Finite Group Databasebeta

← C31×#7678 #21464437

Subgroups — C31×#7678 #21464437

Order 7936.

630 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2854C211yesmaximal, characteristic
2C31×#1728C211yesmaximal, characteristic, complemented
3C2×C31×#336C211yesmaximal, characteristic, complemented
4C31×#1271C212yesmaximal, complemented
5C31×#1270C212yesmaximal, complemented
6C22×M16×C31—21no—
7C23×C8×C31—21no—
8C2×C31×#123—21no—
9C2×C31×#121C411yescharacteristic
10C22×C42×C31C2211yescharacteristic
11C31×#390C411yescharacteristic
12C31×#390C2211yescharacteristic
13C31×#389C411yescharacteristic
14C31×#389C2211yescharacteristic
15C31×#388—21no—
16C31×#388—21no—
17C31×#386—21no—
18C31×#380C411yescharacteristic
19C31×#336—21no—
20C31×#336—21no—
21C31×#336C2212yes—
22C31×#336C2212yes—
23C22×D4×C31—21no—
24C22×D4×C31C2×C411yescharacteristic
25C23×C4×C31C2×C411yescharacteristic
26C23×C4×C31D411yescharacteristic
27C23×C4×C31D411yescharacteristic
28C2×M16×C31—24no—
29C2×M16×C31—42no—
30C2×M16×C31—42no—
31C22×C8×C31—24no—
32C22×C8×C31—22no—
33C22×C8×C31—22no—
34C22×C8×C31—42no—
35C22×C8×C31—42no—
36C31×#123—42no—
37C31×#123—42no—
38C31×#121—42no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31C2×C411yescharacteristic
42C2×C4⋊C4×C31C2×C411yescharacteristic
43C2×C4⋊C4×C31C2×C411yescharacteristic
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C42×C31—42no—
51C2×C42×C31C2×C411yescharacteristic
52C2×C42×C31D411yescharacteristic
53C2×C42×C31D411yescharacteristic
54C2×C42×C31C2311yescharacteristic
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C24×C31—21no—
59C24×C31C22⋊C411yescharacteristic
60C2×D4×C31—22no—
61C2×D4×C31—41no—
62C2×D4×C31—41no—
63C2×D4×C31—42no—
64C2×D4×C31—42no—
65C2×D4×C31—42no—
66C22×C4×C31—22no—
67C22×C4×C31—22no—
68C22×C4×C31—22no—
69C22×C4×C31—22no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31—22no—
73C22×C4×C31—22no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—41no—
78C22×C4×C31C22⋊C411yescharacteristic
79C22×C4×C31C22⋊C411yescharacteristic
80C22×C4×C31C22⋊C411yescharacteristic
81C22×C4×C31D812yes—
82C22×C4×C31SD1612yes—
83C22×C4×C31C22×C411yescharacteristic
84C22×C4×C31C2×D411yescharacteristic
85C22×C4×C31C2×D411yescharacteristic
86M16×C31—48no—
87C2×C8×C31—24no—
88C2×C8×C31—24no—
89C2×C8×C31—48no—
90C2×C8×C31—42no—
91C2×C8×C31—42no—
92C2×C8×C31—42no—
93C2×C8×C31—42no—
94C4⋊C4×C31—22no—
95C4⋊C4×C31—41no—
96C4⋊C4×C31—41no—
97C4⋊C4×C31—42no—
98C4⋊C4×C31—41no—
99C4⋊C4×C31—41no—
100C22⋊C4×C31—42no—
101C22⋊C4×C31—42no—
102C22⋊C4×C31—42no—
103C22⋊C4×C31—42no—
104C22⋊C4×C31—41no—
105C22⋊C4×C31—41no—
106C22⋊C4×C31—41no—
107C22⋊C4×C31—41no—
108C42×C31—41no—
109C42×C31—41no—
110C42×C31—41no—
111C42×C31—41no—
112#7678C3111yesmaximal, characteristic, direct product factor, Sylow
113C23×C31—21no—
114C23×C31—21no—
115C23×C31—21no—
116C23×C31—21no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—41no—
120C23×C31—41no—
121C23×C31—41no—
122C23×C31—41no—
123C23×C31—41no—
124C23×C31#9912yes—
125C23×C31#10212yes—
126C23×C31#10412yes—
127C23×C31C2×C22⋊C411yescharacteristic
128D4×C31—82no—
129D4×C31—84no—
130C2×C4×C31—24no—
131C2×C4×C31—24no—
132C2×C4×C31—24no—
