Finite Group Databasebeta

← C31×#8464 #21465223

Subgroups — C31×#8464 #21465223

Order 7936.

1002 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 150 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×#3487C211yesmaximal, characteristic
2C31×#1782C211yesmaximal, characteristic, complemented
3C31×#1780C211yesmaximal, characteristic, complemented
4C31×#1516C214yesmaximal, complemented
5C2×C31×2−1+4C411yescharacteristic, complemented
6C22×#41×C31C2211yescharacteristic
7C31×#550—24no—
8C31×#548—24no—
9C31×#535—22no—
10C31×#535C412yescomplemented
11C31×#525—22no—
12C31×#525C2212yes—
13C31×#519C411yescharacteristic, complemented
14C2×C31×#99—22no—
15C2×C31×C22⋊C8—22no—
16C2×C31×C22⋊C8C2212yes—
17C31×#381C2212yes—
18C31×#328—24no—
19C31×#327—24no—
20C31×2−1+4—44no—
21C2×#41×C31—22no—
22C2×#41×C31—22no—
23C2×#41×C31—24no—
24C2×#41×C31—44no—
25C2×#41×C31C2×C412yes—
26C22×Q8×C31—22no—
27C22×Q8×C31C2×C411yescharacteristic
28C22×D4×C31C2×C411yescharacteristic
29C22×D4×C31D412yes—
30C23×C4×C31D412yes—
31C23×C4×C31C2311yescharacteristic
32C22×C8×C31—22no—
33C31×#127—24no—
34C31×#127—22no—
35C31×#124—24no—
36C31×#124—22no—
37C31×#123—22no—
38C31×C22⋊Q8—24no—
39C31×C22⋊Q8D414yes—
40C31×#121—24no—
41C31×#121—28no—
42C31×#121—24no—
43C31×#121D414yes—
44C31×#120—24no—
45C31×#120—22no—
46C4×Q8×C31—24no—
47C4×D4×C31—28no—
48C4×D4×C31—24no—
49C31×#117—22no—
50C31×#117C2×C412yes—
51C2×C22⋊C4×C31—22no—
52C31×#99—44no—
53C31×C22⋊C8—24no—
54C31×C22⋊C8—44no—
55C31×#95—24no—
56C24×C31C22⋊C412yes—
57C24×C31C2×D411yescharacteristic
58#41×C31—44no—
59#41×C31—44no—
60#41×C31—416no—
61#41×C31—48no—
62#41×C31—44no—
63C2×Q8×C31—22no—
64C2×Q8×C31—24no—
65C2×Q8×C31—22no—
66C2×Q8×C31—48no—
67C2×Q8×C31C22⋊C412yes—
68C2×D4×C31—22no—
69C2×D4×C31—22no—
70C2×D4×C31—24no—
71C2×D4×C31—28no—
72C2×D4×C31—24no—
73C2×D4×C31—24no—
74C2×D4×C31—22no—
75C2×D4×C31—42no—
76C2×D4×C31—42no—
77C2×D4×C31C22⋊C412yes—
78C22×C4×C31—24no—
79C22×C4×C31—24no—
80C22×C4×C31—22no—
81C22×C4×C31—24no—
82C22×C4×C31—42no—
83C22×C4×C31—42no—
84C22×C4×C31—42no—
85C22×C4×C31C22⋊C412yes—
86C22×C4×C31C22×C411yescharacteristic
87C22×C4×C31C2×D411yescharacteristic
88C22×C4×C31C2×D414yes—
89C2×C8×C31—44no—
90C2×C8×C31—42no—
91C4⋊C4×C31—24no—
92C4⋊C4×C31—24no—
93C4⋊C4×C31C22⋊C414yes—
94C22⋊C4×C31—28no—
95C22⋊C4×C31—22no—
96C22⋊C4×C31—24no—
97C22⋊C4×C31—24no—
98C22⋊C4×C31—44no—
99C42×C31—24no—
100C42×C31—22no—
101#8464C3111yesmaximal, characteristic, direct product factor, Sylow
102C23×C31—22no—
103C23×C31—22no—
104C23×C31—22no—
105C23×C31—22no—
106C23×C31—22no—
107C23×C31—42no—
108C23×C31—44no—
109C23×C31C2×C22⋊C411yescharacteristic
110C23×C31#12012yes—
111Q8×C31—28no—
112Q8×C31—42no—
113Q8×C31—44no—
114D4×C31—48no—
115D4×C31—42no—
116D4×C31—44no—
117D4×C31—44no—
118D4×C31—44no—
119D4×C31—84no—
120D4×C31—88no—
121C2×C4×C31—22no—
122C2×C4×C31—22no—
123C2×C4×C31—22no—
124C2×C4×C31—22no—
125C2×C4×C31—24no—
126C2×C4×C31—22no—
127C2×C4×C31—44no—
128C2×C4×C31—48no—
129C2×C4×C31—48no—
130C2×C4×C31—44no—
131C2×C4×C31—82no—
132C2×C4×C31—84no—
133C2×C4×C31#9914yes—
134C2×C4×C31C2×C22⋊C412yes—
135C2×C4×C31#12012yes—
136C8×C31—82no—
137#3487C2×C3111yescharacteristic
138#1782C2×C3111yescharacteristic, complemented
139#1780C2×C3111yescharacteristic, complemented
140#1516C2×C3114yescomplemented
141C22×C31—22no—
142C22×C31—24no—
143C22×C31—22no—
144C22×C31—44no—
145C22×C31—44no—
146C22×C31—44no—
147C22×C31—42no—
148C22×C31—42no—
149C22×C31—82no—
