Finite Group Databasebeta

← C31×#13358 #21470117

Subgroups — C31×#13358 #21470117

Order 7936.

722 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C2×C31×#456C211yesmaximal, characteristic, complemented
2C2×C31×#454C211yesmaximal, characteristic
3C31×#2731C211yesmaximal, characteristic, complemented
4C2×C31×#356C211yesmaximal, characteristic, complemented
5C2×C31×#355C211yesmaximal, characteristic, complemented
6C31×#1793C211yesmaximal, characteristic, complemented
7C31×#1394C211yesmaximal, characteristic, complemented
8C2×C31×2−1+4C2211yescharacteristic
9C2×C31×#137—21no—
10C2×C31×AGL(1,ℤ/8)—21no—
11C2×C31×C4⋊Q8C2211yescharacteristic
12C2×C31×#124C2211yescharacteristic
13C31×#456—22no—
14C31×#456—22no—
15C31×#454—22no—
16C31×#454—22no—
17C2×C31×#104—21no—
18C2×C31×#104—21no—
19C2×C31×#104—21no—
20C31×#417—21no—
21C2×C31×#101C2211yescharacteristic
22C2×C31×#101C2211yescharacteristic
23C2×C31×#101C2211yescharacteristic
24C31×#396—21no—
25C31×#394—21no—
26C31×#393—21no—
27C31×#391C2211yescharacteristic
28C31×#388—21no—
29C31×#388—21no—
30C31×#387—21no—
31C31×#356—22no—
32C31×#355—22no—
33C31×2−1+4—21no—
34C31×2−1+4—21no—
35C31×2−1+4—21no—
36C31×2−1+4—42no—
37C31×2−1+4D412yes—
38C2×#41×C31—21no—
39C2×#41×C31—21no—
40C2×#41×C31—21no—
41C2×#41×C31—41no—
42C22×Q8×C31—41no—
43C22×Q8×C31D411yescharacteristic
44C22×Q8×C31C2311yescharacteristic
45C22×D4×C31D411yescharacteristic
46C31×#137—24no—
47C31×#137—42no—
48C31×AGL(1,ℤ/8)—24no—
49C31×AGL(1,ℤ/8)—42no—
50C2×Q16×C31—41no—
51C2×SD16×C31—41no—
52C2×SD16×C31—41no—
53C2×D8×C31—41no—
54C2×M16×C31—21no—
55C2×M16×C31—21no—
56C2×M16×C31—21no—
57C31×C4⋊Q8—21no—
58C31×C4⋊Q8—21no—
59C31×C4⋊Q8—21no—
60C31×C4⋊Q8—21no—
61C31×#124—21no—
62C31×#124—21no—
63C31×#124—21no—
64C31×#124—21no—
65C2×C4⋊C4×C31—21no—
66C2×C4⋊C4×C31—21no—
67C2×C4⋊C4×C31—21no—
68C2×C4⋊C4×C31D411yescharacteristic
69C2×C22⋊C4×C31—21no—
70C2×C22⋊C4×C31—21no—
71C2×C22⋊C4×C31—21no—
72C2×C22⋊C4×C31—21no—
73C2×C22⋊C4×C31—21no—
74C2×C22⋊C4×C31—21no—
75C2×C42×C31D411yescharacteristic
76C2×C42×C31D411yescharacteristic
77C2×C42×C31D411yescharacteristic
78C31×#104—24no—
79C31×#104—24no—
80C31×#104—42no—
81C31×#103—41no—
82C31×#102—41no—
83C31×#101—21no—
84C31×#101—21no—
85C31×#101—21no—
86C31×#101—21no—
87C31×#101D412yes—
88C31×#101D412yes—
89C31×#95—21no—
90C31×#95—21no—
91C31×#95—21no—
92C31×#95—21no—
93C31×#95—21no—
94C24×C31—21no—
95#41×C31—22no—
96#41×C31—22no—
97#41×C31—22no—
98#41×C31—22no—
99#41×C31—22no—
100#41×C31—22no—
101#41×C31—41no—
102#41×C31—41no—
103#41×C31—41no—
104#41×C31—42no—
105#41×C31—42no—
106#41×C31—42no—
107#41×C31—42no—
108#41×C31—41no—
109#41×C31—41no—
110#41×C31—41no—
111C2×Q8×C31—21no—
112C2×Q8×C31—21no—
113C2×Q8×C31—21no—
114C2×Q8×C31—21no—
115C2×Q8×C31—21no—
116C2×Q8×C31—41no—
117C2×Q8×C31—41no—
118C2×Q8×C31—41no—
119C2×Q8×C31—41no—
120C2×Q8×C31—42no—
121C2×Q8×C31—42no—
122C2×Q8×C31—42no—
123C2×Q8×C31—42no—
124C2×Q8×C31C2×D411yescharacteristic
125C2×Q8×C31C2×D411yescharacteristic
126C2×Q8×C31C2×D411yescharacteristic
127C2×Q8×C31#4111yescharacteristic
128C2×D4×C31—21no—
129C2×D4×C31—21no—
130C2×D4×C31—21no—
131C2×D4×C31—21no—
132C2×D4×C31—21no—
133C2×D4×C31—41no—
134C2×D4×C31—41no—
135C2×D4×C31—41no—
136C22×C4×C31—21no—
137C22×C4×C31—21no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—41no—
142C22×C4×C31—41no—
143C22×C4×C31—41no—
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147Q16×C31—44no—
148SD16×C31—44no—
149SD16×C31—44no—
150D8×C31—44no—
