Finite Group Databasebeta

← C31×#10320 #21467079

Subgroups — C31×#10320 #21467079

Order 7936.

1066 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3186C211yesmaximal, characteristic, complemented
2C31×#2847C212yesmaximal, complemented
3C31×#1779C211yesmaximal, characteristic, complemented
4C31×#1519C212yesmaximal, complemented
5C31×#1289C211yesmaximal, characteristic, complemented
6C24×C4×C31C2211yescharacteristic
7C2×C31×#121—22no—
8C2×C31×#121C2212yes—
9C2×C31×#120—22no—
10C31×#480—22no—
11C31×#466—22no—
12C31×#447—22no—
13C2×C31×C22⋊C8—22no—
14C2×C31×C22⋊C8C2211yescharacteristic
15C2×C31×C22⋊C8C2211yescharacteristic
16C31×#386—22no—
17C31×#386—24no—
18C31×#380—22no—
19C31×#380C2212yes—
20C31×#327—24no—
21C25×C31D411yescharacteristic
22C22×D4×C31—24no—
23C22×D4×C31—22no—
24C22×D4×C31D412yescomplemented
25C23×C4×C31—21no—
26C23×C4×C31—22no—
27C23×C4×C31—22no—
28C23×C4×C31—21no—
29C23×C4×C31—44no—
30C23×C4×C31D411yescharacteristic
31C23×C4×C31C2311yescharacteristic
32C2×D8×C31—22no—
33C22×C8×C31—21no—
34C22×C8×C31D412yescomplemented
35C31×#121—22no—
36C31×#121—22no—
37C31×#121—22no—
38C31×#121—22no—
39C31×#121—48no—
40C31×#120—44no—
41C31×#120—44no—
42C2×C4⋊C4×C31—22no—
43C2×C4⋊C4×C31D412yescomplemented
44C2×C22⋊C4×C31—24no—
45C2×C22⋊C4×C31—24no—
46C2×C22⋊C4×C31—22no—
47C2×C22⋊C4×C31—22no—
48C31×#107—22no—
49C31×#102—24no—
50C31×C22⋊C8—22no—
51C31×C22⋊C8—22no—
52C31×C22⋊C8—44no—
53C31×#95—24no—
54C31×#95D414yescomplemented
55C24×C31—22no—
56C24×C31—21no—
57C24×C31—21no—
58C24×C31—22no—
59C24×C31—22no—
60C24×C31—42no—
61C24×C31—42no—
62C24×C31D812yescomplemented
63C24×C31C2×D411yescharacteristic
64C2×D4×C31—22no—
65C2×D4×C31—22no—
66C2×D4×C31—44no—
67C2×D4×C31—48no—
68C2×D4×C31—48no—
69C2×D4×C31—48no—
70C2×D4×C31—42no—
71C2×D4×C31—42no—
72C2×D4×C31—42no—
73C2×D4×C31—42no—
74C2×D4×C31—42no—
75C2×D4×C31—42no—
76C22×C4×C31—24no—
77C22×C4×C31—28no—
78C22×C4×C31—22no—
79C22×C4×C31—21no—
80C22×C4×C31—22no—
81C22×C4×C31—24no—
82C22×C4×C31—42no—
83C22×C4×C31—42no—
84C22×C4×C31—44no—
85C22×C4×C31—42no—
86C22×C4×C31—44no—
87C22×C4×C31—42no—
88C22×C4×C31—44no—
89C22×C4×C31—42no—
90C22×C4×C31—41no—
91C22×C4×C31—42no—
92C22×C4×C31—41no—
93C22×C4×C31C2×D412yes—
94C22×C4×C31C2×D411yescharacteristic
95C22×C4×C31C2×D412yes—
96C22×C4×C31#4111yescharacteristic
97D8×C31—82no—
98C2×C8×C31—22no—
99C2×C8×C31—42no—
100C2×C8×C31—42no—
101C2×C8×C31—41no—
102C4⋊C4×C31—22no—
103C4⋊C4×C31—22no—
104C4⋊C4×C31—44no—
105C22⋊C4×C31—48no—
106C22⋊C4×C31—48no—
107C22⋊C4×C31—42no—
108C22⋊C4×C31—42no—
109C22⋊C4×C31—42no—
110C22⋊C4×C31—42no—
111#10320C3111yesmaximal, characteristic, direct product factor, Sylow
112C23×C31—24no—
113C23×C31—22no—
114C23×C31—24no—
115C23×C31—22no—
116C23×C31—24no—
117C23×C31—22no—
118C23×C31—41no—
119C23×C31—44no—
120C23×C31—41no—
121C23×C31—42no—
122C23×C31—41no—
123C23×C31—42no—
124C23×C31—41no—
125C23×C31—48no—
126C23×C31—44no—
