Finite Group Databasebeta

← C31×#9477 #21466236

Subgroups — C31×#9477 #21466236

Order 7936.

734 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2482C211yesmaximal, characteristic
2C31×#2273C211yesmaximal, characteristic
3C31×#1826C212yesmaximal, complemented
4C31×#1462C211yesmaximal, characteristic
5C31×#1284C212yesmaximal, complemented
6C2×C31×#117C2211yescharacteristic
7C22×C22⋊C4×C31—21no—
8C22×C22⋊C4×C31C2211yescharacteristic
9C31×#407C2212yes—
10C31×#399C412yescomplemented
11C31×#397C412yescomplemented
12C31×#388—22no—
13C31×#385—21no—
14C31×#381C2211yescharacteristic
15C31×#379—21no—
16C4×C22⋊C4×C31—21no—
17C4×C22⋊C4×C31—24no—
18C4×C22⋊C4×C31C2212yes—
19C31×#339—24no—
20C31×#323—22no—
21C25×C31D411yescharacteristic
22C23×C4×C31—22no—
23C23×C4×C31D411yescharacteristic
24C23×C4×C31C2311yescharacteristic
25C2×M16×C31—22no—
26C31×#117—22no—
27C31×#117—22no—
28C2×C4⋊C4×C31—22no—
29C2×C4⋊C4×C31C2×C411yescharacteristic, complemented
30C2×C4⋊C4×C31C2×C411yescharacteristic, complemented
31C2×C22⋊C4×C31—28no—
32C2×C22⋊C4×C31—22no—
33C2×C22⋊C4×C31—22no—
34C2×C22⋊C4×C31—22no—
35C2×C22⋊C4×C31—22no—
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—21no—
38C2×C22⋊C4×C31—21no—
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31C2×C412yes—
41C2×C22⋊C4×C31D412yes—
42C2×C42×C31—21no—
43C2×C42×C31—22no—
44C2×C42×C31D414yes—
45C31×C22⋊C8—22no—
46C31×C22⋊C8—22no—
47C31×#95—22no—
48C31×#95—22no—
49C31×#95—22no—
50C31×#95—21no—
51C31×#95C2×C412yescomplemented
52C24×C31—21no—
53C24×C31—24no—
54C24×C31—21no—
55C24×C31—42no—
56C24×C31—81no—
57C24×C31C22⋊C412yes—
58C24×C31C2×D411yescharacteristic
59C22×C4×C31—21no—
60C22×C4×C31—24no—
61C22×C4×C31—28no—
62C22×C4×C31—21no—
63C22×C4×C31—22no—
64C22×C4×C31—21no—
65C22×C4×C31—21no—
66C22×C4×C31—42no—
67C22×C4×C31—81no—
68C22×C4×C31—82no—
69C22×C4×C31—82no—
70C22×C4×C31C22⋊C412yes—
71C22×C4×C31C22×C411yescharacteristic
72C22×C4×C31C2×D412yes—
73C22×C4×C31C2×D411yescharacteristic
74C22×C4×C31#4112yes—
75M16×C31—44no—
76C2×C8×C31—42no—
77C4⋊C4×C31—22no—
78C4⋊C4×C31—22no—
79C4⋊C4×C31—82no—
80C22⋊C4×C31—28no—
81C22⋊C4×C31—28no—
82C22⋊C4×C31—24no—
83C22⋊C4×C31—24no—
84C22⋊C4×C31—42no—
85C22⋊C4×C31—44no—
86C22⋊C4×C31—41no—
87C22⋊C4×C31—41no—
88C22⋊C4×C31—82no—
89C42×C31—44no—
90C42×C31—42no—
91C42×C31—82no—
92#9477C3111yesmaximal, characteristic, direct product factor, Sylow
93C23×C31—22no—
94C23×C31—22no—
95C23×C31—24no—
96C23×C31—22no—
97C23×C31—22no—
98C23×C31—22no—
99C23×C31—22no—
100C23×C31—41no—
101C23×C31—42no—
102C23×C31—42no—
103C23×C31—42no—
104C23×C31—42no—
105C23×C31—81no—
106C23×C31—81no—
107C23×C31—88no—
108C23×C31#10414yes—
109C23×C31C2×C22⋊C411yescharacteristic
110C23×C31#12011yescharacteristic
111C23×C31#12311yescharacteristic
112C2×C4×C31—24no—
113C2×C4×C31—28no—
114C2×C4×C31—22no—
115C2×C4×C31—41no—
116C2×C4×C31—41no—
117C2×C4×C31—42no—
118C2×C4×C31—44no—
119C2×C4×C31—42no—
120C2×C4×C31—44no—
121C2×C4×C31—42no—
122C2×C4×C31—42no—
123C2×C4×C31—42no—
124C2×C4×C31—42no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
127C2×C4×C31—42no—
128C2×C4×C31—42no—
129C2×C4×C31—81no—
130C2×C4×C31—88no—
131C2×C4×C31C4×D412yes—
132C2×C4×C31#12112yes—
133C8×C31—82no—
134#2482C2×C3111yescharacteristic
135#2273C2×C3111yescharacteristic
136#1826C2×C3112yescomplemented
137#1462C2×C3111yescharacteristic
138#1284C2×C3112yescomplemented
139C22×C31—21no—
140C22×C31—21no—
141C22×C31—21no—
142C22×C31—21no—
143C22×C31—21no—
144C22×C31—24no—
145C22×C31—21no—
146C22×C31—21no—
147C22×C31—44no—
148C22×C31—44no—
149C22×C31—44no—
150C22×C31—44no—
151C22×C31—42no—
152C22×C31—42no—
153C22×C31—82no—
154C22×C31—84no—
155C22×C31#38611yescharacteristic
