Finite Group Databasebeta

← C31×#10185 #21466944

Subgroups — C31×#10185 #21466944

Order 7936.

550 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3226C211yesmaximal, characteristic
2C31×#3037C211yesmaximal, characteristic
3C2×C31×#462C211yesmaximal, characteristic, complemented
4C31×#1546C211yesmaximal, characteristic, complemented
5C31×#1480C211yesmaximal, characteristic
6C31×#1461C211yesmaximal, characteristic, complemented
7C31×#1301C211yesmaximal, characteristic, complemented
8C2×C31×#137—21no—
9C22×Q16×C31—21no—
10C31×#532C2211yescharacteristic
11C2×C31×C4⋊Q8C2211yescharacteristic
12C2×C4×Q8×C31C2211yescharacteristic
13C31×#500—21no—
14C31×#486—21no—
15C31×#480—22no—
16C31×#474—22no—
17C31×#464—22no—
18C31×#462—22no—
19C31×#462—21no—
20C31×#462—22no—
21C31×#460—21no—
22C31×#431C2211yescharacteristic
23C31×#431C2211yescharacteristic
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—24no—
37C2×Q16×C31—22no—
38C2×Q16×C31—42no—
39C2×Q16×C31—42no—
40C2×SD16×C31—22no—
41C2×M16×C31—21no—
42C22×C8×C31—21no—
43C31×C4⋊Q8—21no—
44C31×C4⋊Q8—21no—
45C31×C4⋊Q8—21no—
46C31×C4⋊Q8D412yes—
47C31×C4⋊Q8D411yescharacteristic
48C31×#127D411yescharacteristic
49C31×#124—21no—
50C31×#121—21no—
51C31×#121—21no—
52C4×Q8×C31—21no—
53C4×Q8×C31—21no—
54C4×Q8×C31—21no—
55C4×Q8×C31—21no—
56C4×D4×C31—21no—
57C4×D4×C31—21no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31—21no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31D411yescharacteristic
62C2×C42×C31—21no—
63C2×C42×C31C2311yescharacteristic
64C31×#107—24no—
65C31×C4⋊C8—21no—
66C31×C4⋊C8—21no—
67C31×C4⋊C8—21no—
68C31×C4⋊C8—21no—
69C31×C4⋊C8D412yes—
70C31×C4⋊C8D412yes—
71C31×#103—24no—
72C31×#103—22no—
73C31×#102—22no—
74C31×#102—24no—
75C31×C22⋊C8—21no—
76C31×C22⋊C8D412yes—
77C4×C8×C31—21no—
78C4×C8×C31—21no—
79#41×C31—41no—
80#41×C31—41no—
81#41×C31—41no—
82#41×C31—41no—
83C2×Q8×C31—21no—
84C2×Q8×C31—21no—
85C2×Q8×C31—21no—
86C2×Q8×C31—21no—
87C2×Q8×C31—22no—
88C2×Q8×C31—21no—
89C2×Q8×C31—21no—
90C2×Q8×C31—41no—
91C2×Q8×C31—41no—
92C2×Q8×C31—41no—
93C2×Q8×C31—41no—
94C2×Q8×C31—41no—
95C2×Q8×C31D812yes—
96C2×D4×C31—21no—
97C2×D4×C31—21no—
98C2×D4×C31—21no—
99C2×D4×C31—41no—
100C22×C4×C31—21no—
101C22×C4×C31—21no—
102C22×C4×C31—41no—
103C22×C4×C31—41no—
104C22×C4×C31C2×D411yescharacteristic
105C22×C4×C31C2×D411yescharacteristic
106C22×C4×C31#4111yescharacteristic
107Q16×C31—48no—
108Q16×C31—44no—
109SD16×C31—44no—
110M16×C31—41no—
111M16×C31—41no—
112C2×C8×C31—22no—
113C2×C8×C31—22no—
114C2×C8×C31—22no—
115C2×C8×C31—41no—
116C2×C8×C31—41no—
117C2×C8×C31—41no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—22no—
120C4⋊C4×C31—21no—
121C4⋊C4×C31—21no—
122C4⋊C4×C31—22no—
123C4⋊C4×C31—21no—
124C4⋊C4×C31—41no—
125C4⋊C4×C31—41no—
126C4⋊C4×C31—41no—
127C4⋊C4×C31—41no—
128C22⋊C4×C31—41no—
129C22⋊C4×C31—41no—
130C42×C31—41no—
131C42×C31—41no—
