Finite Group Databasebeta

← C31×#8226 #21464985

Subgroups — C31×#8226 #21464985

Order 7936.

572 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C31×#124C211yesmaximal, characteristic
2C31×#1745C211yesmaximal, characteristic, complemented
3C4×C31×#97C211yesmaximal, characteristic, complemented
4C2×C31×#329C212yesmaximal
5C31×#1295C212yesmaximal, complemented
6C2×C31×#124C2211yescharacteristic
7C2×C4×Q8×C31C411yescharacteristic
8C2×C4×D4×C31C411yescharacteristic
9C2×C31×#97—21no—
10C2×C31×#97C2212yes—
11C2×C31×#97C2212yes—
12C2×C4×C8×C31—21no—
13C31×#391C411yescharacteristic
14C31×#390C411yescharacteristic
15C31×#388—21no—
16C31×#388—21no—
17C31×#382C2211yescharacteristic
18C4×C22⋊C4×C31—21no—
19C4×C22⋊C4×C31—21no—
20C43×C31C2211yescharacteristic
21C31×#336—21no—
22C31×#336—21no—
23C31×#329C418yescomplemented
24C22×Q8×C31C2×C411yescharacteristic
25C22×D4×C31C2×C411yescharacteristic
26C23×C4×C31—21no—
27C22×C8×C31—22no—
28C22×C8×C31—22no—
29C31×#124—22no—
30C31×#124C2×C414yes—
31C4×Q8×C31—42no—
32C4×D4×C31—42no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31C2×C411yescharacteristic
35C2×C4⋊C4×C31C2×C411yescharacteristic
36C2×C22⋊C4×C31—21no—
37C2×C22⋊C4×C31—21no—
38C2×C22⋊C4×C31—21no—
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C42×C31—21no—
42C2×C42×C31C2×C411yescharacteristic
43C2×C42×C31C2×C411yescharacteristic
44C2×C42×C31D411yescharacteristic
45C2×C42×C31D411yescharacteristic
46C2×C42×C31D411yescharacteristic
47C2×C42×C31D411yescharacteristic
48C2×C42×C31C2311yescharacteristic
49C31×#97—24no—
50C31×#97—22no—
51C31×#97—22no—
52C31×#97C2×C414yescomplemented
53C31×#97C2×C414yescomplemented
54C4×C8×C31—24no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C24×C31—21no—
60C2×Q8×C31—21no—
61C2×Q8×C31—41no—
62C2×Q8×C31—41no—
63C2×Q8×C31C4212yes—
64C2×D4×C31—21no—
65C2×D4×C31—41no—
66C2×D4×C31—41no—
67C2×D4×C31C4212yes—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—42no—
72C22×C4×C31—42no—
73C22×C4×C31—42no—
74C22×C4×C31C22⋊C411yescharacteristic
75C22×C4×C31C22⋊C411yescharacteristic
76C22×C4×C31C22⋊C411yescharacteristic
77C22×C4×C31C22⋊C411yescharacteristic
78C22×C4×C31C22×C411yescharacteristic
79C22×C4×C31C2×D411yescharacteristic
80C22×C4×C31C2×D411yescharacteristic
81C2×C8×C31—24no—
82C2×C8×C31—24no—
83C2×C8×C31—24no—
84C2×C8×C31—24no—
85C2×C8×C31—42no—
86C2×C8×C31—42no—
87C4⋊C4×C31—41no—
88C4⋊C4×C31—41no—
89C4⋊C4×C31—42no—
90C22⋊C4×C31—24no—
91C22⋊C4×C31—24no—
92C22⋊C4×C31—42no—
93C22⋊C4×C31—42no—
94C22⋊C4×C31—42no—
95C42×C31—22no—
96C42×C31—22no—
97C42×C31—22no—
98C42×C31—41no—
99C42×C31—41no—
100C42×C31—41no—
101C42×C31—42no—
102C42×C31C22×C412yes—
103C42×C31#4112yes—
104#8226C3111yesmaximal, characteristic, direct product factor, Sylow
105C23×C31—22no—
106C23×C31—41no—
107C23×C31—41no—
108C23×C31—41no—
109C23×C31—41no—
110C23×C31—41no—
111C23×C31—41no—
112C23×C31C2×C22⋊C411yescharacteristic
113Q8×C31—42no—
114Q8×C31—42no—
115D4×C31—42no—
116D4×C31—42no—
117C2×C4×C31—21no—
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—21no—
129C2×C4×C31—22no—
130C2×C4×C31—21no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—41no—
134C2×C4×C31—41no—
135C2×C4×C31—41no—
136C2×C4×C31—41no—
137C2×C4×C31—41no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—84no—
142C2×C4×C31#10012yes—
143C2×C4×C31#10412yes—
144C2×C4×C31#10412yes—
145C2×C4×C31C2×C4211yescharacteristic
146C2×C4×C31#11711yescharacteristic
147C2×C4×C31C4×D412yes—
148C2×C4×C31C4×D412yes—
149C8×C31—28no—
150C8×C31—44no—
151C4×#124C2×C3111yescharacteristic
152#1745C2×C3111yescharacteristic, complemented
153C4×#97C2×C3111yescharacteristic, complemented
154C2×#329C2×C3112yes—
155#1295C2×C3112yescomplemented
156C22×C31—42no—
157C22×C31—42no—
158C22×C31—42no—
159C22×C31—81no—
160C22×C31—81no—
161C22×C31—81no—
162C22×C31—81no—
163C22×C31C4×C22⋊C411yescharacteristic
164C22×C31C2×#10011yescharacteristic
165C22×C31#41311yescharacteristic
166C22×C31C2×#10411yescharacteristic
167C22×C31C2×#10411yescharacteristic
168C22×C31#42111yescharacteristic
169C22×C31#42111yescharacteristic
170C4×C31—21no—
171C4×C31—21no—
172C4×C31—22no—