127C23×C31—44no—
128C23×C31—44no—
129C23×C31—41no—
130C23×C31—42no—
131C23×C31—41no—
132C23×C31—42no—
133C23×C31—42no—
134C23×C31—42no—
135C23×C31#12012yes—
136C23×C31#12111yescharacteristic
137C23×C31C2×D811yescharacteristic
138C23×C31#13511yescharacteristic
139C23×C31AGL(1,ℤ/8)12yes—
140D4×C31—44no—
141D4×C31—82no—
142D4×C31—816no—
143D4×C31—82no—
144D4×C31—82no—
145D4×C31—82no—
146D4×C31—82no—
147C2×C4×C31—24no—
148C2×C4×C31—22no—
149C2×C4×C31—24no—
150C2×C4×C31—22no—
151C2×C4×C31—42no—
152C2×C4×C31—41no—
153C2×C4×C31—44no—
154C2×C4×C31—48no—
155C2×C4×C31—42no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—44no—
159C2×C4×C31—44no—
160C2×C4×C31—44no—
161C2×C4×C31—42no—
162C2×C4×C31—44no—
163C2×C4×C31—42no—
164C2×C4×C31—42no—
165C2×C4×C31#12011yescharacteristic
166C2×C4×C31#12112yes—
167C2×C4×C31#12711yescharacteristic
168C8×C31—42no—
169C8×C31—81no—
170#3186C2×C3111yescharacteristic, complemented
171#2847C2×C3112yescomplemented
172#1779C2×C3111yescharacteristic, complemented
173#1519C2×C3112yescomplemented
174#1289C2×C3111yescharacteristic, complemented
175C22×C31—22no—
176C22×C31—21no—
177C22×C31—22no—
178C22×C31—21no—
179C22×C31—28no—
180C22×C31—24no—
181C22×C31—41no—
182C22×C31—41no—
183C22×C31—42no—
184C22×C31—44no—
185C22×C31—42no—
186C22×C31—44no—
187C22×C31—44no—
188C22×C31—42no—
189C22×C31—42no—
190C22×C31—42no—
191C22×C31—42no—
192C22×C31—42no—
193C22×C31—84no—
194C22×C31—84no—
195C22×C31—82no—
196C22×C31—82no—
197C22×C31—82no—
198C22×C31#39211yescharacteristic
199C22×C31#44712yes—
200C22×C31#44912yes—
201C22×C31#46611yescharacteristic
202C22×C31#46911yescharacteristic
203C4×C31—21no—
204C4×C31—22no—
205C4×C31—42no—
206C4×C31—84no—
207C4×C31—82no—
208C4×C31B(4,𝔽2)12yes—
209C24×C4C22×C3111yescharacteristic
210C2×#121—22no—
211C2×#121C22×C3112yes—
212C2×#120—22no—
213#480—22no—
214#466—22no—
215#447—22no—
216C2×C22⋊C8—22no—
217C2×C22⋊C8C22×C3111yescharacteristic
218C2×C22⋊C8C22×C3111yescharacteristic
219#386—22no—
220#386—24no—
221#380—22no—
222#380C22×C3112yes—
223#327—24no—
224C2×C31—22no—
225C2×C31—21no—
226C2×C31—22no—
227C2×C31—21no—
228C2×C31—44no—
229C2×C31—162no—
230C2×C31#199911yescharacteristic
231C2×C31#201011yescharacteristic
232C2×C31#202111yescharacteristic
233C25D4×C3111yescharacteristic
234C22×D4—24no—
235C22×D4—22no—
236C22×D4D4×C3112yescomplemented
237C23×C4—22no—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—44no—
242C23×C4D4×C3111yescharacteristic
243C23×C4C23×C3111yescharacteristic
244C2×D8—22no—
245C22×C8—21no—
246C22×C8D4×C3112yescomplemented
247#121—22no—
248#121—22no—
249#121—22no—
250#121—22no—
251#121—48no—
252#120—44no—
253#120—44no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4D4×C3112yescomplemented
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260#107—22no—
261#102—24no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—44no—
265#95—24no—
266#95D4×C3114yescomplemented
