Finite Group Databasebeta

← C31×#11785 #21468544

Subgroups — C31×#11785 #21468544

Order 7936.

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

normal subgroups only

This group has 590 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×#2409C211yesmaximal, characteristic
2C4×C31×#99C211yesmaximal, characteristic
3C31×#1466C211yesmaximal, characteristic
4C31×#1351C214yesmaximal, complemented
5C23×Q8×C31C411yescharacteristic, complemented
6C2×C31×C22⋊Q8—21no—
7C2×C4×D4×C31C2211yescharacteristic
8C31×#432—21no—
9C2×C31×#99—21no—
10C31×#395C412yescomplemented
11C31×#393C2212yes—
12C31×#391—21no—
13C31×#386—21no—
14C31×#385—21no—
15C31×#380C411yescharacteristic, complemented
16C4×C22⋊C4×C31—21no—
17C31×#342C2212yes—
18C31×#323—21no—
19C31×#323C2212yes—
20C22×Q8×C31—21no—
21C22×Q8×C31—41no—
22C22×Q8×C31—41no—
23C22×Q8×C31—44no—
24C22×Q8×C31C2×C412yes—
25C22×D4×C31D411yescharacteristic
26C23×C4×C31—21no—
27C23×C4×C31C2×C411yescharacteristic
28C23×C4×C31D411yescharacteristic
29C31×C22⋊Q8—44no—
30C4×D4×C31—41no—
31C4×D4×C31—41no—
32C2×C4⋊C4×C31—22no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31C2×C411yescharacteristic
35C2×C22⋊C4×C31—22no—
36C2×C22⋊C4×C31—21no—
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31D411yescharacteristic
39C2×C22⋊C4×C31C2311yescharacteristic
40C2×C42×C31—41no—
41C2×C42×C31D411yescharacteristic
42C31×C4⋊C8—41no—
43C31×#99—24no—
44C31×C22⋊C8—22no—
45C31×#97—41no—
46C31×#95—22no—
47C31×#95—22no—
48C31×#95—41no—
49C31×#95C2×C412yes—
50C24×C31C22⋊C411yescharacteristic
51C24×C31C2×D411yescharacteristic
52C2×Q8×C31—24no—
53C2×Q8×C31—42no—
54C2×Q8×C31—44no—
55C2×Q8×C31—42no—
56C2×Q8×C31—48no—
57C2×Q8×C31—44no—
58C2×Q8×C31—44no—
59C2×Q8×C31—41no—
60C2×Q8×C31—41no—
61C2×D4×C31—22no—
62C2×D4×C31—22no—
63C2×D4×C31—41no—
64C22×C4×C31—22no—
65C22×C4×C31—21no—
66C22×C4×C31—21no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—22no—
70C22×C4×C31—44no—
71C22×C4×C31—42no—
72C22×C4×C31—42no—
73C22×C4×C31—42no—
74C22×C4×C31—42no—
75C22×C4×C31C22⋊C411yescharacteristic
76C22×C4×C31C22⋊C411yescharacteristic
77C22×C4×C31C22⋊C411yescharacteristic
78C22×C4×C31C22×C411yescharacteristic
79C22×C4×C31C2×D411yescharacteristic
80C2×C8×C31—42no—
81C4⋊C4×C31—41no—
82C4⋊C4×C31—42no—
83C4⋊C4×C31—44no—
84C22⋊C4×C31—24no—
85C22⋊C4×C31—41no—
86C22⋊C4×C31—41no—
87C22⋊C4×C31—44no—
88C22⋊C4×C31—42no—
89C22⋊C4×C31—42no—
90C42×C31—21no—
91C42×C31—21no—
92C42×C31—82no—
93#11785C3111yesmaximal, characteristic, direct product factor, Sylow
94C23×C31—21no—
95C23×C31—21no—
96C23×C31—21no—
97C23×C31—21no—
98C23×C31—41no—
99C23×C31—41no—
100C23×C31—41no—
101C23×C31—41no—
102C23×C31#10412yes—
103C23×C31C2×C22⋊C411yescharacteristic
104C23×C31#13712yes—
105Q8×C31—44no—
106Q8×C31—44no—
107Q8×C31—44no—
108Q8×C31—44no—
109D4×C31—44no—
110C2×C4×C31—22no—
111C2×C4×C31—24no—
112C2×C4×C31—21no—
113C2×C4×C31—21no—
114C2×C4×C31—21no—
115C2×C4×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—42no—
119C2×C4×C31—42no—
120C2×C4×C31—48no—
121C2×C4×C31—42no—
122C2×C4×C31—42no—
123C2×C4×C31—42no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
127C2×C4×C31—44no—
128C2×C4×C31—42no—
129C2×C4×C31—42no—
130C2×C4×C31—42no—
131C2×C4×C31—81no—
132C2×C4×C31—82no—
133C2×C4×C31—82no—
134C2×C4×C31#9911yescharacteristic
135C2×C4×C31#9911yescharacteristic
136C8×C31—82no—
137#2409C2×C3111yescharacteristic
138C4×#99C2×C3111yescharacteristic
139#1466C2×C3111yescharacteristic
140#1351C2×C3114yescomplemented
141C22×C31—21no—
142C22×C31—21no—
143C22×C31—22no—
