Finite Group Databasebeta

← C31×#12771 #21469530

Subgroups — C31×#12771 #21469530

Order 7936.

728 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3550C211yesmaximal, characteristic
2C31×#2132C211yesmaximal, characteristic
3C31×#2123C212yesmaximal, complemented
4C31×#1915C212yesmaximal, complemented
5C31×#1884C211yesmaximal, characteristic
6C31×#568C2211yescharacteristic
7C31×#563—21no—
8C31×#563C412yescomplemented
9C31×#559C412yescomplemented
10Q82×C31—21no—
11C31×#556—22no—
12D4×Q8×C31—22no—
13C31×#542—24no—
14C31×#536—21no—
15C31×#536C2212yes—
16C31×#533—21no—
17C31×#532C2211yescharacteristic
18C31×#421—21no—
19C31×#413C2212yes—
20C2×C31×#101—22no—
21C31×#410C2211yescharacteristic
22C31×#356—21no—
23C31×#355—21no—
24C31×#352—24no—
25C31×#344—22no—
26C31×#339—21no—
27C2×#41×C31D411yescharacteristic
28C2×#41×C31C2311yescharacteristic
29C22×Q8×C31—21no—
30C22×Q8×C31D412yes—
31C31×#131—42no—
32C2×M16×C31—22no—
33C2×M16×C31—21no—
34C2×M16×C31—22no—
35C31×C4⋊Q8—21no—
36C31×C4⋊Q8—21no—
37C31×C4⋊Q8—24no—
38C31×C4⋊Q8—22no—
39C31×C4⋊Q8—22no—
40C31×C4⋊Q8D411yescharacteristic
41C31×#127C2×C411yescharacteristic
42C31×#126—24no—
43C31×#126—24no—
44C31×#125—22no—
45C31×#124—22no—
46C31×#124—22no—
47C31×#124—24no—
48C31×#124C2×C411yescharacteristic
49C31×#124D411yescharacteristic
50C31×#123—22no—
51C31×#123—22no—
52C31×#123C2×C414yes—
53C31×#123D414yes—
54C31×C22⋊Q8—22no—
55C31×C22⋊Q8—22no—
56C31×C22⋊Q8—24no—
57C31×C22⋊Q8—24no—
58C31×C22⋊Q8—22no—
59C31×C22⋊Q8—22no—
60C31×#121—22no—
61C4×Q8×C31—22no—
62C4×Q8×C31—24no—
63C4×Q8×C31—22no—
64C4×D4×C31—24no—
65C4×D4×C31—22no—
66C31×#117—22no—
67C31×#117—21no—
68C31×#117—21no—
69C31×#117D412yes—
70C2×C42×C31D411yescharacteristic
71C31×#104—42no—
72C31×#101—22no—
73C31×#101—44no—
74C31×#100—22no—
75C31×#99—22no—
76#41×C31—41no—
77#41×C31—41no—
78#41×C31—41no—
79#41×C31—41no—
80C2×Q8×C31—22no—
81C2×Q8×C31—22no—
82C2×Q8×C31—22no—
83C2×Q8×C31—22no—
84C2×Q8×C31—42no—
85C2×Q8×C31—42no—
86C2×Q8×C31—81no—
87C2×Q8×C31—81no—
88C2×Q8×C31C22⋊C411yescharacteristic
89C2×Q8×C31C2×D411yescharacteristic
90C2×D4×C31—22no—
91C2×D4×C31—21no—
92C2×D4×C31C22⋊C411yescharacteristic
93C2×D4×C31C22×C411yescharacteristic
94C2×D4×C31C2×D412yes—
95C22×C4×C31—21no—
96C22×C4×C31—22no—
97C22×C4×C31—21no—
98C22×C4×C31C22⋊C414yes—
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31C2×D412yes—
101M16×C31—24no—
102M16×C31—22no—
103M16×C31—41no—
104M16×C31—42no—
105M16×C31—42no—
106C2×C8×C31—41no—
107C2×C8×C31—42no—
108C2×C8×C31—42no—
109C4⋊C4×C31—22no—
110C4⋊C4×C31—22no—
111C4⋊C4×C31—22no—
112C4⋊C4×C31—24no—
113C4⋊C4×C31—24no—
114C4⋊C4×C31—24no—
115C4⋊C4×C31—22no—
116C4⋊C4×C31—24no—
117C4⋊C4×C31—22no—
118C4⋊C4×C31—41no—
119C22⋊C4×C31—22no—
120C22⋊C4×C31—24no—
121C22⋊C4×C31—22no—
122C22⋊C4×C31—24no—
123C22⋊C4×C31—24no—
124C22⋊C4×C31—22no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31C22⋊C414yes—
127C42×C31—24no—
128C42×C31—22no—
129C42×C31—22no—
130C42×C31—21no—
131C42×C31—41no—
132C42×C31C22⋊C411yescharacteristic
133C42×C31C22⋊C411yescharacteristic
134#12771C3111yesmaximal, characteristic, direct product factor, Sylow
135C23×C31—21no—
136C23×C31C2×C22⋊C412yes—
137C23×C31#12011yescharacteristic
138Q8×C31—41no—
139Q8×C31—41no—
140Q8×C31—44no—
141Q8×C31—42no—
142Q8×C31—42no—
143Q8×C31—82no—
144Q8×C31—82no—
145D4×C31—41no—
146D4×C31—42no—
147D4×C31—41no—
148D4×C31—44no—
149D4×C31—81no—
150C2×C4×C31—21no—
151C2×C4×C31—21no—
152C2×C4×C31—22no—
153C2×C4×C31—21no—
154C2×C4×C31—22no—
155C2×C4×C31—24no—
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—24no—
159C2×C4×C31—42no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—44no—
163C2×C4×C31—82no—
164C2×C4×C31—81no—
165C2×C4×C31C2×C22⋊C411yescharacteristic
166C2×C4×C31#12012yes—
167C2×C4×C31#12011yescharacteristic
168C8×C31—42no—
169C8×C31—82no—
170#3550C2×C3111yescharacteristic
171#2132C2×C3111yescharacteristic
172#2123C2×C3112yescomplemented
173#1915C2×C3112yescomplemented
174#1884C2×C3111yescharacteristic
175C22×C31—22no—
176C22×C31—21no—
177C22×C31—41no—
