Finite Group Databasebeta

← C31×#11772 #21468531

Subgroups — C31×#11772 #21468531

Order 7936.

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

normal subgroups only

This group has 786 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×#2408C211yesmaximal, characteristic
2C4×C31×#99C211yesmaximal, characteristic, complemented
3C31×#1466C211yesmaximal, characteristic, complemented
4C31×#1343C214yesmaximal, complemented
5C23×D4×C31C411yescharacteristic, complemented
6C2×C31×#121—21no—
7C2×C31×#120—21no—
8C2×C4×D4×C31C2211yescharacteristic
9C31×#432—21no—
10C2×C31×#99—21no—
11C31×#394C412yescomplemented
12C31×#392C2212yes—
13C31×#390—21no—
14C31×#380C411yescharacteristic, complemented
15C31×#379—21no—
16C4×C22⋊C4×C31—21no—
17C31×#342C2212yescomplemented
18C31×#323—21no—
19C31×#323C2212yescomplemented
20C25×C31—21no—
21C22×D4×C31—22no—
22C22×D4×C31—21no—
23C22×D4×C31—41no—
24C22×D4×C31—41no—
25C22×D4×C31—44no—
26C22×D4×C31C2×C412yes—
27C22×D4×C31D411yescharacteristic
28C23×C4×C31C2×C411yescharacteristic
29C23×C4×C31D411yescharacteristic
30C31×#121—44no—
31C31×#120—44no—
32C4×D4×C31—41no—
33C4×D4×C31—41no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31C2×C411yescharacteristic
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—21no—
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31—22no—
41C2×C22⋊C4×C31D411yescharacteristic
42C2×C22⋊C4×C31C2311yescharacteristic
43C2×C42×C31—41no—
44C2×C42×C31D411yescharacteristic
45C31×C4⋊C8—41no—
46C31×#99—24no—
47C31×C22⋊C8—22no—
48C31×#97—41no—
49C31×#95—41no—
50C31×#95C2×C412yes—
51C24×C31—22no—
52C24×C31—42no—
53C24×C31—44no—
54C24×C31—42no—
55C24×C31—42no—
56C24×C31—82no—
57C24×C31C22⋊C411yescharacteristic
58C24×C31C2×D411yescharacteristic
59C2×D4×C31—24no—
60C2×D4×C31—22no—
61C2×D4×C31—22no—
62C2×D4×C31—41no—
63C2×D4×C31—42no—
64C2×D4×C31—42no—
65C2×D4×C31—48no—
66C2×D4×C31—44no—
67C2×D4×C31—44no—
68C2×D4×C31—44no—
69C2×D4×C31—44no—
70C2×D4×C31—44no—
71C2×D4×C31—41no—
72C2×D4×C31—41no—
73C2×D4×C31—82no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—22no—
79C22×C4×C31—42no—
80C22×C4×C31—42no—
81C22×C4×C31—42no—
82C22×C4×C31C22⋊C411yescharacteristic
83C22×C4×C31C22⋊C411yescharacteristic
84C22×C4×C31C22⋊C411yescharacteristic
85C22×C4×C31C22×C411yescharacteristic
86C22×C4×C31C2×D411yescharacteristic
87C2×C8×C31—42no—
88C4⋊C4×C31—41no—
89C4⋊C4×C31—42no—
90C22⋊C4×C31—24no—
91C22⋊C4×C31—41no—
92C22⋊C4×C31—41no—
93C22⋊C4×C31—44no—
94C22⋊C4×C31—42no—
95C22⋊C4×C31—44no—
96C22⋊C4×C31—42no—
97C22⋊C4×C31—82no—
98C42×C31—21no—
99C42×C31—21no—
100C42×C31—82no—
101#11772C3111yesmaximal, characteristic, direct product factor, Sylow
102C23×C31—21no—
103C23×C31—21no—
104C23×C31—21no—
105C23×C31—21no—
106C23×C31—24no—
107C23×C31—41no—
108C23×C31—41no—
109C23×C31—41no—
110C23×C31—41no—
111C23×C31—42no—
112C23×C31—42no—
113C23×C31—42no—
114C23×C31—48no—
115C23×C31—42no—
116C23×C31—42no—
117C23×C31—42no—
118C23×C31—84no—
119C23×C31—88no—
120C23×C31—88no—
121C23×C31—84no—
122C23×C31#10412yes—
123C23×C31C2×C22⋊C411yescharacteristic
124C23×C31AGL(1,ℤ/8)12yes—
125D4×C31—44no—
126D4×C31—44no—
127D4×C31—44no—
128D4×C31—44no—
129D4×C31—44no—
130D4×C31—88no—
131C2×C4×C31—22no—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—42no—
139C2×C4×C31—42no—
140C2×C4×C31—42no—
141C2×C4×C31—41no—
142C2×C4×C31—41no—
143C2×C4×C31—41no—
144C2×C4×C31—44no—
145C2×C4×C31—42no—
146C2×C4×C31—42no—
147C2×C4×C31—42no—
148C2×C4×C31—81no—
149C2×C4×C31—82no—
150C2×C4×C31—82no—
151C2×C4×C31#9911yescharacteristic
152C2×C4×C31#9911yescharacteristic
153C8×C31—82no—
154#2408C2×C3111yescharacteristic
155C4×#99C2×C3111yescharacteristic, complemented
156#1466C2×C3111yescharacteristic, complemented
157#1343C2×C3114yescomplemented
158C22×C31—21no—
159C22×C31—21no—
160C22×C31—22no—
161C22×C31—22no—
162C22×C31—42no—
163C22×C31—42no—
164C22×C31—42no—
165C22×C31—44no—
166C22×C31—44no—
167C22×C31—41no—
168C22×C31—41no—
169C22×C31—41no—
170C22×C31—41no—
171C22×C31—44no—
172C22×C31—81no—
173C22×C31—82no—
174C22×C31—82no—
175C22×C31—816no—
176C22×C31—82no—
177C22×C31—82no—
178C22×C31C2×#9911yescharacteristic
