Finite Group Databasebeta

← C31×#7642 #21464401

Subgroups — C31×#7642 #21464401

Order 7936.

478 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2293C211yesmaximal, characteristic
2C31×#1839C211yesmaximal, characteristic, complemented
3C31×#1831C211yesmaximal, characteristic, complemented
4C31×#1271C212yesmaximal, complemented
5C31×#1270C212yesmaximal
6C2×C4×D4×C31C411yescharacteristic
7C2×C31×#117—21no—
8C2×C31×C4⋊C8—21no—
9C31×#390C411yescharacteristic
10C31×#390C2211yescharacteristic
11C31×#389C411yescharacteristic
12C31×#389C2211yescharacteristic
13C31×#388—21no—
14C31×#388—21no—
15C4×C4⋊C4×C31C411yescharacteristic
16C4×C22⋊C4×C31—21no—
17C43×C31C2211yescharacteristic
18C31×#336—21no—
19C31×#336—21no—
20C31×#336—21no—
21C31×#336C2212yes—
22C31×#336C2212yes—
23C22×D4×C31C2×C411yescharacteristic
24C23×C4×C31—21no—
25C22×C8×C31—22no—
26C22×C8×C31—22no—
27C4×D4×C31—22no—
28C4×D4×C31—22no—
29C31×#117—42no—
30C31×#117—42no—
31C2×C4⋊C4×C31—21no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31C2×C411yescharacteristic
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×C4⋊C8—42no—
50C31×#95—21no—
51C31×#95—21no—
52C31×#95—21no—
53C24×C31—21no—
54C2×D4×C31—21no—
55C2×D4×C31—41no—
56C2×D4×C31—41no—
57C2×D4×C31M1612yes—
58C22×C4×C31—24no—
59C22×C4×C31—21no—
60C22×C4×C31—21no—
61C22×C4×C31—21no—
62C22×C4×C31—42no—
63C22×C4×C31—42no—
64C22×C4×C31C22⋊C411yescharacteristic
65C22×C4×C31C22⋊C411yescharacteristic
66C22×C4×C31C22⋊C411yescharacteristic
67C22×C4×C31C22⋊C411yescharacteristic
68C22×C4×C31C22×C411yescharacteristic
69C22×C4×C31C2×D411yescharacteristic
70C22×C4×C31C2×D411yescharacteristic
71C2×C8×C31—42no—
72C2×C8×C31—42no—
73C2×C8×C31—42no—
74C2×C8×C31—42no—
75C2×C8×C31—42no—
76C2×C8×C31—42no—
77C4⋊C4×C31—21no—
78C4⋊C4×C31—41no—
79C4⋊C4×C31—41no—
80C4⋊C4×C31—42no—
81C4⋊C4×C31—41no—
82C4⋊C4×C31—41no—
83C4⋊C4×C31M1612yes—
84C22⋊C4×C31—24no—
85C22⋊C4×C31—42no—
86C22⋊C4×C31—42no—
87C22⋊C4×C31—41no—
88C22⋊C4×C31—41no—
89C22⋊C4×C31—42no—
90C42×C31—21no—
91C42×C31—21no—
92C42×C31—22no—
93C42×C31—41no—
94C42×C31—41no—
95C42×C31—41no—
96C42×C31—41no—
97C42×C31—42no—
98C42×C31D812yes—
99C42×C31SD1612yes—
100#7642C3111yesmaximal, characteristic, direct product factor, Sylow
101C23×C31—22no—
102C23×C31—41no—
103C23×C31—41no—
104C23×C31—41no—
105C23×C31—41no—
106C23×C31—41no—
107C23×C31—41no—
108C23×C31C2×C22⋊C411yescharacteristic
109D4×C31—42no—
110D4×C31—42no—
111C2×C4×C31—21no—
112C2×C4×C31—21no—
113C2×C4×C31—21no—
114C2×C4×C31—21no—
115C2×C4×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—22no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—21no—
128C2×C4×C31—48no—
129C2×C4×C31—41no—
130C2×C4×C31—41no—
131C2×C4×C31—41no—
132C2×C4×C31—41no—
133C2×C4×C31—41no—
134C2×C4×C31—41no—
135C2×C4×C31—41no—
136C2×C4×C31—41no—
137C2×C4×C31#10212yes—
138C2×C4×C31#10212yes—
139C2×C4×C31C2×M1611yescharacteristic
140C2×C4×C31C2×M1611yescharacteristic
141C2×C4×C31C2×D811yescharacteristic
142C2×C4×C31C2×SD1611yescharacteristic
143C2×C4×C31#13512yes—
144C8×C31—82no—
145C8×C31—82no—
146#2293C2×C3111yescharacteristic
147#1839C2×C3111yescharacteristic, complemented
148#1831C2×C3111yescharacteristic, complemented
149#1271C2×C3112yescomplemented
150#1270C2×C3112yes—
151C22×C31—42no—
152C22×C31—42no—
153C22×C31—42no—
154C22×C31—81no—
155C22×C31—81no—
156C22×C31—81no—
157C22×C31—81no—
158C22×C31#40711yescharacteristic
159C22×C31#41011yescharacteristic
160C22×C31#41311yescharacteristic
161C22×C31C2×#10211yescharacteristic
162C22×C31#41611yescharacteristic
163C22×C31#42111yescharacteristic
164C22×C31#42111yescharacteristic
165C4×C31—22no—
166C4×C31—21no—
167C4×C31—21no—
168C4×C31—22no—
169C4×C31—21no—
170C4×C31—21no—
171C4×C31—22no—
172C4×C31—42no—
173C4×C31—41no—
