Finite Group Databasebeta

← C31×#11776 #21468535

Subgroups — C31×#11776 #21468535

Order 7936.

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

normal subgroups only

This group has 652 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×#2541C211yesmaximal, characteristic
2C4×C31×#99C211yesmaximal, characteristic, complemented
3C31×#1465C211yesmaximal, characteristic, complemented
4C31×#1343C214yesmaximal, complemented
5C2×C31×#127—21no—
6C2×C31×#127C411yescharacteristic, complemented
7C2×C31×#121—21no—
8C2×C31×#121—21no—
9C2×C4×D4×C31C2211yescharacteristic
10C2×C31×#99—21no—
11C31×#407—21no—
12C31×#392C412yescomplemented
13C31×#392C2212yes—
14C31×#390—21no—
15C4×C4⋊C4×C31C411yescharacteristic, complemented
16C4×C22⋊C4×C31—21no—
17C31×#342C2212yes—
18C31×#324—21no—
19C31×#323C2212yes—
20C22×D4×C31—22no—
21C22×D4×C31—22no—
22C22×D4×C31—21no—
23C22×D4×C31—21no—
24C22×D4×C31C2×C412yes—
25C22×D4×C31D411yescharacteristic
26C23×C4×C31D411yescharacteristic
27C23×C4×C31D411yescharacteristic
28C2×M16×C31—41no—
29C31×#127—41no—
30C31×#127—41no—
31C31×#127—44no—
32C31×#121—44no—
33C31×#121—44no—
34C4×D4×C31—41no—
35C4×D4×C31—41no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31D411yescharacteristic
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31—22no—
41C2×C22⋊C4×C31C2×C411yescharacteristic
42C2×C22⋊C4×C31C2311yescharacteristic
43C2×C42×C31—21no—
44C2×C42×C31—41no—
45C2×C42×C31C2×C411yescharacteristic
46C31×#99—24no—
47C31×C22⋊C8—22no—
48C31×C22⋊C8—41no—
49C31×#95—41no—
50C31×#95C2×C412yes—
51C24×C31—22no—
52C24×C31C22⋊C411yescharacteristic
53C24×C31C2×D411yescharacteristic
54C2×D4×C31—22no—
55C2×D4×C31—22no—
56C2×D4×C31—41no—
57C2×D4×C31—42no—
58C2×D4×C31—42no—
59C2×D4×C31—44no—
60C2×D4×C31—44no—
61C2×D4×C31—42no—
62C2×D4×C31—42no—
63C2×D4×C31—42no—
64C2×D4×C31—44no—
65C2×D4×C31—44no—
66C2×D4×C31—41no—
67C2×D4×C31—41no—
68C2×D4×C31—42no—
69C2×D4×C31—82no—
70C2×D4×C31—82no—
71C2×D4×C31—82no—
72C22×C4×C31—22no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—42no—
78C22×C4×C31—42no—
79C22×C4×C31—42no—
80C22×C4×C31C22⋊C411yescharacteristic
81C22×C4×C31C22⋊C411yescharacteristic
82C22×C4×C31C22⋊C411yescharacteristic
83C22×C4×C31C22×C411yescharacteristic
84C22×C4×C31C2×D411yescharacteristic
85M16×C31—82no—
86C2×C8×C31—42no—
87C4⋊C4×C31—24no—
88C4⋊C4×C31—41no—
89C22⋊C4×C31—24no—
90C22⋊C4×C31—41no—
91C22⋊C4×C31—41no—
92C22⋊C4×C31—44no—
93C22⋊C4×C31—44no—
94C22⋊C4×C31—42no—
95C42×C31—21no—
96C42×C31—21no—
97C42×C31—42no—
98C42×C31—82no—
99#11776C3111yesmaximal, characteristic, direct product factor, Sylow
100C23×C31—21no—
101C23×C31—21no—
102C23×C31—21no—
103C23×C31—21no—
104C23×C31—21no—
105C23×C31—41no—
106C23×C31—41no—
107C23×C31—41no—
108C23×C31—41no—
109C23×C31—42no—
110C23×C31—42no—
111C23×C31—42no—
112C23×C31—42no—
113C23×C31—42no—
114C23×C31—42no—
115C23×C31—42no—
116C23×C31C2×C22⋊C411yescharacteristic
117C23×C31AGL(1,ℤ/8)12yes—
118D4×C31—44no—
119D4×C31—82no—
120D4×C31—84no—
121D4×C31—88no—
122D4×C31—88no—
123D4×C31—84no—
124D4×C31—82no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—22no—
128C2×C4×C31—24no—
129C2×C4×C31—21no—
130C2×C4×C31—21no—
131C2×C4×C31—21no—
132C2×C4×C31—42no—
133C2×C4×C31—42no—
134C2×C4×C31—42no—
135C2×C4×C31—42no—
136C2×C4×C31—41no—
137C2×C4×C31—41no—
138C2×C4×C31—41no—
139C2×C4×C31—44no—
140C2×C4×C31—81no—
141C2×C4×C31—81no—
142C2×C4×C31—82no—
143C2×C4×C31—82no—
144C2×C4×C31#9911yescharacteristic
145C2×C4×C31#9911yescharacteristic
146C2×C4×C31#10412yes—
147C8×C31—82no—
148#2541C2×C3111yescharacteristic
149C4×#99C2×C3111yescharacteristic, complemented
150#1465C2×C3111yescharacteristic, complemented
151#1343C2×C3114yescomplemented
152C22×C31—21no—
153C22×C31—21no—
154C22×C31—22no—
155C22×C31—42no—
156C22×C31—42no—
