Finite Group Databasebeta

← C31×#9074 #21465833

Subgroups — C31×#9074 #21465833

Order 7936.

694 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#431C211yesmaximal, characteristic, complemented
2C4×C31×#121C211yesmaximal, characteristic
3C31×#1797C211yesmaximal, characteristic, complemented
4C2×C31×D4⋊C8C212yesmaximal, complemented
5C31×#1279C212yesmaximal, complemented
6C2×C31×#121C411yescharacteristic
7C2×C4×D4×C31—21no—
8C2×C4×D4×C31C2211yescharacteristic
9C22×C42×C31C2211yescharacteristic
10C31×#431—24no—
11C2×C31×C4⋊C8—21no—
12C2×C31×C4⋊C8C2212yes—
13C2×C31×C22⋊C8—21no—
14C2×C31×C22⋊C8—21no—
15C2×C4×C8×C31C2212yes—
16C31×#390C411yescharacteristic
17C31×#389C411yescharacteristic
18C31×#388—21no—
19C31×#386—21no—
20C31×#380C411yescharacteristic
21C4×C4⋊C4×C31C2211yescharacteristic
22C4×C22⋊C4×C31—21no—
23C31×#336—21no—
24C31×D4⋊C8—24no—
25C22×D4×C31—21no—
26C22×D4×C31C2×C411yescharacteristic
27C23×C4×C31—21no—
28C23×C4×C31C2×C411yescharacteristic
29C23×C4×C31D411yescharacteristic
30C23×C4×C31D411yescharacteristic
31C22×C8×C31—22no—
32C22×C8×C31—22no—
33C31×#121—22no—
34C31×#121C814yescomplemented
35C4×D4×C31—22no—
36C4×D4×C31—22no—
37C4×D4×C31—44no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31C2×C411yescharacteristic
40C2×C4⋊C4×C31C2×C411yescharacteristic
41C2×C4⋊C4×C31C2×C411yescharacteristic
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C42×C31—22no—
48C2×C42×C31—22no—
49C2×C42×C31—21no—
50C2×C42×C31C2×C411yescharacteristic
51C2×C42×C31D411yescharacteristic
52C2×C42×C31D411yescharacteristic
53C2×C42×C31C2311yescharacteristic
54C31×C4⋊C8—24no—
55C31×C4⋊C8—22no—
56C31×C4⋊C8—22no—
57C31×C22⋊C8—24no—
58C31×C22⋊C8—42no—
59C4×C8×C31—22no—
60C4×C8×C31D414yescomplemented
61C31×#95—21no—
62C31×#95—21no—
63C31×#95—21no—
64C24×C31—21no—
65C24×C31C22⋊C411yescharacteristic
66C2×D4×C31—24no—
67C2×D4×C31—22no—
68C2×D4×C31—21no—
69C2×D4×C31—22no—
70C2×D4×C31—42no—
71C2×D4×C31—42no—
72C2×D4×C31C2×C812yes—
73C22×C4×C31—22no—
74C22×C4×C31—22no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—22no—
78C22×C4×C31—21no—
79C22×C4×C31—21no—
80C22×C4×C31—22no—
81C22×C4×C31—21no—
82C22×C4×C31—21no—
83C22×C4×C31—21no—
84C22×C4×C31—41no—
85C22×C4×C31—41no—
86C22×C4×C31—42no—
87C22×C4×C31—42no—
88C22×C4×C31—41no—
89C22×C4×C31—41no—
90C22×C4×C31C22⋊C411yescharacteristic
91C22×C4×C31C22⋊C411yescharacteristic
92C22×C4×C31C22⋊C411yescharacteristic
93C22×C4×C31C2×C812yes—
94C22×C4×C31M1612yes—
95C22×C4×C31C22×C411yescharacteristic
96C22×C4×C31C2×D411yescharacteristic
97C22×C4×C31C2×D411yescharacteristic
98C2×C8×C31—24no—
99C2×C8×C31—42no—
100C2×C8×C31—42no—
101C2×C8×C31—42no—
102C2×C8×C31—42no—
103C2×C8×C31—42no—
104C4⋊C4×C31—21no—
105C4⋊C4×C31—22no—
106C4⋊C4×C31—22no—
107C4⋊C4×C31—42no—
108C4⋊C4×C31C2×C812yes—
109C22⋊C4×C31—24no—
110C22⋊C4×C31—42no—
111C22⋊C4×C31—42no—
112C22⋊C4×C31—42no—
113C22⋊C4×C31—41no—
114C22⋊C4×C31—41no—
115C42×C31—22no—
116C42×C31—21no—
117C42×C31—22no—
118C42×C31—21no—
119C42×C31—42no—
120C42×C31C2×D412yes—
121C42×C31#4112yes—
122#9074C3111yesmaximal, characteristic, direct product factor, Sylow
123C23×C31—21no—
124C23×C31—21no—
125C23×C31—21no—
126C23×C31—21no—
127C23×C31—21no—
128C23×C31—21no—
