Finite Group Databasebeta

← C31×#19004 #21475763

Subgroups — C31×#19004 #21475763

Order 7936.

896 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3531C211yesmaximal, characteristic, complemented
2C31×#3217C212yesmaximal, complemented
3C31×#3138C211yesmaximal, characteristic, complemented
4C31×#1658C212yesmaximal, complemented
5C31×#1657C214yesmaximal, complemented
6C31×#1656C212yesmaximal, complemented
7C31×#1537C211yesmaximal, characteristic, complemented
8C31×#1529C212yesmaximal, complemented
9C31×#564C2211yescharacteristic, complemented
10C31×#563—22no—
11C31×#563—22no—
12C31×#563C2212yescomplemented
13C31×#562—22no—
14C31×#562C2211yescharacteristic, complemented
15C31×#533—22no—
16C31×#533C2211yescharacteristic
17C2×C31×#127C2211yescharacteristic
18C31×#529—22no—
19C31×#529—22no—
20C31×#529C2212yes—
21C31×#496C2214yes—
22C31×#490C2212yes—
23C31×#490C2212yes—
24C31×#490C2214yes—
25C31×#486—22no—
26C31×#460—22no—
27C31×#459—22no—
28C31×#450—22no—
29C31×#447—22no—
30C31×#432C2212yes—
31C31×#418—22no—
32C4×M16×C31C2211yescharacteristic
33C31×#330C2212yescomplemented
34C31×#330C2212yescomplemented
35C31×#330C2214yescomplemented
36C31×#329C2214yescomplemented
37C22×D4×C31—21no—
38C22×D4×C31—22no—
39C2×SD16×C31—24no—
40C2×D8×C31—24no—
41C2×M16×C31—21no—
42C31×C4⋊Q8—22no—
43C31×C4⋊Q8—22no—
44C31×C4⋊Q8D412yes—
45C31×#127—22no—
46C31×#127—21no—
47C31×#127C2312yes—
48C31×#126—22no—
49C31×#126—22no—
50C31×#126—22no—
51C31×#126—22no—
52C31×#126—22no—
53C31×#125—22no—
54C31×#125—22no—
55C31×#125—22no—
56C31×#125—22no—
57C31×#125C2312yes—
58C31×#125C2311yescharacteristic
59C31×#125C2311yescharacteristic
60C31×#124—22no—
61C31×#124—22no—
62C31×#124C2312yes—
63C31×#123—22no—
64C31×#123—22no—
65C31×#123—22no—
66C31×#123—22no—
67C31×#123—22no—
68C31×C22⋊Q8—22no—
69C31×C22⋊Q8—22no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8—22no—
73C31×C22⋊Q8—22no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8D412yes—
76C31×#121—22no—
77C31×#121—22no—
78C31×#121D412yes—
79C4×Q8×C31—22no—
80C4×Q8×C31—22no—
81C4×Q8×C31D412yes—
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C4×D4×C31D412yes—
85C31×#117—22no—
86C31×#117—22no—
87C31×#117D412yes—
88C2×C42×C31—22no—
89C2×C42×C31C2311yescharacteristic
90C31×C4⋊C8—22no—
91C31×#102—24no—
92C31×#102—24no—
93C31×#102—24no—
94C31×#102—24no—
95C31×C22⋊C8—22no—
96C31×#97C2312yes—
97C31×#97C2314yes—
98C4×C8×C31—21no—
99C24×C31—41no—
100C2×Q8×C31—22no—
101C2×Q8×C31—22no—
102C2×Q8×C31C2×D412yes—
103C2×D4×C31—22no—
104C2×D4×C31—22no—
105C2×D4×C31—22no—
106C2×D4×C31—24no—
107C2×D4×C31—44no—
108C2×D4×C31—42no—
109C2×D4×C31—41no—
110C2×D4×C31—84no—
111C2×D4×C31C2×D412yes—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31C2×D411yescharacteristic
116C22×C4×C31C2×D412yes—
117SD16×C31—84no—
118D8×C31—84no—
119M16×C31—42no—
120C2×C8×C31—22no—
121C2×C8×C31—24no—
122C4⋊C4×C31—22no—
123C4⋊C4×C31—22no—
124C4⋊C4×C31—22no—
125C4⋊C4×C31—22no—
126C4⋊C4×C31—22no—
127C4⋊C4×C31—22no—
128C4⋊C4×C31—22no—
129C4⋊C4×C31—22no—
130C4⋊C4×C31—22no—
131C4⋊C4×C31—22no—
132C4⋊C4×C31—22no—
