Finite Group Databasebeta

← C31×#7631 #21464390

Subgroups — C31×#7631 #21464390

Order 7936.

730 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C31×#2297C211yesmaximal, characteristic
2C31×#1728C212yesmaximal, complemented
3C31×#1272C212yesmaximal, complemented
4C31×#1270C212yesmaximal, complemented
5C22×#41×C31C411yescharacteristic
6C22×M16×C31—22no—
7C2×C31×#117C411yescharacteristic
8C22×C42×C31C2211yescharacteristic
9C31×#391C411yescharacteristic
10C31×#391C2211yescharacteristic
11C31×#390C411yescharacteristic
12C31×#390C2211yescharacteristic
13C31×#386—22no—
14C4×C22⋊C4×C31—22no—
15C31×#336—22no—
16C31×#336C2214yes—
17C2×#41×C31—22no—
18C2×#41×C31—22no—
19C2×#41×C31—42no—
20C2×#41×C31—42no—
21C22×Q8×C31C2×C411yescharacteristic
22C22×D4×C31—21no—
23C22×D4×C31C2×C411yescharacteristic
24C23×C4×C31—21no—
25C23×C4×C31C2×C411yescharacteristic
26C23×C4×C31D412yes—
27C2×M16×C31—28no—
28C2×M16×C31—44no—
29C2×M16×C31—44no—
30C22×C8×C31—24no—
31C31×#117—42no—
32C2×C4⋊C4×C31C2×C411yescharacteristic
33C2×C4⋊C4×C31C2×C411yescharacteristic
34C2×C22⋊C4×C31—21no—
35C2×C22⋊C4×C31—22no—
36C2×C22⋊C4×C31—22no—
37C2×C42×C31—21no—
38C2×C42×C31—42no—
39C2×C42×C31C2×C411yescharacteristic
40C2×C42×C31D412yes—
41C2×C42×C31C2311yescharacteristic
42C31×#95—22no—
43C31×#95—22no—
44C24×C31—21no—
45C24×C31C22⋊C411yescharacteristic
46#41×C31—48no—
47#41×C31—48no—
48C2×Q8×C31—21no—
49C2×Q8×C31—41no—
50C2×Q8×C31—41no—
51C2×Q8×C31M1612yescomplemented
52C2×D4×C31—24no—
53C2×D4×C31—21no—
54C2×D4×C31—41no—
55C2×D4×C31—41no—
56C2×D4×C31—42no—
57C2×D4×C31—42no—
58C2×D4×C31M1612yescomplemented
59C22×C4×C31—24no—
60C22×C4×C31—22no—
61C22×C4×C31—22no—
62C22×C4×C31—24no—
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—22no—
66C22×C4×C31—22no—
67C22×C4×C31—24no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—42no—
72C22×C4×C31—42no—
73C22×C4×C31C22⋊C411yescharacteristic
74C22×C4×C31C22⋊C412yes—
75C22×C4×C31C22×C411yescharacteristic
76C22×C4×C31C2×D412yes—
77M16×C31—416no—
78C2×C8×C31—28no—
79C2×C8×C31—44no—
80C2×C8×C31—44no—
81C4⋊C4×C31—41no—
82C4⋊C4×C31—41no—
83C22⋊C4×C31—42no—
84C22⋊C4×C31—44no—
85C22⋊C4×C31—42no—
86C22⋊C4×C31—42no—
87C42×C31—41no—
88C42×C31—42no—
89C42×C31—41no—
90C42×C31—42no—
91#7631C3111yesmaximal, characteristic, direct product factor, Sylow
92C23×C31—21no—
93C23×C31—21no—
94C23×C31—22no—
95C23×C31—22no—
96C23×C31—22no—
97C23×C31—41no—
98C23×C31—41no—
99C23×C31—42no—
100C23×C31—42no—
101C23×C31#9912yes—
102C23×C31C2×C22⋊C411yescharacteristic
103Q8×C31—42no—
104Q8×C31—42no—
105D4×C31—42no—
106D4×C31—42no—
107D4×C31—48no—
108C2×C4×C31—28no—
109C2×C4×C31—28no—
110C2×C4×C31—24no—
111C2×C4×C31—24no—
112C2×C4×C31—21no—
113C2×C4×C31—22no—
114C2×C4×C31—21no—
115C2×C4×C31—22no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—22no—
119C2×C4×C31—22no—
120C2×C4×C31—22no—
121C2×C4×C31—48no—
122C2×C4×C31—41no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—42no—
127C2×C4×C31—42no—
128C2×C4×C31#10012yes—
129C2×C4×C31#10414yes—
130C2×C4×C31C2×M1611yescharacteristic
131C2×C4×C31C2×M1611yescharacteristic
132C2×C4×C31AGL(1,ℤ/8)12yes—
133C2×C4×C31#13712yes—
134C8×C31—48no—
135#2297C2×C3111yescharacteristic
136#1728C2×C3112yescomplemented
137#1272C2×C3112yescomplemented
138#1270C2×C3112yescomplemented
139C22×C31—22no—
140C22×C31—24no—
141C22×C31—41no—
142C22×C31—42no—
143C22×C31—41no—
144C22×C31—42no—
145C22×C31—44no—
146C22×C31—81no—
147C22×C31—82no—
148C22×C31—81no—
149C22×C31#40711yescharacteristic
150C22×C31C2×#9911yescharacteristic
151C22×C31C2×#10011yescharacteristic
152C22×C31#41712yes—
