Finite Group Databasebeta

← C31×#11822 #21468581

Subgroups — C31×#11822 #21468581

Order 7936.

434 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2395C211yesmaximal, characteristic
2C31×#1799C211yesmaximal, characteristic
3C31×#1466C211yesmaximal, characteristic
4C31×#1352C212yesmaximal, complemented
5C31×#1344C212yesmaximal, complemented
6C2×C4×D4×C31C2211yescharacteristic
7C2×C31×#117C411yescharacteristic, complemented
8C31×#432—21no—
9C2×C31×#99—21no—
10C31×#400C411yescharacteristic, complemented
11C31×#399C2211yescharacteristic
12C31×#398C411yescharacteristic, complemented
13C31×#397—21no—
14C31×#397C2211yescharacteristic
15C31×#394—21no—
16C31×#380—21no—
17C4×C4⋊C4×C31—21no—
18C4×C22⋊C4×C31—21no—
19C4×C22⋊C4×C31C411yescharacteristic, complemented
20C31×#342C2212yes—
21C31×#323—21no—
22C31×#323C2212yes—
23C22×D4×C31D411yescharacteristic
24C23×C4×C31C2×C411yescharacteristic
25C23×C4×C31D411yescharacteristic
26C4×D4×C31—41no—
27C4×D4×C31—41no—
28C31×#117—41no—
29C31×#117—41no—
30C2×C4⋊C4×C31—21no—
31C2×C4⋊C4×C31—21no—
32C2×C4⋊C4×C31—41no—
33C2×C4⋊C4×C31C2×C411yescharacteristic
34C2×C4⋊C4×C31C2×C411yescharacteristic
35C2×C4⋊C4×C31C2×C411yescharacteristic
36C2×C22⋊C4×C31—21no—
37C2×C22⋊C4×C31—21no—
38C2×C22⋊C4×C31—21no—
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—22no—
41C2×C22⋊C4×C31D411yescharacteristic
42C2×C22⋊C4×C31C2311yescharacteristic
43C2×C42×C31—21no—
44C2×C42×C31—21no—
45C2×C42×C31D411yescharacteristic
46C31×C4⋊C8—41no—
47C31×#99—42no—
48C31×C22⋊C8—22no—
49C31×#97—41no—
50C31×#95—21no—
51C31×#95—21no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—41no—
55C31×#95C2×C411yescharacteristic
56C31×#95C2×C411yescharacteristic
57C24×C31C22⋊C411yescharacteristic
58C24×C31C2×D411yescharacteristic
59C2×D4×C31—21no—
60C2×D4×C31—21no—
61C2×D4×C31—41no—
62C2×D4×C31—41no—
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—21no—
66C22×C4×C31—21no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—42no—
72C22×C4×C31—42no—
73C22×C4×C31—42no—
74C22×C4×C31C22⋊C411yescharacteristic
75C22×C4×C31C22⋊C411yescharacteristic
76C22×C4×C31C22⋊C411yescharacteristic
77C22×C4×C31C22×C411yescharacteristic
78C22×C4×C31C2×D411yescharacteristic
79C2×C8×C31—42no—
80C4⋊C4×C31—41no—
81C4⋊C4×C31—42no—
82C4⋊C4×C31—42no—
83C4⋊C4×C31—41no—
84C4⋊C4×C31—41no—
85C4⋊C4×C31—82no—
86C22⋊C4×C31—41no—
87C22⋊C4×C31—41no—
88C22⋊C4×C31—42no—
89C22⋊C4×C31—42no—
90C22⋊C4×C31—42no—
91C22⋊C4×C31—41no—
92C22⋊C4×C31—41no—
93C22⋊C4×C31—42no—
94C22⋊C4×C31—42no—
95C42×C31—21no—
96C42×C31—21no—
97C42×C31—42no—
98C42×C31—41no—
99C42×C31—41no—
100#11822C3111yesmaximal, characteristic, direct product factor, Sylow
101C23×C31—21no—
102C23×C31—21no—
103C23×C31—21no—
104C23×C31—21no—
105C23×C31—21no—
106C23×C31—41no—
107C23×C31—41no—
108C23×C31—41no—
109C23×C31—41no—
110C23×C31C2×C22⋊C411yescharacteristic
111C23×C31AGL(1,ℤ/8)11yescharacteristic
112C23×C31#13711yescharacteristic
113D4×C31—42no—
114D4×C31—42no—
115C2×C4×C31—22no—
116C2×C4×C31—22no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—42no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—42no—
127C2×C4×C31—42no—
128C2×C4×C31—41no—
129C2×C4×C31—41no—
130C2×C4×C31—41no—
131C2×C4×C31—41no—
132C2×C4×C31—41no—
133C2×C4×C31—41no—
134C2×C4×C31—41no—
135C2×C4×C31—41no—
136C2×C4×C31—41no—
137C2×C4×C31—41no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—81no—
141C2×C4×C31—82no—
142C2×C4×C31—82no—
143C2×C4×C31—82no—
144C2×C4×C31#9911yescharacteristic
145C2×C4×C31#9911yescharacteristic
146C8×C31—82no—
147#2395C2×C3111yescharacteristic
148#1799C2×C3111yescharacteristic
149#1466C2×C3111yescharacteristic
150#1352C2×C3112yescomplemented
151#1344C2×C3112yescomplemented
152C22×C31—21no—
153C22×C31—21no—
154C22×C31—21no—
