Finite Group Databasebeta

← C31×#11784 #21468543

Subgroups — C31×#11784 #21468543

Order 7936.

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

normal subgroups only

This group has 460 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×#2580C211yesmaximal, characteristic
2C4×C31×#99C211yesmaximal, characteristic
3C31×#1465C211yesmaximal, characteristic
4C31×#1352C212yesmaximal, complemented
5C31×#1351C212yesmaximal, complemented
6C2×C4×Q8×C31C411yescharacteristic, complemented
7C2×C4×D4×C31C2211yescharacteristic
8C2×C31×#99—21no—
9C31×#407—21no—
10C31×#399C411yescharacteristic, complemented
11C31×#399C2211yescharacteristic
12C31×#393C411yescharacteristic, complemented
13C31×#393C2211yescharacteristic
14C31×#390—21no—
15C31×#389—21no—
16C31×#385—21no—
17C31×#385—21no—
18C31×#382C411yescharacteristic, complemented
19C4×C22⋊C4×C31—21no—
20C31×#342C2212yes—
21C31×#324—21no—
22C31×#323C2212yes—
23C22×Q8×C31C2×C411yescharacteristic
24C22×D4×C31D411yescharacteristic
25C23×C4×C31D411yescharacteristic
26C23×C4×C31D411yescharacteristic
27C2×M16×C31—41no—
28C4×Q8×C31—41no—
29C4×Q8×C31—41no—
30C4×D4×C31—41no—
31C4×D4×C31—41no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31C2×C411yescharacteristic
37C2×C4⋊C4×C31D411yescharacteristic
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31C2×C411yescharacteristic
44C2×C22⋊C4×C31C2311yescharacteristic
45C2×C42×C31—21no—
46C2×C42×C31—41no—
47C2×C42×C31C2×C411yescharacteristic
48C31×#99—24no—
49C31×C22⋊C8—22no—
50C31×C22⋊C8—41no—
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×Q8×C31—42no—
60C2×Q8×C31—41no—
61C2×D4×C31—22no—
62C2×D4×C31—22no—
63C2×D4×C31—41no—
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—21no—
72C22×C4×C31—42no—
73C22×C4×C31—81no—
74C22×C4×C31—81no—
75C22×C4×C31C22⋊C411yescharacteristic
76C22×C4×C31C22⋊C411yescharacteristic
77C22×C4×C31C22⋊C411yescharacteristic
78C22×C4×C31C22×C411yescharacteristic
79C22×C4×C31C2×D411yescharacteristic
80M16×C31—82no—
81C2×C8×C31—42no—
82C4⋊C4×C31—41no—
83C4⋊C4×C31—41no—
84C4⋊C4×C31—41no—
85C4⋊C4×C31—41no—
86C4⋊C4×C31—42no—
87C4⋊C4×C31—81no—
88C4⋊C4×C31—81no—
89C22⋊C4×C31—24no—
90C22⋊C4×C31—41no—
91C22⋊C4×C31—41no—
92C22⋊C4×C31—42no—
93C22⋊C4×C31—42no—
94C22⋊C4×C31—42no—
95C22⋊C4×C31—81no—
96C22⋊C4×C31—81no—
97C42×C31—21no—
98C42×C31—21no—
99C42×C31—42no—
100C42×C31—82no—
101#11784C3111yesmaximal, characteristic, direct product factor, Sylow
102C23×C31—21no—
103C23×C31—21no—
104C23×C31—21no—
105C23×C31—21no—
106C23×C31—21no—
107C23×C31—41no—
108C23×C31—41no—
109C23×C31—41no—
110C23×C31—41no—
111C23×C31C2×C22⋊C411yescharacteristic
112C23×C31#13712yes—
113Q8×C31—42no—
114Q8×C31—42no—
115D4×C31—44no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—22no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—41no—
123C2×C4×C31—42no—
124C2×C4×C31—42no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
127C2×C4×C31—41no—
128C2×C4×C31—41no—
129C2×C4×C31—41no—
130C2×C4×C31—44no—
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—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#2580C2×C3111yescharacteristic
149C4×#99C2×C3111yescharacteristic
150#1465C2×C3111yescharacteristic
151#1352C2×C3112yescomplemented
152#1351C2×C3112yescomplemented
153C22×C31—21no—
154C22×C31—21no—
155C22×C31—22no—
156C22×C31—42no—
157C22×C31—42no—
158C22×C31—41no—
159C22×C31—41no—
160C22×C31—41no—
161C22×C31—41no—
162C22×C31—41no—
163C22×C31—81no—
164C22×C31—81no—
165C22×C31C2×#9911yescharacteristic
166C22×C31#41911yescharacteristic
167C22×C31C2×#10411yescharacteristic
168C4×C31—21no—