173C4×C31—22no—
174C4×C31—22no—
175C4×C31—22no—
176C4×C31—22no—
177C4×C31—42no—
178C4×C31—41no—
179C4×C31—81no—
180C4×C31—81no—
181C2×#124C22×C3111yescharacteristic
182C2×C4×Q8C4×C3111yescharacteristic
183C2×C4×D4C4×C3111yescharacteristic
184C2×#97—21no—
185C2×#97C22×C3112yes—
186C2×#97C22×C3112yes—
187C2×C4×C8—21no—
188#391C4×C3111yescharacteristic
189#390C4×C3111yescharacteristic
190#388—21no—
191#388—21no—
192#382C22×C3111yescharacteristic
193C4×C22⋊C4—21no—
194C4×C22⋊C4—21no—
195C43C22×C3111yescharacteristic
196#336—21no—
197#336—21no—
198#329C4×C3118yescomplemented
199C2×C31—82no—
200C2×C31#152711yescharacteristic
201C2×C31#152911yescharacteristic
202C2×C31#153011yescharacteristic
203C2×C31#153211yescharacteristic
204C2×C31C4×#10011yescharacteristic
205C2×C31C4×#10411yescharacteristic
206C2×C31#175911yescharacteristic
207C22×Q8C2×C4×C3111yescharacteristic
208C22×D4C2×C4×C3111yescharacteristic
209C23×C4—21no—
210C22×C8—22no—
211C22×C8—22no—
212#124—22no—
213#124C2×C4×C3114yes—
214C4×Q8—42no—
215C4×D4—42no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C2×C4×C3111yescharacteristic
218C2×C4⋊C4C2×C4×C3111yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C42—21no—
225C2×C42C2×C4×C3111yescharacteristic
226C2×C42C2×C4×C3111yescharacteristic
227C2×C42D4×C3111yescharacteristic
228C2×C42D4×C3111yescharacteristic
229C2×C42D4×C3111yescharacteristic
230C2×C42D4×C3111yescharacteristic
231C2×C42C23×C3111yescharacteristic
232#97—24no—
233#97—22no—
234#97—22no—
235#97C2×C4×C3114yescomplemented
236#97C2×C4×C3114yescomplemented
237C4×C8—24no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242C31#822611yescharacteristic, direct product factor, Sylow
243C24—21no—
244C2×Q8—21no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8C42×C3112yes—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4C42×C3112yes—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4C22⋊C4×C3111yescharacteristic
259C22×C4C22⋊C4×C3111yescharacteristic
260C22×C4C22⋊C4×C3111yescharacteristic
261C22×C4C22⋊C4×C3111yescharacteristic
262C22×C4C22×C4×C3111yescharacteristic
263C22×C4C2×D4×C3111yescharacteristic
264C22×C4C2×D4×C3111yescharacteristic
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—42no—
270C2×C8—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—41no—
283C42—41no—
284C42—41no—
285C42—42no—
286C42C22×C4×C3112yes—
287C42#41×C3112yes—
288C23—22no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23C2×C22⋊C4×C3111yescharacteristic
296Q8—42no—
297Q8—42no—
298D4—42no—
299D4—42no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—84no—
325C2×C4C31×#10012yes—
326C2×C4C31×#10412yes—
327C2×C4C31×#10412yes—
328C2×C4C2×C42×C3111yescharacteristic
329C2×C4C31×#11711yescharacteristic
330C2×C4C4×D4×C3112yes—
331C2×C4C4×D4×C3112yes—
332C8—28no—
333C8—44no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22C4×C22⋊C4×C3111yescharacteristic
342C22C2×C31×#10011yescharacteristic
343C22C31×#41311yescharacteristic
344C22C2×C31×#10411yescharacteristic
345C22C2×C31×#10411yescharacteristic
346C22C31×#42111yescharacteristic
347C22C31×#42111yescharacteristic
348C4—21no—
349C4—21no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—42no—
356C4—41no—
357C4—81no—
358C4—81no—
359C2—82no—
360C2C31×#152711yescharacteristic
361C2C31×#152911yescharacteristic
362C2C31×#153011yescharacteristic
363C2C31×#153211yescharacteristic
364C2C4×C31×#10011yescharacteristic
365C2C4×C31×#10411yescharacteristic
366C2C31×#175911yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC2×C4C2×C42×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C43×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1745C2semidirect
3. C4×C31×#97C2semidirect
5. C31×#1295C2semidirect
23. C31×#329C4semidirect
52. C31×#97C2×C4semidirect
53. C31×#97C2×C4semidirect
104. #8226C31direct (N × H)
152. #1745C2×C31semidirect
153. C4×#97C2×C31semidirect
155. #1295C2×C31semidirect
198. #329C4×C31semidirect
235. #97C2×C4×C31semidirect
236. #97C2×C4×C31semidirect
242. C31#8226direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C31×#1242
C31×#17452
C4×C31×#972
C2×C31×#3292
C31×#12952
#822631