Finite Group Databasebeta

← C31×#11788 #21468547

Subgroups — C31×#11788 #21468547

Order 7936.

516 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2542C211yesmaximal, characteristic
2C4×C31×#99C211yesmaximal, characteristic
3C31×#1465C211yesmaximal, characteristic
4C31×#1352C214yesmaximal, complemented
5C2×C31×C4⋊Q8C411yescharacteristic, complemented
6C2×C31×C22⋊Q8—21no—
7C2×C31×C22⋊Q8—21no—
8C2×C4×D4×C31C2211yescharacteristic
9C2×C31×#99—21no—
10C31×#407—21no—
11C31×#399C412yescomplemented
12C31×#399C2212yes—
13C31×#390—21no—
14C31×#384—21no—
15C4×C4⋊C4×C31C411yescharacteristic, complemented
16C4×C22⋊C4×C31—21no—
17C31×#342C2212yes—
18C31×#324—21no—
19C31×#323C2212yes—
20C22×Q8×C31—21no—
21C22×D4×C31D411yescharacteristic
22C23×C4×C31D411yescharacteristic
23C23×C4×C31D411yescharacteristic
24C2×M16×C31—41no—
25C31×C4⋊Q8—41no—
26C31×C4⋊Q8—41no—
27C31×C22⋊Q8—44no—
28C31×C22⋊Q8—44no—
29C4×D4×C31—41no—
30C4×D4×C31—41no—
31C2×C4⋊C4×C31—21no—
32C2×C4⋊C4×C31—22no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—22no—
35C2×C4⋊C4×C31C2×C412yes—
36C2×C4⋊C4×C31D411yescharacteristic
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31C2×C411yescharacteristic
41C2×C22⋊C4×C31C2311yescharacteristic
42C2×C42×C31—21no—
43C2×C42×C31—41no—
44C2×C42×C31C2×C411yescharacteristic
45C31×#99—24no—
46C31×C22⋊C8—22no—
47C31×C22⋊C8—41no—
48C31×#95—41no—
49C31×#95C2×C412yes—
50C24×C31C22⋊C411yescharacteristic
51C24×C31C2×D411yescharacteristic
52C2×Q8×C31—42no—
53C2×Q8×C31—42no—
54C2×Q8×C31—41no—
55C2×Q8×C31—41no—
56C2×D4×C31—22no—
57C2×D4×C31—22no—
58C2×D4×C31—41no—
59C22×C4×C31—22no—
60C22×C4×C31—21no—
61C22×C4×C31—21no—
62C22×C4×C31—21no—
63C22×C4×C31—21no—
64C22×C4×C31—22no—
65C22×C4×C31—42no—
66C22×C4×C31—42no—
67C22×C4×C31—42no—
68C22×C4×C31C22⋊C411yescharacteristic
69C22×C4×C31C22⋊C411yescharacteristic
70C22×C4×C31C22⋊C411yescharacteristic
71C22×C4×C31C22×C411yescharacteristic
72C22×C4×C31C2×D411yescharacteristic
73M16×C31—82no—
74C2×C8×C31—42no—
75C4⋊C4×C31—24no—
76C4⋊C4×C31—41no—
77C4⋊C4×C31—44no—
78C4⋊C4×C31—44no—
79C4⋊C4×C31—42no—
80C4⋊C4×C31—42no—
81C22⋊C4×C31—24no—
82C22⋊C4×C31—41no—
83C22⋊C4×C31—41no—
84C22⋊C4×C31—44no—
85C22⋊C4×C31—44no—
86C22⋊C4×C31—42no—
87C42×C31—21no—
88C42×C31—21no—
89C42×C31—42no—
90C42×C31—82no—
91#11788C3111yesmaximal, characteristic, direct product factor, Sylow
92C23×C31—21no—
93C23×C31—21no—
94C23×C31—21no—
95C23×C31—21no—
96C23×C31—21no—
97C23×C31—41no—
98C23×C31—41no—
99C23×C31—41no—
100C23×C31—41no—
101C23×C31C2×C22⋊C411yescharacteristic
102C23×C31#13712yes—
103Q8×C31—82no—
104Q8×C31—82no—
105D4×C31—44no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—22no—
109C2×C4×C31—24no—
110C2×C4×C31—21no—
111C2×C4×C31—21no—
112C2×C4×C31—21no—
113C2×C4×C31—42no—
114C2×C4×C31—42no—
115C2×C4×C31—42no—
116C2×C4×C31—42no—
117C2×C4×C31—41no—
118C2×C4×C31—41no—
119C2×C4×C31—41no—
120C2×C4×C31—44no—
121C2×C4×C31—42no—
122C2×C4×C31—42no—
123C2×C4×C31—42no—
124C2×C4×C31—81no—
125C2×C4×C31—81no—
126C2×C4×C31—82no—
127C2×C4×C31—82no—
128C2×C4×C31#9911yescharacteristic
129C2×C4×C31#9911yescharacteristic
130C2×C4×C31#10412yes—
131C8×C31—82no—
132#2542C2×C3111yescharacteristic
133C4×#99C2×C3111yescharacteristic
134#1465C2×C3111yescharacteristic
135#1352C2×C3114yescomplemented
136C22×C31—21no—
137C22×C31—21no—
138C22×C31—22no—
139C22×C31—42no—
