Finite Group Databasebeta

← C31×#9474 #21466233

Subgroups — C31×#9474 #21466233

Order 7936.

576 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2973C211yesmaximal, characteristic
2C31×#2486C211yesmaximal, characteristic
3C31×#1823C212yesmaximal, complemented
4C31×#1799C211yesmaximal, characteristic
5C31×#1289C212yesmaximal, complemented
6C2×C31×#123—21no—
7C2×C4×D4×C31C2211yescharacteristic
8C22×C4⋊C4×C31C2211yescharacteristic
9C31×#435—24no—
10C31×#432—21no—
11C2×C31×#99—21no—
12C31×#407—21no—
13C2×C31×C22⋊C8C2212yes—
14C31×#398C412yescomplemented
15C31×#397C412yescomplemented
16C31×#389—21no—
17C31×#381—21no—
18C31×#380—21no—
19C31×#380C2212yes—
20C31×#380C2211yescharacteristic
21C31×#342—21no—
22C31×#340—24no—
23C22×D4×C31D411yescharacteristic
24C23×C4×C31—21no—
25C23×C4×C31D411yescharacteristic
26C23×C4×C31C2311yescharacteristic
27C2×M16×C31—22no—
28C22×C8×C31D414yes—
29C31×#123—44no—
30C4×D4×C31—22no—
31C4×D4×C31—22no—
32C2×C4⋊C4×C31—22no—
33C2×C4⋊C4×C31—22no—
34C2×C4⋊C4×C31—22no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—22no—
38C2×C4⋊C4×C31C2×C412yes—
39C2×C4⋊C4×C31C2×C411yescharacteristic
40C2×C22⋊C4×C31—22no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—22no—
43C2×C22⋊C4×C31C2×C411yescharacteristic
44C2×C22⋊C4×C31D411yescharacteristic
45C2×C42×C31D411yescharacteristic
46C31×C4⋊C8—22no—
47C31×#99—42no—
48C31×C22⋊C8—22no—
49C31×C22⋊C8—22no—
50C31×C22⋊C8—22no—
51C31×#97—22no—
52C31×#95—22no—
53C31×#95—22no—
54C31×#95—22no—
55C31×#95C2×C412yes—
56C24×C31C22⋊C411yescharacteristic
57C24×C31C2×D411yescharacteristic
58C2×D4×C31—22no—
59C2×D4×C31—21no—
60C2×D4×C31—22no—
61C2×D4×C31—21no—
62C22×C4×C31—21no—
63C22×C4×C31—21no—
64C22×C4×C31—22no—
65C22×C4×C31—22no—
66C22×C4×C31—42no—
67C22×C4×C31—42no—
68C22×C4×C31—42no—
69C22×C4×C31—42no—
70C22×C4×C31—41no—
71C22×C4×C31—41no—
72C22×C4×C31—42no—
73C22×C4×C31C22⋊C411yescharacteristic
74C22×C4×C31C22⋊C411yescharacteristic
75C22×C4×C31C22⋊C411yescharacteristic
76C22×C4×C31C22×C411yescharacteristic
77C22×C4×C31C2×D412yes—
78C22×C4×C31C2×D411yescharacteristic
79C22×C4×C31#4112yes—
80M16×C31—82no—
81C2×C8×C31—24no—
82C2×C8×C31—44no—
83C4⋊C4×C31—22no—
84C4⋊C4×C31—24no—
85C4⋊C4×C31—42no—
86C4⋊C4×C31—44no—
87C4⋊C4×C31—44no—
88C4⋊C4×C31—41no—
89C4⋊C4×C31—41no—
90C22⋊C4×C31—22no—
91C22⋊C4×C31—22no—
92C22⋊C4×C31—44no—
93C22⋊C4×C31—42no—
94C22⋊C4×C31—42no—
95C22⋊C4×C31—42no—
96C42×C31—21no—
97C42×C31—21no—
98#9474C3111yesmaximal, characteristic, direct product factor, Sylow
99C23×C31—21no—
100C23×C31—21no—
101C23×C31—21no—
102C23×C31—41no—
103C23×C31—41no—
104C23×C31—41no—
105C23×C31—41no—
106C23×C31C2×C22⋊C411yescharacteristic
107C23×C31#12112yes—
108C23×C31AGL(1,ℤ/8)11yescharacteristic
109C23×C31AGL(1,ℤ/8)12yes—
110C23×C31#13711yescharacteristic
111D4×C31—42no—
112D4×C31—42no—
113C2×C4×C31—24no—
114C2×C4×C31—22no—
115C2×C4×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—41no—
121C2×C4×C31—42no—
122C2×C4×C31—42no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—42no—
126C2×C4×C31—42no—
127C2×C4×C31—42no—
128C2×C4×C31—42no—
129C2×C4×C31—42no—
130C2×C4×C31—42no—
131C2×C4×C31—81no—
132C2×C4×C31—82no—
133C2×C4×C31—82no—
134C2×C4×C31—82no—
135C2×C4×C31—82no—
136C2×C4×C31—82no—
137C2×C4×C31C4×D412yes—
138C2×C4×C31#12011yescharacteristic
139C2×C4×C31#12311yescharacteristic
140C8×C31—44no—
141#2973C2×C3111yescharacteristic
142#2486C2×C3111yescharacteristic
143#1823C2×C3112yescomplemented
144#1799C2×C3111yescharacteristic
145#1289C2×C3112yescomplemented
146C22×C31—21no—
147C22×C31—22no—
148C22×C31—21no—