155C22×C31—21no—
156C22×C31—41no—
157C22×C31—41no—
158C22×C31—41no—
159C22×C31—41no—
160C22×C31—41no—
161C22×C31—41no—
162C22×C31—41no—
163C22×C31—81no—
164C22×C31—81no—
165C22×C31—81no—
166C22×C31C2×#9911yescharacteristic
167C22×C31#41711yescharacteristic
168C22×C31#42111yescharacteristic
169C4×C31—21no—
170C4×C31—42no—
171C4×C31—41no—
172C4×C31—41no—
173C4×C31—41no—
174C4×C31—41no—
175C4×C31—81no—
176C4×C31—81no—
177C4×C31—81no—
178C4×C31—162no—
179C4×C31#35212yes—
180C2×C4×D4C22×C3111yescharacteristic
181C2×#117C4×C3111yescharacteristic, complemented
182#432—21no—
183C2×#99—21no—
184#400C4×C3111yescharacteristic, complemented
185#399C22×C3111yescharacteristic
186#398C4×C3111yescharacteristic, complemented
187#397—21no—
188#397C22×C3111yescharacteristic
189#394—21no—
190#380—21no—
191C4×C4⋊C4—21no—
192C4×C22⋊C4—21no—
193C4×C22⋊C4C4×C3111yescharacteristic, complemented
194#342C22×C3112yes—
195#323—21no—
196#323C22×C3112yes—
197C2×C31—21no—
198C2×C31—21no—
199C2×C31—41no—
200C2×C31—41no—
201C2×C31—81no—
202C2×C31#151311yescharacteristic
203C2×C31C2×#35211yescharacteristic
204C2×C31#212611yescharacteristic
205C22×D4D4×C3111yescharacteristic
206C23×C4C2×C4×C3111yescharacteristic
207C23×C4D4×C3111yescharacteristic
208C4×D4—41no—
209C4×D4—41no—
210#117—41no—
211#117—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4C2×C4×C3111yescharacteristic
216C2×C4⋊C4C2×C4×C3111yescharacteristic
217C2×C4⋊C4C2×C4×C3111yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4D4×C3111yescharacteristic
224C2×C22⋊C4C23×C3111yescharacteristic
225C2×C42—21no—
226C2×C42—21no—
227C2×C42D4×C3111yescharacteristic
228C4⋊C8—41no—
229#99—42no—
230C22⋊C8—22no—
231#97—41no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—41no—
237#95C2×C4×C3111yescharacteristic
238#95C2×C4×C3111yescharacteristic
239C31#1182211yescharacteristic, direct product factor, Sylow
240C24C22⋊C4×C3111yescharacteristic
241C24C2×D4×C3111yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—41no—
245C2×D4—41no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4C22⋊C4×C3111yescharacteristic
258C22×C4C22⋊C4×C3111yescharacteristic
259C22×C4C22⋊C4×C3111yescharacteristic
260C22×C4C22×C4×C3111yescharacteristic
261C22×C4C2×D4×C3111yescharacteristic
262C2×C8—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—82no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C42—21no—
279C42—21no—
280C42—42no—
281C42—41no—
282C42—41no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23C2×C22⋊C4×C3111yescharacteristic
293C23C31×AGL(1,ℤ/8)11yescharacteristic
294C23C31×#13711yescharacteristic
295D4—42no—
296D4—42no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4C31×#9911yescharacteristic
327C2×C4C31×#9911yescharacteristic
328C8—82no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22C2×C31×#9911yescharacteristic
344C22C31×#41711yescharacteristic
345C22C31×#42111yescharacteristic
346C4—21no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—162no—
356C4C31×#35212yes—
357C2—21no—
358C2—21no—
359C2—41no—
360C2—41no—
361C2—81no—
362C2C31×#151311yescharacteristic
363C2C2×C31×#35211yescharacteristic
364C2C31×#212611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC22×D4×C31D4
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
4. C31×#1352C2semidirect
5. C31×#1344C2semidirect
7. C2×C31×#117C4semidirect
10. C31×#400C4semidirect
12. C31×#398C4semidirect
19. C4×C22⋊C4×C31C4semidirect
100. #11822C31direct (N × H)
150. #1352C2×C31semidirect
151. #1344C2×C31semidirect
181. C2×#117C4×C31semidirect
184. #400C4×C31semidirect
186. #398C4×C31semidirect
193. C4×C22⋊C4C4×C31semidirect
239. C31#11822direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#23952
C31×#17992
C31×#14662
C31×#13522
C31×#13442
#1182231