132C42×C31C2×D411yescharacteristic
133C42×C31C2×D411yescharacteristic
134C42×C31C2×D411yescharacteristic
135C42×C31C2×D411yescharacteristic
136#10185C3111yesmaximal, characteristic, direct product factor, Sylow
137C23×C31—41no—
138C23×C31#12111yescharacteristic
139Q8×C31—22no—
140Q8×C31—22no—
141Q8×C31—41no—
142Q8×C31—41no—
143Q8×C31—42no—
144Q8×C31—41no—
145Q8×C31—42no—
146Q8×C31—41no—
147Q8×C31—81no—
148Q8×C31—81no—
149D4×C31—42no—
150D4×C31—41no—
151D4×C31—41no—
152D4×C31—42no—
153D4×C31—81no—
154D4×C31—81no—
155C2×C4×C31—22no—
156C2×C4×C31—22no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—42no—
163C2×C4×C31—81no—
164C2×C4×C31—81no—
165C2×C4×C31—81no—
166C2×C4×C31—81no—
167C2×C4×C31—81no—
168C2×C4×C31#12011yescharacteristic
169C2×C4×C31#12011yescharacteristic
170C2×C4×C31#12011yescharacteristic
171C2×C4×C31#12111yescharacteristic
172C2×C4×C31#12111yescharacteristic
173C2×C4×C31#12711yescharacteristic
174C2×C4×C31C2×D811yescharacteristic
175C2×C4×C31#13511yescharacteristic
176C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
178C8×C31—42no—
179C8×C31—42no—
180C8×C31—42no—
181C8×C31—81no—
182#3226C2×C3111yescharacteristic
183#3037C2×C3111yescharacteristic
184C2×#462C2×C3111yescharacteristic, complemented
185#1546C2×C3111yescharacteristic, complemented
186#1480C2×C3111yescharacteristic
187#1461C2×C3111yescharacteristic, complemented
188#1301C2×C3111yescharacteristic, complemented
189C22×C31—21no—
190C22×C31—21no—
191C22×C31—81no—
192C22×C31—81no—
193C22×C31—81no—
194C22×C31#39211yescharacteristic
195C22×C31#46611yescharacteristic
196C22×C31#46911yescharacteristic
197C4×C31—21no—
198C4×C31—21no—
199C4×C31—21no—
200C4×C31—42no—
201C4×C31—81no—
202C4×C31—82no—
203C4×C31—81no—
204C4×C31#44711yescharacteristic
205C4×C31#44711yescharacteristic
206C4×C31#44911yescharacteristic
207C4×C31#44911yescharacteristic
208C4×C31#47312yes—
209C2×#137—21no—
210C22×Q16—21no—
211#532C22×C3111yescharacteristic
212C2×C4⋊Q8C22×C3111yescharacteristic
213C2×C4×Q8C22×C3111yescharacteristic
214#500—21no—
215#486—21no—
216#480—22no—
217#474—22no—
218#464—22no—
219#462—22no—
220#462—21no—
221#462—22no—
222#460—21no—
223#431C22×C3111yescharacteristic
224#431C22×C3111yescharacteristic
225#423C22×C3111yescharacteristic
226C2×C4⋊C8C22×C3111yescharacteristic
227#417—21no—
228C2×#103—21no—
229#332—22no—
230D4⋊C8—22no—
231#324—22no—
232C2×C31—21no—
233C2×C31—21no—
234C2×C31—161no—
235C2×C31#199911yescharacteristic
236C2×C31#200911yescharacteristic
237C2×C31#202011yescharacteristic
238C2×#41—21no—
239C22×Q8—21no—
240C22×Q8D4×C3111yescharacteristic
241#137—42no—
242#137—42no—
243C2×Q16—24no—
244C2×Q16—22no—
245C2×Q16—42no—
246C2×Q16—42no—
247C2×SD16—22no—
248C2×M16—21no—
249C22×C8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8D4×C3112yes—
254C4⋊Q8D4×C3111yescharacteristic
255#127D4×C3111yescharacteristic
256#124—21no—
257#121—21no—