174C4×C31—41no—
175C4×C31—81no—
176C4×C31—81no—
177C2×C4×D4C4×C3111yescharacteristic
178C2×#117—21no—
179C2×C4⋊C8—21no—
180#390C4×C3111yescharacteristic
181#390C22×C3111yescharacteristic
182#389C4×C3111yescharacteristic
183#389C22×C3111yescharacteristic
184#388—21no—
185#388—21no—
186C4×C4⋊C4C4×C3111yescharacteristic
187C4×C22⋊C4—21no—
188C43C22×C3111yescharacteristic
189#336—21no—
190#336—21no—
191#336—21no—
192#336C22×C3112yes—
193#336C22×C3112yes—
194C2×C31—82no—
195C2×C31#146411yescharacteristic
196C2×C31#148611yescharacteristic
197C2×C31#148811yescharacteristic
198C2×C31#151711yescharacteristic
199C2×C31#152111yescharacteristic
200C2×C31#153511yescharacteristic
201C2×C31#154011yescharacteristic
202C22×D4C2×C4×C3111yescharacteristic
203C23×C4—21no—
204C22×C8—22no—
205C22×C8—22no—
206C4×D4—22no—
207C4×D4—22no—
208#117—42no—
209#117—42no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C2×C4×C3111yescharacteristic
213C2×C4⋊C4C2×C4×C3111yescharacteristic
214C2×C4⋊C4C2×C4×C3111yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C42—21no—
221C2×C42C2×C4×C3111yescharacteristic
222C2×C42C2×C4×C3111yescharacteristic
223C2×C42D4×C3111yescharacteristic
224C2×C42D4×C3111yescharacteristic
225C2×C42D4×C3111yescharacteristic
226C2×C42D4×C3111yescharacteristic
227C2×C42C23×C3111yescharacteristic
228C4⋊C8—42no—
229#95—21no—
230#95—21no—
231#95—21no—
232C31#764211yescharacteristic, direct product factor, Sylow
233C24—21no—
234C2×D4—21no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4M16×C3112yes—
238C22×C4—24no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4C22⋊C4×C3111yescharacteristic
245C22×C4C22⋊C4×C3111yescharacteristic
246C22×C4C22⋊C4×C3111yescharacteristic
247C22×C4C22⋊C4×C3111yescharacteristic
248C22×C4C22×C4×C3111yescharacteristic
249C22×C4C2×D4×C3111yescharacteristic
250C22×C4C2×D4×C3111yescharacteristic
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4M16×C3112yes—
264C22⋊C4—24no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—42no—
270C42—21no—
271C42—21no—
272C42—22no—
273C42—41no—
274C42—41no—
275C42—41no—
276C42—41no—
277C42—42no—
278C42D8×C3112yes—
279C42SD16×C3112yes—
280C23—22no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23C2×C22⋊C4×C3111yescharacteristic
288D4—42no—
289D4—42no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—48no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4C31×#10212yes—
317C2×C4C31×#10212yes—
318C2×C4C2×M16×C3111yescharacteristic
319C2×C4C2×M16×C3111yescharacteristic
320C2×C4C2×D8×C3111yescharacteristic
321C2×C4C2×SD16×C3111yescharacteristic
322C2×C4C31×#13512yes—
323C8—82no—
324C8—82no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22C31×#40711yescharacteristic
333C22C31×#41011yescharacteristic
334C22C31×#41311yescharacteristic
335C22C2×C31×#10211yescharacteristic
336C22C31×#41611yescharacteristic
337C22C31×#42111yescharacteristic
338C22C31×#42111yescharacteristic
339C4—22no—
340C4—21no—
341C4—21no—
342C4—22no—
343C4—21no—
344C4—21no—
345C4—22no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—81no—
350C4—81no—
351C2—82no—
352C2C31×#146411yescharacteristic
353C2C31×#148611yescharacteristic
354C2C31×#148811yescharacteristic
355C2C31×#151711yescharacteristic
356C2C31×#152111yescharacteristic
357C2C31×#153511yescharacteristic
358C2C31×#154011yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31C2×C4
LCSG ▷ C22×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×#1839C2semidirect
3. C31×#1831C2semidirect
4. C31×#1271C2semidirect
100. #7642C31direct (N × H)
147. #1839C2×C31semidirect
148. #1831C2×C31semidirect
149. #1271C2×C31semidirect
232. C31#7642direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#22932
C31×#18392
C31×#18312
C31×#12712
C31×#12702
#764231