156C22×C31C2×#10412yes—
157C22×C31#45812yes—
158C4×C31—22no—
159C4×C31—22no—
160C4×C31—48no—
161C4×C31—42no—
162C4×C31—82no—
163C4×C31—81no—
164C4×C31—81no—
165C2×#117C22×C3111yescharacteristic
166C22×C22⋊C4—21no—
167C22×C22⋊C4C22×C3111yescharacteristic
168#407C22×C3112yes—
169#399C4×C3112yescomplemented
170#397C4×C3112yescomplemented
171#388—22no—
172#385—21no—
173#381C22×C3111yescharacteristic
174#379—21no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—24no—
177C4×C22⋊C4C22×C3112yes—
178#339—24no—
179#323—22no—
180C2×C31—22no—
181C2×C31—22no—
182C2×C31—41no—
183C2×C31—44no—
184C2×C31#189011yescharacteristic
185C2×C31#189611yescharacteristic
186C2×C31#191111yescharacteristic
187C25D4×C3111yescharacteristic
188C23×C4—22no—
189C23×C4D4×C3111yescharacteristic
190C23×C4C23×C3111yescharacteristic
191C2×M16—22no—
192#117—22no—
193#117—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4C2×C4×C3111yescharacteristic, complemented
196C2×C4⋊C4C2×C4×C3111yescharacteristic, complemented
197C2×C22⋊C4—22no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C2×C4×C3112yes—
207C2×C22⋊C4D4×C3112yes—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42D4×C3114yes—
211C22⋊C8—22no—
212C22⋊C8—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—21no—
217#95C2×C4×C3112yescomplemented
218C31#947711yescharacteristic, direct product factor, Sylow
219C24—21no—
220C24—24no—
221C24—21no—
222C24—42no—
223C24—81no—
224C24C22⋊C4×C3112yes—
225C24C2×D4×C3111yescharacteristic
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—28no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—81no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4C22⋊C4×C3112yes—
238C22×C4C22×C4×C3111yescharacteristic
239C22×C4C2×D4×C3112yes—
240C22×C4C2×D4×C3111yescharacteristic
241C22×C4#41×C3112yes—
242M16—44no—
243C2×C8—42no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—82no—
247C22⋊C4—28no—
248C22⋊C4—28no—
249C22⋊C4—24no—
250C22⋊C4—24no—
251C22⋊C4—42no—
252C22⋊C4—44no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—82no—
256C42—44no—
257C42—42no—
258C42—82no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—41no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—81no—
272C23—81no—
273C23—88no—
274C23C31×#10414yes—
275C23C2×C22⋊C4×C3111yescharacteristic
276C23C31×#12011yescharacteristic
277C23C31×#12311yescharacteristic
278C2×C4—24no—
279C2×C4—28no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—81no—
296C2×C4—88no—
297C2×C4C4×D4×C3112yes—
298C2×C4C31×#12112yes—
299C8—82no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—44no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—42no—
313C22—42no—
314C22—82no—
315C22—84no—
316C22C31×#38611yescharacteristic
317C22C2×C31×#10412yes—
318C22C31×#45812yes—
319C4—22no—
320C4—22no—
321C4—48no—
322C4—42no—
323C4—82no—
324C4—81no—
325C4—81no—
326C2—22no—
327C2—22no—
328C2—41no—
329C2—44no—
330C2C31×#189011yescharacteristic
331C2C31×#189611yescharacteristic
332C2C31×#191111yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC25×C31D4
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
3. C31×#1826C2semidirect
5. C31×#1284C2semidirect
10. C31×#399C4semidirect
11. C31×#397C4semidirect
29. C2×C4⋊C4×C31C2×C4semidirect
30. C2×C4⋊C4×C31C2×C4semidirect
51. C31×#95C2×C4semidirect
92. #9477C31direct (N × H)
136. #1826C2×C31semidirect
138. #1284C2×C31semidirect
169. #399C4×C31semidirect
170. #397C4×C31semidirect
195. C2×C4⋊C4C2×C4×C31semidirect
196. C2×C4⋊C4C2×C4×C31semidirect
217. #95C2×C4×C31semidirect
218. C31#9477direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#24822
C31×#22732
C31×#18262
C31×#14622
C31×#12842
#947731