Finite Group Databasebeta

← C31×#10022 #21466781

Subgroups — C31×#10022 #21466781

Order 7936.

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

normal subgroups only

This group has 576 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
1C31×#3140C211yesmaximal, characteristic, complemented
2C31×#3110C211yesmaximal, characteristic, complemented
3C31×#2631C211yesmaximal, characteristic
4C31×#1533C211yesmaximal, characteristic, complemented
5C31×#1515C211yesmaximal, characteristic, complemented
6C31×#1490C211yesmaximal, characteristic, complemented
7C31×#1288C211yesmaximal, characteristic, complemented
8C31×#533C2211yescharacteristic
9C2×C31×#124C2211yescharacteristic
10C2×C4×D4×C31C2211yescharacteristic
11C31×#489—21no—
12C31×#489—21no—
13C31×#488—22no—
14C31×#484—21no—
15C31×#474—21no—
16C31×#461—21no—
17C31×#450—21no—
18C31×#440—22no—
19C31×#432C2211yescharacteristic
20C31×#432C2211yescharacteristic
21C31×#419—21no—
22C31×#418—21no—
23C4×M16×C31C2211yescharacteristic
24C31×#397—21no—
25C31×#393—21no—
26C31×#392—21no—
27C31×#392—21no—
28C31×#386—22no—
29C31×#383C2211yescharacteristic
30C31×#329—22no—
31C31×#328—22no—
32C31×D4⋊C8—22no—
33C22×Q8×C31D411yescharacteristic
34C22×D4×C31—22no—
35C22×D4×C31D411yescharacteristic
36C22×D4×C31D411yescharacteristic
37C23×C4×C31—21no—
38C2×SD16×C31—22no—
39C2×M16×C31—21no—
40C31×C4⋊Q8D411yescharacteristic, complemented
41C31×C4⋊Q8D411yescharacteristic, complemented
42C31×#125—21no—
43C31×#124—21no—
44C31×#124—21no—
45C31×#124—21no—
46C31×#124—21no—
47C31×C22⋊Q8—21no—
48C31×C22⋊Q8—21no—
49C4×Q8×C31—21no—
50C4×Q8×C31—21no—
51C4×D4×C31—21no—
52C4×D4×C31—21no—
53C4×D4×C31—21no—
54C4×D4×C31—21no—
55C31×#117—21no—
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C22⋊C4×C31—21no—
58C2×C22⋊C4×C31—21no—
59C2×C22⋊C4×C31—21no—
60C2×C22⋊C4×C31—22no—
61C2×C22⋊C4×C31—22no—
62C2×C42×C31C2311yescharacteristic
63C31×#107—22no—
64C31×C4⋊C8—21no—
65C31×C4⋊C8—21no—
66C31×#103—22no—
67C31×#103—22no—
68C31×#103—22no—
69C31×#102—22no—
70C31×#102—22no—
71C31×#102—22no—
72C31×C22⋊C8—21no—
73C31×C22⋊C8—21no—
74C31×#97—21no—
75C31×#97—21no—
76C31×#97D412yescomplemented
77C4×C8×C31D412yescomplemented
78C31×#95—22no—
79C31×#95—21no—
80C31×#95—21no—
81C31×#95D412yes—
82C24×C31—21no—
83C24×C31—21no—
84C2×Q8×C31—21no—
85C2×Q8×C31—21no—
86C2×Q8×C31—21no—
87C2×Q8×C31—21no—
88C2×Q8×C31—42no—
89C2×D4×C31—21no—
90C2×D4×C31—21no—
91C2×D4×C31—21no—
92C2×D4×C31—21no—
93C2×D4×C31—42no—
94C2×D4×C31—42no—
95C2×D4×C31—44no—
96C2×D4×C31—82no—
97C2×D4×C31—82no—
98C22×C4×C31—22no—
99C22×C4×C31—21no—
100C22×C4×C31—21no—
101C22×C4×C31—41no—
102C22×C4×C31—41no—
103C22×C4×C31—82no—
104C22×C4×C31C2×D411yescharacteristic
105C22×C4×C31C2×D411yescharacteristic
106C22×C4×C31C2×D411yescharacteristic
107SD16×C31—44no—
108M16×C31—42no—
109C2×C8×C31—22no—
110C2×C8×C31—22no—
111C2×C8×C31—41no—
112C4⋊C4×C31—21no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—21no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—21no—
118C4⋊C4×C31—41no—
119C4⋊C4×C31—41no—
120C22⋊C4×C31—41no—
