Finite Group Databasebeta

← C31×#13422 #21470181

Subgroups — C31×#13422 #21470181

Order 7936.

734 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2669C211yesmaximal, characteristic
2C2×C31×#353C211yesmaximal, characteristic, complemented
3C2×C31×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
4C31×#1889C211yesmaximal, characteristic, complemented
5C31×#1887C211yesmaximal, characteristic, complemented
6C31×#1867C211yesmaximal, characteristic, complemented
7C31×#1390C211yesmaximal, characteristic, complemented
8C2×C31×#127C2211yescharacteristic
9C2×C31×#121—21no—
10C2×C31×#120—21no—
11C2×C31×#120C2211yescharacteristic
12C31×#518C2211yescharacteristic
13C31×#418—21no—
14C2×C31×#100C2211yescharacteristic
15C2×C31×#99C2211yescharacteristic
16C2×C31×#99C2211yescharacteristic
17C31×#394—21no—
18C31×#392—21no—
19C31×#392—21no—
20C31×#388—21no—
21C31×#386C2211yescharacteristic
22C31×#385—21no—
23C31×#379—21no—
24C31×#353—22no—
25C31×AΣL(1,𝔽16)—22no—
26C31×#342—21no—
27C31×#342—21no—
28C31×#339—21no—
29C25×C31D411yescharacteristic
30C22×D4×C31—21no—
31C22×D4×C31—21no—
32C22×D4×C31—21no—
33C22×D4×C31—21no—
34C22×D4×C31—21no—
35C22×D4×C31D411yescharacteristic
36C22×D4×C31C2311yescharacteristic
37C23×C4×C31D411yescharacteristic
38C2×M16×C31—21no—
39C31×#127—21no—
40C31×#127—21no—
41C31×#127—21no—
42C31×#127—21no—
43C31×#121—44no—
44C31×#120—21no—
45C31×#120—21no—
46C31×#120—21no—
47C31×#120—21no—
48C31×#120—44no—
49C4×D4×C31—41no—
50C4×D4×C31—41no—
51C31×#117—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—41no—
54C2×C22⋊C4×C31—21no—
55C2×C22⋊C4×C31—21no—
56C2×C22⋊C4×C31—21no—
57C2×C22⋊C4×C31—21no—
58C2×C22⋊C4×C31—21no—
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—21no—
61C2×C22⋊C4×C31—21no—
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31—21no—
64C2×C22⋊C4×C31—41no—
65C2×C22⋊C4×C31—41no—
66C2×C22⋊C4×C31D411yescharacteristic
67C2×C22⋊C4×C31D411yescharacteristic
68C2×C42×C31D411yescharacteristic
69C31×#102—41no—
70C31×#102—41no—
71C31×#100—21no—
72C31×#100—21no—
73C31×#99—21no—
74C31×#99—21no—
75C31×#99—21no—
76C31×#99D412yescomplemented
77C31×#95—21no—
78C31×#95—21no—
79C31×#95—41no—
80C24×C31—21no—
81C24×C31—21no—
82C24×C31—21no—
83C24×C31—21no—
84C24×C31—41no—
85C24×C31—42no—
86C24×C31—41no—
87C24×C31—81no—
88C24×C31C2×D411yescharacteristic
89C24×C31C2×D411yescharacteristic
90C2×D4×C31—21no—
91C2×D4×C31—21no—
92C2×D4×C31—41no—
93C2×D4×C31—41no—
94C2×D4×C31—41no—
95C2×D4×C31—41no—
96C2×D4×C31—42no—
97C2×D4×C31—42no—
98C2×D4×C31—42no—
99C2×D4×C31—42no—
100C2×D4×C31—42no—
101C2×D4×C31—42no—
102C2×D4×C31—41no—
103C2×D4×C31—41no—
104C2×D4×C31—41no—
105C2×D4×C31—41no—
106C2×D4×C31—41no—
107C2×D4×C31—41no—
108C2×D4×C31—81no—
109C2×D4×C31—81no—
110C2×D4×C31—81no—
111C2×D4×C31—81no—
112C2×D4×C31—81no—
113C2×D4×C31—81no—
114C2×D4×C31C2×D411yescharacteristic
115C2×D4×C31#4111yescharacteristic
116C2×D4×C31#4111yescharacteristic
117C2×D4×C31#4111yescharacteristic
118C22×C4×C31—21no—
119C22×C4×C31—21no—
120C22×C4×C31—21no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—42no—
127C22×C4×C31—41no—
128C22×C4×C31—41no—
129C22×C4×C31—41no—
130C22×C4×C31C2×D411yescharacteristic
131M16×C31—41no—
132M16×C31—41no—
133C2×C8×C31—41no—
134C4⋊C4×C31—21no—
135C4⋊C4×C31—21no—
136C4⋊C4×C31—42no—
137C4⋊C4×C31—82no—
138C22⋊C4×C31—24no—
139C22⋊C4×C31—22no—
140C22⋊C4×C31—21no—
141C22⋊C4×C31—21no—
142C22⋊C4×C31—42no—
143C22⋊C4×C31—42no—
144C22⋊C4×C31—42no—
145C22⋊C4×C31—41no—
146C22⋊C4×C31—41no—
147C22⋊C4×C31—41no—
148C22⋊C4×C31—41no—
149C22⋊C4×C31—41no—
150C22⋊C4×C31—41no—