144C22×C31—22no—
145C22×C31—42no—
146C22×C31—42no—
147C22×C31—42no—
148C22×C31—41no—
149C22×C31—41no—
150C22×C31—41no—
151C22×C31—41no—
152C22×C31—81no—
153C22×C31C2×#9911yescharacteristic
154C22×C31#41911yescharacteristic
155C22×C31C2×#10411yescharacteristic
156C4×C31—21no—
157C4×C31—44no—
158C4×C31—41no—
159C4×C31—41no—
160C4×C31—41no—
161C4×C31—41no—
162C4×C31—82no—
163C4×C31—84no—
164C4×C31AΣL(1,𝔽16)12yes—
165C23×Q8C4×C3111yescharacteristic, complemented
166C2×C22⋊Q8—21no—
167C2×C4×D4C22×C3111yescharacteristic
168#432—21no—
169C2×#99—21no—
170#395C4×C3112yescomplemented
171#393C22×C3112yes—
172#391—21no—
173#386—21no—
174#385—21no—
175#380C4×C3111yescharacteristic, complemented
176C4×C22⋊C4—21no—
177#342C22×C3112yes—
178#323—21no—
179#323C22×C3112yes—
180C2×C31—21no—
181C2×C31—21no—
182C2×C31—42no—
183C2×C31—42no—
184C2×C31#150811yescharacteristic
185C2×C31C2×AΣL(1,𝔽16)11yescharacteristic
186C2×C31#212911yescharacteristic
187C22×Q8—21no—
188C22×Q8—41no—
189C22×Q8—41no—
190C22×Q8—44no—
191C22×Q8C2×C4×C3112yes—
192C22×D4D4×C3111yescharacteristic
193C23×C4—21no—
194C23×C4C2×C4×C3111yescharacteristic
195C23×C4D4×C3111yescharacteristic
196C22⋊Q8—44no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C2×C4×C3111yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4D4×C3111yescharacteristic
206C2×C22⋊C4C23×C3111yescharacteristic
207C2×C42—41no—
208C2×C42D4×C3111yescharacteristic
209C4⋊C8—41no—
210#99—24no—
211C22⋊C8—22no—
212#97—41no—
213#95—22no—
214#95—22no—
215#95—41no—
216#95C2×C4×C3112yes—
217C31#1178511yescharacteristic, direct product factor, Sylow
218C24C22⋊C4×C3111yescharacteristic
219C24C2×D4×C3111yescharacteristic
220C2×Q8—24no—
221C2×Q8—44no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—44no—
225C2×Q8—48no—
226C2×Q8—44no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—41no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4C22⋊C4×C3111yescharacteristic
244C22×C4C22⋊C4×C3111yescharacteristic
245C22×C4C22⋊C4×C3111yescharacteristic
246C22×C4C22×C4×C3111yescharacteristic
247C22×C4C2×D4×C3111yescharacteristic
248C2×C8—42no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C22⋊C4—24no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—44no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C42—21no—
259C42—21no—
260C42—82no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23C31×#10412yes—
270C23C2×C22⋊C4×C3111yescharacteristic
271C23C31×#13712yes—
272Q8—44no—
273Q8—44no—
274Q8—44no—
275Q8—44no—
276D4—44no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—48no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4C31×#9911yescharacteristic
302C2×C4C31×#9911yescharacteristic
303C8—82no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22C2×C31×#9911yescharacteristic
317C22C31×#41911yescharacteristic
318C22C2×C31×#10411yescharacteristic
319C4—21no—
320C4—44no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—82no—
326C4—84no—
327C4C31×AΣL(1,𝔽16)12yes—
328C2—21no—
329C2—21no—
330C2—42no—
331C2—42no—
332C2C31×#150811yescharacteristic
333C2C2×C31×AΣL(1,𝔽16)11yescharacteristic
334C2C31×#212911yescharacteristic
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×#1351C2semidirect
5. C23×Q8×C31C4semidirect
10. C31×#395C4semidirect
15. C31×#380C4semidirect
93. #11785C31direct (N × H)
140. #1351C2×C31semidirect
165. C23×Q8C4×C31semidirect
170. #395C4×C31semidirect
175. #380C4×C31semidirect
217. C31#11785direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#24092
C4×C31×#992
C31×#14662
C31×#13512
#1178531