Finite Group Databasebeta

← C31×#9989 #21466748

Subgroups — C31×#9989 #21466748

Order 7936.

724 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3193C211yesmaximal, characteristic
2C2×C31×#451C211yesmaximal, characteristic, complemented
3C2×C31×#448C211yesmaximal, characteristic
4C31×#1784C211yesmaximal, characteristic, complemented
5C31×#1472C211yesmaximal, characteristic, complemented
6C31×#1471C211yesmaximal, characteristic
7C31×#1298C211yesmaximal, characteristic, complemented
8C23×Q8×C31C2211yescharacteristic
9C22×Q16×C31—21no—
10C22×SD16×C31—21no—
11C2×C31×C22⋊Q8C2211yescharacteristic
12C2×C31×#121C2211yescharacteristic
13C31×#484—21no—
14C31×#480—21no—
15C31×#468—22no—
16C31×#451—22no—
17C31×#451—22no—
18C31×#450—21no—
19C31×#448—21no—
20C31×#448—22no—
21C31×#448—22no—
22C2×C31×#103—21no—
23C2×C31×#103—21no—
24C2×C31×#101—22no—
25C31×#407C2211yescharacteristic
26C31×#407C2211yescharacteristic
27C2×C31×C22⋊C8C2211yescharacteristic
28C2×C31×C22⋊C8C2211yescharacteristic
29C31×#324—22no—
30C31×#323—22no—
31C22×Q8×C31—21no—
32C22×Q8×C31—21no—
33C22×Q8×C31—21no—
34C22×Q8×C31—21no—
35C22×Q8×C31—21no—
36C22×Q8×C31—44no—
37C22×Q8×C31D412yes—
38C22×Q8×C31D411yescharacteristic
39C22×D4×C31—21no—
40C22×D4×C31D411yescharacteristic
41C23×C4×C31—21no—
42C23×C4×C31C2311yescharacteristic
43C2×Q16×C31—24no—
44C2×Q16×C31—44no—
45C2×SD16×C31—24no—
46C2×SD16×C31—22no—
47C2×SD16×C31—44no—
48C2×M16×C31—21no—
49C2×M16×C31D412yes—
50C22×C8×C31—21no—
51C22×C8×C31—21no—
52C31×C22⋊Q8—21no—
53C31×C22⋊Q8—21no—
54C31×C22⋊Q8—21no—
55C31×C22⋊Q8—21no—
56C31×#121—21no—
57C31×#121—21no—
58C31×#121—21no—
59C31×#121—21no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31D411yescharacteristic
62C2×C4⋊C4×C31D411yescharacteristic
63C2×C22⋊C4×C31—21no—
64C2×C22⋊C4×C31—21no—
65C31×#107—22no—
66C31×#103—24no—
67C31×#103—24no—
68C31×#103—22no—
69C31×#102—22no—
70C31×#101—44no—
71C31×C22⋊C8—21no—
72C31×C22⋊C8—21no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8—21no—
75C31×C22⋊C8—21no—
76C31×C22⋊C8—21no—
77C31×C22⋊C8D412yes—
78C31×C22⋊C8D412yes—
79C24×C31—21no—
80C24×C31#4111yescharacteristic
81C2×Q8×C31—24no—
82C2×Q8×C31—24no—
83C2×Q8×C31—24no—
84C2×Q8×C31—22no—
85C2×Q8×C31—22no—
86C2×Q8×C31—21no—
87C2×Q8×C31—21no—
88C2×Q8×C31—44no—
89C2×Q8×C31—42no—
90C2×Q8×C31—42no—
91C2×Q8×C31—42no—
92C2×Q8×C31—42no—
93C2×Q8×C31—44no—
94C2×Q8×C31—44no—
95C2×Q8×C31—41no—
96C2×Q8×C31—41no—
97C2×D4×C31—21no—
98C2×D4×C31—21no—
99C2×D4×C31—42no—
100C2×D4×C31—41no—
101C2×D4×C31—41no—
102C2×D4×C31—82no—
103C22×C4×C31—22no—
104C22×C4×C31—21no—
105C22×C4×C31—21no—
106C22×C4×C31—21no—
107C22×C4×C31—42no—
108C22×C4×C31—44no—
109C22×C4×C31—41no—
110C22×C4×C31—41no—
111C22×C4×C31C2×D411yescharacteristic
112C22×C4×C31C2×D411yescharacteristic
113C22×C4×C31C2×D411yescharacteristic
114C22×C4×C31C2×D411yescharacteristic
115C22×C4×C31C2×D411yescharacteristic
116C22×C4×C31C2×D411yescharacteristic
117Q16×C31—48no—
118SD16×C31—48no—
