Finite Group Databasebeta

← C31×#9172 #21465931

Subgroups — C31×#9172 #21465931

Order 7936.

464 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3039C211yesmaximal, characteristic
2C31×#2925C211yesmaximal, characteristic
3C31×#1546C211yesmaximal, characteristic, complemented
4C2×C31×#332C211yesmaximal, characteristic, complemented
5C31×#1477C211yesmaximal, characteristic
6C31×#1301C211yesmaximal, characteristic, complemented
7C31×#1292C211yesmaximal, characteristic, complemented
8C2×C31×#137—21no—
9C2×C31×C4⋊Q8C2211yescharacteristic
10C4×#41×C31C2211yescharacteristic
11C31×#462—21no—
12C31×#462C412yes—
13C31×#461—21no—
14C31×#461C412yes—
15C31×#433—22no—
16C31×#432—21no—
17C31×#431—21no—
18C31×#431C2211yescharacteristic
19C31×#423C2211yescharacteristic
20C31×#423C2211yescharacteristic
21C2×C31×C4⋊C8C2211yescharacteristic
22C2×C31×C4⋊C8C2211yescharacteristic
23C31×#419—21no—
24C31×#408—21no—
25C31×#405—21no—
26C31×#332—21no—
27C31×#332—21no—
28C31×#332—22no—
29C31×Q8⋊C8—21no—
30C31×D4⋊C8—21no—
31C2×#41×C31D411yescharacteristic
32C22×Q8×C31—21no—
33C31×#137—42no—
34C31×#137—42no—
35C2×Q16×C31—22no—
36C2×SD16×C31—22no—
37C2×M16×C31—21no—
38C2×M16×C31—21no—
39C22×C8×C31—21no—
40C22×C8×C31—21no—
41C31×C4⋊Q8—21no—
42C31×C4⋊Q8—21no—
43C31×C4⋊Q8—21no—
44C31×C4⋊Q8C2×C412yes—
45C4×Q8×C31C2×C411yescharacteristic
46C4×Q8×C31D411yescharacteristic
47C4×D4×C31—21no—
48C4×D4×C31—21no—
49C4×D4×C31C2×C411yescharacteristic
50C4×D4×C31D411yescharacteristic
51C31×#117—21no—
52C31×#117D411yescharacteristic
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C42×C31—21no—
56C2×C42×C31C2311yescharacteristic
57C31×C4⋊C8—21no—
58C31×C4⋊C8—21no—
59C31×C4⋊C8—21no—
60C31×C4⋊C8—21no—
61C31×C4⋊C8—21no—
62C31×C4⋊C8—21no—
63C31×C4⋊C8C2×C412yes—
64C31×C4⋊C8D411yescharacteristic
65C31×C4⋊C8D411yescharacteristic
66C31×C4⋊C8D412yes—
67C31×#103—22no—
68C31×#103—22no—
69C31×C22⋊C8—22no—
70C31×C22⋊C8—22no—
71C31×C22⋊C8—21no—
72C31×#97—22no—
73C4×C8×C31—22no—
74C4×C8×C31—21no—
75#41×C31—21no—
76#41×C31—21no—
77#41×C31—21no—
78#41×C31—21no—
79C2×Q8×C31—22no—
80C2×Q8×C31—41no—
81C2×Q8×C31—41no—
82C2×Q8×C31—41no—
83C2×Q8×C31—41no—
84C2×Q8×C31—41no—
85C2×Q8×C31C22⋊C411yescharacteristic
86C2×D4×C31—21no—
87C2×D4×C31C22⋊C411yescharacteristic
88C22×C4×C31—21no—
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—41no—
92C22×C4×C31C2×D411yescharacteristic
93C22×C4×C31#4111yescharacteristic
94C22×C4×C31#4111yescharacteristic
95Q16×C31—44no—
96SD16×C31—44no—
97M16×C31—41no—
98M16×C31—41no—
99M16×C31—41no—
100M16×C31—41no—
101C2×C8×C31—22no—
102C2×C8×C31—21no—
103C2×C8×C31—22no—
104C2×C8×C31—21no—
105C2×C8×C31—41no—
106C2×C8×C31—41no—
107C2×C8×C31—41no—
108C2×C8×C31—41no—
109C2×C8×C31—81no—
110C4⋊C4×C31—21no—
111C4⋊C4×C31—21no—
112C4⋊C4×C31—22no—
113C4⋊C4×C31—22no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31—41no—
116C4⋊C4×C31C22⋊C411yescharacteristic
117C4⋊C4×C31C22⋊C411yescharacteristic
118C22⋊C4×C31—21no—
119C22⋊C4×C31—21no—
120C22⋊C4×C31—21no—
121C42×C31—21no—
