Finite Group Databasebeta

← C31×#13356 #21470115

Subgroups — C31×#13356 #21470115

Order 7936.

676 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3202C211yesmaximal, characteristic, complemented
2C31×#3028C211yesmaximal, characteristic
3C31×#3022C211yesmaximal, characteristic
4C31×#2135C211yesmaximal, characteristic, complemented
5C31×#2123C211yesmaximal, characteristic, complemented
6C31×#1791C211yesmaximal, characteristic
7C31×#1395C211yesmaximal, characteristic, complemented
8C31×#585C2211yescharacteristic
9C2×C31×#137—21no—
10C31×#536—21no—
11C31×#536C2211yescharacteristic
12C31×#536C2211yescharacteristic
13C31×#485—21no—
14C31×#485—21no—
15C31×#470—21no—
16C31×#467—21no—
17C31×#458—22no—
18C31×#458—22no—
19C31×#456—22no—
20C31×#455—22no—
21C31×#451—21no—
22C31×#451—21no—
23C31×#421—21no—
24C31×#413—21no—
25C31×#413C2211yescharacteristic
26C31×#413C2211yescharacteristic
27C31×#410—21no—
28C31×#410C2211yescharacteristic
29C31×#408—21no—
30C31×#408C2211yescharacteristic
31C31×#356—22no—
32C31×#352—22no—
33C31×2−1+4—21no—
34C31×2−1+4—41no—
35C31×H(5,𝔽2)—21no—
36C31×H(5,𝔽2)—21no—
37C31×H(5,𝔽2)—41no—
38C31×H(5,𝔽2)D412yes—
39C2×#41×C31—21no—
40C2×#41×C31—21no—
41C2×#41×C31—21no—
42C2×#41×C31—42no—
43C2×#41×C31C2311yescharacteristic
44C22×Q8×C31D411yescharacteristic
45C22×Q8×C31D411yescharacteristic
46C31×#137—24no—
47C31×#137—42no—
48C2×Q16×C31—21no—
49C2×Q16×C31—21no—
50C2×Q16×C31—41no—
51C2×SD16×C31—41no—
52C2×M16×C31—21no—
53C2×M16×C31—21no—
54C2×M16×C31D411yescharacteristic
55C22×C8×C31D411yescharacteristic
56C31×C4⋊Q8—21no—
57C31×C4⋊Q8—21no—
58C31×C4⋊Q8—41no—
59C31×#124—21no—
60C31×#124—21no—
61C31×#124—41no—
62C31×#123—24no—
63C31×#123—21no—
64C31×#123—21no—
65C31×#123—21no—
66C31×#123—21no—
67C31×C22⋊Q8—21no—
68C31×C22⋊Q8—21no—
69C31×C22⋊Q8—21no—
70C31×C22⋊Q8—21no—
71C31×C22⋊Q8—42no—
72C31×#117—21no—
73C31×#117D411yescharacteristic
74C31×#117D411yescharacteristic
75C31×#107—21no—
76C31×#106—21no—
77C31×#104—24no—
78C31×#103—21no—
79C31×#103—21no—
80C31×#103—21no—
81C31×#103—21no—
82C31×#101—21no—
83C31×#101—41no—
84C31×#101D412yes—
85C31×#100—21no—
86C31×#100—21no—
87C31×#100—41no—
88C31×#99—24no—
89C31×#99—21no—
90C31×#99D412yes—
91C31×C22⋊C8—21no—
92C31×C22⋊C8—21no—
93C31×C22⋊C8—42no—
94#41×C31—22no—
95#41×C31—22no—
96#41×C31—41no—
97#41×C31—41no—
98#41×C31—42no—
99#41×C31—41no—
100#41×C31—41no—
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—41no—
108#41×C31—41no—
109#41×C31—41no—
110C2×Q8×C31—21no—
111C2×Q8×C31—21no—
112C2×Q8×C31—21no—
113C2×Q8×C31—21no—
114C2×Q8×C31—21no—
115C2×Q8×C31—21no—
116C2×Q8×C31—21no—
117C2×Q8×C31—41no—
118C2×Q8×C31—41no—
119C2×Q8×C31—41no—
120C2×Q8×C31—41no—
121C2×Q8×C31—42no—
122C2×Q8×C31C2×D411yescharacteristic
123C2×D4×C31—21no—
124C2×D4×C31—21no—
125C2×D4×C31—21no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—42no—
129C2×D4×C31—42no—
130C2×D4×C31—42no—
131C2×D4×C31—41no—
132C2×D4×C31C2×D411yescharacteristic
133C2×D4×C31C2×D411yescharacteristic
134C2×D4×C31#4111yescharacteristic
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—21no—
138C22×C4×C31—21no—
139C22×C4×C31—41no—
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2×D411yescharacteristic
143Q16×C31—44no—
144Q16×C31—81no—
145Q16×C31—81no—
146SD16×C31—44no—
147M16×C31—22no—
148M16×C31—41no—
149M16×C31—41no—
