Finite Group Databasebeta

← C31×#8491 #21465250

Subgroups — C31×#8491 #21465250

Order 7936.

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

normal subgroups only

This group has 820 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×#3549C211yesmaximal, characteristic
2C31×#1963C211yesmaximal, characteristic, complemented
3C31×#1557C212yesmaximal, complemented
4C31×#1536C211yesmaximal, characteristic, complemented
5C31×#1505C212yesmaximal, complemented
6C31×H(3,𝔽4)—22no—
7C31×#555—22no—
8C31×#555C412yescomplemented
9D42×C31—22no—
10D42×C31C412yescomplemented
11C31×#535—22no—
12C31×#535—24no—
13C31×#532C2211yescharacteristic
14C31×#532C2212yes—
15C31×#432C2212yescomplemented
16C31×#423C2211yescharacteristic, complemented
17C2×C31×#104—22no—
18C31×#376C2211yescharacteristic, complemented
19C31×#332—22no—
20C31×#331—22no—
21C31×#329—22no—
22C31×#327—24no—
23C2×#41×C31—22no—
24C2×#41×C31D412yes—
25C22×D4×C31—22no—
26C22×D4×C31—24no—
27C2×M16×C31—21no—
28C31×C4⋊Q8C2×C411yescharacteristic
29C31×C4⋊Q8D412yes—
30C31×#127—24no—
31C31×#127C2×C411yescharacteristic
32C31×#127D412yes—
33C31×#124—28no—
34C31×#124—24no—
35C31×#124—22no—
36C31×#124D412yes—
37C31×#121—24no—
38C31×#121—24no—
39C31×#121—22no—
40C31×#121—22no—
41C31×#121—24no—
42C31×#121D414yes—
43C31×#120—24no—
44C31×#120—28no—
45C31×#120—24no—
46C4×D4×C31—22no—
47C4×D4×C31—22no—
48C4×D4×C31C2×C414yes—
49C31×#117—22no—
50C31×#117—24no—
51C2×C42×C31—21no—
52C2×C42×C31C2311yescharacteristic
53C31×C4⋊C8—22no—
54C31×C4⋊C8—22no—
55C31×#104—44no—
56C31×C22⋊C8—22no—
57C31×#97—22no—
58C31×#95—22no—
59C24×C31—24no—
60#41×C31—42no—
61#41×C31—42no—
62#41×C31—84no—
63C2×Q8×C31—22no—
64C2×Q8×C31C22⋊C412yes—
65C2×D4×C31—24no—
66C2×D4×C31—28no—
67C2×D4×C31—22no—
68C2×D4×C31—24no—
69C2×D4×C31—22no—
70C2×D4×C31—22no—
71C2×D4×C31—22no—
72C2×D4×C31—88no—
73C2×D4×C31—82no—
74C2×D4×C31—82no—
75C2×D4×C31C22⋊C414yes—
76C2×D4×C31C22⋊C412yes—
77C22×C4×C31—24no—
78C22×C4×C31—22no—
79C22×C4×C31—41no—
80C22×C4×C31C2×D411yescharacteristic
81C22×C4×C31C2×D412yes—
82M16×C31—42no—
83C2×C8×C31—42no—
84C2×C8×C31—41no—
85C4⋊C4×C31—22no—
86C4⋊C4×C31—22no—
87C4⋊C4×C31C22⋊C414yes—
88C22⋊C4×C31—28no—
89C22⋊C4×C31—24no—
90C22⋊C4×C31—24no—
91C22⋊C4×C31—24no—
92C22⋊C4×C31—24no—
93C42×C31—24no—
94C42×C31—22no—
95C42×C31—41no—
96C42×C31C22×C411yescharacteristic
97C42×C31C2×D411yescharacteristic
98C42×C31C2×D412yes—
99#8491C3111yesmaximal, characteristic, direct product factor, Sylow
100C23×C31—22no—
101C23×C31—24no—
102C23×C31—84no—
103C23×C31—88no—
104C23×C31#12011yescharacteristic
105Q8×C31—42no—
106Q8×C31—82no—
107D4×C31—44no—
108D4×C31—42no—
109D4×C31—44no—
110D4×C31—42no—
111D4×C31—42no—
112D4×C31—82no—
113D4×C31—82no—
114D4×C31—84no—
115D4×C31—84no—
116D4×C31—88no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—22no—
120C2×C4×C31—24no—
121C2×C4×C31—42no—
122C2×C4×C31—48no—
123C2×C4×C31—42no—
124C2×C4×C31—42no—
125C2×C4×C31—42no—
126C2×C4×C31—82no—
127C2×C4×C31C2×C22⋊C411yescharacteristic
128C2×C4×C31C2×C22⋊C412yes—
129C2×C4×C31#12012yes—
130C2×C4×C31#12011yescharacteristic
131C8×C31—82no—
132C8×C31—81no—
133#3549C2×C3111yescharacteristic
134#1963C2×C3111yescharacteristic, complemented
135#1557C2×C3112yescomplemented
136#1536C2×C3111yescharacteristic, complemented
137#1505C2×C3112yescomplemented
138C22×C31—21no—
139C22×C31—22no—
140C22×C31—42no—
141C22×C31—44no—
142C22×C31—44no—
143C22×C31—44no—
144C22×C31—44no—
145C22×C31—168no—
146C22×C31#37911yescharacteristic
147C4×C31—21no—
148C4×C31—22no—
149C4×C31—22no—
150C4×C31—42no—
151C4×C31—44no—
152C4×C31—82no—
153C4×C31#45311yescharacteristic