149C22×C31—22no—
150C22×C31—42no—
151C22×C31—41no—
152C22×C31—41no—
153C22×C31—42no—
154C22×C31—41no—
155C22×C31—41no—
156C22×C31—41no—
157C22×C31—81no—
158C22×C31#38611yescharacteristic
159C22×C31#41711yescharacteristic
160C22×C31#41811yescharacteristic
161C22×C31#46812yes—
162C22×C31#46912yes—
163C4×C31—21no—
164C4×C31—22no—
165C4×C31—41no—
166C4×C31—41no—
167C4×C31—41no—
168C4×C31—82no—
169C4×C31—82no—
170C4×C31—162no—
171C4×C31#45812yes—
172C2×#123—21no—
173C2×C4×D4C22×C3111yescharacteristic
174C22×C4⋊C4C22×C3111yescharacteristic
175#435—24no—
176#432—21no—
177C2×#99—21no—
178#407—21no—
179C2×C22⋊C8C22×C3112yes—
180#398C4×C3112yescomplemented
181#397C4×C3112yescomplemented
182#389—21no—
183#381—21no—
184#380—21no—
185#380C22×C3112yes—
186#380C22×C3111yescharacteristic
187#342—21no—
188#340—24no—
189C2×C31—21no—
190C2×C31—22no—
191C2×C31—21no—
192C2×C31—22no—
193C2×C31—81no—
194C2×C31#189011yescharacteristic
195C2×C31#189511yescharacteristic
196C2×C31#190911yescharacteristic
197C22×D4D4×C3111yescharacteristic
198C23×C4—21no—
199C23×C4D4×C3111yescharacteristic
200C23×C4C23×C3111yescharacteristic
201C2×M16—22no—
202C22×C8D4×C3114yes—
203#123—44no—
204C4×D4—22no—
205C4×D4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4C2×C4×C3112yes—
213C2×C4⋊C4C2×C4×C3111yescharacteristic
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4C2×C4×C3111yescharacteristic
218C2×C22⋊C4D4×C3111yescharacteristic
219C2×C42D4×C3111yescharacteristic
220C4⋊C8—22no—
221#99—42no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225#97—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95C2×C4×C3112yes—
230C31#947411yescharacteristic, direct product factor, Sylow
231C24C22⋊C4×C3111yescharacteristic
232C24C2×D4×C3111yescharacteristic
233C2×D4—22no—
234C2×D4—21no—
235C2×D4—22no—
236C2×D4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4C22⋊C4×C3111yescharacteristic
249C22×C4C22⋊C4×C3111yescharacteristic
250C22×C4C22⋊C4×C3111yescharacteristic
251C22×C4C22×C4×C3111yescharacteristic
252C22×C4C2×D4×C3112yes—
253C22×C4C2×D4×C3111yescharacteristic
254C22×C4#41×C3112yes—
255M16—82no—
256C2×C8—24no—
257C2×C8—44no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—44no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C42—21no—
272C42—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×C22⋊C4×C3111yescharacteristic
281C23C31×#12112yes—
282C23C31×AGL(1,ℤ/8)11yescharacteristic
283C23C31×AGL(1,ℤ/8)12yes—
284C23C31×#13711yescharacteristic
285D4—42no—
286D4—42no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4C4×D4×C3112yes—
312C2×C4C31×#12011yescharacteristic
313C2×C4C31×#12311yescharacteristic
314C8—44no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22C31×#38611yescharacteristic
328C22C31×#41711yescharacteristic
329C22C31×#41811yescharacteristic
330C22C31×#46812yes—
331C22C31×#46912yes—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—82no—
338C4—82no—
339C4—162no—
340C4C31×#45812yes—
341C2—21no—
342C2—22no—
343C2—21no—
344C2—22no—
345C2—81no—
346C2C31×#189011yescharacteristic
347C2C31×#189511yescharacteristic
348C2C31×#190911yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC22×D4×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×#1823C2semidirect
5. C31×#1289C2semidirect
14. C31×#398C4semidirect
15. C31×#397C4semidirect
98. #9474C31direct (N × H)
143. #1823C2×C31semidirect
145. #1289C2×C31semidirect
180. #398C4×C31semidirect
181. #397C4×C31semidirect
230. C31#9474direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#29732
C31×#24862
C31×#18232
C31×#17992
C31×#12892
#947431