129C23×C31—22no—
130C23×C31—41no—
131C23×C31—41no—
132C23×C31—41no—
133C23×C31—41no—
134C23×C31—41no—
135C23×C31—41no—
136C23×C31C22⋊C812yes—
137C23×C31C2×C22⋊C411yescharacteristic
138D4×C31—24no—
139D4×C31—48no—
140D4×C31—42no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—21no—
145C2×C4×C31—24no—
146C2×C4×C31—24no—
147C2×C4×C31—21no—
148C2×C4×C31—21no—
149C2×C4×C31—22no—
150C2×C4×C31—21no—
151C2×C4×C31—21no—
152C2×C4×C31—21no—
153C2×C4×C31—21no—
154C2×C4×C31—42no—
155C2×C4×C31—42no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—42no—
159C2×C4×C31—42no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—82no—
167C2×C4×C31—82no—
168C2×C4×C31C22⋊C812yes—
169C2×C4×C31C4×D412yes—
170C2×C4×C31#12011yescharacteristic
171C2×C4×C31#12112yes—
172C2×C4×C31#12311yescharacteristic
173C2×C4×C31C22×C811yescharacteristic
174C2×C4×C31C2×M1611yescharacteristic
175C2×C4×C31#13112yes—
176C2×C4×C31AGL(1,ℤ/8)12yes—
177C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C31#13711yescharacteristic
179C8×C31—44no—
180C8×C31—82no—
181C2×#431C2×C3111yescharacteristic, complemented
182C4×#121C2×C3111yescharacteristic
183#1797C2×C3111yescharacteristic, complemented
184C2×D4⋊C8C2×C3112yescomplemented
185#1279C2×C3112yescomplemented
186C22×C31—22no—
187C22×C31—22no—
188C22×C31—22no—
189C22×C31—41no—
190C22×C31—41no—
191C22×C31—41no—
192C22×C31—41no—
193C22×C31—42no—
194C22×C31—42no—
195C22×C31—42no—
196C22×C31—81no—
197C22×C31—81no—
198C22×C31—81no—
199C22×C31—81no—
200C22×C31#38611yescharacteristic
201C22×C31C2×C22⋊C811yescharacteristic
202C22×C31#40811yescharacteristic
203C22×C31#41711yescharacteristic
204C22×C31#41811yescharacteristic
205C22×C31#42111yescharacteristic
206C22×C31#42111yescharacteristic
207C4×C31—22no—
208C4×C31—21no—
209C4×C31—21no—
210C4×C31—41no—
211C4×C31—41no—
212C4×C31—42no—
213C4×C31—81no—
214C4×C31—81no—
215C4×C31C8×D412yes—
216C4×C31#43512yes—
217C4×C31#46812yes—
218C4×C31#46912yes—
219C2×#121C4×C3111yescharacteristic
220C2×C4×D4—21no—
221C2×C4×D4C22×C3111yescharacteristic
222C22×C42C22×C3111yescharacteristic
223#431—24no—
224C2×C4⋊C8—21no—
225C2×C4⋊C8C22×C3112yes—
226C2×C22⋊C8—21no—
227C2×C22⋊C8—21no—
228C2×C4×C8C22×C3112yes—
229#390C4×C3111yescharacteristic
230#389C4×C3111yescharacteristic
231#388—21no—
232#386—21no—
233#380C4×C3111yescharacteristic
234C4×C4⋊C4C22×C3111yescharacteristic
235C4×C22⋊C4—21no—
236#336—21no—
237D4⋊C8—24no—
238C2×C31—42no—
239C2×C31—82no—
240C2×C31#147711yescharacteristic
241C2×C31#147811yescharacteristic
242C2×C31#149511yescharacteristic
243C2×C31#149611yescharacteristic
244C2×C31#186911yescharacteristic
245C2×C31#189511yescharacteristic
246C2×C31#189711yescharacteristic
247C22×D4—21no—
248C22×D4C2×C4×C3111yescharacteristic
249C23×C4—21no—
250C23×C4C2×C4×C3111yescharacteristic
251C23×C4D4×C3111yescharacteristic
252C23×C4D4×C3111yescharacteristic
253C22×C8—22no—
254C22×C8—22no—
255#121—22no—
256#121C8×C3114yescomplemented
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—44no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C2×C4×C3111yescharacteristic
262C2×C4⋊C4C2×C4×C3111yescharacteristic
263C2×C4⋊C4C2×C4×C3111yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42C2×C4×C3111yescharacteristic