154C4×C31#45511yescharacteristic
155H(3,𝔽4)—22no—
156#555—22no—
157#555C4×C3112yescomplemented
158D42—22no—
159D42C4×C3112yescomplemented
160#535—22no—
161#535—24no—
162#532C22×C3111yescharacteristic
163#532C22×C3112yes—
164#432C22×C3112yescomplemented
165#423C22×C3111yescharacteristic, complemented
166C2×#104—22no—
167#376C22×C3111yescharacteristic, complemented
168#332—22no—
169#331—22no—
170#329—22no—
171#327—24no—
172C2×C31—41no—
173C2×C31—82no—
174C2×C31—84no—
175C2×C31#179211yescharacteristic
176C2×C31#179512yes—
177C2×#41—22no—
178C2×#41D4×C3112yes—
179C22×D4—22no—
180C22×D4—24no—
181C2×M16—21no—
182C4⋊Q8C2×C4×C3111yescharacteristic
183C4⋊Q8D4×C3112yes—
184#127—24no—
185#127C2×C4×C3111yescharacteristic
186#127D4×C3112yes—
187#124—28no—
188#124—24no—
189#124—22no—
190#124D4×C3112yes—
191#121—24no—
192#121—24no—
193#121—22no—
194#121—22no—
195#121—24no—
196#121D4×C3114yes—
197#120—24no—
198#120—28no—
199#120—24no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4C2×C4×C3114yes—
203#117—22no—
204#117—24no—
205C2×C42—21no—
206C2×C42C23×C3111yescharacteristic
207C4⋊C8—22no—
208C4⋊C8—22no—
209#104—44no—
210C22⋊C8—22no—
211#97—22no—
212#95—22no—
213C31#849111yescharacteristic, direct product factor, Sylow
214C24—24no—
215#41—42no—
216#41—42no—
217#41—84no—
218C2×Q8—22no—
219C2×Q8C22⋊C4×C3112yes—
220C2×D4—24no—
221C2×D4—28no—
222C2×D4—22no—
223C2×D4—24no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—88no—
228C2×D4—82no—
229C2×D4—82no—
230C2×D4C22⋊C4×C3114yes—
231C2×D4C22⋊C4×C3112yes—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4C2×D4×C3111yescharacteristic
236C22×C4C2×D4×C3112yes—
237M16—42no—
238C2×C8—42no—
239C2×C8—41no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4C22⋊C4×C3114yes—
243C22⋊C4—28no—
244C22⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C42—24no—
249C42—22no—
250C42—41no—
251C42C22×C4×C3111yescharacteristic
252C42C2×D4×C3111yescharacteristic
253C42C2×D4×C3112yes—
254C23—24no—
255C23—22no—
256C23—88no—
257C23—84no—
258C23C31×#12011yescharacteristic
259Q8—42no—
260Q8—82no—
261D4—44no—
262D4—42no—
263D4—44no—
264D4—42no—
265D4—42no—
266D4—82no—
267D4—82no—
268D4—84no—
269D4—88no—
270D4—84no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—42no—
276C2×C4—48no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—82no—
281C2×C4C2×C22⋊C4×C3111yescharacteristic
282C2×C4C2×C22⋊C4×C3112yes—
283C2×C4C31×#12012yes—
284C2×C4C31×#12011yescharacteristic
285C8—82no—
286C8—81no—
287C22—22no—
288C22—21no—
289C22—44no—
290C22—44no—
291C22—44no—
292C22—42no—
293C22—44no—
294C22—168no—
295C22C31×#37911yescharacteristic
296C4—22no—
297C4—22no—
298C4—21no—
299C4—44no—
300C4—42no—
301C4—82no—
302C4C31×#45311yescharacteristic
303C4C31×#45511yescharacteristic
304C2—41no—
305C2—84no—
306C2—82no—
307C2C31×#179211yescharacteristic
308C2C31×#179512yes—
subgroupG/H
center, socleC22×C31#379
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3549C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1963C2semidirect
3. C31×#1557C2semidirect
4. C31×#1536C2semidirect
5. C31×#1505C2semidirect
8. C31×#555C4semidirect
10. D42×C31C4semidirect
15. C31×#432C22semidirect
16. C31×#423C22semidirect
18. C31×#376C22semidirect
99. #8491C31direct (N × H)
134. #1963C2×C31semidirect
135. #1557C2×C31semidirect
136. #1536C2×C31semidirect
137. #1505C2×C31semidirect
157. #555C4×C31semidirect
159. D42C4×C31semidirect
164. #432C22×C31semidirect
165. #423C22×C31semidirect
167. #376C22×C31semidirect
213. C31#8491direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35492
C31×#19632
C31×#15572
C31×#15362
C31×#15052
#849131