Finite Group Databasebeta

← C31×#12798 #21469557

Subgroups — C31×#12798 #21469557

Order 7936.

544 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 134 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
1C4×C31×#124C211yesmaximal, characteristic
2C2×C31×#354C212yesmaximal
3C31×#1799C212yesmaximal
4C31×#1393C212yesmaximal
5C2×C31×#124C2211yescharacteristic
6C2×C4×Q8×C31C411yescharacteristic, complemented
7C2×C4×D4×C31C2211yescharacteristic
8C2×C31×#99C2214yes—
9C31×#391C411yescharacteristic, complemented
10C31×#390C2211yescharacteristic
11C31×#388—22no—
12C31×#382C411yescharacteristic, complemented
13C31×#380—22no—
14C4×C22⋊C4×C31—22no—
15C43×C31C411yescharacteristic, complemented
16C31×#354—24no—
17C31×#342—22no—
18C22×Q8×C31C2×C411yescharacteristic
19C22×D4×C31C2311yescharacteristic
20C23×C4×C31D412yes—
21C31×#124—22no—
22C31×#124D412yes—
23C31×#124Q812yes—
24C4×Q8×C31—41no—
25C4×Q8×C31—41no—
26C4×D4×C31—41no—
27C4×D4×C31—41no—
28C2×C4⋊C4×C31—21no—
29C2×C4⋊C4×C31—42no—
30C2×C4⋊C4×C31C2×C411yescharacteristic
31C2×C4⋊C4×C31C2×C411yescharacteristic
32C2×C22⋊C4×C31—22no—
33C2×C22⋊C4×C31—22no—
34C2×C22⋊C4×C31—24no—
35C2×C22⋊C4×C31D412yes—
36C2×C42×C31—21no—
37C2×C42×C31—21no—
38C2×C42×C31—21no—
39C2×C42×C31C2×C411yescharacteristic
40C2×C42×C31C2×C411yescharacteristic
41C2×C42×C31C2×C411yescharacteristic
42C31×#99—24no—
43C31×#99—24no—
44C31×#95—22no—
45C31×#95—22no—
46C31×#95—42no—
47C24×C31C2×D412yes—
48C2×Q8×C31—21no—
49C2×Q8×C31—41no—
50C2×Q8×C31—41no—
51C2×Q8×C31C4⋊C412yescomplemented
52C2×D4×C31—41no—
53C2×D4×C31—41no—
54C2×D4×C31C2×D411yescharacteristic
55C2×D4×C31C2×Q811yescharacteristic
56C2×D4×C31#4112yes—
57C22×C4×C31—21no—
58C22×C4×C31—21no—
59C22×C4×C31—21no—
60C22×C4×C31—22no—
61C22×C4×C31—22no—
62C22×C4×C31—21no—
63C22×C4×C31—44no—
64C22×C4×C31—44no—
65C22×C4×C31C22⋊C412yes—
66C22×C4×C31C22⋊C411yescharacteristic
67C22×C4×C31C22⋊C411yescharacteristic
68C22×C4×C31C22×C411yescharacteristic
69C4⋊C4×C31—41no—
70C4⋊C4×C31—42no—
71C4⋊C4×C31—41no—
72C4⋊C4×C31—84no—
73C22⋊C4×C31—24no—
74C22⋊C4×C31—24no—
75C22⋊C4×C31—42no—
76C22⋊C4×C31—44no—
77C22⋊C4×C31—44no—
78C22⋊C4×C31—44no—
79C42×C31—21no—
80C42×C31—21no—
81C42×C31—21no—
82C42×C31—42no—
83C42×C31—42no—
84C42×C31—41no—
85C42×C31—41no—
86C42×C31—41no—
87C42×C31C4⋊C412yescomplemented
88#12798C3111yesmaximal, characteristic, direct product factor, Sylow
89C23×C31—22no—
90C23×C31—22no—
91C23×C31—42no—
92C23×C31—42no—
93C23×C31C2×C22⋊C411yescharacteristic
94C23×C31#12112yes—
95C23×C31C22⋊Q812yes—
96Q8×C31—42no—
97Q8×C31—42no—
98D4×C31—41no—
99D4×C31—41no—
100D4×C31—42no—
101C2×C4×C31—21no—
102C2×C4×C31—21no—
103C2×C4×C31—22no—
104C2×C4×C31—22no—
105C2×C4×C31—22no—
106C2×C4×C31—22no—
107C2×C4×C31—21no—
108C2×C4×C31—22no—
109C2×C4×C31—41no—
110C2×C4×C31—41no—
111C2×C4×C31—41no—
112C2×C4×C31—41no—
113C2×C4×C31—41no—
114C2×C4×C31—41no—
115C2×C4×C31—44no—
116C2×C4×C31—41no—
117C2×C4×C31—42no—
118C2×C4×C31—42no—
119C2×C4×C31—41no—
120C2×C4×C31—82no—
121C2×C4×C31—84no—
122C2×C4×C31—84no—
123C2×C4×C31—84no—
124C2×C4×C31#9911yescharacteristic
125C2×C4×C31#9911yescharacteristic
126C2×C4×C31C2×C4⋊C411yescharacteristic
127C2×C4×C31#11711yescharacteristic
128C4×#124C2×C3111yescharacteristic
129C2×#354C2×C3112yes—
130#1799C2×C3112yes—
131#1393C2×C3112yes—
132C22×C31—21no—
133C22×C31—21no—
134C22×C31—22no—
135C22×C31—22no—
136C22×C31—42no—
137C22×C31—42no—
138C22×C31—42no—
139C22×C31—42no—
140C22×C31—82no—
141C22×C31#38011yescharacteristic
142C22×C31C2×#9911yescharacteristic
143C22×C31#41011yescharacteristic
144C4×C31—21no—
145C4×C31—21no—
146C4×C31—21no—
147C4×C31—42no—
148C4×C31—42no—
149C4×C31—42no—
150C4×C31—42no—
151C4×C31—41no—
152C4×C31—81no—
153C4×C31—164no—
