Finite Group Databasebeta

← C31×#18303 #21475062

Subgroups — C31×#18303 #21475062

Order 7936.

912 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3458C211yesmaximal, characteristic
2C31×#3209C212yesmaximal, complemented
3C31×#2628C211yesmaximal, characteristic
4C31×#1605C214yesmaximal, complemented
5C31×#1602C214yesmaximal
6C31×#1518C212yesmaximal, complemented
7C31×#1458C211yesmaximal, characteristic
8C31×#544—22no—
9C31×#544—22no—
10C31×#544—21no—
11C31×#544C2212yes—
12C31×#543—21no—
13C31×#543C2212yes—
14C31×#533—24no—
15C31×#533—21no—
16C31×#533C2212yes—
17C2×C31×C4⋊Q8C2211yescharacteristic
18C2×C31×#117—21no—
19C2×C31×#117C2211yescharacteristic
20C31×#494—22no—
21C31×#487—22no—
22C31×#462—24no—
23C31×#451C2218yes—
24C31×#431C2212yes—
25C31×#400—21no—
26C31×#399—21no—
27C31×#395C2212yes—
28C31×#393C2212yes—
29C31×#389—21no—
30C31×#382C2211yescharacteristic
31C31×#380—21no—
32C31×#328C2218yes—
33C31×#324C2212yes—
34C31×#323C2212yes—
35C22×Q8×C31C2312yes—
36C23×C4×C31D411yescharacteristic
37C2×Q16×C31—28no—
38C31×C4⋊Q8—24no—
39C31×C4⋊Q8—21no—
40C31×C4⋊Q8—21no—
41C31×C4⋊Q8—24no—
42C31×C4⋊Q8—22no—
43C31×C4⋊Q8—22no—
44C31×C4⋊Q8D412yes—
45C31×C4⋊Q8D412yes—
46C31×#125—24no—
47C31×#125—22no—
48C31×#125—24no—
49C31×#125—22no—
50C31×C22⋊Q8—24no—
51C31×C22⋊Q8—24no—
52C31×C22⋊Q8—24no—
53C31×C22⋊Q8—22no—
54C31×C22⋊Q8—22no—
55C31×C22⋊Q8—22no—
56C31×C22⋊Q8—24no—
57C31×C22⋊Q8C2314yes—
58C4×Q8×C31—22no—
59C4×Q8×C31—22no—
60C4×Q8×C31—22no—
61C4×Q8×C31D414yes—
62C31×#117—22no—
63C31×#117—22no—
64C31×#117—22no—
65C31×#117—22no—
66C31×#117—22no—
67C31×#117—22no—
68C2×C4⋊C4×C31—21no—
69C2×C4⋊C4×C31—21no—
70C2×C4⋊C4×C31—21no—
71C2×C4⋊C4×C31—21no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31—22no—
74C2×C4⋊C4×C31D411yescharacteristic
75C2×C4⋊C4×C31C2311yescharacteristic
76C2×C22⋊C4×C31—22no—
77C2×C22⋊C4×C31—22no—
78C2×C22⋊C4×C31D411yescharacteristic
79C2×C22⋊C4×C31C2311yescharacteristic
80C2×C42×C31—22no—
81C2×C42×C31D411yescharacteristic
82C2×C42×C31C2311yescharacteristic
83C31×C4⋊C8—22no—
84C31×#103—28no—
85C31×C22⋊C8C2314yes—
86C4×C8×C31—22no—
87C31×#95—22no—
88C31×#95—21no—
89C31×#95—22no—
90C31×#95C2312yes—
91C24×C31C2×D411yescharacteristic
92C2×Q8×C31—22no—
93C2×Q8×C31—22no—
94C2×Q8×C31—22no—
95C2×Q8×C31—24no—
96C2×Q8×C31—82no—
97C2×Q8×C31C2×D414yes—
98C22×C4×C31—21no—
99C22×C4×C31—21no—
100C22×C4×C31—22no—
101C22×C4×C31—22no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—22no—
105C22×C4×C31Q1614yescomplemented
106C22×C4×C31C2×D411yescharacteristic
107C22×C4×C31C2×D411yescharacteristic
108C22×C4×C31C2×D411yescharacteristic
109C22×C4×C31C2×D411yescharacteristic
110C22×C4×C31C2×D411yescharacteristic
111C22×C4×C31C2411yescharacteristic
112Q16×C31—88no—
113C2×C8×C31—24no—
114C4⋊C4×C31—22no—
115C4⋊C4×C31—22no—
116C4⋊C4×C31—22no—
117C4⋊C4×C31—24no—
