Finite Group Databasebeta

← C31×#8333 #21465092

Subgroups — C31×#8333 #21465092

Order 7936.

876 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#533C211yesmaximal, characteristic
2C31×#1817C211yesmaximal, characteristic
3C2×C31×#328C214yesmaximal
4C31×#1434C211yesmaximal, characteristic
5C31×#533—28no—
6C2×C31×C4⋊Q8C411yescharacteristic, complemented
7C2×C31×C4⋊Q8C411yescharacteristic, complemented
8C2×C31×#125—21no—
9C2×C31×C22⋊Q8—21no—
10C2×C31×C22⋊Q8C2212yes—
11C2×C4×Q8×C31—21no—
12C2×C4×Q8×C31C412yescomplemented
13C2×C31×#117—21no—
14C22×C42×C31C2211yescharacteristic
15C2×C31×C22⋊C8C2212yes—
16C2×C31×#95—21no—
17C2×C31×#95—21no—
18C2×C31×#95C2212yes—
19C31×#336—21no—
20C31×#336—21no—
21C31×#328—28no—
22C22×Q8×C31—21no—
23C22×Q8×C31C2×C412yes—
24C23×C4×C31—22no—
25C23×C4×C31D411yescharacteristic
26C23×C4×C31D411yescharacteristic
27C23×C4×C31C2311yescharacteristic
28C22×C8×C31—22no—
29C31×C4⋊Q8—24no—
30C31×C4⋊Q8—24no—
31C31×#125—28no—
32C31×C22⋊Q8—28no—
33C31×C22⋊Q8—24no—
34C31×C22⋊Q8D418yes—
35C4×Q8×C31—28no—
36C4×Q8×C31—24no—
37C4×Q8×C31—24no—
38C31×#117—28no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31—21no—
42C2×C4⋊C4×C31—22no—
43C2×C4⋊C4×C31—22no—
44C2×C4⋊C4×C31C2×C412yes—
45C2×C22⋊C4×C31—22no—
46C2×C42×C31—22no—
47C2×C42×C31—24no—
48C2×C42×C31C2×C411yescharacteristic
49C2×C42×C31C2×C411yescharacteristic
50C2×C42×C31D411yescharacteristic
51C2×C42×C31D411yescharacteristic
52C31×C22⋊C8—24no—
53C31×#95—24no—
54C31×#95—24no—
55C31×#95—42no—
56C24×C31C2×D411yescharacteristic
57C2×Q8×C31—24no—
58C2×Q8×C31—22no—
59C2×Q8×C31—42no—
60C2×Q8×C31—42no—
61C2×Q8×C31—42no—
62C2×Q8×C31—42no—
63C2×Q8×C31C22⋊C414yes—
64C22×C4×C31—22no—
65C22×C4×C31—22no—
66C22×C4×C31—22no—
67C22×C4×C31—22no—
68C22×C4×C31—22no—
69C22×C4×C31—22no—
70C22×C4×C31—24no—
71C22×C4×C31—41no—
72C22×C4×C31—41no—
73C22×C4×C31—41no—
74C22×C4×C31—42no—
75C22×C4×C31—42no—
76C22×C4×C31—42no—
77C22×C4×C31—42no—
78C22×C4×C31—42no—
79C22×C4×C31—42no—
80C22×C4×C31C22⋊C411yescharacteristic
81C22×C4×C31C22⋊C411yescharacteristic
82C22×C4×C31C22⋊C411yescharacteristic
83C22×C4×C31C22⋊C411yescharacteristic
84C22×C4×C31SD1612yes—
85C22×C4×C31Q1612yes—
86C22×C4×C31C22×C411yescharacteristic
87C22×C4×C31C2×D411yescharacteristic
88C22×C4×C31C2×D414yes—
89C2×C8×C31—42no—
90C2×C8×C31—42no—
91C2×C8×C31—42no—
92C4⋊C4×C31—24no—
93C4⋊C4×C31—24no—
94C4⋊C4×C31—28no—
95C4⋊C4×C31—24no—
96C4⋊C4×C31—22no—
97C4⋊C4×C31—28no—
98C4⋊C4×C31C22⋊C414yes—
99C22⋊C4×C31—28no—
100C42×C31—28no—
101C42×C31—22no—
102C42×C31—22no—
103C42×C31—22no—
104C42×C31—22no—
105#8333C3111yesmaximal, characteristic, direct product factor, Sylow
106C23×C31—21no—
107C23×C31—21no—
108C23×C31—21no—
109C23×C31—21no—
110C23×C31C2×C22⋊C411yescharacteristic
111C23×C31#12012yes—
112C23×C31C2×SD1611yescharacteristic
113C23×C31C2×Q1611yescharacteristic
114C23×C31#13712yes—
115Q8×C31—44no—
116Q8×C31—44no—
117Q8×C31—48no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—21no—
128C2×C4×C31—24no—
129C2×C4×C31—24no—
130C2×C4×C31—21no—
131C2×C4×C31—24no—
132C2×C4×C31—24no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—44no—
136C2×C4×C31—44no—
137C2×C4×C31—42no—
138C2×C4×C31—42no—
139C2×C4×C31—42no—
140C2×C4×C31—42no—
141C2×C4×C31—44no—
142C2×C4×C31—44no—
143C2×C4×C31—44no—
144C2×C4×C31—82no—
145C2×C4×C31—82no—
146C2×C4×C31—82no—
147C2×C4×C31—82no—
148C2×C4×C31#10312yes—
149C2×C4×C31#10312yes—
150C2×C4×C31#10412yes—
151C2×C4×C31C2×C22⋊C412yes—
152C2×C4×C31#12012yes—
153C8×C31—82no—
154C2×#533C2×C3111yescharacteristic
155#1817C2×C3111yescharacteristic
156C2×#328C2×C3114yes—
157#1434C2×C3111yescharacteristic
158C22×C31—21no—
159C22×C31—21no—
160C22×C31—22no—
161C22×C31—22no—
162C22×C31—21no—
163C22×C31—21no—
164C22×C31—22no—
165C22×C31—22no—
166C22×C31#37911yescharacteristic
167C22×C31#41011yescharacteristic
168C22×C31#41011yescharacteristic
169C22×C31C2×#10311yescharacteristic