157C22×C31—41no—
158C22×C31—41no—
159C22×C31—41no—
160C22×C31—41no—
161C22×C31—41no—
162C22×C31—81no—
163C22×C31—81no—
164C22×C31—82no—
165C22×C31—82no—
166C22×C31—82no—
167C22×C31—82no—
168C22×C31—82no—
169C22×C31—82no—
170C22×C31—82no—
171C22×C31C2×#9911yescharacteristic
172C22×C31#41811yescharacteristic
173C22×C31C2×#10411yescharacteristic
174C4×C31—21no—
175C4×C31—22no—
176C4×C31—44no—
177C4×C31—41no—
178C4×C31—41no—
179C4×C31—41no—
180C4×C31—84no—
181C4×C31#35312yes—
182C2×#127—21no—
183C2×#127C4×C3111yescharacteristic, complemented
184C2×#121—21no—
185C2×#121—21no—
186C2×C4×D4C22×C3111yescharacteristic
187C2×#99—21no—
188#407—21no—
189#392C4×C3112yescomplemented
190#392C22×C3112yes—
191#390—21no—
192C4×C4⋊C4C4×C3111yescharacteristic, complemented
193C4×C22⋊C4—21no—
194#342C22×C3112yes—
195#324—21no—
196#323C22×C3112yes—
197C2×C31—21no—
198C2×C31—21no—
199C2×C31—42no—
200C2×C31—81no—
201C2×C31—162no—
202C2×C31#150711yescharacteristic
203C2×C31#212211yescharacteristic
204C2×C31C2×#35311yescharacteristic
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4C2×C4×C3112yes—
210C22×D4D4×C3111yescharacteristic
211C23×C4D4×C3111yescharacteristic
212C23×C4D4×C3111yescharacteristic
213C2×M16—41no—
214#127—41no—
215#127—41no—
216#127—44no—
217#121—44no—
218#121—44no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4D4×C3111yescharacteristic
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4C2×C4×C3111yescharacteristic
227C2×C22⋊C4C23×C3111yescharacteristic
228C2×C42—21no—
229C2×C42—41no—
230C2×C42C2×C4×C3111yescharacteristic
231#99—24no—
232C22⋊C8—22no—
233C22⋊C8—41no—
234#95—41no—
235#95C2×C4×C3112yes—
236C31#1177611yescharacteristic, direct product factor, Sylow
237C24—22no—
238C24C22⋊C4×C3111yescharacteristic
239C24C2×D4×C3111yescharacteristic
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—82no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4C22⋊C4×C3111yescharacteristic
267C22×C4C22⋊C4×C3111yescharacteristic
268C22×C4C22⋊C4×C3111yescharacteristic
269C22×C4C22×C4×C3111yescharacteristic
270C22×C4C2×D4×C3111yescharacteristic
271M16—82no—
272C2×C8—42no—
273C4⋊C4—24no—
274C4⋊C4—41no—
275C22⋊C4—24no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—42no—
281C42—21no—
282C42—21no—
283C42—42no—
284C42—82no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23C2×C22⋊C4×C3111yescharacteristic
302C23C31×AGL(1,ℤ/8)12yes—
303D4—44no—
304D4—82no—
305D4—84no—
306D4—88no—
307D4—88no—
308D4—84no—
309D4—82no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4C31×#9911yescharacteristic
330C2×C4C31×#9911yescharacteristic
331C2×C4C31×#10412yes—
332C8—82no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22—42no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22—82no—
346C22—82no—
347C22—82no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—82no—
352C22C2×C31×#9911yescharacteristic
353C22C31×#41811yescharacteristic
354C22C2×C31×#10411yescharacteristic
355C4—21no—
356C4—22no—
357C4—44no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—84no—
362C4C31×#35312yes—
363C2—21no—
364C2—21no—
365C2—42no—
366C2—81no—
367C2—162no—
368C2C31×#150711yescharacteristic
369C2C31×#212211yescharacteristic
370C2C2×C31×#35311yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC31×#2541C2
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×#1465C2semidirect
4. C31×#1343C2semidirect
6. C2×C31×#127C4semidirect
12. C31×#392C4semidirect
15. C4×C4⋊C4×C31C4semidirect
99. #11776C31direct (N × H)
149. C4×#99C2×C31semidirect
150. #1465C2×C31semidirect
151. #1343C2×C31semidirect
183. C2×#127C4×C31semidirect
189. #392C4×C31semidirect
192. C4×C4⋊C4C4×C31semidirect
236. C31#11776direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25412
C4×C31×#992
C31×#14652
C31×#13432
#1177631