118C4⋊C4×C31—24no—
119C4⋊C4×C31—22no—
120C4⋊C4×C31—22no—
121C4⋊C4×C31—22no—
122C4⋊C4×C31—22no—
123C4⋊C4×C31—22no—
124C4⋊C4×C31—24no—
125C4⋊C4×C31—22no—
126C4⋊C4×C31—24no—
127C4⋊C4×C31—42no—
128C4⋊C4×C31—41no—
129C4⋊C4×C31—41no—
130C4⋊C4×C31—42no—
131C4⋊C4×C31—82no—
132C4⋊C4×C31C2×D414yes—
133C22⋊C4×C31—24no—
134C22⋊C4×C31—24no—
135C22⋊C4×C31—22no—
136C22⋊C4×C31—22no—
137C22⋊C4×C31—82no—
138C42×C31—24no—
139C42×C31—24no—
140C42×C31—21no—
141C42×C31—21no—
142C42×C31C2×D412yes—
143C42×C31C2×D412yes—
144#18303C3111yesmaximal, characteristic, direct product factor, Sylow
145C23×C31—21no—
146C23×C31—21no—
147C23×C31—41no—
148C23×C31—41no—
149C23×C31C2×Q1612yes—
150C23×C31C22×D411yescharacteristic
151Q8×C31—44no—
152Q8×C31—48no—
153Q8×C31—82no—
154Q8×C31—82no—
155Q8×C31—82no—
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—24no—
159C2×C4×C31—24no—
160C2×C4×C31—24no—
161C2×C4×C31—21no—
162C2×C4×C31—21no—
163C2×C4×C31—24no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—42no—
169C2×C4×C31—42no—
170C2×C4×C31—42no—
171C2×C4×C31—42no—
172C2×C4×C31—42no—
173C2×C4×C31—42no—
174C2×C4×C31—82no—
175C2×C4×C31—82no—
176C2×C4×C31#12012yes—
177C2×C4×C31#12011yescharacteristic
178C2×C4×C31#12011yescharacteristic
179C2×C4×C31C2×Q1612yes—
180C2×C4×C31C2×Q1612yes—
181C2×C4×C31#13712yes—
182C2×C4×C31C22×D412yes—
183C8×C31—44no—
184#3458C2×C3111yescharacteristic
185#3209C2×C3112yescomplemented
186#2628C2×C3111yescharacteristic
187#1605C2×C3114yescomplemented
188#1602C2×C3114yes—
189#1518C2×C3112yescomplemented
190#1458C2×C3111yescharacteristic
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—22no—
194C22×C31—42no—
195C22×C31—41no—
196C22×C31—41no—
197C22×C31—81no—
198C22×C31C2×#12011yescharacteristic
199C22×C31C22×Q1611yescharacteristic
200C22×C31C2×#13711yescharacteristic
201C4×C31—22no—
202C4×C31—22no—
203C4×C31—22no—
204C4×C31—41no—
205C4×C31—44no—
206C4×C31—41no—
207C4×C31—44no—
208C4×C31—82no—
209C4×C31—82no—
210C4×C31#45112yes—
211C4×C31#45112yes—
212#544—22no—
213#544—22no—
214#544—21no—
215#544C22×C3112yes—
216#543—21no—
217#543C22×C3112yes—
218#533—24no—
219#533—21no—
220#533C22×C3112yes—
221C2×C4⋊Q8C22×C3111yescharacteristic
222C2×#117—21no—
223C2×#117C22×C3111yescharacteristic
224#494—22no—
225#487—22no—
226#462—24no—
227#451C22×C3118yes—
228#431C22×C3112yes—
229#400—21no—
230#399—21no—
231#395C22×C3112yes—
232#393C22×C3112yes—
233#389—21no—
234#382C22×C3111yescharacteristic
235#380—21no—
236#328C22×C3118yes—
237#324C22×C3112yes—
238#323C22×C3112yes—
239C2×C31—21no—
240C2×C31—21no—
241C2×C31—42no—
242C2×C31C2×#45111yescharacteristic
243C2×C31#302211yescharacteristic
244C2×C31#302711yescharacteristic
245C22×Q8C23×C3112yes—
246C23×C4D4×C3111yescharacteristic
247C2×Q16—28no—
248C4⋊Q8—24no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8D4×C3112yes—
