Finite Group Databasebeta

← C31×#32708 #21489467

Subgroups — C31×#32708 #21489467

Order 7936.

1062 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 226 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×#2848C211yesmaximal, characteristic, complemented
2C31×#2847C211yesmaximal, characteristic
3C31×#2077C213yesmaximal, complemented
4C31×#2035C213yesmaximal, complemented
5C31×#2034C213yesmaximal, complemented
6C31×#2000C213yesmaximal, complemented
7C31×#1779C211yesmaximal, characteristic, complemented
8C24×C4×C31C2211yescharacteristic
9C22×SD16×C31C2211yescharacteristic, complemented
10C2×C31×C22⋊Q8C2213yes—
11C2×C31×#121C2213yes—
12C2×C31×#120—21no—
13C2×C31×#106C2213yescomplemented
14C2×C31×#103C2213yescomplemented
15C2×C31×#102C2213yescomplemented
16C2×C31×C22⋊C8C2213yes—
17C2×C31×C22⋊C8C2213yes—
18C31×#386—23no—
19C31×#385—23no—
20C31×#380C2213yes—
21C31×#380C2213yes—
22C31×#379—21no—
23C31×#340C2213yescomplemented
24C31×#340C2213yescomplemented
25C25×C31D411yescharacteristic
26C22×Q8×C31C2311yescharacteristic
27C22×D4×C31—23no—
28C22×D4×C31C2311yescharacteristic
29C23×C4×C31—43no—
30C23×C4×C31—43no—
31C23×C4×C31D413yes—
32C23×C4×C31C2313yes—
33C2×SD16×C31—43no—
34C22×C8×C31C2313yes—
35C22×C8×C31C2311yescharacteristic
36C31×C22⋊Q8—43no—
37C31×C22⋊Q8—43no—
38C31×#121—43no—
39C31×#121—43no—
40C31×#120—44no—
41C2×C4⋊C4×C31—23no—
42C2×C4⋊C4×C31C2313yes—
43C2×C4⋊C4×C31C2313yes—
44C2×C22⋊C4×C31—23no—
45C2×C22⋊C4×C31—23no—
46C2×C22⋊C4×C31—23no—
47C2×C22⋊C4×C31—23no—
48C31×#106—43no—
49C31×#103—43no—
50C31×#102—43no—
51C31×C22⋊C8—43no—
52C31×C22⋊C8—43no—
53C31×#95—23no—
54C31×#95—23no—
55C24×C31—21no—
56C24×C31—43no—
57C24×C31—43no—
58C24×C31SD1614yescomplemented
59C24×C31C2×D413yes—
60C2×Q8×C31—23no—
61C2×Q8×C31—23no—
62C2×D4×C31—23no—
63C2×D4×C31—23no—
64C2×D4×C31—46no—
65C2×D4×C31—46no—
66C2×D4×C31—46no—
67C22×C4×C31—23no—
68C22×C4×C31—23no—
69C22×C4×C31—23no—
70C22×C4×C31—23no—
71C22×C4×C31—23no—
72C22×C4×C31—23no—
73C22×C4×C31—23no—
74C22×C4×C31—23no—
75C22×C4×C31—21no—
76C22×C4×C31—43no—
77C22×C4×C31—43no—
78C22×C4×C31—43no—
79C22×C4×C31—412no—
80C22×C4×C31—44no—
81C22×C4×C31C2×D413yes—
82C22×C4×C31C2411yescharacteristic
83SD16×C31—44no—
84C2×C8×C31—23no—
85C2×C8×C31—23no—
86C2×C8×C31—23no—
87C2×C8×C31—23no—
88C4⋊C4×C31—23no—
89C4⋊C4×C31—23no—
90C4⋊C4×C31—23no—
91C4⋊C4×C31—23no—
92C4⋊C4×C31—46no—
93C22⋊C4×C31—46no—
94C22⋊C4×C31—46no—
95C22⋊C4×C31—46no—
96C22⋊C4×C31—46no—
97#32708C3111yesmaximal, characteristic, direct product factor, Sylow
98C23×C31—23no—
99C23×C31—23no—
100C23×C31—23no—
101C23×C31—23no—