121C22⋊C4×C31—41no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—41no—
124C22⋊C4×C31—41no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31—41no—
127C22⋊C4×C31—82no—
128C22⋊C4×C31—82no—
129C42×C31—41no—
130C42×C31C2×D411yescharacteristic
131C42×C31C2×D411yescharacteristic
132C42×C31C2×D411yescharacteristic
133C42×C31#4111yescharacteristic
134#10022C3111yesmaximal, characteristic, direct product factor, Sylow
135C23×C31—22no—
136C23×C31—42no—
137C23×C31—41no—
138C23×C31—41no—
139C23×C31—42no—
140C23×C31—41no—
141C23×C31—41no—
142C23×C31—42no—
143C23×C31—41no—
144C23×C31—42no—
145C23×C31#12011yescharacteristic
146Q8×C31—42no—
147Q8×C31—42no—
148Q8×C31—81no—
149Q8×C31—81no—
150D4×C31—22no—
151D4×C31—22no—
152D4×C31—41no—
153D4×C31—41no—
154D4×C31—42no—
155D4×C31—81no—
156D4×C31—88no—
157C2×C4×C31—22no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—42no—
161C2×C4×C31—42no—
162C2×C4×C31—42no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—42no—
168C2×C4×C31—42no—
169C2×C4×C31—82no—
170C2×C4×C31—82no—
171C2×C4×C31#12011yescharacteristic
172C2×C4×C31#12011yescharacteristic
173C2×C4×C31#12111yescharacteristic
174C2×C4×C31#12111yescharacteristic
175C2×C4×C31#12111yescharacteristic
176C2×C4×C31#12711yescharacteristic
177C8×C31—42no—
178C8×C31—42no—
179C8×C31—81no—
180#3140C2×C3111yescharacteristic, complemented
181#3110C2×C3111yescharacteristic, complemented
182#2631C2×C3111yescharacteristic
183#1533C2×C3111yescharacteristic, complemented
184#1515C2×C3111yescharacteristic, complemented
185#1490C2×C3111yescharacteristic, complemented
186#1288C2×C3111yescharacteristic, complemented
187C22×C31—21no—
188C22×C31—21no—
189C22×C31—44no—
190C22×C31—42no—
191C22×C31—82no—
192C22×C31—82no—
193C22×C31—82no—
194C22×C31—81no—
195C22×C31—81no—
196C22×C31—81no—
197C22×C31—81no—
198C22×C31—81no—
199C22×C31#39211yescharacteristic
200C22×C31#45512yes—
201C4×C31—21no—
202C4×C31—21no—
203C4×C31—21no—
204C4×C31—21no—
205C4×C31—21no—
206C4×C31—21no—
207C4×C31—82no—
208C4×C31—81no—
209C4×C31—81no—
210C4×C31—81no—
211C4×C31—81no—
212#533C22×C3111yescharacteristic
213C2×#124C22×C3111yescharacteristic
214C2×C4×D4C22×C3111yescharacteristic
215#489—21no—
216#489—21no—
217#488—22no—
218#484—21no—
219#474—21no—
220#461—21no—
221#450—21no—
222#440—22no—
223#432C22×C3111yescharacteristic
224#432C22×C3111yescharacteristic
225#419—21no—
226#418—21no—
227C4×M16C22×C3111yescharacteristic
228#397—21no—
229#393—21no—
230#392—21no—
231#392—21no—
232#386—22no—
233#383C22×C3111yescharacteristic
234#329—22no—
235#328—22no—
236D4⋊C8—22no—
237C2×C31—22no—
238C2×C31—82no—
239C2×C31—161no—
240C2×C31#200411yescharacteristic
241C2×C31#200811yescharacteristic
242C2×C31#201011yescharacteristic
243C22×Q8D4×C3111yescharacteristic
244C22×D4—22no—
245C22×D4D4×C3111yescharacteristic
246C22×D4D4×C3111yescharacteristic
247C23×C4—21no—
248C2×SD16—22no—
249C2×M16—21no—
250C4⋊Q8D4×C3111yescharacteristic, complemented
251C4⋊Q8D4×C3111yescharacteristic, complemented
252#125—21no—
253#124—21no—
254#124—21no—
255#124—21no—