170C22×C31#41911yescharacteristic
171C22×C31C2×#10411yescharacteristic
172C22×C31#42111yescharacteristic
173C22×C31#44812yes—
174C22×C31#45112yes—
175C4×C31—22no—
176C4×C31—22no—
177C4×C31—22no—
178C4×C31—22no—
179C4×C31—22no—
180C4×C31—44no—
181C4×C31—41no—
182C4×C31—44no—
183C4×C31—84no—
184#533—28no—
185C2×C4⋊Q8C4×C3111yescharacteristic, complemented
186C2×C4⋊Q8C4×C3111yescharacteristic, complemented
187C2×#125—21no—
188C2×C22⋊Q8—21no—
189C2×C22⋊Q8C22×C3112yes—
190C2×C4×Q8—21no—
191C2×C4×Q8C4×C3112yescomplemented
192C2×#117—21no—
193C22×C42C22×C3111yescharacteristic
194C2×C22⋊C8C22×C3112yes—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95C22×C3112yes—
198#336—21no—
199#336—21no—
200#328—28no—
201C2×C31—22no—
202C2×C31—22no—
203C2×C31#150311yescharacteristic
204C2×C31#150611yescharacteristic
205C2×C31#151511yescharacteristic
206C2×C31#151811yescharacteristic
207C2×C31#178211yescharacteristic
208C2×C31#178711yescharacteristic
209C2×C31#178911yescharacteristic
210C22×Q8—21no—
211C22×Q8C2×C4×C3112yes—
212C23×C4—22no—
213C23×C4D4×C3111yescharacteristic
214C23×C4D4×C3111yescharacteristic
215C23×C4C23×C3111yescharacteristic
216C22×C8—22no—
217C4⋊Q8—24no—
218C4⋊Q8—24no—
219#125—28no—
220C22⋊Q8—28no—
221C22⋊Q8—24no—
222C22⋊Q8D4×C3118yes—
223C4×Q8—28no—
224C4×Q8—24no—
225C4×Q8—24no—
226#117—28no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4C2×C4×C3112yes—
233C2×C22⋊C4—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42C2×C4×C3111yescharacteristic
237C2×C42C2×C4×C3111yescharacteristic
238C2×C42D4×C3111yescharacteristic
239C2×C42D4×C3111yescharacteristic
240C22⋊C8—24no—
241#95—24no—
242#95—24no—
243#95—42no—
244C31#833311yescharacteristic, direct product factor, Sylow
245C24C2×D4×C3111yescharacteristic
246C2×Q8—24no—
247C2×Q8—22no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8C22⋊C4×C3114yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4C22⋊C4×C3111yescharacteristic
270C22×C4C22⋊C4×C3111yescharacteristic
271C22×C4C22⋊C4×C3111yescharacteristic
272C22×C4C22⋊C4×C3111yescharacteristic
273C22×C4SD16×C3112yes—
274C22×C4Q16×C3112yes—
275C22×C4C22×C4×C3111yescharacteristic
276C22×C4C2×D4×C3111yescharacteristic
277C22×C4C2×D4×C3114yes—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—28no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—28no—
287C4⋊C4C22⋊C4×C3114yes—
288C22⋊C4—28no—
289C42—28no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23C2×C22⋊C4×C3111yescharacteristic
299C23C31×#12012yes—
300C23C2×SD16×C3111yescharacteristic
301C23C2×Q16×C3111yescharacteristic
302C23C31×#13712yes—
303Q8—44no—
304Q8—44no—
305Q8—48no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4C31×#10312yes—
337C2×C4C31×#10312yes—
338C2×C4C31×#10412yes—
339C2×C4C2×C22⋊C4×C3112yes—
340C2×C4C31×#12012yes—
341C8—82no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—22no—
346C22—21no—
347C22—21no—
348C22—22no—
349C22—22no—
350C22C31×#37911yescharacteristic
351C22C31×#41011yescharacteristic
352C22C31×#41011yescharacteristic
353C22C2×C31×#10311yescharacteristic
354C22C31×#41911yescharacteristic
355C22C2×C31×#10411yescharacteristic
356C22C31×#42111yescharacteristic
357C22C31×#44812yes—
358C22C31×#45112yes—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—22no—
364C4—44no—
365C4—41no—
366C4—44no—
367C4—84no—
368C2—22no—
369C2—22no—
370C2C31×#150311yescharacteristic
371C2C31×#150611yescharacteristic
372C2C31×#151511yescharacteristic
373C2C31×#151811yescharacteristic
374C2C31×#178211yescharacteristic
375C2C31×#178711yescharacteristic
376C2C31×#178911yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×C31×C4⋊Q8C4semidirect
7. C2×C31×C4⋊Q8C4semidirect
12. C2×C4×Q8×C31C4semidirect
105. #8333C31direct (N × H)
185. C2×C4⋊Q8C4×C31semidirect
186. C2×C4⋊Q8C4×C31semidirect
191. C2×C4×Q8C4×C31semidirect
244. C31#8333direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5332
C31×#18172
C2×C31×#3282
C31×#14342
#833331