150M16×C31—41no—
151M16×C31—41no—
152C2×C8×C31—21no—
153C2×C8×C31—21no—
154C2×C8×C31—41no—
155C2×C8×C31—41no—
156C2×C8×C31—41no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—41no—
160C4⋊C4×C31—44no—
161C4⋊C4×C31—41no—
162C4⋊C4×C31—41no—
163C4⋊C4×C31—41no—
164C4⋊C4×C31—41no—
165C22⋊C4×C31—21no—
166C22⋊C4×C31—22no—
167C22⋊C4×C31—21no—
168C22⋊C4×C31—44no—
169C22⋊C4×C31—41no—
170C22⋊C4×C31—41no—
171C22⋊C4×C31—41no—
172C22⋊C4×C31—41no—
173C42×C31—21no—
174C42×C31—21no—
175C42×C31—41no—
176#13356C3111yesmaximal, characteristic, direct product factor, Sylow
177C23×C31—22no—
178C23×C31—22no—
179C23×C31—41no—
180C23×C31#12011yescharacteristic
181C23×C31#12111yescharacteristic
182C23×C31#12111yescharacteristic
183Q8×C31—22no—
184Q8×C31—41no—
185Q8×C31—41no—
186Q8×C31—41no—
187Q8×C31—42no—
188Q8×C31—41no—
189Q8×C31—41no—
190Q8×C31—41no—
191Q8×C31—41no—
192Q8×C31—41no—
193Q8×C31—81no—
194D4×C31—22no—
195D4×C31—41no—
196D4×C31—41no—
197D4×C31—41no—
198D4×C31—41no—
199D4×C31—41no—
200D4×C31—42no—
201D4×C31—42no—
202D4×C31—42no—
203D4×C31—42no—
204D4×C31—41no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—21no—
208C2×C4×C31—21no—
209C2×C4×C31—21no—
210C2×C4×C31—21no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—41no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—81no—
225C2×C4×C31#12011yescharacteristic
226C2×C4×C31#12011yescharacteristic
227C2×C4×C31#12111yescharacteristic
228C2×C4×C31#12711yescharacteristic
229C8×C31—42no—
230C8×C31—41no—
231C8×C31—41no—
232C8×C31—81no—
233#3202C2×C3111yescharacteristic, complemented
234#3028C2×C3111yescharacteristic
235#3022C2×C3111yescharacteristic
236#2135C2×C3111yescharacteristic, complemented
237#2123C2×C3111yescharacteristic, complemented
238#1791C2×C3111yescharacteristic
239#1395C2×C3111yescharacteristic, complemented
240C22×C31—21no—
241C22×C31—21no—
242C22×C31—21no—
243C22×C31—41no—
244C22×C31—41no—
245C22×C31—41no—
246C22×C31—42no—
247C22×C31—82no—
248C22×C31—82no—
249C22×C31—161no—
250C22×C31#39211yescharacteristic
251C4×C31—21no—
252C4×C31—21no—
253C4×C31—21no—
254C4×C31—42no—
255C4×C31—81no—
256C4×C31—81no—
257C4×C31—82no—
258C4×C31—81no—
259C4×C31—81no—
260C4×C31B(4,𝔽2)11yescharacteristic
261C4×C31B(4,𝔽2)11yescharacteristic
262#585C22×C3111yescharacteristic
263C2×#137—21no—
264#536—21no—
265#536C22×C3111yescharacteristic
266#536C22×C3111yescharacteristic
267#485—21no—
268#485—21no—
269#470—21no—
270#467—21no—
271#458—22no—
272#458—22no—
273#456—22no—
274#455—22no—
275#451—21no—
276#451—21no—
277#421—21no—
278#413—21no—
279#413C22×C3111yescharacteristic
280#413C22×C3111yescharacteristic
281#410—21no—
282#410C22×C3111yescharacteristic
283#408—21no—
284#408C22×C3111yescharacteristic
285#356—22no—
286#352—22no—
287C2×C31—21no—
288C2×C31—41no—
289C2×C31—41no—
290C2×C31—41no—
291C2×C31—82no—
292C2×C31#202111yescharacteristic
2932−1+4—21no—
2942−1+4—41no—
295H(5,𝔽2)—21no—
296H(5,𝔽2)—21no—
297H(5,𝔽2)—41no—
298H(5,𝔽2)D4×C3112yes—
299C2×#41—21no—
300C2×#41—21no—
301C2×#41—21no—
302C2×#41—42no—
303C2×#41C23×C3111yescharacteristic
304C22×Q8D4×C3111yescharacteristic
305C22×Q8D4×C3111yescharacteristic
306#137—24no—
307#137—42no—
308C2×Q16—21no—
309C2×Q16—21no—
310C2×Q16—41no—
311C2×SD16—41no—
312C2×M16—21no—
313C2×M16—21no—
314C2×M16D4×C3111yescharacteristic