255C4⋊Q8D4×C3112yes—
256#125—24no—
257#125—22no—
258#125—24no—
259#125—22no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266C22⋊Q8—24no—
267C22⋊Q8C23×C3114yes—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8D4×C3114yes—
272#117—22no—
273#117—22no—
274#117—22no—
275#117—22no—
276#117—22no—
277#117—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4D4×C3111yescharacteristic
285C2×C4⋊C4C23×C3111yescharacteristic
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4D4×C3111yescharacteristic
289C2×C22⋊C4C23×C3111yescharacteristic
290C2×C42—22no—
291C2×C42D4×C3111yescharacteristic
292C2×C42C23×C3111yescharacteristic
293C4⋊C8—22no—
294#103—28no—
295C22⋊C8C23×C3114yes—
296C4×C8—22no—
297#95—22no—
298#95—21no—
299#95—22no—
300#95C23×C3112yes—
301C31#1830311yescharacteristic, direct product factor, Sylow
302C24C2×D4×C3111yescharacteristic
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—82no—
308C2×Q8C2×D4×C3114yes—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4Q16×C3114yescomplemented
317C22×C4C2×D4×C3111yescharacteristic
318C22×C4C2×D4×C3111yescharacteristic
319C22×C4C2×D4×C3111yescharacteristic
320C22×C4C2×D4×C3111yescharacteristic
321C22×C4C2×D4×C3111yescharacteristic
322C22×C4C24×C3111yescharacteristic
323Q16—88no—
324C2×C8—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—82no—
343C4⋊C4C2×D4×C3114yes—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—82no—
349C42—24no—
350C42—24no—
351C42—21no—
352C42—21no—
353C42C2×D4×C3112yes—
354C42C2×D4×C3112yes—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—41no—
359C23C2×Q16×C3112yes—
360C23C22×D4×C3111yescharacteristic
361Q8—44no—
362Q8—48no—
363Q8—82no—
364Q8—82no—
365Q8—82no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4C31×#12012yes—
387C2×C4C31×#12011yescharacteristic
388C2×C4C31×#12011yescharacteristic
389C2×C4C2×Q16×C3112yes—
390C2×C4C2×Q16×C3112yes—
391C2×C4C31×#13712yes—
392C2×C4C22×D4×C3112yes—
393C8—44no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22C2×C31×#12011yescharacteristic
402C22C22×Q16×C3111yescharacteristic
403C22C2×C31×#13711yescharacteristic
404C4—22no—
405C4—22no—
406C4—22no—
407C4—41no—
408C4—44no—
409C4—41no—
410C4—44no—
411C4—82no—
412C4—82no—
413C4C31×#45112yes—
414C4C31×#45112yes—
415C2—21no—
416C2—21no—
417C2—42no—
418C2C2×C31×#45111yescharacteristic
419C2C31×#302211yescharacteristic
420C2C31×#302711yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3209C2semidirect
4. C31×#1605C2semidirect
6. C31×#1518C2semidirect
105. C22×C4×C31Q16semidirect
144. #18303C31direct (N × H)
185. #3209C2×C31semidirect
187. #1605C2×C31semidirect
189. #1518C2×C31semidirect
301. C31#18303direct (N × H)
316. C22×C4Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34582
C31×#32092
C31×#26282
C31×#16052
C31×#16022
C31×#15182
C31×#14582
#1830331