Finite Group Databasebeta

← C31×#11756 #21468515

Subgroups — C31×#11756 #21468515

Order 7936.

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

normal subgroups only

This group has 444 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×#2599C211yesmaximal, characteristic
2C31×#1740C211yesmaximal, characteristic
3C31×#1464C211yesmaximal, characteristic
4C31×#1351C212yesmaximal
5C31×#1344C212yesmaximal
6C2×C4×Q8×C31C411yescharacteristic, complemented
7C2×C31×#117—21no—
8C2×C31×#117C2211yescharacteristic
9C31×#432—21no—
10C31×#398C411yescharacteristic, complemented
11C31×#397C2211yescharacteristic
12C31×#395C411yescharacteristic, complemented
13C31×#393C2211yescharacteristic
14C31×#388—21no—
15C31×#387—21no—
16C31×#387—21no—
17C31×#385—21no—
18C31×#383C411yescharacteristic, complemented
19C31×#342—21no—
20C31×#342C2212yes—
21C31×#324—21no—
22C31×#323C2212yes—
23C22×Q8×C31C2×C411yescharacteristic
24C23×C4×C31D411yescharacteristic
25C4×Q8×C31—41no—
26C4×Q8×C31—41no—
27C31×#117—24no—
28C31×#117—41no—
29C31×#117—41no—
30C31×#117—42no—
31C2×C4⋊C4×C31—21no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—41no—
35C2×C4⋊C4×C31C2×C411yescharacteristic
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×C31D411yescharacteristic
42C2×C22⋊C4×C31C2311yescharacteristic
43C2×C42×C31—21no—
44C2×C42×C31—41no—
45C2×C42×C31C2×C411yescharacteristic
46C2×C42×C31D411yescharacteristic
47C31×C4⋊C8—41no—
48C31×C22⋊C8—22no—
49C31×#97—41no—
50C31×#95—21no—
51C31×#95—21no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95C2×C411yescharacteristic
57C31×#95C2×C411yescharacteristic
58C24×C31C2×D411yescharacteristic
59C2×Q8×C31—42no—
60C2×Q8×C31—41no—
61C22×C4×C31—21no—
62C22×C4×C31—21no—
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—22no—
66C22×C4×C31—21no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—42no—
71C22×C4×C31C22⋊C411yescharacteristic
72C22×C4×C31C22⋊C411yescharacteristic
73C22×C4×C31C22⋊C411yescharacteristic
74C22×C4×C31C22⋊C411yescharacteristic
75C22×C4×C31C22×C411yescharacteristic
76C22×C4×C31C2×D411yescharacteristic
77C2×C8×C31—42no—
78C4⋊C4×C31—41no—
79C4⋊C4×C31—41no—
80C4⋊C4×C31—42no—
81C4⋊C4×C31—44no—
82C4⋊C4×C31—41no—
83C4⋊C4×C31—41no—
84C4⋊C4×C31—41no—
85C4⋊C4×C31—42no—
86C22⋊C4×C31—24no—
87C22⋊C4×C31—41no—
88C22⋊C4×C31—41no—
89C22⋊C4×C31—42no—
90C22⋊C4×C31—41no—
91C22⋊C4×C31—42no—
92C22⋊C4×C31—41no—
93C42×C31—21no—
94C42×C31—21no—
95C42×C31—21no—
96C42×C31—21no—
97C42×C31—42no—
98C42×C31—44no—
99#11756C3111yesmaximal, characteristic, direct product factor, Sylow
100C23×C31—21no—
101C23×C31—21no—
102C23×C31—41no—
103C23×C31—41no—
104C23×C31C2×C22⋊C411yescharacteristic
105C23×C31AGL(1,ℤ/8)11yescharacteristic
106C23×C31#13711yescharacteristic
107Q8×C31—42no—
108Q8×C31—42no—
109C2×C4×C31—22no—
110C2×C4×C31—22no—
111C2×C4×C31—22no—
112C2×C4×C31—22no—
113C2×C4×C31—21no—
114C2×C4×C31—21no—
115C2×C4×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—22no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—41no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
127C2×C4×C31—41no—
128C2×C4×C31—44no—
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—82no—
139C2×C4×C31—82no—
140C2×C4×C31#9911yescharacteristic
141C2×C4×C31#9911yescharacteristic
142C2×C4×C31#10412yes—
143C8×C31—82no—
144#2599C2×C3111yescharacteristic
145#1740C2×C3111yescharacteristic
146#1464C2×C3111yescharacteristic
147#1351C2×C3112yes—
148#1344C2×C3112yes—
149C22×C31—21no—
150C22×C31—21no—
151C22×C31—21no—
152C22×C31—21no—
153C22×C31—41no—
154C22×C31—41no—
155C22×C31—41no—
156C22×C31—41no—
