Finite Group Databasebeta

← C31×#12801 #21469560

Subgroups — C31×#12801 #21469560

Order 7936.

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

normal subgroups only

This group has 500 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×#355C212yesmaximal
3C31×#1801C212yesmaximal, complemented
4C31×#1394C212yesmaximal, complemented
5C2×C31×#124C2211yescharacteristic
6C2×C4×Q8×C31C2211yescharacteristic
7C2×C4×D4×C31C411yescharacteristic
8C31×#423—22no—
9C2×C31×#101C2214yes—
10C31×#391C2211yescharacteristic
11C31×#390C411yescharacteristic
12C31×#388—22no—
13C31×#382C411yescharacteristic
14C4×C22⋊C4×C31—22no—
15C43×C31C411yescharacteristic
16C31×#355—24no—
17C31×#343—22no—
18C22×Q8×C31C2311yescharacteristic
19C22×D4×C31C2×C411yescharacteristic
20C23×C4×C31—21no—
21C2×M16×C31—24no—
22C31×#124—22no—
23C31×#124D412yes—
24C31×#124Q812yes—
25C4×Q8×C31—41no—
26C4×Q8×C31—41no—
27C4×D4×C31—41no—
28C4×D4×C31—41no—
29C2×C4⋊C4×C31C2×C411yescharacteristic
30C2×C4⋊C4×C31C2×C411yescharacteristic
31C2×C4⋊C4×C31D412yes—
32C2×C22⋊C4×C31—22no—
33C2×C22⋊C4×C31—21no—
34C2×C22⋊C4×C31—22no—
35C2×C42×C31—21no—
36C2×C42×C31—21no—
37C2×C42×C31C2×C411yescharacteristic
38C2×C42×C31C2×C411yescharacteristic
39C2×C42×C31C2×C411yescharacteristic
40C2×C42×C31D412yes—
41C31×C4⋊C8—42no—
42C31×C4⋊C8—42no—
43C31×#101—24no—
44C31×#101—24no—
45C31×#95—22no—
46C31×#95—22no—
47C24×C31—21no—
48C2×Q8×C31—41no—
49C2×Q8×C31—41no—
50C2×Q8×C31C2×D411yescharacteristic
51C2×Q8×C31C2×Q811yescharacteristic
52C2×Q8×C31#4112yes—
53C2×D4×C31—21no—
54C2×D4×C31—41no—
55C2×D4×C31—41no—
56C2×D4×C31C4⋊C412yes—
57C22×C4×C31—21no—
58C22×C4×C31—21no—
59C22×C4×C31—21no—
60C22×C4×C31—44no—
61C22×C4×C31—41no—
62C22×C4×C31—41no—
63C22×C4×C31C22⋊C412yes—
64C22×C4×C31C22⋊C411yescharacteristic
65C22×C4×C31C22⋊C411yescharacteristic
66C22×C4×C31C22×C411yescharacteristic
67C22×C4×C31C2×D412yes—
68M16×C31—44no—
69M16×C31—44no—
70C2×C8×C31—44no—
71C4⋊C4×C31—41no—
72C4⋊C4×C31—42no—
73C4⋊C4×C31—41no—
74C22⋊C4×C31—24no—
75C22⋊C4×C31—24no—
76C22⋊C4×C31—44no—
77C22⋊C4×C31—42no—
78C42×C31—22no—
79C42×C31—22no—
80C42×C31—21no—
81C42×C31—21no—
82C42×C31—21no—
83C42×C31—42no—
84C42×C31—42no—
85C42×C31—41no—
86C42×C31C4⋊C412yes—
87#12801C3111yesmaximal, characteristic, direct product factor, Sylow
88C23×C31—22no—
89C23×C31—42no—
90C23×C31—42no—
91C23×C31—41no—
92C23×C31—41no—
93C23×C31C2×C22⋊C411yescharacteristic
94Q8×C31—41no—
95Q8×C31—41no—
96Q8×C31—42no—
97D4×C31—42no—
98D4×C31—42no—
99C2×C4×C31—21no—
100C2×C4×C31—22no—
101C2×C4×C31—21no—
102C2×C4×C31—22no—
103C2×C4×C31—22no—
104C2×C4×C31—22no—
105C2×C4×C31—22no—
106C2×C4×C31—21no—
107C2×C4×C31—22no—
108C2×C4×C31—41no—
109C2×C4×C31—41no—
110C2×C4×C31—41no—
111C2×C4×C31—41no—
112C2×C4×C31—41no—
113C2×C4×C31—42no—
114C2×C4×C31—42no—
115C2×C4×C31—41no—
116C2×C4×C31—82no—
117C2×C4×C31—82no—
118C2×C4×C31#9911yescharacteristic
119C2×C4×C31#9911yescharacteristic
120C2×C4×C31C2×C4⋊C411yescharacteristic
121C2×C4×C31#11711yescharacteristic
122C2×C4×C31#12112yes—
123C2×C4×C31C22⋊Q812yes—
124C8×C31—84no—
125C4×#124C2×C3111yescharacteristic
126C2×#355C2×C3112yes—
127#1801C2×C3112yescomplemented
128#1394C2×C3112yescomplemented
129C22×C31—21no—
130C22×C31—21no—
131C22×C31—44no—
132C22×C31—42no—
133C22×C31—82no—
134C22×C31—81no—
135C22×C31—81no—
136C22×C31#38011yescharacteristic
137C22×C31C2×#9911yescharacteristic
138C22×C31#41011yescharacteristic
139C4×C31—22no—
140C4×C31—21no—