179C22×C31#41811yescharacteristic
180C22×C31C2×#10411yescharacteristic
181C4×C31—21no—
182C4×C31—41no—
183C4×C31—41no—
184C4×C31—41no—
185C4×C31—41no—
186C4×C31—82no—
187C4×C31—84no—
188C4×C31AΣL(1,𝔽16)12yes—
189C23×D4C4×C3111yescharacteristic, complemented
190C2×#121—21no—
191C2×#120—21no—
192C2×C4×D4C22×C3111yescharacteristic
193#432—21no—
194C2×#99—21no—
195#394C4×C3112yescomplemented
196#392C22×C3112yes—
197#390—21no—
198#380C4×C3111yescharacteristic, complemented
199#379—21no—
200C4×C22⋊C4—21no—
201#342C22×C3112yescomplemented
202#323—21no—
203#323C22×C3112yescomplemented
204C2×C31—21no—
205C2×C31—21no—
206C2×C31—42no—
207C2×C31—42no—
208C2×C31—84no—
209C2×C31#150711yescharacteristic
210C2×C31C2×AΣL(1,𝔽16)11yescharacteristic
211C2×C31#212811yescharacteristic
212C25—21no—
213C22×D4—22no—
214C22×D4—21no—
215C22×D4—41no—
216C22×D4—41no—
217C22×D4—44no—
218C22×D4C2×C4×C3112yes—
219C22×D4D4×C3111yescharacteristic
220C23×C4C2×C4×C3111yescharacteristic
221C23×C4D4×C3111yescharacteristic
222#121—44no—
223#120—44no—
224C4×D4—41no—
225C4×D4—41no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C2×C4×C3111yescharacteristic
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4D4×C3111yescharacteristic
234C2×C22⋊C4C23×C3111yescharacteristic
235C2×C42—41no—
236C2×C42D4×C3111yescharacteristic
237C4⋊C8—41no—
238#99—24no—
239C22⋊C8—22no—
240#97—41no—
241#95—41no—
242#95C2×C4×C3112yes—
243C31#1177211yescharacteristic, direct product factor, Sylow
244C24—22no—
245C24—42no—
246C24—44no—
247C24—42no—
248C24—42no—
249C24—82no—
250C24C22⋊C4×C3111yescharacteristic
251C24C2×D4×C3111yescharacteristic
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—48no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—82no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4C22⋊C4×C3111yescharacteristic
276C22×C4C22⋊C4×C3111yescharacteristic
277C22×C4C22⋊C4×C3111yescharacteristic
278C22×C4C22×C4×C3111yescharacteristic
279C22×C4C2×D4×C3111yescharacteristic
280C2×C8—42no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C22⋊C4—24no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—44no—
287C22⋊C4—42no—
288C22⋊C4—44no—
289C22⋊C4—42no—
290C22⋊C4—82no—
291C42—21no—
292C42—21no—
293C42—82no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—24no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—48no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—84no—
311C23—88no—
312C23—88no—
313C23—84no—
314C23C31×#10412yes—
315C23C2×C22⋊C4×C3111yescharacteristic
316C23C31×AGL(1,ℤ/8)12yes—
317D4—44no—
318D4—44no—
319D4—44no—
320D4—44no—
321D4—44no—
322D4—88no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—81no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4C31×#9911yescharacteristic
344C2×C4C31×#9911yescharacteristic
345C8—82no—
346C22—21no—
347C22—21no—
348C22—22no—
349C22—22no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—44no—
354C22—44no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—44no—
360C22—81no—
361C22—82no—
362C22—82no—
363C22—816no—
364C22—82no—
365C22—82no—
366C22C2×C31×#9911yescharacteristic
367C22C31×#41811yescharacteristic
368C22C2×C31×#10411yescharacteristic
369C4—21no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—82no—
375C4—84no—
376C4C31×AΣL(1,𝔽16)12yes—
377C2—21no—
378C2—21no—
379C2—42no—
380C2—42no—
381C2—84no—
382C2C31×#150711yescharacteristic
383C2C2×C31×AΣL(1,𝔽16)11yescharacteristic
384C2C31×#212811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC31×#2408C2
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
2. C4×C31×#99C2semidirect
3. C31×#1466C2semidirect
4. C31×#1343C2semidirect
5. C23×D4×C31C4semidirect
11. C31×#394C4semidirect
14. C31×#380C4semidirect
17. C31×#342C22semidirect
19. C31×#323C22semidirect
101. #11772C31direct (N × H)
155. C4×#99C2×C31semidirect
156. #1466C2×C31semidirect
157. #1343C2×C31semidirect
189. C23×D4C4×C31semidirect
195. #394C4×C31semidirect
198. #380C4×C31semidirect
201. #342C22×C31semidirect
203. #323C22×C31semidirect
243. C31#11772direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#24082
C4×C31×#992
C31×#14662
C31×#13432
#1177231