119SD16×C31—82no—
120M16×C31—42no—
121M16×C31—42no—
122C2×C8×C31—22no—
123C2×C8×C31—22no—
124C2×C8×C31—22no—
125C2×C8×C31—42no—
126C2×C8×C31—42no—
127C2×C8×C31—41no—
128C4⋊C4×C31—21no—
129C4⋊C4×C31—21no—
130C4⋊C4×C31—21no—
131C4⋊C4×C31—21no—
132C4⋊C4×C31—41no—
133C4⋊C4×C31—41no—
134C22⋊C4×C31—41no—
135C22⋊C4×C31—41no—
136C22⋊C4×C31—41no—
137C22⋊C4×C31—41no—
138#9989C3111yesmaximal, characteristic, direct product factor, Sylow
139C23×C31—22no—
140C23×C31—22no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31—42no—
144C23×C31—81no—
145C23×C31—81no—
146C23×C31#12111yescharacteristic
147C23×C31#12111yescharacteristic
148C23×C31#12111yescharacteristic
149Q8×C31—24no—
150Q8×C31—24no—
151Q8×C31—42no—
152Q8×C31—42no—
153Q8×C31—44no—
154Q8×C31—42no—
155Q8×C31—44no—
156Q8×C31—42no—
157D4×C31—42no—
158D4×C31—42no—
159D4×C31—82no—
160D4×C31—82no—
161C2×C4×C31—22no—
162C2×C4×C31—22no—
163C2×C4×C31—22no—
164C2×C4×C31—24no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—42no—
169C2×C4×C31—44no—
170C2×C4×C31—44no—
171C2×C4×C31—42no—
172C2×C4×C31—41no—
173C2×C4×C31—41no—
174C2×C4×C31—41no—
175C2×C4×C31—42no—
176C2×C4×C31—81no—
177C2×C4×C31—81no—
178C2×C4×C31#12011yescharacteristic
179C2×C4×C31#12011yescharacteristic
180C2×C4×C31#12011yescharacteristic
181C2×C4×C31#12711yescharacteristic
182C8×C31—42no—
183C8×C31—42no—
184C8×C31—42no—
185C8×C31—81no—
186#3193C2×C3111yescharacteristic
187C2×#451C2×C3111yescharacteristic, complemented
188C2×#448C2×C3111yescharacteristic
189#1784C2×C3111yescharacteristic, complemented
190#1472C2×C3111yescharacteristic, complemented
191#1471C2×C3111yescharacteristic
192#1298C2×C3111yescharacteristic, complemented
193C22×C31—21no—
194C22×C31—21no—
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—21no—
198C22×C31—21no—
199C22×C31—21no—
200C22×C31—44no—
201C22×C31—81no—
202C22×C31—81no—
203C22×C31—81no—
204C22×C31—162no—
205C22×C31#39211yescharacteristic
206C22×C31#47112yes—
207C22×C31#47312yes—
208C4×C31—21no—
209C4×C31—21no—
210C4×C31—21no—
211C4×C31—44no—
212C4×C31—81no—
213C4×C31—81no—
214C4×C31—81no—
215C4×C31B(4,𝔽2)12yes—
216C23×Q8C22×C3111yescharacteristic
217C22×Q16—21no—
218C22×SD16—21no—
219C2×C22⋊Q8C22×C3111yescharacteristic
220C2×#121C22×C3111yescharacteristic
221#484—21no—
222#480—21no—
223#468—22no—
224#451—22no—
225#451—22no—
226#450—21no—
227#448—21no—
228#448—22no—
229#448—22no—
230C2×#103—21no—
231C2×#103—21no—
232C2×#101—22no—
233#407C22×C3111yescharacteristic
234#407C22×C3111yescharacteristic
235C2×C22⋊C8C22×C3111yescharacteristic
236C2×C22⋊C8C22×C3111yescharacteristic
237#324—22no—
238#323—22no—
239C2×C31—22no—
240C2×C31—22no—
241C2×C31—41no—
242C2×C31—161no—
243C2×C31#201911yescharacteristic
244C2×C31#202011yescharacteristic
245C2×C31#202111yescharacteristic
246C22×Q8—21no—
247C22×Q8—21no—
248C22×Q8—21no—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×Q8—44no—
252C22×Q8D4×C3111yescharacteristic
253C22×Q8D4×C3112yes—
254C22×D4—21no—