157C22×C31—81no—
158C22×C31C2×#9911yescharacteristic
159C22×C31#41711yescharacteristic
160C22×C31C2×#10411yescharacteristic
161C4×C31—21no—
162C4×C31—21no—
163C4×C31—22no—
164C4×C31—41no—
165C4×C31—42no—
166C4×C31—41no—
167C4×C31—41no—
168C4×C31—41no—
169C4×C31—81no—
170C4×C31—81no—
171C4×C31—84no—
172C4×C31—81no—
173C2×C4×Q8C4×C3111yescharacteristic, complemented
174C2×#117—21no—
175C2×#117C22×C3111yescharacteristic
176#432—21no—
177#398C4×C3111yescharacteristic, complemented
178#397C22×C3111yescharacteristic
179#395C4×C3111yescharacteristic, complemented
180#393C22×C3111yescharacteristic
181#388—21no—
182#387—21no—
183#387—21no—
184#385—21no—
185#383C4×C3111yescharacteristic, complemented
186#342—21no—
187#342C22×C3112yes—
188#324—21no—
189#323C22×C3112yes—
190C2×C31—21no—
191C2×C31—21no—
192C2×C31—41no—
193C2×C31—41no—
194C2×C31#150411yescharacteristic
195C2×C31#212011yescharacteristic
196C2×C31#212911yescharacteristic
197C22×Q8C2×C4×C3111yescharacteristic
198C23×C4D4×C3111yescharacteristic
199C4×Q8—41no—
200C4×Q8—41no—
201#117—24no—
202#117—41no—
203#117—41no—
204#117—42no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4C2×C4×C3111yescharacteristic
210C2×C4⋊C4C2×C4×C3111yescharacteristic
211C2×C4⋊C4D4×C3111yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4D4×C3111yescharacteristic
216C2×C22⋊C4C23×C3111yescharacteristic
217C2×C42—21no—
218C2×C42—41no—
219C2×C42C2×C4×C3111yescharacteristic
220C2×C42D4×C3111yescharacteristic
221C4⋊C8—41no—
222C22⋊C8—22no—
223#97—41no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95C2×C4×C3111yescharacteristic
231#95C2×C4×C3111yescharacteristic
232C31#1175611yescharacteristic, direct product factor, Sylow
233C24C2×D4×C3111yescharacteristic
234C2×Q8—42no—
235C2×Q8—41no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4C22⋊C4×C3111yescharacteristic
247C22×C4C22⋊C4×C3111yescharacteristic
248C22×C4C22⋊C4×C3111yescharacteristic
249C22×C4C22⋊C4×C3111yescharacteristic
250C22×C4C22×C4×C3111yescharacteristic
251C22×C4C2×D4×C3111yescharacteristic
252C2×C8—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C22⋊C4—24no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—42no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—41no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—42no—
273C42—44no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23C2×C22⋊C4×C3111yescharacteristic
279C23C31×AGL(1,ℤ/8)11yescharacteristic
280C23C31×#13711yescharacteristic
281Q8—42no—
282Q8—42no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—44no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4C31×#9911yescharacteristic
315C2×C4C31×#9911yescharacteristic
316C2×C4C31×#10412yes—
317C8—82no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22C2×C31×#9911yescharacteristic
328C22C31×#41711yescharacteristic
329C22C2×C31×#10411yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—42no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—84no—
341C4—81no—
342C2—21no—
343C2—21no—
344C2—41no—
345C2—41no—
346C2C31×#150411yescharacteristic
347C2C31×#212011yescharacteristic
348C2C31×#212911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C31×#117 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. C2×C4×Q8×C31C4semidirect
10. C31×#398C4semidirect
12. C31×#395C4semidirect
18. C31×#383C4semidirect
99. #11756C31direct (N × H)
173. C2×C4×Q8C4×C31semidirect
177. #398C4×C31semidirect
179. #395C4×C31semidirect
185. #383C4×C31semidirect
232. C31#11756direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25992
C31×#17402
C31×#14642
C31×#13512
C31×#13442
#1175631