133C4⋊C4×C31—22no—
134C4⋊C4×C31—22no—
135C4⋊C4×C31C2×D412yes—
136C4⋊C4×C31C2×D412yes—
137C4⋊C4×C31C2×D412yes—
138C4⋊C4×C31C2×D412yes—
139C22⋊C4×C31—22no—
140C22⋊C4×C31—22no—
141C22⋊C4×C31—22no—
142C22⋊C4×C31—22no—
143C22⋊C4×C31—22no—
144C22⋊C4×C31—22no—
145C22⋊C4×C31—22no—
146C22⋊C4×C31—22no—
147C22⋊C4×C31—22no—
148C22⋊C4×C31—22no—
149C42×C31—22no—
150C42×C31—22no—
151C42×C31—22no—
152C42×C31—22no—
153C42×C31C2×D411yescharacteristic
154C42×C31C2×D412yes—
155C42×C31C2411yescharacteristic
156#19004C3111yesmaximal, characteristic, direct product factor, Sylow
157C23×C31—22no—
158C23×C31—42no—
159C23×C31—82no—
160C23×C31—84no—
161C23×C31#12011yescharacteristic
162Q8×C31—42no—
163Q8×C31—82no—
164Q8×C31—82no—
165D4×C31—42no—
166D4×C31—44no—
167D4×C31—44no—
168D4×C31—44no—
169D4×C31—84no—
170D4×C31—82no—
171D4×C31—82no—
172D4×C31—82no—
173C2×C4×C31—22no—
174C2×C4×C31—22no—
175C2×C4×C31—22no—
176C2×C4×C31—22no—
177C2×C4×C31—22no—
178C2×C4×C31—22no—
179C2×C4×C31—42no—
180C2×C4×C31—44no—
181C2×C4×C31—82no—
182C2×C4×C31—82no—
183C2×C4×C31#12012yes—
184C2×C4×C31#12011yescharacteristic
185C2×C4×C31C22×D411yescharacteristic
186C2×C4×C31C22×D412yes—
187C8×C31—42no—
188C8×C31—44no—
189#3531C2×C3111yescharacteristic, complemented
190#3217C2×C3112yescomplemented
191#3138C2×C3111yescharacteristic, complemented
192#1658C2×C3112yescomplemented
193#1657C2×C3114yescomplemented
194#1656C2×C3112yescomplemented
195#1537C2×C3111yescharacteristic, complemented
196#1529C2×C3112yescomplemented
197C22×C31—21no—
198C22×C31—22no—
199C22×C31—42no—
200C22×C31—42no—
201C22×C31—42no—
202C22×C31—82no—
203C22×C31—84no—
204C22×C31—164no—
205C22×C31C2×#12011yescharacteristic
206C4×C31—21no—
207C4×C31—22no—
208C4×C31—22no—
209C4×C31—42no—
210C4×C31—42no—
211C4×C31—42no—
212C4×C31—42no—
213C4×C31—42no—
214C4×C31#45412yes—
215#564C22×C3111yescharacteristic, complemented
216#563—22no—
217#563—22no—
218#563C22×C3112yescomplemented
219#562—22no—
220#562C22×C3111yescharacteristic, complemented
221#533—22no—
222#533C22×C3111yescharacteristic
223C2×#127C22×C3111yescharacteristic
224#529—22no—
225#529—22no—
226#529C22×C3112yes—
227#496C22×C3114yes—
228#490C22×C3112yes—
229#490C22×C3112yes—
230#490C22×C3114yes—
231#486—22no—
232#460—22no—
233#459—22no—
234#450—22no—
235#447—22no—
236#432C22×C3112yes—
237#418—22no—
238C4×M16C22×C3111yescharacteristic
239#330C22×C3112yescomplemented
240#330C22×C3112yescomplemented
241#330C22×C3114yescomplemented
242#329C22×C3114yescomplemented
243C2×C31—41no—
244C2×C31—82no—
245C2×C31—162no—
246C2×C31C2×#45411yescharacteristic
247C2×C31#302112yes—
248C22×D4—21no—
249C22×D4—22no—
250C2×SD16—24no—
251C2×D8—24no—
252C2×M16—21no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8D4×C3112yes—
256#127—22no—
257#127—21no—
258#127C23×C3112yes—
259#126—22no—
260#126—22no—
261#126—22no—
262#126—22no—
263#126—22no—
264#125—22no—
265#125—22no—
266#125—22no—
267#125—22no—
268#125C23×C3112yes—
269#125C23×C3111yescharacteristic
270#125C23×C3111yescharacteristic
271#124—22no—
272#124—22no—
273#124C23×C3112yes—
274#123—22no—
275#123—22no—
276#123—22no—