255C22×D4D4×C3111yescharacteristic
256C23×C4—21no—
257C23×C4C23×C3111yescharacteristic
258C2×Q16—24no—
259C2×Q16—44no—
260C2×SD16—24no—
261C2×SD16—22no—
262C2×SD16—44no—
263C2×M16—21no—
264C2×M16D4×C3112yes—
265C22×C8—21no—
266C22×C8—21no—
267C22⋊Q8—21no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271#121—21no—
272#121—21no—
273#121—21no—
274#121—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4D4×C3111yescharacteristic
277C2×C4⋊C4D4×C3111yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280#107—22no—
281#103—24no—
282#103—24no—
283#103—22no—
284#102—22no—
285#101—44no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8D4×C3112yes—
293C22⋊C8D4×C3112yes—
294C31#998911yescharacteristic, direct product factor, Sylow
295C24—21no—
296C24#41×C3111yescharacteristic
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—44no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—42no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—82no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4C2×D4×C3111yescharacteristic
328C22×C4C2×D4×C3111yescharacteristic
329C22×C4C2×D4×C3111yescharacteristic
330C22×C4C2×D4×C3111yescharacteristic
331C22×C4C2×D4×C3111yescharacteristic
332C22×C4C2×D4×C3111yescharacteristic
333Q16—48no—
334SD16—48no—
335SD16—82no—
336M16—42no—
337M16—42no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—41no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C23—22no—
355C23—22no—
356C23—41no—
357C23—41no—
358C23—42no—
359C23—81no—
360C23—81no—
361C23C31×#12111yescharacteristic
362C23C31×#12111yescharacteristic
363C23C31×#12111yescharacteristic
364Q8—24no—
365Q8—24no—
366Q8—42no—
367Q8—42no—
368Q8—44no—
369Q8—42no—
370Q8—44no—
371Q8—42no—
372D4—42no—
373D4—42no—
374D4—82no—
375D4—82no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4C31×#12011yescharacteristic
394C2×C4C31×#12011yescharacteristic
395C2×C4C31×#12011yescharacteristic
396C2×C4C31×#12711yescharacteristic
397C8—42no—
398C8—42no—
399C8—42no—
400C8—81no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—44no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—162no—
413C22C31×#39211yescharacteristic
414C22C31×#47112yes—
415C22C31×#47312yes—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—44no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4C31×B(4,𝔽2)12yes—
424C2—22no—
425C2—22no—
426C2—41no—
427C2—161no—
428C2C31×#201911yescharacteristic
429C2C31×#202011yescharacteristic
430C2C31×#202111yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC2×C31×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#451C2semidirect
4. C31×#1784C2semidirect
5. C31×#1472C2semidirect
7. C31×#1298C2semidirect
138. #9989C31direct (N × H)
187. C2×#451C2×C31semidirect
189. #1784C2×C31semidirect
190. #1472C2×C31semidirect
192. #1298C2×C31semidirect
294. C31#9989direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31932
C2×C31×#4512
C2×C31×#4482
C31×#17842
C31×#14722
C31×#14712
C31×#12982
#998931