169C4×C31—21no—
170C4×C31—22no—
171C4×C31—42no—
172C4×C31—41no—
173C4×C31—41no—
174C4×C31—41no—
175C4×C31—81no—
176C4×C31—81no—
177C4×C31—84no—
178C4×C31—81no—
179C2×C4×Q8C4×C3111yescharacteristic, complemented
180C2×C4×D4C22×C3111yescharacteristic
181C2×#99—21no—
182#407—21no—
183#399C4×C3111yescharacteristic, complemented
184#399C22×C3111yescharacteristic
185#393C4×C3111yescharacteristic, complemented
186#393C22×C3111yescharacteristic
187#390—21no—
188#389—21no—
189#385—21no—
190#385—21no—
191#382C4×C3111yescharacteristic, complemented
192C4×C22⋊C4—21no—
193#342C22×C3112yes—
194#324—21no—
195#323C22×C3112yes—
196C2×C31—21no—
197C2×C31—21no—
198C2×C31—42no—
199C2×C31—81no—
200C2×C31#150811yescharacteristic
201C2×C31#212011yescharacteristic
202C2×C31#212711yescharacteristic
203C22×Q8C2×C4×C3111yescharacteristic
204C22×D4D4×C3111yescharacteristic
205C23×C4D4×C3111yescharacteristic
206C23×C4D4×C3111yescharacteristic
207C2×M16—41no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×D4—41no—
211C4×D4—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C2×C4×C3111yescharacteristic
217C2×C4⋊C4D4×C3111yescharacteristic
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4C2×C4×C3111yescharacteristic
224C2×C22⋊C4C23×C3111yescharacteristic
225C2×C42—21no—
226C2×C42—41no—
227C2×C42C2×C4×C3111yescharacteristic
228#99—24no—
229C22⋊C8—22no—
230C22⋊C8—41no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—41no—
235#95C2×C4×C3111yescharacteristic
236#95C2×C4×C3111yescharacteristic
237C31#1178411yescharacteristic, direct product factor, Sylow
238C24C22⋊C4×C3111yescharacteristic
239C24C2×D4×C3111yescharacteristic
240C2×Q8—42no—
241C2×Q8—41no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—41no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—42no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4C22⋊C4×C3111yescharacteristic
257C22×C4C22⋊C4×C3111yescharacteristic
258C22×C4C22⋊C4×C3111yescharacteristic
259C22×C4C22×C4×C3111yescharacteristic
260C22×C4C2×D4×C3111yescharacteristic
261M16—82no—
262C2×C8—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C22⋊C4—24no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—81no—
277C22⋊C4—81no—
278C42—21no—
279C42—21no—
280C42—42no—
281C42—82no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C2×C22⋊C4×C3111yescharacteristic
292C23C31×#13712yes—
293Q8—42no—
294Q8—42no—
295D4—44no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—44no—
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—81no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4C31×#9911yescharacteristic
325C2×C4C31×#9911yescharacteristic
326C2×C4C31×#10412yes—
327C8—82no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22C2×C31×#9911yescharacteristic
341C22C31×#41911yescharacteristic
342C22C2×C31×#10411yescharacteristic
343C4—21no—
344C4—21no—
345C4—22no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—84no—
353C4—81no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—81no—
358C2C31×#150811yescharacteristic
359C2C31×#212011yescharacteristic
360C2C31×#212711yescharacteristic
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×#1351C2semidirect
6. C2×C4×Q8×C31C4semidirect
10. C31×#399C4semidirect
12. C31×#393C4semidirect
18. C31×#382C4semidirect
101. #11784C31direct (N × H)
151. #1352C2×C31semidirect
152. #1351C2×C31semidirect
179. C2×C4×Q8C4×C31semidirect
183. #399C4×C31semidirect
185. #393C4×C31semidirect
191. #382C4×C31semidirect
237. C31#11784direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25802
C4×C31×#992
C31×#14652
C31×#13522
C31×#13512
#1178431