154C4×C31#35412yescomplemented
155C2×#124C22×C3111yescharacteristic
156C2×C4×Q8C4×C3111yescharacteristic, complemented
157C2×C4×D4C22×C3111yescharacteristic
158C2×#99C22×C3114yes—
159#391C4×C3111yescharacteristic, complemented
160#390C22×C3111yescharacteristic
161#388—22no—
162#382C4×C3111yescharacteristic, complemented
163#380—22no—
164C4×C22⋊C4—22no—
165C43C4×C3111yescharacteristic, complemented
166#354—24no—
167#342—22no—
168C2×C31—21no—
169C2×C31—21no—
170C2×C31—42no—
171C2×C31—42no—
172C2×C31#179911yescharacteristic
173C2×C31C2×#35411yescharacteristic
174C2×C31#212611yescharacteristic
175C22×Q8C2×C4×C3111yescharacteristic
176C22×D4C23×C3111yescharacteristic
177C23×C4D4×C3112yes—
178#124—22no—
179#124D4×C3112yes—
180#124Q8×C3112yes—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4C2×C4×C3111yescharacteristic
188C2×C4⋊C4C2×C4×C3111yescharacteristic
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4D4×C3112yes—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42C2×C4×C3111yescharacteristic
197C2×C42C2×C4×C3111yescharacteristic
198C2×C42C2×C4×C3111yescharacteristic
199#99—24no—
200#99—24no—
201#95—22no—
202#95—22no—
203#95—42no—
204C31#1279811yescharacteristic, direct product factor, Sylow
205C24C2×D4×C3112yes—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8C4⋊C4×C3112yescomplemented
210C2×D4—41no—
211C2×D4—41no—
212C2×D4C2×D4×C3111yescharacteristic
213C2×D4C2×Q8×C3111yescharacteristic
214C2×D4#41×C3112yes—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4C22⋊C4×C3111yescharacteristic
224C22×C4C22⋊C4×C3112yes—
225C22×C4C22⋊C4×C3111yescharacteristic
226C22×C4C22×C4×C3111yescharacteristic
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—84no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C22⋊C4—42no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—44no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—41no—
241C42—42no—
242C42—42no—
243C42—41no—
244C42—41no—
245C42C4⋊C4×C3112yescomplemented
246C23—22no—
247C23—22no—
248C23—42no—
249C23—42no—
250C23C2×C22⋊C4×C3111yescharacteristic
251C23C31×#12112yes—
252C23C31×C22⋊Q812yes—
253Q8—42no—
254Q8—42no—
255D4—41no—
256D4—41no—
257D4—42no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—44no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—82no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4—84no—
281C2×C4C31×#9911yescharacteristic
282C2×C4C31×#9911yescharacteristic
283C2×C4C2×C4⋊C4×C3111yescharacteristic
284C2×C4C31×#11711yescharacteristic
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—82no—
294C22C31×#38011yescharacteristic
295C22C2×C31×#9911yescharacteristic
296C22C31×#41011yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—42no—
301C4—42no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—81no—
306C4—164no—
307C4C31×#35412yescomplemented
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2C31×#179911yescharacteristic
313C2C2×C31×#35411yescharacteristic
314C2C31×#212611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC22×D4C23×C31
prime radicalC22×D4×C31C23
LCSG ▷ C22×C4 ▷ C22 ▷ 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
6. C2×C4×Q8×C31C4semidirect
9. C31×#391C4semidirect
12. C31×#382C4semidirect
15. C43×C31C4semidirect
51. C2×Q8×C31C4⋊C4semidirect
87. C42×C31C4⋊C4semidirect
88. #12798C31direct (N × H)
154. C4×C31#354semidirect
156. C2×C4×Q8C4×C31semidirect
159. #391C4×C31semidirect
162. #382C4×C31semidirect
165. C43C4×C31semidirect
204. C31#12798direct (N × H)
209. C2×Q8C4⋊C4×C31semidirect
245. C42C4⋊C4×C31semidirect
307. C4C31×#354semidirect

Maximal subgroups

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

subgroupindex
C4×C31×#1242
C2×C31×#3542
C31×#17992
C31×#13932
#1279831