Finite Group Databasebeta

← C31×#10181 #21466940

Subgroups — C31×#10181 #21466940

Order 7936.

530 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3206C211yesmaximal, characteristic
2C2×C31×#477C211yesmaximal, characteristic, complemented
3C31×#3037C211yesmaximal, characteristic
4C31×#1549C211yesmaximal, characteristic, complemented
5C31×#1480C211yesmaximal, characteristic
6C31×#1461C211yesmaximal, characteristic, complemented
7C31×#1301C211yesmaximal, characteristic, complemented
8C2×C31×#137—21no—
9C31×#532C2211yescharacteristic
10C2×C31×C4⋊Q8C2211yescharacteristic
11C2×C4×Q8×C31C2211yescharacteristic
12C31×#492—21no—
13C31×#487—21no—
14C31×#477—22no—
15C31×#477—22no—
16C31×#464—22no—
17C31×#462—21no—
18C31×#461—22no—
19C31×#460—21no—
20C31×#448—22no—
21C31×#431C2211yescharacteristic
22C31×#431C2211yescharacteristic
23C2×C31×#107—21no—
24C31×#423C2211yescharacteristic
25C2×C31×C4⋊C8C2211yescharacteristic
26C31×#417—21no—
27C2×C31×#103—21no—
28C31×#332—22no—
29C31×D4⋊C8—22no—
30C31×#324—22no—
31C2×#41×C31—21no—
32C22×Q8×C31—21no—
33C22×Q8×C31D411yescharacteristic
34C31×#137—42no—
35C31×#137—42no—
36C2×Q16×C31—22no—
37C2×SD16×C31—22no—
38C2×SD16×C31—24no—
39C2×M16×C31—21no—
40C22×C8×C31—21no—
41C31×C4⋊Q8—21no—
42C31×C4⋊Q8—21no—
43C31×C4⋊Q8—21no—
44C31×C4⋊Q8D412yes—
45C31×C4⋊Q8D411yescharacteristic
46C31×#127D411yescharacteristic
47C31×#124—21no—
48C31×#121—21no—
49C31×#121—21no—
50C4×Q8×C31—21no—
51C4×Q8×C31—21no—
52C4×Q8×C31—21no—
53C4×Q8×C31—21no—
54C4×D4×C31—21no—
55C4×D4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31D411yescharacteristic
60C2×C42×C31—21no—
61C2×C42×C31C2311yescharacteristic
62C31×#107—24no—
63C31×C4⋊C8—21no—
64C31×C4⋊C8—21no—
65C31×C4⋊C8—21no—
66C31×C4⋊C8—21no—
67C31×C4⋊C8D412yes—
68C31×C4⋊C8D412yes—
69C31×#103—24no—
70C31×#103—22no—
71C31×#103—24no—
72C31×#102—22no—
73C31×C22⋊C8—21no—
74C31×C22⋊C8D412yes—
75C4×C8×C31—21no—
76C4×C8×C31—21no—
77#41×C31—41no—
78#41×C31—41no—
79#41×C31—41no—
80#41×C31—41no—
81C2×Q8×C31—21no—
82C2×Q8×C31—21no—
83C2×Q8×C31—21no—
84C2×Q8×C31—21no—
85C2×Q8×C31—22no—
86C2×Q8×C31—21no—
87C2×Q8×C31—21no—
88C2×Q8×C31—41no—
89C2×Q8×C31—82no—
90C2×Q8×C31SD1612yescomplemented
91C2×D4×C31—21no—
92C2×D4×C31—21no—
93C2×D4×C31—21no—
94C2×D4×C31—41no—
95C22×C4×C31—21no—
96C22×C4×C31—21no—
97C22×C4×C31—41no—
98C22×C4×C31—41no—
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31C2×D411yescharacteristic
101C22×C4×C31#4111yescharacteristic
102Q16×C31—44no—
103SD16×C31—44no—
104SD16×C31—84no—
105M16×C31—41no—
106M16×C31—41no—
107C2×C8×C31—22no—
108C2×C8×C31—22no—
109C2×C8×C31—22no—
110C2×C8×C31—41no—
111C2×C8×C31—41no—
112C2×C8×C31—41no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—22no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—22no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—41no—
120C4⋊C4×C31—41no—
121C4⋊C4×C31—41no—
122C4⋊C4×C31—41no—
123C22⋊C4×C31—41no—