151M16×C31—22no—
152M16×C31—22no—
153M16×C31—41no—
154M16×C31—41no—
155M16×C31—41no—
156M16×C31—41no—
157C2×C8×C31—41no—
158C2×C8×C31—41no—
159C2×C8×C31—41no—
160C4⋊C4×C31—42no—
161C4⋊C4×C31—41no—
162C4⋊C4×C31—41no—
163C4⋊C4×C31—41no—
164C4⋊C4×C31—41no—
165C4⋊C4×C31—41no—
166C22⋊C4×C31—42no—
167C22⋊C4×C31—42no—
168C22⋊C4×C31—41no—
169C22⋊C4×C31—41no—
170C22⋊C4×C31—41no—
171C22⋊C4×C31—41no—
172C22⋊C4×C31—81no—
173C22⋊C4×C31—81no—
174C42×C31—21no—
175C42×C31—21no—
176C42×C31—21no—
177C42×C31—21no—
178C42×C31—41no—
179#13358C3111yesmaximal, characteristic, direct product factor, Sylow
180C23×C31—41no—
181C23×C31—41no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31—41no—
185C23×C31—41no—
186C23×C31—41no—
187C23×C31—41no—
188C23×C31#12011yescharacteristic
189Q8×C31—22no—
190Q8×C31—22no—
191Q8×C31—22no—
192Q8×C31—41no—
193Q8×C31—41no—
194Q8×C31—41no—
195Q8×C31—41no—
196Q8×C31—41no—
197Q8×C31—41no—
198Q8×C31—41no—
199Q8×C31—42no—
200Q8×C31—81no—
201D4×C31—22no—
202D4×C31—22no—
203D4×C31—22no—
204D4×C31—41no—
205D4×C31—41no—
206D4×C31—41no—
207D4×C31—42no—
208D4×C31—42no—
209D4×C31—42no—
210D4×C31—81no—
211C2×C4×C31—21no—
212C2×C4×C31—21no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—22no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31#12011yescharacteristic
240C2×C4×C31#12011yescharacteristic
241C2×C4×C31#12111yescharacteristic
242C2×C4×C31#12111yescharacteristic
243C2×C4×C31#12111yescharacteristic
244C2×C4×C31#12711yescharacteristic
245C8×C31—42no—
246C8×C31—42no—
247C8×C31—81no—
248C2×#456C2×C3111yescharacteristic, complemented
249C2×#454C2×C3111yescharacteristic
250#2731C2×C3111yescharacteristic, complemented
251C2×#356C2×C3111yescharacteristic, complemented
252C2×#355C2×C3111yescharacteristic, complemented
253#1793C2×C3111yescharacteristic, complemented
254#1394C2×C3111yescharacteristic, complemented
255C22×C31—21no—
256C22×C31—21no—
257C22×C31—42no—
258C22×C31—81no—
259C22×C31—81no—
260C22×C31—81no—
261C22×C31—81no—
262C22×C31—81no—
263C22×C31—81no—
264C22×C31—81no—
265C22×C31—81no—
266C22×C31—81no—
267C22×C31#39211yescharacteristic
268C22×C31B(4,𝔽2)11yescharacteristic
269C22×C31B(4,𝔽2)11yescharacteristic
270C4×C31—21no—
271C4×C31—21no—
272C4×C31—21no—
273C4×C31—21no—
274C4×C31—21no—
275C4×C31—21no—
276C4×C31—42no—
277C4×C31—42no—
278C4×C31—81no—
279C4×C31—81no—
280C4×C31—81no—
281C4×C31—81no—
282C2×2−1+4C22×C3111yescharacteristic
283C2×#137—21no—
284C2×AGL(1,ℤ/8)—21no—
285C2×C4⋊Q8C22×C3111yescharacteristic
286C2×#124C22×C3111yescharacteristic
287#456—22no—
288#456—22no—
289#454—22no—
290#454—22no—
291C2×#104—21no—
292C2×#104—21no—
293C2×#104—21no—
294#417—21no—
295C2×#101C22×C3111yescharacteristic
296C2×#101C22×C3111yescharacteristic
297C2×#101C22×C3111yescharacteristic
298#396—21no—
299#394—21no—
300#393—21no—
301#391C22×C3111yescharacteristic
302#388—21no—
303#388—21no—
304#387—21no—
305#356—22no—
306#355—22no—
307C2×C31—21no—
308C2×C31—21no—
309C2×C31—82no—
310C2×C31—161no—
311C2×C31#202111yescharacteristic
312C2×C31#220311yescharacteristic
313C2×C31#220411yescharacteristic
3142−1+4—21no—
3152−1+4—21no—
3162−1+4—21no—
3172−1+4—42no—
3182−1+4D4×C3112yes—
319C2×#41—21no—
320C2×#41—21no—
321C2×#41—21no—
322C2×#41—41no—
323C22×Q8—41no—
324C22×Q8D4×C3111yescharacteristic
325C22×Q8C23×C3111yescharacteristic
326C22×D4D4×C3111yescharacteristic
327#137—24no—
328#137—42no—
329AGL(1,ℤ/8)—24no—
330AGL(1,ℤ/8)—42no—
331C2×Q16—41no—
332C2×SD16—41no—
333C2×SD16—41no—
334C2×D8—41no—
