Finite Group Databasebeta

← C31×#18180 #21474939

Subgroups — C31×#18180 #21474939

Order 7936.

1106 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3501C211yesmaximal, characteristic, complemented
2C31×#3110C212yesmaximal, complemented
3C31×#2397C211yesmaximal, characteristic
4C31×#1614C214yesmaximal, complemented
5C31×#1596C214yesmaximal, complemented
6C31×#1506C212yesmaximal, complemented
7C31×#1466C211yesmaximal, characteristic, complemented
8C31×#554—22no—
9C31×#554—24no—
10D4×Q8×C31—22no—
11C31×#544—21no—
12C31×#543C2214yes—
13C31×#533C2212yes—
14C2×C31×C22⋊Q8—22no—
15C31×#518—21no—
16C2×C4×D4×C31—22no—
17C2×C4×D4×C31C2211yescharacteristic
18C2×C4×D4×C31C2211yescharacteristic
19C31×#484C2214yes—
20C31×#474—22no—
21C31×#461—22no—
22C31×#450C2214yes—
23C31×#440—24no—
24C31×#432C2212yes—
25C31×#399—21no—
26C31×#399C2212yes—
27C31×#392—21no—
28C31×#392C2212yes—
29C31×#376C2211yescharacteristic
30C31×#328C2218yes—
31C31×#323C2214yes—
32C22×Q8×C31—22no—
33C22×D4×C31—22no—
34C22×D4×C31D411yescharacteristic
35C22×D4×C31C2311yescharacteristic
36C23×C4×C31—21no—
37C23×C4×C31D412yes—
38C2×SD16×C31—24no—
39C31×C4⋊Q8—24no—
40C31×C4⋊Q8D412yescomplemented
41C31×C4⋊Q8D412yescomplemented
42C31×#125—24no—
43C31×#125—22no—
44C31×C22⋊Q8—24no—
45C31×C22⋊Q8—24no—
46C31×C22⋊Q8—28no—
47C31×C22⋊Q8—22no—
48C31×C22⋊Q8—24no—
49C31×C22⋊Q8—28no—
50C31×C22⋊Q8C2314yes—
51C4×Q8×C31—22no—
52C4×Q8×C31D414yescomplemented
53C4×D4×C31—24no—
54C4×D4×C31—24no—
55C4×D4×C31—22no—
56C4×D4×C31—22no—
57C4×D4×C31—22no—
58C4×D4×C31—22no—
59C4×D4×C31—48no—
60C31×#117—22no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31—24no—
63C2×C4⋊C4×C31—22no—
64C2×C4⋊C4×C31D411yescharacteristic
65C2×C4⋊C4×C31C2311yescharacteristic
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—22no—
69C2×C22⋊C4×C31—21no—
70C2×C22⋊C4×C31C2312yes—
71C2×C42×C31—21no—
72C2×C42×C31C2311yescharacteristic
73C31×#107—24no—
74C31×C4⋊C8—22no—
75C31×#103—24no—
76C31×#102—24no—
77C31×C22⋊C8C2314yes—
78C31×#97—22no—
79C31×#95—21no—
80C31×#95C2312yes—
81C24×C31—21no—
82C24×C31C2×D412yes—
83C2×Q8×C31—24no—
84C2×Q8×C31—22no—
85C2×Q8×C31—84no—
86C2×Q8×C31C2×D414yes—
87C2×D4×C31—24no—
88C2×D4×C31—22no—
89C2×D4×C31—22no—
90C2×D4×C31—44no—
91C2×D4×C31—42no—
92C2×D4×C31—44no—
93C2×D4×C31—44no—
94C22×C4×C31—22no—
95C22×C4×C31—24no—
96C22×C4×C31—24no—
97C22×C4×C31—22no—
98C22×C4×C31—22no—
99C22×C4×C31—21no—
100C22×C4×C31—44no—
101C22×C4×C31—44no—
102C22×C4×C31—41no—
103C22×C4×C31—41no—
104C22×C4×C31C2×D411yescharacteristic
105C22×C4×C31C2×D411yescharacteristic
106C22×C4×C31C2×D412yes—
107C22×C4×C31C2411yescharacteristic
108SD16×C31—48no—
109C2×C8×C31—24no—
110C4⋊C4×C31—24no—
111C4⋊C4×C31—28no—
112C4⋊C4×C31—22no—
113C4⋊C4×C31—24no—
114C4⋊C4×C31—22no—
115C4⋊C4×C31—22no—
116C4⋊C4×C31—22no—
117C4⋊C4×C31—24no—
118C4⋊C4×C31—22no—
