C2×C3112yes—
150C22×C31—21no—
151C22×C31—21no—
152C22×C31C22×C22⋊C411yescharacteristic
153C22×C31C2×#13711yescharacteristic
154C22×C31C2×#13711yescharacteristic
155C4×C31—21no—
156C4×C31—21no—
157C4×C31—24no—
158C4×C31—24no—
159C4×C31—22no—
160C4×C31—42no—
161C4×C31—44no—
162C4×C31—44no—
163C4×C31—81no—
164C4×C31—82no—
165C4×C31#41914yes—
166Q82C4×C3114yescomplemented
167#557—24no—
168#557C4×C3114yescomplemented
169#544—21no—
170#544—22no—
171#533C22×C3112yes—
172#533C22×C3114yes—
173#519—21no—
174C2×C4×Q8C22×C3111yescharacteristic
175#446—24no—
176#432C22×C3114yes—
177#431C22×C3112yes—
178#391—21no—
179#387—22no—
180#383C22×C3112yes—
181#382C22×C3111yescharacteristic
182C4×C4⋊C4—21no—
183C4×C4⋊C4C22×C3111yescharacteristic
184#376C22×C3112yes—
185#328C22×C3114yes—
186#328C22×C3114yes—
187#328C22×C3118yes—
188C2×C31—21no—
189C2×C31—21no—
190C2×C31#288411yescharacteristic
191C2×C31C2×#41911yescharacteristic
192C2×C31#290111yescharacteristic
193C22×Q8D4×C3111yescharacteristic
194C4⋊Q8—28no—
195C4⋊Q8—24no—
196C4⋊Q8C2×C4×C3112yes—
197C4⋊Q8C2×C4×C3114yes—
198C4⋊Q8C2×C4×C3112yes—
199C4⋊Q8C2×C4×C3114yes—
200#125—28no—
201#125—24no—
202#125—22no—
203#125—24no—
204C22⋊Q8—24no—
205C22⋊Q8—22no—
206C22⋊Q8C23×C3114yes—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—22no—
210C4×Q8—24no—
211C4×Q8C2×C4×C3114yes—
212C4×Q8C2×C4×C3118yes—
213C4×Q8C2×C4×C3114yes—
214C4×Q8D4×C3114yes—
215#117—24no—
216#117—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4D4×C3111yescharacteristic
220C2×C4⋊C4D4×C3112yes—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42C23×C3112yes—
224C2×C42C23×C3111yescharacteristic
225C4⋊C8—22no—
226C4⋊C8—24no—
227C22⋊C8C23×C3114yes—
228#97—24no—
229C4×C8—22no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95C23×C3111yescharacteristic
235#95C23×C3112yes—
236#95C23×C3111yescharacteristic
237C31#1763211yescharacteristic, direct product factor, Sylow
238C2×Q8—22no—
239C2×Q8—24no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8C22⋊C4×C3114yes—
243C2×Q8C22×C4×C3114yes—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4C2×D4×C3112yes—
248C22×C4C2×D4×C3112yes—
249C22×C4C2×D4×C3111yescharacteristic
250C22×C4C2×D4×C3111yescharacteristic
251C22×C4C24×C3111yescharacteristic
252C2×C8—24no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—28no—
261C4⋊C4—24no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4C22⋊C4×C3118yes—
265C4⋊C4C22⋊C4×C3114yes—
266C4⋊C4C22×C4×C3114yes—
267C22⋊C4—24no—
268C42—28no—
269C42—24no—
270C42—44no—
271C42—42no—
272C42C22×C4×C3114yes—
273C42C22×C4×C3112yes—
274C42C2×D4×C3112yes—
275C42C2×D4×C3114yes—
276C23C22×D4×C3111yescharacteristic
277Q8—28no—
278Q8—44no—
279Q8—84no—
280Q8—82no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4C2×C22⋊C4×C3114yes—
298C2×C4C2×C22⋊C4×C3112yes—
299C2×C4C2×C22⋊C4×C3114yes—
300C2×C4C2×C22⋊C4×C3112yes—
301C2×C4C31×#13712yes—
302C2×C4C31×#13712yes—
303C2×C4C23×C4×C3111yescharacteristic
304C2×C4C22×D4×C3111yescharacteristic
305C8—44no—
306C22—21no—
307C22—21no—
308C22C22×C22⋊C4×C3111yescharacteristic
309C22C2×C31×#13711yescharacteristic
310C22C2×C31×#13711yescharacteristic
311C4—21no—
312C4—21no—
313C4—24no—
314C4—24no—
315C4—22no—
316C4—42no—
317C4—44no—
318C4—44no—
319C4—81no—
320C4—82no—
321C4C31×#41914yes—
322C2—21no—
323C2—21no—
324C2C31×#288411yescharacteristic
325C2C2×C31×#41911yescharacteristic
326C2C31×#290111yescharacteristic
subgroupG/H
center, socleC22×C31C22×C22⋊C4
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC23×C31C22×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
8. Q82×C31C4semidirect
10. C31×#557C4semidirect
112. #17632C31direct (N × H)
166. Q82C4×C31semidirect
168. #557C4×C31semidirect
237. C31#17632direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35022
C31×#29832
C31×#23562
C31×#15182
C31×#15152
C31×#15062
C31×#15032
#1763231