151C22⋊C4×C31—41no—
152C22⋊C4×C31—41no—
153C22⋊C4×C31—41no—
154C22⋊C4×C31—81no—
155C22⋊C4×C31—82no—
156C22⋊C4×C31—82no—
157C42×C31—21no—
158C42×C31—21no—
159C42×C31—41no—
160#13422C3111yesmaximal, characteristic, direct product factor, Sylow
161C23×C31—21no—
162C23×C31—21no—
163C23×C31—21no—
164C23×C31—21no—
165C23×C31—22no—
166C23×C31—41no—
167C23×C31—41no—
168C23×C31—41no—
169C23×C31—41no—
170C23×C31—41no—
171C23×C31—41no—
172C23×C31—41no—
173C23×C31—42no—
174C23×C31—41no—
175C23×C31—41no—
176C23×C31—41no—
177C23×C31—41no—
178C23×C31—42no—
179C23×C31—42no—
180C23×C31—41no—
181C23×C31—41no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31—41no—
185C23×C31—41no—
186C23×C31—81no—
187C23×C31—82no—
188C23×C31—81no—
189C23×C31—82no—
190C23×C31—84no—
191C23×C31—81no—
192C23×C31#12011yescharacteristic
193C23×C31#12111yescharacteristic
194C23×C31#12111yescharacteristic
195C23×C31#12311yescharacteristic
196C23×C31#12411yescharacteristic
197D4×C31—41no—
198D4×C31—41no—
199D4×C31—41no—
200D4×C31—41no—
201D4×C31—42no—
202D4×C31—81no—
203D4×C31—82no—
204D4×C31—82no—
205D4×C31—82no—
206D4×C31—82no—
207D4×C31—84no—
208D4×C31—82no—
209D4×C31—82no—
210D4×C31—84no—
211C2×C4×C31—21no—
212C2×C4×C31—21no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—41no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—81no—
230C2×C4×C31—81no—
231C2×C4×C31—81no—
232C2×C4×C31—81no—
233C2×C4×C31—81no—
234C2×C4×C31—81no—
235C2×C4×C31—81no—
236C2×C4×C31—81no—
237C2×C4×C31—81no—
238C2×C4×C31—81no—
239C2×C4×C31#12111yescharacteristic
240C2×C4×C31#12311yescharacteristic
241C8×C31—81no—
242#2669C2×C3111yescharacteristic
243C2×#353C2×C3111yescharacteristic, complemented
244C2×AΣL(1,𝔽16)C2×C3111yescharacteristic, complemented
245#1889C2×C3111yescharacteristic, complemented
246#1887C2×C3111yescharacteristic, complemented
247#1867C2×C3111yescharacteristic, complemented
248#1390C2×C3111yescharacteristic, complemented
249C22×C31—21no—
250C22×C31—21no—
251C22×C31—21no—
252C22×C31—21no—
253C22×C31—21no—
254C22×C31—21no—
255C22×C31—41no—
256C22×C31—41no—
257C22×C31—41no—
258C22×C31—41no—
259C22×C31—42no—
260C22×C31—42no—
261C22×C31—41no—
262C22×C31—41no—
263C22×C31—41no—
264C22×C31—41no—
265C22×C31—44no—
266C22×C31—42no—
267C22×C31—81no—
268C22×C31—81no—
269C22×C31—81no—
270C22×C31—81no—
271C22×C31—81no—
272C22×C31—81no—
273C22×C31—81no—
274C22×C31—81no—
275C22×C31—82no—
276C22×C31—82no—
277C22×C31—82no—
278C22×C31—81no—
279C22×C31—81no—
280C22×C31—81no—
281C22×C31—81no—
282C22×C31—81no—
283C22×C31#39411yescharacteristic
284C22×C31B(4,𝔽2)11yescharacteristic
285C22×C31#45811yescharacteristic
286C4×C31—21no—
287C4×C31—21no—
288C4×C31—42no—
289C4×C31—42no—
290C4×C31—81no—
291C4×C31—81no—
292C4×C31—82no—
293C4×C31—81no—
294C4×C31—81no—
295C4×C31—161no—
296C2×#127C22×C3111yescharacteristic
297C2×#121—21no—
298C2×#120—21no—
299C2×#120C22×C3111yescharacteristic
300#518C22×C3111yescharacteristic
301#418—21no—
302C2×#100C22×C3111yescharacteristic
303C2×#99C22×C3111yescharacteristic
304C2×#99C22×C3111yescharacteristic
305#394—21no—
306#392—21no—
307#392—21no—
308#388—21no—
309#386C22×C3111yescharacteristic
310#385—21no—
311#379—21no—
312#353—22no—
313AΣL(1,𝔽16)—22no—
314#342—21no—
315#342—21no—
316#339—21no—
317C2×C31—21no—
318C2×C31—21no—
319C2×C31—41no—
320C2×C31—41no—
321C2×C31—41no—
322C2×C31—41no—
323C2×C31—82no—
324C2×C31—161no—
325C2×C31#204411yescharacteristic
326C2×C31#219611yescharacteristic
327C2×C31#220011yescharacteristic
328C25D4×C3111yescharacteristic
329C22×D4—21no—
330C22×D4—21no—
331C22×D4—21no—