277#123—22no—
278#123—22no—
279C22⋊Q8—22no—
280C22⋊Q8—22no—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8D4×C3112yes—
287#121—22no—
288#121—22no—
289#121D4×C3112yes—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8D4×C3112yes—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4D4×C3112yes—
296#117—22no—
297#117—22no—
298#117D4×C3112yes—
299C2×C42—22no—
300C2×C42C23×C3111yescharacteristic
301C4⋊C8—22no—
302#102—24no—
303#102—24no—
304#102—24no—
305#102—24no—
306C22⋊C8—22no—
307#97C23×C3112yes—
308#97C23×C3114yes—
309C4×C8—21no—
310C31#1900411yescharacteristic, direct product factor, Sylow
311C24—41no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8C2×D4×C3112yes—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—24no—
319C2×D4—44no—
320C2×D4—42no—
321C2×D4—41no—
322C2×D4—84no—
323C2×D4C2×D4×C3112yes—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4C2×D4×C3111yescharacteristic
328C22×C4C2×D4×C3112yes—
329SD16—84no—
330D8—84no—
331M16—42no—
332C2×C8—22no—
333C2×C8—24no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4C2×D4×C3112yes—
348C4⋊C4C2×D4×C3112yes—
349C4⋊C4C2×D4×C3112yes—
350C4⋊C4C2×D4×C3112yes—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42C2×D4×C3111yescharacteristic
366C42C2×D4×C3112yes—
367C42C24×C3111yescharacteristic
368C23—22no—
369C23—42no—
370C23—82no—
371C23—84no—
372C23C31×#12011yescharacteristic
373Q8—42no—
374Q8—82no—
375Q8—82no—
376D4—42no—
377D4—44no—
378D4—44no—
379D4—44no—
380D4—84no—
381D4—82no—
382D4—82no—
383D4—82no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4C31×#12012yes—
395C2×C4C31×#12011yescharacteristic
396C2×C4C22×D4×C3111yescharacteristic
397C2×C4C22×D4×C3112yes—
398C8—42no—
399C8—44no—
400C22—21no—
401C22—22no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22—84no—
407C22—164no—
408C22C2×C31×#12011yescharacteristic
409C4—21no—
410C4—22no—
411C4—22no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4C31×#45412yes—
418C2—41no—
419C2—82no—
420C2—162no—
421C2C2×C31×#45411yescharacteristic
422C2C31×#302112yes—
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3531C2semidirect
2. C31×#3217C2semidirect
3. C31×#3138C2semidirect
4. C31×#1658C2semidirect
5. C31×#1657C2semidirect
6. C31×#1656C2semidirect
7. C31×#1537C2semidirect
8. C31×#1529C2semidirect
9. C31×#564C22semidirect
12. C31×#563C22semidirect
14. C31×#562C22semidirect
33. C31×#330C22semidirect
34. C31×#330C22semidirect
35. C31×#330C22semidirect
36. C31×#329C22semidirect
156. #19004C31direct (N × H)
189. #3531C2×C31semidirect
190. #3217C2×C31semidirect
191. #3138C2×C31semidirect
192. #1658C2×C31semidirect
193. #1657C2×C31semidirect
194. #1656C2×C31semidirect
195. #1537C2×C31semidirect
196. #1529C2×C31semidirect
215. #564C22×C31semidirect
218. #563C22×C31semidirect
220. #562C22×C31semidirect
239. #330C22×C31semidirect
240. #330C22×C31semidirect
241. #330C22×C31semidirect
242. #329C22×C31semidirect
310. C31#19004direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35312
C31×#32172
C31×#31382
C31×#16582
C31×#16572
C31×#16562
C31×#15372
C31×#15292
#1900431