122C42×C31—21no—
123C42×C31—21no—
124C42×C31C22×C411yescharacteristic
125C42×C31C2×D411yescharacteristic
126C42×C31C2×D411yescharacteristic
127C42×C31C2×D411yescharacteristic
128#9172C3111yesmaximal, characteristic, direct product factor, Sylow
129C23×C31—21no—
130C23×C31#12311yescharacteristic
131Q8×C31—21no—
132Q8×C31—21no—
133Q8×C31—42no—
134Q8×C31—42no—
135Q8×C31—81no—
136Q8×C31—81no—
137D4×C31—21no—
138D4×C31—21no—
139D4×C31—41no—
140D4×C31—41no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—41no—
150C2×C4×C31—41no—
151C2×C4×C31—42no—
152C2×C4×C31—81no—
153C2×C4×C31—81no—
154C2×C4×C31—81no—
155C2×C4×C31C2×C22⋊C411yescharacteristic
156C2×C4×C31C4×D411yescharacteristic
157C2×C4×C31C4×D411yescharacteristic
158C2×C4×C31#12011yescharacteristic
159C2×C4×C31#12111yescharacteristic
160C2×C4×C31#12111yescharacteristic
161C2×C4×C31#13511yescharacteristic
162C2×C4×C31#13511yescharacteristic
163C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
165C8×C31—42no—
166C8×C31—41no—
167C8×C31—41no—
168C8×C31—42no—
169C8×C31—81no—
170C8×C31—81no—
171#3039C2×C3111yescharacteristic
172#2925C2×C3111yescharacteristic
173#1546C2×C3111yescharacteristic, complemented
174C2×#332C2×C3111yescharacteristic, complemented
175#1477C2×C3111yescharacteristic
176#1301C2×C3111yescharacteristic, complemented
177#1292C2×C3111yescharacteristic, complemented
178C22×C31—21no—
179C22×C31—21no—
180C22×C31—41no—
181C22×C31—41no—
182C22×C31—41no—
183C22×C31#38611yescharacteristic
184C22×C31#48211yescharacteristic
185C22×C31#48211yescharacteristic
186C4×C31—21no—
187C4×C31—21no—
188C4×C31—21no—
189C4×C31—41no—
190C4×C31—41no—
191C4×C31—41no—
192C4×C31—82no—
193C4×C31#41611yescharacteristic
194C4×C31#41811yescharacteristic
195C4×C31#44911yescharacteristic
196C4×C31#44911yescharacteristic
197C4×C31#47111yescharacteristic
198C4×C31#47311yescharacteristic
199C2×#137—21no—
200C2×C4⋊Q8C22×C3111yescharacteristic
201C4×#41C22×C3111yescharacteristic
202#462—21no—
203#462C4×C3112yes—
204#461—21no—
205#461C4×C3112yes—
206#433—22no—
207#432—21no—
208#431—21no—
209#431C22×C3111yescharacteristic
210#423C22×C3111yescharacteristic
211#423C22×C3111yescharacteristic
212C2×C4⋊C8C22×C3111yescharacteristic
213C2×C4⋊C8C22×C3111yescharacteristic
214#419—21no—
215#408—21no—
216#405—21no—
217#332—21no—
218#332—21no—
219#332—22no—
220Q8⋊C8—21no—
221D4⋊C8—21no—
222C2×C31—21no—
223C2×C31—21no—
224C2×C31—81no—
225C2×C31#187511yescharacteristic
226C2×C31#190211yescharacteristic
227C2×C31#190811yescharacteristic
228C2×#41D4×C3111yescharacteristic
229C22×Q8—21no—
230#137—42no—
231#137—42no—
232C2×Q16—22no—
233C2×SD16—22no—
234C2×M16—21no—
235C2×M16—21no—
236C22×C8—21no—
237C22×C8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8C2×C4×C3112yes—
242C4×Q8C2×C4×C3111yescharacteristic
243C4×Q8D4×C3111yescharacteristic
244C4×D4—21no—
245C4×D4—21no—
246C4×D4C2×C4×C3111yescharacteristic
247C4×D4D4×C3111yescharacteristic
248#117—21no—
249#117D4×C3111yescharacteristic
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C42—21no—