332C22×D4—21no—
333C22×D4—21no—
334C22×D4D4×C3111yescharacteristic
335C22×D4C23×C3111yescharacteristic
336C23×C4D4×C3111yescharacteristic
337C2×M16—21no—
338#127—21no—
339#127—21no—
340#127—21no—
341#127—21no—
342#121—44no—
343#120—21no—
344#120—21no—
345#120—21no—
346#120—21no—
347#120—44no—
348C4×D4—41no—
349C4×D4—41no—
350#117—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—41no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—41no—
364C2×C22⋊C4—41no—
365C2×C22⋊C4D4×C3111yescharacteristic
366C2×C22⋊C4D4×C3111yescharacteristic
367C2×C42D4×C3111yescharacteristic
368#102—41no—
369#102—41no—
370#100—21no—
371#100—21no—
372#99—21no—
373#99—21no—
374#99—21no—
375#99D4×C3112yescomplemented
376#95—21no—
377#95—21no—
378#95—41no—
379C31#1342211yescharacteristic, direct product factor, Sylow
380C24—21no—
381C24—21no—
382C24—21no—
383C24—21no—
384C24—41no—
385C24—42no—
386C24—41no—
387C24—81no—
388C24C2×D4×C3111yescharacteristic
389C24C2×D4×C3111yescharacteristic
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—81no—
409C2×D4—81no—
410C2×D4—81no—
411C2×D4—81no—
412C2×D4—81no—
413C2×D4—81no—
414C2×D4C2×D4×C3111yescharacteristic
415C2×D4#41×C3111yescharacteristic
416C2×D4#41×C3111yescharacteristic
417C2×D4#41×C3111yescharacteristic
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4C2×D4×C3111yescharacteristic
431M16—41no—
432M16—41no—
433C2×C8—41no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—42no—
437C4⋊C4—82no—
438C22⋊C4—24no—
439C22⋊C4—22no—
440C22⋊C4—21no—
441C22⋊C4—21no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—81no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C42—21no—
458C42—21no—
459C42—41no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—81no—
486C23—82no—
487C23—81no—
488C23—82no—
489C23—84no—
490C23—81no—
491C23C31×#12011yescharacteristic
492C23C31×#12111yescharacteristic
493C23C31×#12111yescharacteristic
494C23C31×#12311yescharacteristic
495C23C31×#12411yescharacteristic
496D4—41no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—42no—
501D4—81no—
502D4—82no—
503D4—82no—
504D4—82no—
505D4—82no—
506D4—84no—
507D4—82no—
508D4—82no—
509D4—84no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4C31×#12111yescharacteristic
539C2×C4C31×#12311yescharacteristic
540C8—81no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—44no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22C31×#39411yescharacteristic
576C22C31×B(4,𝔽2)11yescharacteristic
577C22C31×#45811yescharacteristic
578C4—21no—
579C4—21no—
580C4—42no—
581C4—42no—
582C4—81no—
583C4—81no—
584C4—82no—
585C4—81no—
586C4—81no—
587C4—161no—
588C2—21no—
589C2—21no—
590C2—41no—
591C2—41no—
592C2—41no—
593C2—41no—
594C2—82no—
595C2—161no—
596C2C31×#204411yescharacteristic
597C2C31×#219611yescharacteristic
598C2C31×#220011yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C31
prime radicalC31×#2669C2
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C22×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#353C2semidirect
3. C2×C31×AΣL(1,𝔽16)C2semidirect
4. C31×#1889C2semidirect
5. C31×#1887C2semidirect
6. C31×#1867C2semidirect
7. C31×#1390C2semidirect
76. C31×#99D4semidirect
160. #13422C31direct (N × H)
243. C2×#353C2×C31semidirect
244. C2×AΣL(1,𝔽16)C2×C31semidirect
245. #1889C2×C31semidirect
246. #1887C2×C31semidirect
247. #1867C2×C31semidirect
248. #1390C2×C31semidirect
375. #99D4×C31semidirect
379. C31#13422direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26692
C2×C31×#3532
C2×C31×AΣL(1,𝔽16)2
C31×#18892
C31×#18872
C31×#18672
C31×#13902
#1342231