119C4⋊C4×C31—44no—
120C4⋊C4×C31C2×D414yes—
121C22⋊C4×C31—28no—
122C22⋊C4×C31—24no—
123C22⋊C4×C31—24no—
124C22⋊C4×C31—44no—
125C22⋊C4×C31—44no—
126C22⋊C4×C31—42no—
127C42×C31—24no—
128C42×C31—24no—
129C42×C31C2×D412yes—
130C42×C31C2×D412yes—
131#18180C3111yesmaximal, characteristic, direct product factor, Sylow
132C23×C31—22no—
133C23×C31—22no—
134C23×C31—22no—
135C23×C31—42no—
136C23×C31—42no—
137C23×C31—42no—
138C23×C31—41no—
139C23×C31—41no—
140C23×C31—42no—
141C23×C31#13714yes—
142C23×C31C22×D411yescharacteristic
143Q8×C31—44no—
144Q8×C31—44no—
145Q8×C31—84no—
146D4×C31—28no—
147D4×C31—24no—
148D4×C31—42no—
149D4×C31—416no—
150C2×C4×C31—22no—
151C2×C4×C31—22no—
152C2×C4×C31—24no—
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—44no—
160C2×C4×C31—42no—
161C2×C4×C31—42no—
162C2×C4×C31—41no—
163C2×C4×C31—82no—
164C2×C4×C31—82no—
165C2×C4×C31—84no—
166C2×C4×C31—84no—
167C2×C4×C31—82no—
168C2×C4×C31#12012yes—
169C2×C4×C31#12011yescharacteristic
170C2×C4×C31#12011yescharacteristic
171C2×C4×C31C22×D412yes—
172C8×C31—44no—
173#3501C2×C3111yescharacteristic, complemented
174#3110C2×C3112yescomplemented
175#2397C2×C3111yescharacteristic
176#1614C2×C3114yescomplemented
177#1596C2×C3114yescomplemented
178#1506C2×C3112yescomplemented
179#1466C2×C3111yescharacteristic, complemented
180C22×C31—21no—
181C22×C31—21no—
182C22×C31—24no—
183C22×C31—44no—
184C22×C31—44no—
185C22×C31—42no—
186C22×C31—42no—
187C22×C31—42no—
188C22×C31—82no—
189C22×C31—82no—
190C22×C31—81no—
191C22×C31—81no—
192C22×C31C2×#12011yescharacteristic
193C22×C31C2×#13712yes—
194C4×C31—22no—
195C4×C31—21no—
196C4×C31—22no—
197C4×C31—44no—
198C4×C31—42no—
199C4×C31—44no—
200C4×C31—44no—
201C4×C31—81no—
202C4×C31#45312yes—
203#554—22no—
204#554—24no—
205D4×Q8—22no—
206#544—21no—
207#543C22×C3114yes—
208#533C22×C3112yes—
209C2×C22⋊Q8—22no—
210#518—21no—
211C2×C4×D4—22no—
212C2×C4×D4C22×C3111yescharacteristic
213C2×C4×D4C22×C3111yescharacteristic
214#484C22×C3114yes—
215#474—22no—
216#461—22no—
217#450C22×C3114yes—
218#440—24no—
219#432C22×C3112yes—
220#399—21no—
221#399C22×C3112yes—
222#392—21no—
223#392C22×C3112yes—
224#376C22×C3111yescharacteristic
225#328C22×C3118yes—
226#323C22×C3114yes—
227C2×C31—22no—
228C2×C31—44no—
229C2×C31—82no—
230C2×C31#300411yescharacteristic
231C2×C31C2×#45311yescharacteristic
232C2×C31#302511yescharacteristic
233C22×Q8—22no—
234C22×D4—22no—
235C22×D4D4×C3111yescharacteristic
236C22×D4C23×C3111yescharacteristic
237C23×C4—21no—
238C23×C4D4×C3112yes—
239C2×SD16—24no—
240C4⋊Q8—24no—
241C4⋊Q8D4×C3112yescomplemented
242C4⋊Q8D4×C3112yescomplemented
243#125—24no—
244#125—22no—
245C22⋊Q8—24no—
246C22⋊Q8—24no—
247C22⋊Q8—28no—
248C22⋊Q8—22no—
249C22⋊Q8—24no—
250C22⋊Q8—28no—
251C22⋊Q8C23×C3114yes—
252C4×Q8—22no—
253C4×Q8D4×C3114yescomplemented
254C4×D4—24no—