335C2×M16—21no—
336C2×M16—21no—
337C2×M16—21no—
338C4⋊Q8—21no—
339C4⋊Q8—21no—
340C4⋊Q8—21no—
341C4⋊Q8—21no—
342#124—21no—
343#124—21no—
344#124—21no—
345#124—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4D4×C3111yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C42D4×C3111yescharacteristic
357C2×C42D4×C3111yescharacteristic
358C2×C42D4×C3111yescharacteristic
359#104—24no—
360#104—24no—
361#104—42no—
362#103—41no—
363#102—41no—
364#101—21no—
365#101—21no—
366#101—21no—
367#101—21no—
368#101D4×C3112yes—
369#101D4×C3112yes—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375C31#1335811yescharacteristic, direct product factor, Sylow
376C24—21no—
377#41—22no—
378#41—22no—
379#41—22no—
380#41—22no—
381#41—22no—
382#41—22no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41—42no—
387#41—42no—
388#41—42no—
389#41—42no—
390#41—41no—
391#41—41no—
392#41—41no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—41no—
399C2×Q8—41no—
400C2×Q8—41no—
401C2×Q8—41no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8C2×D4×C3111yescharacteristic
407C2×Q8C2×D4×C3111yescharacteristic
408C2×Q8C2×D4×C3111yescharacteristic
409C2×Q8#41×C3111yescharacteristic
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—41no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4C2×D4×C3111yescharacteristic
427C22×C4C2×D4×C3111yescharacteristic
428C22×C4C2×D4×C3111yescharacteristic
429Q16—44no—
430SD16—44no—
431SD16—44no—
432D8—44no—
433M16—22no—
434M16—22no—
435M16—41no—
436M16—41no—
437M16—41no—
438M16—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C4⋊C4—41no—
443C4⋊C4—42no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—81no—
455C22⋊C4—81no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23C31×#12011yescharacteristic
470Q8—22no—
471Q8—22no—
472Q8—22no—
473Q8—41no—
474Q8—41no—
475Q8—41no—
476Q8—41no—
477Q8—41no—
478Q8—41no—
479Q8—41no—
480Q8—42no—
481Q8—81no—
482D4—22no—
483D4—22no—
484D4—22no—
485D4—41no—
486D4—41no—
487D4—41no—
488D4—42no—
489D4—42no—
490D4—42no—
491D4—81no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4C31×#12011yescharacteristic
521C2×C4C31×#12011yescharacteristic
522C2×C4C31×#12111yescharacteristic
523C2×C4C31×#12111yescharacteristic
524C2×C4C31×#12111yescharacteristic
525C2×C4C31×#12711yescharacteristic
526C8—42no—
527C8—42no—
528C8—81no—
529C22—21no—
530C22—21no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22C31×#39211yescharacteristic
542C22C31×B(4,𝔽2)11yescharacteristic
543C22C31×B(4,𝔽2)11yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—42no—
551C4—42no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—81no—
556C2—21no—
557C2—21no—
558C2—82no—
559C2—161no—
560C2C31×#202111yescharacteristic
561C2C31×#220311yescharacteristic
562C2C31×#220411yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC22×Q8C23×C31
prime radicalC2×C31×#454C2
LCSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C22×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#456C2semidirect
3. C31×#2731C2semidirect
4. C2×C31×#356C2semidirect
5. C2×C31×#355C2semidirect
6. C31×#1793C2semidirect
7. C31×#1394C2semidirect
179. #13358C31direct (N × H)
248. C2×#456C2×C31semidirect
250. #2731C2×C31semidirect
251. C2×#356C2×C31semidirect
252. C2×#355C2×C31semidirect
253. #1793C2×C31semidirect
254. #1394C2×C31semidirect
375. C31#13358direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4562
C2×C31×#4542
C31×#27312
C2×C31×#3562
C2×C31×#3552
C31×#17932
C31×#13942
#1335831