140C22×C31—42no—
141C22×C31—41no—
142C22×C31—41no—
143C22×C31—41no—
144C22×C31—41no—
145C22×C31—41no—
146C22×C31—81no—
147C22×C31—81no—
148C22×C31C2×#9911yescharacteristic
149C22×C31#41911yescharacteristic
150C22×C31C2×#10411yescharacteristic
151C4×C31—21no—
152C4×C31—22no—
153C4×C31—44no—
154C4×C31—41no—
155C4×C31—41no—
156C4×C31—41no—
157C4×C31—84no—
158C4×C31—82no—
159C4×C31#35312yes—
160C2×C4⋊Q8C4×C3111yescharacteristic, complemented
161C2×C22⋊Q8—21no—
162C2×C22⋊Q8—21no—
163C2×C4×D4C22×C3111yescharacteristic
164C2×#99—21no—
165#407—21no—
166#399C4×C3112yescomplemented
167#399C22×C3112yes—
168#390—21no—
169#384—21no—
170C4×C4⋊C4C4×C3111yescharacteristic, complemented
171C4×C22⋊C4—21no—
172#342C22×C3112yes—
173#324—21no—
174#323C22×C3112yes—
175C2×C31—21no—
176C2×C31—21no—
177C2×C31—42no—
178C2×C31—81no—
179C2×C31#150811yescharacteristic
180C2×C31#212311yescharacteristic
181C2×C31C2×#35311yescharacteristic
182C22×Q8—21no—
183C22×D4D4×C3111yescharacteristic
184C23×C4D4×C3111yescharacteristic
185C23×C4D4×C3111yescharacteristic
186C2×M16—41no—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189C22⋊Q8—44no—
190C22⋊Q8—44no—
191C4×D4—41no—
192C4×D4—41no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C2×C4×C3112yes—
198C2×C4⋊C4D4×C3111yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C2×C4×C3111yescharacteristic
203C2×C22⋊C4C23×C3111yescharacteristic
204C2×C42—21no—
205C2×C42—41no—
206C2×C42C2×C4×C3111yescharacteristic
207#99—24no—
208C22⋊C8—22no—
209C22⋊C8—41no—
210#95—41no—
211#95C2×C4×C3112yes—
212C31#1178811yescharacteristic, direct product factor, Sylow
213C24C22⋊C4×C3111yescharacteristic
214C24C2×D4×C3111yescharacteristic
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—41no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C22⋊C4×C3111yescharacteristic
232C22×C4C22⋊C4×C3111yescharacteristic
233C22×C4C22⋊C4×C3111yescharacteristic
234C22×C4C22×C4×C3111yescharacteristic
235C22×C4C2×D4×C3111yescharacteristic
236M16—82no—
237C2×C8—42no—
238C4⋊C4—24no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C22⋊C4—24no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4—42no—
250C42—21no—
251C42—21no—
252C42—42no—
253C42—82no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23C2×C22⋊C4×C3111yescharacteristic
264C23C31×#13712yes—
265Q8—82no—
266Q8—82no—
267D4—44no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4C31×#9911yescharacteristic
291C2×C4C31×#9911yescharacteristic
292C2×C4C31×#10412yes—
293C8—82no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22—81no—
306C22C2×C31×#9911yescharacteristic
307C22C31×#41911yescharacteristic
308C22C2×C31×#10411yescharacteristic
309C4—21no—
310C4—22no—
311C4—44no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—84no—
316C4—82no—
317C4C31×#35312yes—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—81no—
322C2C31×#150811yescharacteristic
323C2C31×#212311yescharacteristic
324C2C2×C31×#35311yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1352C2semidirect
5. C2×C31×C4⋊Q8C4semidirect
11. C31×#399C4semidirect
15. C4×C4⋊C4×C31C4semidirect
91. #11788C31direct (N × H)
135. #1352C2×C31semidirect
160. C2×C4⋊Q8C4×C31semidirect
166. #399C4×C31semidirect
170. C4×C4⋊C4C4×C31semidirect
212. C31#11788direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25422
C4×C31×#992
C31×#14652
C31×#13522
#1178831