267C31#1032011yescharacteristic, direct product factor, Sylow
268C24—22no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—42no—
274C24—42no—
275C24D8×C3112yescomplemented
276C24C2×D4×C3111yescharacteristic
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—44no—
280C2×D4—48no—
281C2×D4—48no—
282C2×D4—48no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C22×C4—24no—
290C22×C4—28no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4C2×D4×C3112yes—
307C22×C4C2×D4×C3111yescharacteristic
308C22×C4C2×D4×C3112yes—
309C22×C4#41×C3111yescharacteristic
310D8—82no—
311C2×C8—22no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—41no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—44no—
318C22⋊C4—48no—
319C22⋊C4—48no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C23—24no—
325C23—22no—
326C23—24no—
327C23—22no—
328C23—24no—
329C23—22no—
330C23—41no—
331C23—44no—
332C23—41no—
333C23—42no—
334C23—41no—
335C23—42no—
336C23—41no—
337C23—48no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23C31×#12012yes—
348C23C31×#12111yescharacteristic
349C23C2×D8×C3111yescharacteristic
350C23C31×#13511yescharacteristic
351C23C31×AGL(1,ℤ/8)12yes—
352D4—44no—
353D4—82no—
354D4—816no—
355D4—82no—
356D4—82no—
357D4—82no—
358D4—82no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—48no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4C31×#12011yescharacteristic
378C2×C4C31×#12112yes—
379C2×C4C31×#12711yescharacteristic
380C8—42no—
381C8—81no—
382C22—22no—
383C22—21no—
384C22—22no—
385C22—21no—
386C22—28no—
387C22—24no—
388C22—41no—
389C22—41no—
390C22—42no—
391C22—44no—
392C22—42no—
393C22—44no—
394C22—44no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—84no—
401C22—84no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22C31×#39211yescharacteristic
406C22C31×#44712yes—
407C22C31×#44912yes—
408C22C31×#46611yescharacteristic
409C22C31×#46911yescharacteristic
410C4—21no—
411C4—22no—
412C4—42no—
413C4—84no—
414C4—82no—
415C4C31×B(4,𝔽2)12yes—
416C2—22no—
417C2—21no—
418C2—22no—
419C2—21no—
420C2—44no—
421C2—162no—
422C2C31×#199911yescharacteristic
423C2C31×#201011yescharacteristic
424C2C31×#202111yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3186C2semidirect
2. C31×#2847C2semidirect
3. C31×#1779C2semidirect
4. C31×#1519C2semidirect
5. C31×#1289C2semidirect
24. C22×D4×C31D4semidirect
34. C22×C8×C31D4semidirect
43. C2×C4⋊C4×C31D4semidirect
54. C31×#95D4semidirect
62. C24×C31D8semidirect
111. #10320C31direct (N × H)
170. #3186C2×C31semidirect
171. #2847C2×C31semidirect
172. #1779C2×C31semidirect
173. #1519C2×C31semidirect
174. #1289C2×C31semidirect
236. C22×D4D4×C31semidirect
246. C22×C8D4×C31semidirect
255. C2×C4⋊C4D4×C31semidirect
266. #95D4×C31semidirect
267. C31#10320direct (N × H)
275. C24D8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31862
C31×#28472
C31×#17792
C31×#15192
C31×#12892
#1032031