255C4×D4—24no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—48no—
261#117—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4D4×C3111yescharacteristic
266C2×C4⋊C4C23×C3111yescharacteristic
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4C23×C3112yes—
272C2×C42—21no—
273C2×C42C23×C3111yescharacteristic
274#107—24no—
275C4⋊C8—22no—
276#103—24no—
277#102—24no—
278C22⋊C8C23×C3114yes—
279#97—22no—
280#95—21no—
281#95C23×C3112yes—
282C31#1818011yescharacteristic, direct product factor, Sylow
283C24—21no—
284C24C2×D4×C3112yes—
285C2×Q8—24no—
286C2×Q8—22no—
287C2×Q8—84no—
288C2×Q8C2×D4×C3114yes—
289C2×D4—24no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—44no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—44no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4C2×D4×C3111yescharacteristic
307C22×C4C2×D4×C3111yescharacteristic
308C22×C4C2×D4×C3112yes—
309C22×C4C24×C3111yescharacteristic
310SD16—48no—
311C2×C8—24no—
312C4⋊C4—24no—
313C4⋊C4—28no—
314C4⋊C4—22no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—44no—
322C4⋊C4C2×D4×C3114yes—
323C22⋊C4—28no—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—42no—
329C42—24no—
330C42—24no—
331C42C2×D4×C3112yes—
332C42C2×D4×C3112yes—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23C31×#13714yes—
343C23C22×D4×C3111yescharacteristic
344Q8—44no—
345Q8—44no—
346Q8—84no—
347D4—28no—
348D4—24no—
349D4—42no—
350D4—416no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—82no—
369C2×C4C31×#12012yes—
370C2×C4C31×#12011yescharacteristic
371C2×C4C31×#12011yescharacteristic
372C2×C4C22×D4×C3112yes—
373C8—44no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—44no—
378C22—44no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—82no—
383C22—82no—
384C22—81no—
385C22—81no—
386C22C2×C31×#12011yescharacteristic
387C22C2×C31×#13712yes—
388C4—22no—
389C4—21no—
390C4—22no—
391C4—44no—
392C4—42no—
393C4—44no—
394C4—44no—
395C4—81no—
396C4C31×#45312yes—
397C2—22no—
398C2—44no—
399C2—82no—
400C2C31×#300411yescharacteristic
401C2C2×C31×#45311yescharacteristic
402C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2397C2
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
1. C31×#3501C2semidirect
2. C31×#3110C2semidirect
4. C31×#1614C2semidirect
5. C31×#1596C2semidirect
6. C31×#1506C2semidirect
7. C31×#1466C2semidirect
40. C31×C4⋊Q8D4semidirect
41. C31×C4⋊Q8D4semidirect
52. C4×Q8×C31D4semidirect
131. #18180C31direct (N × H)
173. #3501C2×C31semidirect
174. #3110C2×C31semidirect
176. #1614C2×C31semidirect
177. #1596C2×C31semidirect
178. #1506C2×C31semidirect
179. #1466C2×C31semidirect
241. C4⋊Q8D4×C31semidirect
242. C4⋊Q8D4×C31semidirect
253. C4×Q8D4×C31semidirect
282. C31#18180direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35012
C31×#31102
C31×#23972
C31×#16142
C31×#15962
C31×#15062
C31×#14662
#1818031