253C2×C42C23×C3111yescharacteristic
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8C2×C4×C3112yes—
261C4⋊C8D4×C3111yescharacteristic
262C4⋊C8D4×C3111yescharacteristic
263C4⋊C8D4×C3112yes—
264#103—22no—
265#103—22no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—21no—
269#97—22no—
270C4×C8—22no—
271C4×C8—21no—
272C31#917211yescharacteristic, direct product factor, Sylow
273#41—21no—
274#41—21no—
275#41—21no—
276#41—21no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8C22⋊C4×C3111yescharacteristic
284C2×D4—21no—
285C2×D4C22⋊C4×C3111yescharacteristic
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4C2×D4×C3111yescharacteristic
291C22×C4#41×C3111yescharacteristic
292C22×C4#41×C3111yescharacteristic
293Q16—44no—
294SD16—44no—
295M16—41no—
296M16—41no—
297M16—41no—
298M16—41no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—21no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—81no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4C22⋊C4×C3111yescharacteristic
315C4⋊C4C22⋊C4×C3111yescharacteristic
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42C22×C4×C3111yescharacteristic
323C42C2×D4×C3111yescharacteristic
324C42C2×D4×C3111yescharacteristic
325C42C2×D4×C3111yescharacteristic
326C23—21no—
327C23C31×#12311yescharacteristic
328Q8—21no—
329Q8—21no—
330Q8—42no—
331Q8—42no—
332Q8—81no—
333Q8—81no—
334D4—21no—
335D4—21no—
336D4—41no—
337D4—41no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4C2×C22⋊C4×C3111yescharacteristic
353C2×C4C4×D4×C3111yescharacteristic
354C2×C4C4×D4×C3111yescharacteristic
355C2×C4C31×#12011yescharacteristic
356C2×C4C31×#12111yescharacteristic
357C2×C4C31×#12111yescharacteristic
358C2×C4C31×#13511yescharacteristic
359C2×C4C31×#13511yescharacteristic
360C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
361C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
362C8—42no—
363C8—41no—
364C8—41no—
365C8—42no—
366C8—81no—
367C8—81no—
368C22—21no—
369C22—21no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22C31×#38611yescharacteristic
374C22C31×#48211yescharacteristic
375C22C31×#48211yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—82no—
383C4C31×#41611yescharacteristic
384C4C31×#41811yescharacteristic
385C4C31×#44911yescharacteristic
386C4C31×#44911yescharacteristic
387C4C31×#47111yescharacteristic
388C4C31×#47311yescharacteristic
389C2—21no—
390C2—21no—
391C2—81no—
392C2C31×#187511yescharacteristic
393C2C31×#190211yescharacteristic
394C2C31×#190811yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC2×#41×C31D4
LCSG ▷ 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. C31×#1546C2semidirect
4. C2×C31×#332C2semidirect
6. C31×#1301C2semidirect
7. C31×#1292C2semidirect
128. #9172C31direct (N × H)
173. #1546C2×C31semidirect
174. C2×#332C2×C31semidirect
176. #1301C2×C31semidirect
177. #1292C2×C31semidirect
272. C31#9172direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#30392
C31×#29252
C31×#15462
C2×C31×#3322
C31×#14772
C31×#13012
C31×#12922
#917231