Finite Group Databasebeta

← C31×#18326 #21475085

Subgroups — C31×#18326 #21475085

Order 7936.

976 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#544C211yesmaximal, characteristic, complemented
2C31×#3193C212yesmaximal
3C31×#2591C211yesmaximal, characteristic
4C31×#1617C214yesmaximal, complemented
5C31×#1599C214yesmaximal, complemented
6C31×#1508C212yesmaximal, complemented
7C31×#1463C211yesmaximal, characteristic, complemented
8C31×#544—24no—
9C31×#544—22no—
10C31×#544C2214yes—
11C2×C31×C4⋊Q8—22no—
12C2×C31×#125—22no—
13C2×C31×C22⋊Q8—21no—
14C2×C31×C22⋊Q8C2212yes—
15C2×C31×#121C2211yescharacteristic
16C2×C31×#117—21no—
17C22×C4⋊C4×C31C2211yescharacteristic
18C31×#484C2214yes—
19C31×#480—22no—
20C31×#468—24no—
21C31×#450C2214yes—
22C31×#448—22no—
23C31×#407C2212yes—
24C31×#400C2212yes—
25C31×#394C2212yes—
26C31×#388—22no—
27C31×#386—21no—
28C31×#386—21no—
29C4×C22⋊C4×C31C2211yescharacteristic
30C31×#328C2218yes—
31C31×#324C2214yes—
32C22×Q8×C31—21no—
33C22×Q8×C31D412yes—
34C22×D4×C31—21no—
35C22×D4×C31C2311yescharacteristic
36C23×C4×C31D411yescharacteristic
37C23×C4×C31D411yescharacteristic
38C23×C4×C31C2311yescharacteristic
39C2×SD16×C31—24no—
40C2×M16×C31—22no—
41C31×C4⋊Q8—28no—
42C31×C4⋊Q8—28no—
43C31×#125—28no—
44C31×#125—28no—
45C31×C22⋊Q8—24no—
46C31×C22⋊Q8—24no—
47C31×C22⋊Q8—22no—
48C31×C22⋊Q8—22no—
49C31×C22⋊Q8D414yes—
50C31×C22⋊Q8C2314yes—
51C31×#121—22no—
52C31×#121—22no—
53C31×#117—24no—
54C31×#117—24no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31—22no—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—22no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31D412yes—
63C2×C4⋊C4×C31D412yes—
64C2×C4⋊C4×C31C2311yescharacteristic
65C2×C4⋊C4×C31C2312yes—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—21no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—22no—
70C2×C22⋊C4×C31—22no—
71C2×C42×C31—22no—
72C2×C42×C31—21no—
73C31×#107—24no—
74C31×#103—24no—
75C31×#102—24no—
76C31×C22⋊C8—22no—
77C31×C22⋊C8C2314yes—
78C31×#95—22no—
79C31×#95—22no—
80C31×#95C2312yes—
81C24×C31—21no—
82C24×C31C2×D411yescharacteristic
83C2×Q8×C31—24no—
84C2×Q8×C31—22no—
85C2×Q8×C31—82no—
86C2×Q8×C31—82no—
87C2×Q8×C31C2×D414yes—
88C2×D4×C31—22no—
89C2×D4×C31—44no—
90C2×D4×C31—41no—
91C2×D4×C31—41no—
92C2×D4×C31—81no—
93C22×C4×C31—21no—
94C22×C4×C31—21no—
95C22×C4×C31—24no—
96C22×C4×C31—22no—
97C22×C4×C31—22no—
98C22×C4×C31—21no—
99C22×C4×C31—21no—
100C22×C4×C31—21no—
101C22×C4×C31—22no—
102C22×C4×C31—81no—
103C22×C4×C31—81no—
104C22×C4×C31C2×D412yes—
105C22×C4×C31C2×D412yes—
106C22×C4×C31C2×D411yescharacteristic
107C22×C4×C31C2×D412yes—
108C22×C4×C31C2×D412yes—
109C22×C4×C31C2411yescharacteristic
110SD16×C31—84no—
111M16×C31—82no—
112C2×C8×C31—24no—
113C4⋊C4×C31—24no—