153C22×C31C2×#10412yes—
154C4×C31—22no—
155C4×C31—21no—
156C4×C31—21no—
157C4×C31—22no—
158C4×C31—24no—
159C4×C31—22no—
160C4×C31—42no—
161C4×C31—81no—
162C4×C31—81no—
163C22×#41C4×C3111yescharacteristic
164C22×M16—22no—
165C2×#117C4×C3111yescharacteristic
166C22×C42C22×C3111yescharacteristic
167#391C4×C3111yescharacteristic
168#391C22×C3111yescharacteristic
169#390C4×C3111yescharacteristic
170#390C22×C3111yescharacteristic
171#386—22no—
172C4×C22⋊C4—22no—
173#336—22no—
174#336C22×C3114yes—
175C2×C31—42no—
176C2×C31—82no—
177C2×C31#146211yescharacteristic
178C2×C31#148912yes—
179C2×C31#150412yes—
180C2×C31#152511yescharacteristic
181C2×C31#156011yescharacteristic
182C2×#41—22no—
183C2×#41—22no—
184C2×#41—42no—
185C2×#41—42no—
186C22×Q8C2×C4×C3111yescharacteristic
187C22×D4—21no—
188C22×D4C2×C4×C3111yescharacteristic
189C23×C4—21no—
190C23×C4C2×C4×C3111yescharacteristic
191C23×C4D4×C3112yes—
192C2×M16—28no—
193C2×M16—44no—
194C2×M16—44no—
195C22×C8—24no—
196#117—42no—
197C2×C4⋊C4C2×C4×C3111yescharacteristic
198C2×C4⋊C4C2×C4×C3111yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C42—21no—
203C2×C42—42no—
204C2×C42C2×C4×C3111yescharacteristic
205C2×C42D4×C3112yes—
206C2×C42C23×C3111yescharacteristic
207#95—22no—
208#95—22no—
209C31#763111yescharacteristic, direct product factor, Sylow
210C24—21no—
211C24C22⋊C4×C3111yescharacteristic
212#41—48no—
213#41—48no—
214C2×Q8—21no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8M16×C3112yescomplemented
218C2×D4—24no—
219C2×D4—21no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4M16×C3112yescomplemented
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C22⋊C4×C3111yescharacteristic
240C22×C4C22⋊C4×C3112yes—
241C22×C4C22×C4×C3111yescharacteristic
242C22×C4C2×D4×C3112yes—
243M16—416no—
244C2×C8—28no—
245C2×C8—44no—
246C2×C8—44no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—44no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C42—41no—
254C42—42no—
255C42—41no—
256C42—42no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—41no—
263C23—41no—
264C23—42no—
265C23—42no—
266C23C31×#9912yes—
267C23C2×C22⋊C4×C3111yescharacteristic
268Q8—42no—
269Q8—42no—
270D4—42no—
271D4—42no—
272D4—48no—
273C2×C4—28no—
274C2×C4—28no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—48no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4C31×#10012yes—
294C2×C4C31×#10414yes—
295C2×C4C2×M16×C3111yescharacteristic
296C2×C4C2×M16×C3111yescharacteristic
297C2×C4C31×AGL(1,ℤ/8)12yes—
298C2×C4C31×#13712yes—
299C8—48no—
300C22—22no—
301C22—24no—
302C22—41no—
303C22—42no—
304C22—41no—
305C22—42no—
306C22—44no—
307C22—81no—
308C22—82no—
309C22—81no—
310C22C31×#40711yescharacteristic
311C22C2×C31×#9911yescharacteristic
312C22C2×C31×#10011yescharacteristic
313C22C31×#41712yes—
314C22C2×C31×#10412yes—
315C4—22no—
316C4—21no—
317C4—21no—
318C4—22no—
319C4—24no—
320C4—22no—
321C4—42no—
322C4—81no—
323C4—81no—
324C2—42no—
325C2—82no—
326C2C31×#146211yescharacteristic
327C2C31×#148912yes—
328C2C31×#150412yes—
329C2C31×#152511yescharacteristic
330C2C31×#156011yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC22×#41×C31C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1728C2semidirect
3. C31×#1272C2semidirect
4. C31×#1270C2semidirect
51. C2×Q8×C31M16semidirect
58. C2×D4×C31M16semidirect
91. #7631C31direct (N × H)
136. #1728C2×C31semidirect
137. #1272C2×C31semidirect
138. #1270C2×C31semidirect
209. C31#7631direct (N × H)
217. C2×Q8M16×C31semidirect
224. C2×D4M16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#22972
C31×#17282
C31×#12722
C31×#12702
#763131