256#124—21no—
257C22⋊Q8—21no—
258C22⋊Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265#117—21no—
266C2×C4⋊C4D4×C3111yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C42C23×C3111yescharacteristic
273#107—22no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276#103—22no—
277#103—22no—
278#103—22no—
279#102—22no—
280#102—22no—
281#102—22no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284#97—21no—
285#97—21no—
286#97D4×C3112yescomplemented
287C4×C8D4×C3112yescomplemented
288#95—22no—
289#95—21no—
290#95—21no—
291#95D4×C3112yes—
292C31#1002211yescharacteristic, direct product factor, Sylow
293C24—21no—
294C24—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—42no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—44no—
307C2×D4—82no—
308C2×D4—82no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—82no—
315C22×C4C2×D4×C3111yescharacteristic
316C22×C4C2×D4×C3111yescharacteristic
317C22×C4C2×D4×C3111yescharacteristic
318SD16—44no—
319M16—42no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—41no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C42—41no—
341C42C2×D4×C3111yescharacteristic
342C42C2×D4×C3111yescharacteristic
343C42C2×D4×C3111yescharacteristic
344C42#41×C3111yescharacteristic
345C23—22no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—41no—
351C23—42no—
352C23—41no—
353C23—42no—
354C23—41no—
355C23C31×#12011yescharacteristic
356Q8—42no—
357Q8—42no—
358Q8—81no—
359Q8—81no—
360D4—22no—
361D4—22no—
362D4—41no—
363D4—41no—
364D4—42no—
365D4—81no—
366D4—88no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4C31×#12011yescharacteristic
382C2×C4C31×#12011yescharacteristic
383C2×C4C31×#12111yescharacteristic
384C2×C4C31×#12111yescharacteristic
385C2×C4C31×#12111yescharacteristic
386C2×C4C31×#12711yescharacteristic
387C8—42no—
388C8—42no—
389C8—81no—
390C22—21no—
391C22—21no—
392C22—44no—
393C22—42no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22C31×#39211yescharacteristic
403C22C31×#45512yes—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—82no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—81no—
415C2—22no—
416C2—82no—
417C2—161no—
418C2C31×#200411yescharacteristic
419C2C31×#200811yescharacteristic
420C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#2631C2
LCSG ▷ C2×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×#3140C2semidirect
2. C31×#3110C2semidirect
4. C31×#1533C2semidirect
5. C31×#1515C2semidirect
6. C31×#1490C2semidirect
7. C31×#1288C2semidirect
40. C31×C4⋊Q8D4semidirect
41. C31×C4⋊Q8D4semidirect
76. C31×#97D4semidirect
77. C4×C8×C31D4semidirect
134. #10022C31direct (N × H)
180. #3140C2×C31semidirect
181. #3110C2×C31semidirect
183. #1533C2×C31semidirect
184. #1515C2×C31semidirect
185. #1490C2×C31semidirect
186. #1288C2×C31semidirect
250. C4⋊Q8D4×C31semidirect
251. C4⋊Q8D4×C31semidirect
286. #97D4×C31semidirect
287. C4×C8D4×C31semidirect
292. C31#10022direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31402
C31×#31102
C31×#26312
C31×#15332
C31×#15152
C31×#14902
C31×#12882
#1002231