258#121—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×D4—21no—
264C4×D4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4D4×C3111yescharacteristic
269C2×C42—21no—
270C2×C42C23×C3111yescharacteristic
271#107—24no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8D4×C3112yes—
277C4⋊C8D4×C3112yes—
278#103—24no—
279#103—22no—
280#102—22no—
281#102—24no—
282C22⋊C8—21no—
283C22⋊C8D4×C3112yes—
284C4×C8—21no—
285C4×C8—21no—
286C31#1018511yescharacteristic, direct product factor, Sylow
287#41—41no—
288#41—41no—
289#41—41no—
290#41—41no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—22no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8D8×C3112yes—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—41no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4C2×D4×C3111yescharacteristic
313C22×C4C2×D4×C3111yescharacteristic
314C22×C4#41×C3111yescharacteristic
315Q16—48no—
316Q16—44no—
317SD16—44no—
318M16—41no—
319M16—41no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42—41no—
339C42—41no—
340C42C2×D4×C3111yescharacteristic
341C42C2×D4×C3111yescharacteristic
342C42C2×D4×C3111yescharacteristic
343C42C2×D4×C3111yescharacteristic
344C23—41no—
345C23C31×#12111yescharacteristic
346Q8—22no—
347Q8—22no—
348Q8—41no—
349Q8—41no—
350Q8—42no—
351Q8—41no—
352Q8—42no—
353Q8—41no—
354Q8—81no—
355Q8—81no—
356D4—42no—
357D4—41no—
358D4—41no—
359D4—42no—
360D4—81no—
361D4—81no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4C31×#12011yescharacteristic
376C2×C4C31×#12011yescharacteristic
377C2×C4C31×#12011yescharacteristic
378C2×C4C31×#12111yescharacteristic
379C2×C4C31×#12111yescharacteristic
380C2×C4C31×#12711yescharacteristic
381C2×C4C2×D8×C3111yescharacteristic
382C2×C4C31×#13511yescharacteristic
383C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
384C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
385C8—42no—
386C8—42no—
387C8—42no—
388C8—81no—
389C22—21no—
390C22—21no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22C31×#39211yescharacteristic
395C22C31×#46611yescharacteristic
396C22C31×#46911yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—81no—
402C4—82no—
403C4—81no—
404C4C31×#44711yescharacteristic
405C4C31×#44711yescharacteristic
406C4C31×#44911yescharacteristic
407C4C31×#44911yescharacteristic
408C4C31×#47312yes—
409C2—21no—
410C2—21no—
411C2—161no—
412C2C31×#199911yescharacteristic
413C2C31×#200911yescharacteristic
414C2C31×#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
3. C2×C31×#462C2semidirect
4. C31×#1546C2semidirect
6. C31×#1461C2semidirect
7. C31×#1301C2semidirect
136. #10185C31direct (N × H)
184. C2×#462C2×C31semidirect
185. #1546C2×C31semidirect
187. #1461C2×C31semidirect
188. #1301C2×C31semidirect
286. C31#10185direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32262
C31×#30372
C2×C31×#4622
C31×#15462
C31×#14802
C31×#14612
C31×#13012
#1018531