114C4⋊C4×C31—24no—
115C4⋊C4×C31—24no—
116C4⋊C4×C31—28no—
117C4⋊C4×C31—24no—
118C4⋊C4×C31—24no—
119C4⋊C4×C31—22no—
120C4⋊C4×C31—24no—
121C4⋊C4×C31—24no—
122C4⋊C4×C31—22no—
123C4⋊C4×C31C2×D414yes—
124C22⋊C4×C31—24no—
125C22⋊C4×C31—24no—
126C22⋊C4×C31—42no—
127C22⋊C4×C31—42no—
128C22⋊C4×C31—44no—
129C22⋊C4×C31—82no—
130C42×C31—28no—
131C42×C31—81no—
132#18326C3111yesmaximal, characteristic, direct product factor, Sylow
133C23×C31—22no—
134C23×C31—22no—
135C23×C31—41no—
136C23×C31—42no—
137C23×C31—42no—
138C23×C31—41no—
139C23×C31—42no—
140C23×C31#12011yescharacteristic
141C23×C31#12011yescharacteristic
142C23×C31C22×D411yescharacteristic
143Q8×C31—44no—
144Q8×C31—82no—
145Q8×C31—84no—
146D4×C31—44no—
147D4×C31—82no—
148D4×C31—82no—
149C2×C4×C31—24no—
150C2×C4×C31—24no—
151C2×C4×C31—22no—
152C2×C4×C31—22no—
153C2×C4×C31—24no—
154C2×C4×C31—24no—
155C2×C4×C31—22no—
156C2×C4×C31—41no—
157C2×C4×C31—42no—
158C2×C4×C31—42no—
159C2×C4×C31—41no—
160C2×C4×C31—42no—
161C2×C4×C31—44no—
162C2×C4×C31—44no—
163C2×C4×C31—42no—
164C2×C4×C31—42no—
165C2×C4×C31—42no—
166C2×C4×C31—42no—
167C2×C4×C31—41no—
168C2×C4×C31—42no—
169C2×C4×C31—81no—
170C2×C4×C31#12012yes—
171C2×C4×C31#13714yes—
172C2×C4×C31C22×D412yes—
173C8×C31—44no—
174C2×#544C2×C3111yescharacteristic, complemented
175#3193C2×C3112yes—
176#2591C2×C3111yescharacteristic
177#1617C2×C3114yescomplemented
178#1599C2×C3114yescomplemented
179#1508C2×C3112yescomplemented
180#1463C2×C3111yescharacteristic, complemented
181C22×C31—21no—
182C22×C31—21no—
183C22×C31—21no—
184C22×C31—21no—
185C22×C31—21no—
186C22×C31—21no—
187C22×C31—21no—
188C22×C31—44no—
189C22×C31—84no—
190C22×C31—81no—
191C22×C31—81no—
192C22×C31—81no—
193C22×C31#45412yes—
194C22×C31C2×#12011yescharacteristic
195C22×C31C2×#13712yes—
196C4×C31—24no—
197C4×C31—22no—
198C4×C31—22no—
199C4×C31—44no—
200C4×C31—42no—
201C4×C31—44no—
202C4×C31—82no—
203C4×C31—81no—
204#544—24no—
205#544—22no—
206#544C22×C3114yes—
207C2×C4⋊Q8—22no—
208C2×#125—22no—
209C2×C22⋊Q8—21no—
210C2×C22⋊Q8C22×C3112yes—
211C2×#121C22×C3111yescharacteristic
212C2×#117—21no—
213C22×C4⋊C4C22×C3111yescharacteristic
214#484C22×C3114yes—
215#480—22no—
216#468—24no—
217#450C22×C3114yes—
218#448—22no—
219#407C22×C3112yes—
220#400C22×C3112yes—
221#394C22×C3112yes—
222#388—22no—
223#386—21no—
224#386—21no—
225C4×C22⋊C4C22×C3111yescharacteristic
226#328C22×C3118yes—
227#324C22×C3114yes—
228C2×C31—22no—
229C2×C31—22no—
230C2×C31—41no—
231C2×C31—161no—
232C2×C31#300411yescharacteristic
233C2×C31C2×#45411yescharacteristic
234C2×C31#302611yescharacteristic
235C22×Q8—21no—
236C22×Q8D4×C3112yes—
237C22×D4—21no—
238C22×D4C23×C3111yescharacteristic
239C23×C4D4×C3111yescharacteristic
240C23×C4D4×C3111yescharacteristic
241C23×C4C23×C3111yescharacteristic