178C22×C31—42no—
179C22×C31—42no—
180C22×C31—41no—
181C22×C31#37911yescharacteristic
182C4×C31—22no—
183C4×C31—22no—
184C4×C31—21no—
185C4×C31—41no—
186C4×C31—44no—
187C4×C31—41no—
188C4×C31—44no—
189C4×C31—161no—
190C4×C31B(4,𝔽2)11yescharacteristic
191C4×C31#45811yescharacteristic
192#568C22×C3111yescharacteristic
193#563—21no—
194#563C4×C3112yescomplemented
195#559C4×C3112yescomplemented
196Q82—21no—
197#556—22no—
198D4×Q8—22no—
199#542—24no—
200#536—21no—
201#536C22×C3112yes—
202#533—21no—
203#532C22×C3111yescharacteristic
204#421—21no—
205#413C22×C3112yes—
206C2×#101—22no—
207#410C22×C3111yescharacteristic
208#356—21no—
209#355—21no—
210#352—24no—
211#344—22no—
212#339—21no—
213C2×C31—21no—
214C2×C31—41no—
215C2×C31—42no—
216C2×C31—81no—
217C2×C31#179011yescharacteristic
218C2×#41D4×C3111yescharacteristic
219C2×#41C23×C3111yescharacteristic
220C22×Q8—21no—
221C22×Q8D4×C3112yes—
222#131—42no—
223C2×M16—21no—
224C2×M16—22no—
225C2×M16—22no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—24no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8D4×C3111yescharacteristic
232#127C2×C4×C3111yescharacteristic
233#126—24no—
234#126—24no—
235#125—22no—
236#124—22no—
237#124—22no—
238#124—24no—
239#124C2×C4×C3111yescharacteristic
240#124D4×C3111yescharacteristic
241#123—22no—
242#123—22no—
243#123C2×C4×C3114yes—
244#123D4×C3114yes—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8—24no—
248C22⋊Q8—24no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251#121—22no—
252C4×Q8—22no—
253C4×Q8—24no—
254C4×Q8—22no—
255C4×D4—24no—
256C4×D4—22no—
257#117—22no—
258#117—21no—
259#117—21no—
260#117D4×C3112yes—
261C2×C42D4×C3111yescharacteristic
262#104—42no—
263#101—22no—
264#101—44no—
265#100—22no—
266#99—22no—
267C31#1277111yescharacteristic, direct product factor, Sylow
268#41—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—81no—
279C2×Q8—81no—
280C2×Q8C22⋊C4×C3111yescharacteristic
281C2×Q8C2×D4×C3111yescharacteristic
282C2×D4—22no—
283C2×D4—21no—
284C2×D4C22⋊C4×C3111yescharacteristic
285C2×D4C22×C4×C3111yescharacteristic
286C2×D4C2×D4×C3112yes—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4C22⋊C4×C3114yes—
291C22×C4C2×D4×C3111yescharacteristic
292C22×C4C2×D4×C3112yes—
293M16—24no—
294M16—22no—
295M16—41no—
296M16—42no—
297M16—42no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—41no—
311C22⋊C4—22no—
312C22⋊C4—24no—
313C22⋊C4—22no—
314C22⋊C4—24no—
315C22⋊C4—24no—
316C22⋊C4—22no—
317C22⋊C4—41no—
318C22⋊C4C22⋊C4×C3114yes—
319C42—24no—
320C42—22no—
321C42—21no—
322C42—22no—
323C42—41no—
324C42C22⋊C4×C3111yescharacteristic
325C42C22⋊C4×C3111yescharacteristic
326C23—21no—
327C23C2×C22⋊C4×C3112yes—
328C23C31×#12011yescharacteristic
329Q8—41no—
330Q8—41no—
331Q8—44no—
332Q8—42no—
333Q8—42no—
334Q8—82no—
335Q8—82no—
336D4—41no—
337D4—42no—
338D4—41no—
339D4—44no—
340D4—81no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—24no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—44no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4C2×C22⋊C4×C3111yescharacteristic
357C2×C4C31×#12012yes—
358C2×C4C31×#12011yescharacteristic
359C8—42no—
360C8—82no—
361C22—22no—
362C22—21no—
363C22—41no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22C31×#37911yescharacteristic
368C4—22no—
369C4—22no—
370C4—21no—
371C4—41no—
372C4—44no—
373C4—41no—
374C4—44no—
375C4—161no—
376C4C31×B(4,𝔽2)11yescharacteristic
377C4C31×#45811yescharacteristic
378C2—21no—
379C2—41no—
380C2—42no—
381C2—81no—
382C2C31×#179011yescharacteristic
subgroupG/H
center, socleC2×C31#1790
commutatorC2×D4C22×C4×C31
Frattini, cosocleC2×#41C23×C31
prime radicalC31×#532C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#2123C2semidirect
4. C31×#1915C2semidirect
8. C31×#563C4semidirect
9. C31×#559C4semidirect
134. #12771C31direct (N × H)
172. #2123C2×C31semidirect
173. #1915C2×C31semidirect
194. #563C4×C31semidirect
195. #559C4×C31semidirect
267. C31#12771direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35502
C31×#21322
C31×#21232
C31×#19152
C31×#18842
#1277131