150C22×C31—82no—
151C22×C31#37911yescharacteristic
152C22×C31C2×#9912yes—
153C4×C31—22no—
154C4×C31—24no—
155C4×C31—22no—
156C4×C31—44no—
157C4×C31—42no—
158C4×C31—162no—
159C2×2−1+4C4×C3111yescharacteristic, complemented
160C22×#41C22×C3111yescharacteristic
161#550—24no—
162#548—24no—
163#535—22no—
164#535C4×C3112yescomplemented
165#525—22no—
166#525C22×C3112yes—
167#519C4×C3111yescharacteristic, complemented
168C2×#99—22no—
169C2×C22⋊C8—22no—
170C2×C22⋊C8C22×C3112yes—
171#381C22×C3112yes—
172#328—24no—
173#327—24no—
174C2×C31—22no—
175C2×C31—42no—
176C2×C31—82no—
177C2×C31—82no—
178C2×C31#178111yescharacteristic
179C2×C31#179412yes—
1802−1+4—44no—
181C2×#41—22no—
182C2×#41—22no—
183C2×#41—24no—
184C2×#41—44no—
185C2×#41C2×C4×C3112yes—
186C22×Q8—22no—
187C22×Q8C2×C4×C3111yescharacteristic
188C22×D4C2×C4×C3111yescharacteristic
189C22×D4D4×C3112yes—
190C23×C4D4×C3112yes—
191C23×C4C23×C3111yescharacteristic
192C22×C8—22no—
193#127—24no—
194#127—22no—
195#124—24no—
196#124—22no—
197#123—22no—
198C22⋊Q8—24no—
199C22⋊Q8D4×C3114yes—
200#121—24no—
201#121—28no—
202#121—24no—
203#121D4×C3114yes—
204#120—24no—
205#120—22no—
206C4×Q8—24no—
207C4×D4—28no—
208C4×D4—24no—
209#117—22no—
210#117C2×C4×C3112yes—
211C2×C22⋊C4—22no—
212#99—44no—
213C22⋊C8—24no—
214C22⋊C8—44no—
215#95—24no—
216C31#846411yescharacteristic, direct product factor, Sylow
217C24C22⋊C4×C3112yes—
218C24C2×D4×C3111yescharacteristic
219#41—44no—
220#41—44no—
221#41—416no—
222#41—48no—
223#41—44no—
224C2×Q8—22no—
225C2×Q8—24no—
226C2×Q8—22no—
227C2×Q8—48no—
228C2×Q8C22⋊C4×C3112yes—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—24no—
232C2×D4—28no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4C22⋊C4×C3112yes—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4C22⋊C4×C3112yes—
247C22×C4C22×C4×C3111yescharacteristic
248C22×C4C2×D4×C3111yescharacteristic
249C22×C4C2×D4×C3114yes—
250C2×C8—44no—
251C2×C8—42no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4C22⋊C4×C3114yes—
255C22⋊C4—28no—
256C22⋊C4—22no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—44no—
260C42—24no—
261C42—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—42no—
268C23—44no—
269C23C2×C22⋊C4×C3111yescharacteristic
270C23C31×#12012yes—
271Q8—28no—
272Q8—42no—
273Q8—44no—
274D4—48no—
275D4—42no—
276D4—44no—
277D4—44no—
278D4—44no—
279D4—84no—
280D4—88no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—82no—
292C2×C4—84no—
293C2×C4C31×#9914yes—
294C2×C4C2×C22⋊C4×C3112yes—
295C2×C4C31×#12012yes—
296C8—82no—
297C22—22no—
298C22—22no—
299C22—24no—
300C22—44no—
301C22—42no—
302C22—42no—
303C22—44no—
304C22—44no—
305C22—82no—
306C22—82no—
307C22C31×#37911yescharacteristic
308C22C2×C31×#9912yes—
309C4—22no—
310C4—24no—
311C4—22no—
312C4—42no—
313C4—44no—
314C4—162no—
315C2—22no—
316C2—42no—
317C2—82no—
318C2—82no—
319C2C31×#178111yescharacteristic
320C2C31×#179412yes—
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC31×#3487C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1782C2semidirect
3. C31×#1780C2semidirect
4. C31×#1516C2semidirect
5. C2×C31×2−1+4C4semidirect
10. C31×#535C4semidirect
13. C31×#519C4semidirect
101. #8464C31direct (N × H)
138. #1782C2×C31semidirect
139. #1780C2×C31semidirect
140. #1516C2×C31semidirect
159. C2×2−1+4C4×C31semidirect
164. #535C4×C31semidirect
167. #519C4×C31semidirect
216. C31#8464direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34872
C31×#17822
C31×#17802
C31×#15162
#846431