242C2×SD16—24no—
243C2×M16—22no—
244C4⋊Q8—28no—
245C4⋊Q8—28no—
246#125—28no—
247#125—28no—
248C22⋊Q8—24no—
249C22⋊Q8—24no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8D4×C3114yes—
253C22⋊Q8C23×C3114yes—
254#121—22no—
255#121—22no—
256#117—24no—
257#117—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4D4×C3112yes—
266C2×C4⋊C4D4×C3112yes—
267C2×C4⋊C4C23×C3111yescharacteristic
268C2×C4⋊C4C23×C3112yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C42—22no—
275C2×C42—21no—
276#107—24no—
277#103—24no—
278#102—24no—
279C22⋊C8—22no—
280C22⋊C8C23×C3114yes—
281#95—22no—
282#95—22no—
283#95C23×C3112yes—
284C31#1832611yescharacteristic, direct product factor, Sylow
285C24—21no—
286C24C2×D4×C3111yescharacteristic
287C2×Q8—24no—
288C2×Q8—22no—
289C2×Q8—82no—
290C2×Q8—82no—
291C2×Q8C2×D4×C3114yes—
292C2×D4—22no—
293C2×D4—44no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—81no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—81no—
307C22×C4—81no—
308C22×C4C2×D4×C3112yes—
309C22×C4C2×D4×C3112yes—
310C22×C4C2×D4×C3111yescharacteristic
311C22×C4C2×D4×C3112yes—
312C22×C4C2×D4×C3112yes—
313C22×C4C24×C3111yescharacteristic
314SD16—84no—
315M16—82no—
316C2×C8—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—28no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4C2×D4×C3114yes—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—44no—
333C22⋊C4—82no—
334C42—28no—
335C42—81no—
336C23—22no—
337C23—22no—
338C23—41no—
339C23—42no—
340C23—42no—
341C23—41no—
342C23—42no—
343C23C31×#12011yescharacteristic
344C23C31×#12011yescharacteristic
345C23C22×D4×C3111yescharacteristic
346Q8—44no—
347Q8—82no—
348Q8—84no—
349D4—44no—
350D4—82no—
351D4—82no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—81no—
373C2×C4C31×#12012yes—
374C2×C4C31×#13714yes—
375C2×C4C22×D4×C3112yes—
376C8—44no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—44no—
385C22—84no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22C31×#45412yes—
390C22C2×C31×#12011yescharacteristic
391C22C2×C31×#13712yes—
392C4—24no—
393C4—22no—
394C4—22no—
395C4—44no—
396C4—42no—
397C4—44no—
398C4—82no—
399C4—81no—
400C2—22no—
401C2—22no—
402C2—41no—
403C2—161no—
404C2C31×#300411yescharacteristic
405C2C2×C31×#45411yescharacteristic
406C2C31×#302611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#544C2semidirect
4. C31×#1617C2semidirect
5. C31×#1599C2semidirect
6. C31×#1508C2semidirect
7. C31×#1463C2semidirect
132. #18326C31direct (N × H)
174. C2×#544C2×C31semidirect
177. #1617C2×C31semidirect
178. #1599C2×C31semidirect
179. #1508C2×C31semidirect
180. #1463C2×C31semidirect
284. C31#18326direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5442
C31×#31932
C31×#25912
C31×#16172
C31×#15992
C31×#15082
C31×#14632
#1832631