102C23×C31—23no—
103C23×C31—23no—
104C23×C31—412no—
105C23×C31—44no—
106C23×C31—412no—
107C23×C31—43no—
108C23×C31—41no—
109C23×C31—43no—
110C23×C31C2×SD1616yes—
111C23×C31C22×D411yescharacteristic
112Q8×C31—43no—
113Q8×C31—41no—
114D4×C31—43no—
115D4×C31—41no—
116D4×C31—812no—
117C2×C4×C31—23no—
118C2×C4×C31—23no—
119C2×C4×C31—23no—
120C2×C4×C31—412no—
121C2×C4×C31—43no—
122C2×C4×C31—43no—
123C2×C4×C31—43no—
124C2×C4×C31—43no—
125C2×C4×C31—43no—
126C2×C4×C31—43no—
127C2×C4×C31—43no—
128C2×C4×C31—43no—
129C2×C4×C31—46no—
130C2×C4×C31—43no—
131C2×C4×C31H(5,𝔽2)13yes—
132C2×C4×C31H(5,𝔽2)13yes—
133C8×C31—43no—
134C8×C31—41no—
135#2848C2×C3111yescharacteristic, complemented
136#2847C2×C3111yescharacteristic
137#2077C2×C3113yescomplemented
138#2035C2×C3113yescomplemented
139#2034C2×C3113yescomplemented
140#2000C2×C3113yescomplemented
141#1779C2×C3111yescharacteristic, complemented
142C22×C31—26no—
143C22×C31—26no—
144C22×C31—26no—
145C22×C31—416no—
146C22×C31—43no—
147C22×C31—43no—
148C22×C31—43no—
149C22×C31—43no—
150C22×C31—83no—
151C22×C31—83no—
152C22×C31—81no—
153C22×C31#53413yes—
154C22×C31C22×SD1611yescharacteristic
155C22×C31#57513yes—
156C4×C31—43no—
157C4×C31—83no—
158C4×C31—83no—
159C4×C31—81no—
160C4×C31#56013yes—
161C4×C31H(3,𝔽4)11yescharacteristic
162C24×C4C22×C3111yescharacteristic
163C22×SD16C22×C3111yescharacteristic, complemented
164C2×C22⋊Q8C22×C3113yes—
165C2×#121C22×C3113yes—
166C2×#120—21no—
167C2×#106C22×C3113yescomplemented
168C2×#103C22×C3113yescomplemented
169C2×#102C22×C3113yescomplemented
170C2×C22⋊C8C22×C3113yes—
171C2×C22⋊C8C22×C3113yes—
172#386—23no—
173#385—23no—
174#380C22×C3113yes—
175#380C22×C3113yes—
176#379—21no—
177#340C22×C3113yescomplemented
178#340C22×C3113yescomplemented
179C2×C31—46no—
180C2×C31—161no—
181C2×C31#281211yescharacteristic
182C2×C31#318713yes—
183C2×C31#319713yes—
184C25D4×C3111yescharacteristic
185C22×Q8C23×C3111yescharacteristic
186C22×D4—23no—
187C22×D4C23×C3111yescharacteristic
188C23×C4—43no—
189C23×C4—43no—
190C23×C4D4×C3113yes—
191C23×C4C23×C3113yes—
192C2×SD16—43no—
193C22×C8C23×C3111yescharacteristic
194C22×C8C23×C3113yes—
195C22⋊Q8—43no—
196C22⋊Q8—43no—
197#121—43no—
198#121—43no—
199#120—44no—
200C2×C4⋊C4—23no—
201C2×C4⋊C4C23×C3113yes—
202C2×C4⋊C4C23×C3113yes—
203C2×C22⋊C4—23no—
204C2×C22⋊C4—23no—
205C2×C22⋊C4—23no—
206C2×C22⋊C4—23no—
207#106—43no—
208#103—43no—
209#102—43no—
210C22⋊C8—43no—
211C22⋊C8—43no—
212#95—23no—
213#95—23no—
214C31#3270811yescharacteristic, direct product factor, Sylow
215C24—21no—
216C24—43no—
217C24—43no—
218C24SD16×C3114yescomplemented