315C22×C8D4×C3111yescharacteristic
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8—41no—
319#124—21no—
320#124—21no—
321#124—41no—
322#123—24no—
323#123—21no—
324#123—21no—
325#123—21no—
326#123—21no—
327C22⋊Q8—21no—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C22⋊Q8—42no—
332#117—21no—
333#117D4×C3111yescharacteristic
334#117D4×C3111yescharacteristic
335#107—21no—
336#106—21no—
337#104—24no—
338#103—21no—
339#103—21no—
340#103—21no—
341#103—21no—
342#101—21no—
343#101—41no—
344#101D4×C3112yes—
345#100—21no—
346#100—21no—
347#100—41no—
348#99—24no—
349#99—21no—
350#99D4×C3112yes—
351C22⋊C8—21no—
352C22⋊C8—21no—
353C22⋊C8—42no—
354C31#1335611yescharacteristic, direct product factor, Sylow
355#41—22no—
356#41—22no—
357#41—41no—
358#41—41no—
359#41—42no—
360#41—41no—
361#41—41no—
362#41—41no—
363#41—41no—
364#41—41no—
365#41—42no—
366#41—42no—
367#41—42no—
368#41—41no—
369#41—41no—
370#41—41no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—42no—
383C2×Q8C2×D4×C3111yescharacteristic
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—41no—
393C2×D4C2×D4×C3111yescharacteristic
394C2×D4C2×D4×C3111yescharacteristic
395C2×D4#41×C3111yescharacteristic
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—41no—
401C22×C4C2×D4×C3111yescharacteristic
402C22×C4C2×D4×C3111yescharacteristic
403C22×C4C2×D4×C3111yescharacteristic
404Q16—44no—
405Q16—81no—
406Q16—81no—
407SD16—44no—
408M16—22no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—41no—
421C4⋊C4—44no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C22⋊C4—21no—
427C22⋊C4—22no—
428C22⋊C4—21no—
429C22⋊C4—44no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C42—21no—
435C42—21no—
436C42—41no—
437C23—22no—
438C23—22no—
439C23—41no—
440C23C31×#12011yescharacteristic
441C23C31×#12111yescharacteristic
442C23C31×#12111yescharacteristic
443Q8—22no—
444Q8—41no—
445Q8—41no—
446Q8—41no—
447Q8—42no—
448Q8—41no—
449Q8—41no—
450Q8—41no—
451Q8—41no—
452Q8—41no—
453Q8—81no—
454D4—22no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—41no—
459D4—41no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—41no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—81no—
485C2×C4C31×#12011yescharacteristic
486C2×C4C31×#12011yescharacteristic
487C2×C4C31×#12111yescharacteristic
488C2×C4C31×#12711yescharacteristic
489C8—42no—
490C8—41no—
491C8—41no—
492C8—81no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—42no—
500C22—82no—
501C22—82no—
502C22—161no—
503C22C31×#39211yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—42no—
508C4—81no—
509C4—81no—
510C4—82no—
511C4—81no—
512C4—81no—
513C4C31×B(4,𝔽2)11yescharacteristic
514C4C31×B(4,𝔽2)11yescharacteristic
515C2—21no—
516C2—41no—
517C2—41no—
518C2—41no—
519C2—82no—
520C2C31×#202111yescharacteristic
subgroupG/H
center, socleC2×C31#2021
commutator, Frattini, cosocleC2×#41C23×C31
prime radicalC31×#585C22
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3202C2semidirect
4. C31×#2135C2semidirect
5. C31×#2123C2semidirect
7. C31×#1395C2semidirect
176. #13356C31direct (N × H)
233. #3202C2×C31semidirect
236. #2135C2×C31semidirect
237. #2123C2×C31semidirect
239. #1395C2×C31semidirect
354. C31#13356direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32022
C31×#30282
C31×#30222
C31×#21352
C31×#21232
C31×#17912
C31×#13952
#1335631