273C2×C42D4×C3111yescharacteristic
274C2×C42D4×C3111yescharacteristic
275C2×C42C23×C3111yescharacteristic
276C4⋊C8—24no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C22⋊C8—24no—
280C22⋊C8—42no—
281C4×C8—22no—
282C4×C8D4×C3114yescomplemented
283#95—21no—
284#95—21no—
285#95—21no—
286C31#907411yescharacteristic, direct product factor, Sylow
287C24—21no—
288C24C22⋊C4×C3111yescharacteristic
289C2×D4—24no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—22no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4C2×C8×C3112yes—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C22⋊C4×C3111yescharacteristic
314C22×C4C22⋊C4×C3111yescharacteristic
315C22×C4C22⋊C4×C3111yescharacteristic
316C22×C4C2×C8×C3112yes—
317C22×C4M16×C3112yes—
318C22×C4C22×C4×C3111yescharacteristic
319C22×C4C2×D4×C3111yescharacteristic
320C22×C4C2×D4×C3111yescharacteristic
321C2×C8—24no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—42no—
331C4⋊C4C2×C8×C3112yes—
332C22⋊C4—24no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42—22no—
339C42—21no—
340C42—22no—
341C42—21no—
342C42—42no—
343C42C2×D4×C3112yes—
344C42#41×C3112yes—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—22no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23C31×C22⋊C812yes—
359C23C2×C22⋊C4×C3111yescharacteristic
360D4—24no—
361D4—48no—
362D4—42no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4C31×C22⋊C812yes—
391C2×C4C4×D4×C3112yes—
392C2×C4C31×#12011yescharacteristic
393C2×C4C31×#12112yes—
394C2×C4C31×#12311yescharacteristic
395C2×C4C22×C8×C3111yescharacteristic
396C2×C4C2×M16×C3111yescharacteristic
397C2×C4C31×#13112yes—
398C2×C4C31×AGL(1,ℤ/8)12yes—
399C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
400C2×C4C31×#13711yescharacteristic
401C8—44no—
402C8—82no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22C31×#38611yescharacteristic
418C22C2×C31×C22⋊C811yescharacteristic
419C22C31×#40811yescharacteristic
420C22C31×#41711yescharacteristic
421C22C31×#41811yescharacteristic
422C22C31×#42111yescharacteristic
423C22C31×#42111yescharacteristic
424C4—22no—
425C4—21no—
426C4—21no—
427C4—41no—
428C4—41no—
429C4—42no—
430C4—81no—
431C4—81no—
432C4C8×D4×C3112yes—
433C4C31×#43512yes—
434C4C31×#46812yes—
435C4C31×#46912yes—
436C2—42no—
437C2—82no—
438C2C31×#147711yescharacteristic
439C2C31×#147811yescharacteristic
440C2C31×#149511yescharacteristic
441C2C31×#149611yescharacteristic
442C2C31×#186911yescharacteristic
443C2C31×#189511yescharacteristic
444C2C31×#189711yescharacteristic
subgroupG/H
centerC22×C4×C31C2×D4
commutatorC2×C4C22×C8×C31
Frattini, cosocleC2×C42C23×C31
socleC23×C31C2×C22⋊C4
prime radicalC2×C31×#121C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#431C2semidirect
3. C31×#1797C2semidirect
4. C2×C31×D4⋊C8C2semidirect
5. C31×#1279C2semidirect
34. C31×#121C8semidirect
60. C4×C8×C31D4semidirect
122. #9074C31direct (N × H)
181. C2×#431C2×C31semidirect
183. #1797C2×C31semidirect
184. C2×D4⋊C8C2×C31semidirect
185. #1279C2×C31semidirect
256. #121C8×C31semidirect
282. C4×C8D4×C31semidirect
286. C31#9074direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4312
C4×C31×#1212
C31×#17972
C2×C31×D4⋊C82
C31×#12792
#907431