219C24C2×D4×C3113yes—
220C2×Q8—23no—
221C2×Q8—23no—
222C2×D4—23no—
223C2×D4—23no—
224C2×D4—46no—
225C2×D4—46no—
226C2×D4—46no—
227C22×C4—23no—
228C22×C4—23no—
229C22×C4—23no—
230C22×C4—23no—
231C22×C4—23no—
232C22×C4—23no—
233C22×C4—23no—
234C22×C4—23no—
235C22×C4—21no—
236C22×C4—43no—
237C22×C4—43no—
238C22×C4—43no—
239C22×C4—412no—
240C22×C4—44no—
241C22×C4C2×D4×C3113yes—
242C22×C4C24×C3111yescharacteristic
243SD16—44no—
244C2×C8—23no—
245C2×C8—23no—
246C2×C8—23no—
247C2×C8—23no—
248C4⋊C4—23no—
249C4⋊C4—23no—
250C4⋊C4—23no—
251C4⋊C4—23no—
252C4⋊C4—46no—
253C22⋊C4—46no—
254C22⋊C4—46no—
255C22⋊C4—46no—
256C22⋊C4—46no—
257C23—23no—
258C23—23no—
259C23—23no—
260C23—23no—
261C23—23no—
262C23—23no—
263C23—412no—
264C23—44no—
265C23—412no—
266C23—43no—
267C23—41no—
268C23—43no—
269C23C2×SD16×C3116yes—
270C23C22×D4×C3111yescharacteristic
271Q8—43no—
272Q8—41no—
273D4—43no—
274D4—41no—
275D4—812no—
276C2×C4—23no—
277C2×C4—23no—
278C2×C4—23no—
279C2×C4—412no—
280C2×C4—43no—
281C2×C4—43no—
282C2×C4—43no—
283C2×C4—43no—
284C2×C4—43no—
285C2×C4—43no—
286C2×C4—43no—
287C2×C4—43no—
288C2×C4—46no—
289C2×C4—43no—
290C2×C4C31×H(5,𝔽2)13yes—
291C2×C4C31×H(5,𝔽2)13yes—
292C8—43no—
293C8—41no—
294C22—26no—
295C22—26no—
296C22—26no—
297C22—416no—
298C22—43no—
299C22—43no—
300C22—43no—
301C22—43no—
302C22—83no—
303C22—83no—
304C22—81no—
305C22C31×#53413yes—
306C22C22×SD16×C3111yescharacteristic
307C22C31×#57513yes—
308C4—43no—
309C4—83no—
310C4—83no—
311C4—81no—
312C4C31×#56013yes—
313C4C31×H(3,𝔽4)11yescharacteristic
314C2—46no—
315C2—161no—
316C2C31×#281211yescharacteristic
317C2C31×#318713yes—
318C2C31×#319713yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2848C2semidirect
3. C31×#2077C2semidirect
4. C31×#2035C2semidirect
5. C31×#2034C2semidirect
6. C31×#2000C2semidirect
7. C31×#1779C2semidirect
9. C22×SD16×C31C22semidirect
13. C2×C31×#106C22semidirect
14. C2×C31×#103C22semidirect
15. C2×C31×#102C22semidirect
23. C31×#340C22semidirect
24. C31×#340C22semidirect
58. C24×C31SD16semidirect
97. #32708C31direct (N × H)
135. #2848C2×C31semidirect
137. #2077C2×C31semidirect
138. #2035C2×C31semidirect
139. #2034C2×C31semidirect
140. #2000C2×C31semidirect
141. #1779C2×C31semidirect
163. C22×SD16C22×C31semidirect
167. C2×#106C22×C31semidirect
168. C2×#103C22×C31semidirect
169. C2×#102C22×C31semidirect
177. #340C22×C31semidirect
178. #340C22×C31semidirect
214. C31#32708direct (N × H)
218. C24SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#28482
C31×#28472
C31×#20772
C31×#20352
C31×#20342
C31×#20002
C31×#17792
#3270831