124C22⋊C4×C31—41no—
125C42×C31—41no—
126C42×C31—41no—
127C42×C31C2×D411yescharacteristic
128C42×C31C2×D411yescharacteristic
129C42×C31C2×D411yescharacteristic
130C42×C31C2×D411yescharacteristic
131#10181C3111yesmaximal, characteristic, direct product factor, Sylow
132C23×C31—41no—
133C23×C31#12111yescharacteristic
134Q8×C31—22no—
135Q8×C31—22no—
136Q8×C31—44no—
137Q8×C31—41no—
138Q8×C31—41no—
139Q8×C31—41no—
140Q8×C31—41no—
141Q8×C31—82no—
142D4×C31—42no—
143D4×C31—41no—
144D4×C31—41no—
145D4×C31—42no—
146D4×C31—81no—
147D4×C31—81no—
148C2×C4×C31—22no—
149C2×C4×C31—22no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—42no—
156C2×C4×C31—82no—
157C2×C4×C31—81no—
158C2×C4×C31—81no—
159C2×C4×C31—81no—
160C2×C4×C31#12011yescharacteristic
161C2×C4×C31#12011yescharacteristic
162C2×C4×C31#12011yescharacteristic
163C2×C4×C31#12111yescharacteristic
164C2×C4×C31#12111yescharacteristic
165C2×C4×C31#12711yescharacteristic
166C2×C4×C31C2×SD1611yescharacteristic
167C2×C4×C31#13511yescharacteristic
168C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C31#13711yescharacteristic
170C8×C31—42no—
171C8×C31—42no—
172C8×C31—42no—
173C8×C31—81no—
174#3206C2×C3111yescharacteristic
175C2×#477C2×C3111yescharacteristic, complemented
176#3037C2×C3111yescharacteristic
177#1549C2×C3111yescharacteristic, complemented
178#1480C2×C3111yescharacteristic
179#1461C2×C3111yescharacteristic, complemented
180#1301C2×C3111yescharacteristic, complemented
181C22×C31—21no—
182C22×C31—21no—
183C22×C31—81no—
184C22×C31—81no—
185C22×C31—81no—
186C22×C31#39211yescharacteristic
187C22×C31#46511yescharacteristic
188C22×C31#46811yescharacteristic
189C4×C31—21no—
190C4×C31—21no—
191C4×C31—21no—
192C4×C31—42no—
193C4×C31—81no—
194C4×C31—81no—
195C4×C31—82no—
196C4×C31#44811yescharacteristic
197C4×C31#44911yescharacteristic
198C4×C31#45011yescharacteristic
199C4×C31#45211yescharacteristic
200C4×C31#47312yes—
201C2×#137—21no—
202#532C22×C3111yescharacteristic
203C2×C4⋊Q8C22×C3111yescharacteristic
204C2×C4×Q8C22×C3111yescharacteristic
205#492—21no—
206#487—21no—
207#477—22no—
208#477—22no—
209#464—22no—
210#462—21no—
211#461—22no—
212#460—21no—
213#448—22no—
214#431C22×C3111yescharacteristic
215#431C22×C3111yescharacteristic
216C2×#107—21no—
217#423C22×C3111yescharacteristic
218C2×C4⋊C8C22×C3111yescharacteristic
219#417—21no—
220C2×#103—21no—
221#332—22no—
222D4⋊C8—22no—
223#324—22no—
224C2×C31—21no—
225C2×C31—21no—
226C2×C31—161no—
227C2×C31#200011yescharacteristic
228C2×C31#200811yescharacteristic
229C2×C31#202011yescharacteristic
230C2×#41—21no—
231C22×Q8—21no—
232C22×Q8D4×C3111yescharacteristic
233#137—42no—
234#137—42no—
235C2×Q16—22no—
236C2×SD16—22no—
237C2×SD16—24no—
238C2×M16—21no—
239C22×C8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8D4×C3112yes—
244C4⋊Q8D4×C3111yescharacteristic
245#127D4×C3111yescharacteristic
246#124—21no—
247#121—21no—
248#121—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×D4—21no—