141C4×C31—21no—
142C4×C31—22no—
143C4×C31—21no—
144C4×C31—42no—
145C4×C31—42no—
146C4×C31—41no—
147C4×C31—42no—
148C4×C31—81no—
149C4×C31#35512yescomplemented
150C2×#124C22×C3111yescharacteristic
151C2×C4×Q8C22×C3111yescharacteristic
152C2×C4×D4C4×C3111yescharacteristic
153#423—22no—
154C2×#101C22×C3114yes—
155#391C22×C3111yescharacteristic
156#390C4×C3111yescharacteristic
157#388—22no—
158#382C4×C3111yescharacteristic
159C4×C22⋊C4—22no—
160C43C4×C3111yescharacteristic
161#355—24no—
162#343—22no—
163C2×C31—21no—
164C2×C31—21no—
165C2×C31—82no—
166C2×C31#179911yescharacteristic
167C2×C31C2×#35511yescharacteristic
168C2×C31#213211yescharacteristic
169C22×Q8C23×C3111yescharacteristic
170C22×D4C2×C4×C3111yescharacteristic
171C23×C4—21no—
172C2×M16—24no—
173#124—22no—
174#124D4×C3112yes—
175#124Q8×C3112yes—
176C4×Q8—41no—
177C4×Q8—41no—
178C4×D4—41no—
179C4×D4—41no—
180C2×C4⋊C4C2×C4×C3111yescharacteristic
181C2×C4⋊C4C2×C4×C3111yescharacteristic
182C2×C4⋊C4D4×C3112yes—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42C2×C4×C3111yescharacteristic
189C2×C42C2×C4×C3111yescharacteristic
190C2×C42C2×C4×C3111yescharacteristic
191C2×C42D4×C3112yes—
192C4⋊C8—42no—
193C4⋊C8—42no—
194#101—24no—
195#101—24no—
196#95—22no—
197#95—22no—
198C31#1280111yescharacteristic, direct product factor, Sylow
199C24—21no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8C2×D4×C3111yescharacteristic
203C2×Q8C2×Q8×C3111yescharacteristic
204C2×Q8#41×C3112yes—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4C4⋊C4×C3112yes—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—44no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4C22⋊C4×C3111yescharacteristic
216C22×C4C22⋊C4×C3112yes—
217C22×C4C22⋊C4×C3111yescharacteristic
218C22×C4C22×C4×C3111yescharacteristic
219C22×C4C2×D4×C3112yes—
220M16—44no—
221M16—44no—
222C2×C8—44no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C22⋊C4—24no—
227C22⋊C4—24no—
228C22⋊C4—44no—
229C22⋊C4—42no—
230C42—22no—
231C42—22no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—41no—
236C42—42no—
237C42—42no—
238C42C4⋊C4×C3112yes—
239C23—22no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23C2×C22⋊C4×C3111yescharacteristic
245Q8—41no—
246Q8—41no—
247Q8—42no—
248D4—42no—
249D4—42no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4C31×#9911yescharacteristic
270C2×C4C31×#9911yescharacteristic
271C2×C4C2×C4⋊C4×C3111yescharacteristic
272C2×C4C31×#11711yescharacteristic
273C2×C4C31×#12112yes—
274C2×C4C31×C22⋊Q812yes—
275C8—84no—
276C22—21no—
277C22—21no—
278C22—44no—
279C22—42no—
280C22—82no—
281C22—81no—
282C22—81no—
283C22C31×#38011yescharacteristic
284C22C2×C31×#9911yescharacteristic
285C22C31×#41011yescharacteristic
286C4—22no—
287C4—21no—
288C4—21no—
289C4—22no—
290C4—21no—
291C4—42no—
292C4—42no—
293C4—41no—
294C4—42no—
295C4—81no—
296C4C31×#35512yescomplemented
297C2—21no—
298C2—21no—
299C2—82no—
300C2C31×#179911yescharacteristic
301C2C2×C31×#35511yescharacteristic
302C2C31×#213211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC22×Q8C23×C31
prime radicalC22×D4×C31C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1801C2semidirect
4. C31×#1394C2semidirect
87. #12801C31direct (N × H)
127. #1801C2×C31semidirect
128. #1394C2×C31semidirect
149. C4×C31#355semidirect
198. C31#12801direct (N × H)
296. C4C31×#355semidirect

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×#3552
C31×#18012
C31×#13942
#1280131