254C4×D4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4D4×C3111yescharacteristic
259C2×C42—21no—
260C2×C42C23×C3111yescharacteristic
261#107—24no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8D4×C3112yes—
267C4⋊C8D4×C3112yes—
268#103—24no—
269#103—22no—
270#103—24no—
271#102—22no—
272C22⋊C8—21no—
273C22⋊C8D4×C3112yes—
274C4×C8—21no—
275C4×C8—21no—
276C31#1018111yescharacteristic, direct product factor, Sylow
277#41—41no—
278#41—41no—
279#41—41no—
280#41—41no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—41no—
289C2×Q8—82no—
290C2×Q8SD16×C3112yescomplemented
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4C2×D4×C3111yescharacteristic
300C22×C4C2×D4×C3111yescharacteristic
301C22×C4#41×C3111yescharacteristic
302Q16—44no—
303SD16—44no—
304SD16—84no—
305M16—41no—
306M16—41no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—21no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C42—41no—
326C42—41no—
327C42C2×D4×C3111yescharacteristic
328C42C2×D4×C3111yescharacteristic
329C42C2×D4×C3111yescharacteristic
330C42C2×D4×C3111yescharacteristic
331C23—41no—
332C23C31×#12111yescharacteristic
333Q8—22no—
334Q8—22no—
335Q8—44no—
336Q8—41no—
337Q8—41no—
338Q8—41no—
339Q8—41no—
340Q8—82no—
341D4—42no—
342D4—41no—
343D4—41no—
344D4—42no—
345D4—81no—
346D4—81no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4C31×#12011yescharacteristic
360C2×C4C31×#12011yescharacteristic
361C2×C4C31×#12011yescharacteristic
362C2×C4C31×#12111yescharacteristic
363C2×C4C31×#12111yescharacteristic
364C2×C4C31×#12711yescharacteristic
365C2×C4C2×SD16×C3111yescharacteristic
366C2×C4C31×#13511yescharacteristic
367C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
368C2×C4C31×#13711yescharacteristic
369C8—42no—
370C8—42no—
371C8—42no—
372C8—81no—
373C22—21no—
374C22—21no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22C31×#39211yescharacteristic
379C22C31×#46511yescharacteristic
380C22C31×#46811yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—42no—
385C4—81no—
386C4—82no—
387C4—81no—
388C4C31×#44811yescharacteristic
389C4C31×#44911yescharacteristic
390C4C31×#45011yescharacteristic
391C4C31×#45211yescharacteristic
392C4C31×#47312yes—
393C2—21no—
394C2—21no—
395C2—161no—
396C2C31×#200011yescharacteristic
397C2C31×#200811yescharacteristic
398C2C31×#202011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#532C22
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
2. C2×C31×#477C2semidirect
4. C31×#1549C2semidirect
6. C31×#1461C2semidirect
7. C31×#1301C2semidirect
90. C2×Q8×C31SD16semidirect
131. #10181C31direct (N × H)
175. C2×#477C2×C31semidirect
177. #1549C2×C31semidirect
179. #1461C2×C31semidirect
180. #1301C2×C31semidirect
276. C31#10181direct (N × H)
290. C2×Q8SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#32062
C2×C31×#4772
C31×#30372
C31×#15492
C31×#14802
C31×#14612
C31×#13012
#1018131