133C2×C4×C31—24no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31—21no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—21no—
147C2×C4×C31—44no—
148C2×C4×C31—41no—
149C2×C4×C31—41no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31#10212yes—
158C2×C4×C31#10412yes—
159C2×C4×C31#13112yes—
160C2×C4×C31C2×D811yescharacteristic
161C2×C4×C31C2×SD1611yescharacteristic
162C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C31#13711yescharacteristic
164C8×C31—44no—
165C8×C31—44no—
166#2854C2×C3111yescharacteristic
167#1728C2×C3111yescharacteristic, complemented
168C2×#336C2×C3111yescharacteristic, complemented
169#1271C2×C3112yescomplemented
170#1270C2×C3112yescomplemented
171C22×C31—22no—
172C22×C31—22no—
173C22×C31—22no—
174C22×C31—22no—
175C22×C31—22no—
176C22×C31—22no—
177C22×C31—81no—
178C22×C31—81no—
179C22×C31—81no—
180C22×C31—81no—
181C22×C31—81no—
182C22×C31—81no—
183C22×C31—81no—
184C22×C31#32714yes—
185C22×C31#40811yescharacteristic
186C22×C31C2×#9911yescharacteristic
187C22×C31#41311yescharacteristic
188C22×C31C2×#10211yescharacteristic
189C22×C31#41711yescharacteristic
190C22×C31C2×#10411yescharacteristic
191C22×C31C2×#10411yescharacteristic
192C4×C31—22no—
193C4×C31—21no—
194C4×C31—22no—
195C4×C31—21no—
196C4×C31—21no—
197C4×C31—22no—
198C4×C31—21no—
199C4×C31—41no—
200C4×C31—81no—
201C4×C31—81no—
202C4×C31—81no—
203C22×M16—21no—
204C23×C8—21no—
205C2×#123—21no—
206C2×#121C4×C3111yescharacteristic
207C22×C42C22×C3111yescharacteristic
208#390C4×C3111yescharacteristic
209#390C22×C3111yescharacteristic
210#389C4×C3111yescharacteristic
211#389C22×C3111yescharacteristic
212#388—21no—
213#388—21no—
214#386—21no—
215#380C4×C3111yescharacteristic
216#336—21no—
217#336—21no—
218#336C22×C3112yes—
219#336C22×C3112yes—
220C2×C31—24no—
221C2×C31—161no—
222C2×C31#146811yescharacteristic
223C2×C31#148311yescharacteristic
224C2×C31#149211yescharacteristic
225C2×C31C2×#32711yescharacteristic
226C2×C31#150411yescharacteristic
227C2×C31#152411yescharacteristic
228C2×C31#154711yescharacteristic
229C22×D4—21no—
230C22×D4C2×C4×C3111yescharacteristic
231C23×C4C2×C4×C3111yescharacteristic
232C23×C4D4×C3111yescharacteristic
233C23×C4D4×C3111yescharacteristic
234C2×M16—24no—
235C2×M16—42no—
236C2×M16—42no—
237C22×C8—24no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—42no—
241C22×C8—42no—
242#123—42no—
243#123—42no—
244#121—42no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C2×C4×C3111yescharacteristic
248C2×C4⋊C4C2×C4×C3111yescharacteristic
249C2×C4⋊C4C2×C4×C3111yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C42—42no—
257C2×C42C2×C4×C3111yescharacteristic
258C2×C42D4×C3111yescharacteristic
259C2×C42D4×C3111yescharacteristic
260C2×C42C23×C3111yescharacteristic
261#95—21no—
262#95—21no—
263#95—21no—
264C31#767811yescharacteristic, direct product factor, Sylow
265C24—21no—
266C24C22⋊C4×C3111yescharacteristic
267C2×D4—22no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4C22⋊C4×C3111yescharacteristic
286C22×C4C22⋊C4×C3111yescharacteristic
287C22×C4C22⋊C4×C3111yescharacteristic
288C22×C4D8×C3112yes—
289C22×C4SD16×C3112yes—
290C22×C4C22×C4×C3111yescharacteristic
291C22×C4C2×D4×C3111yescharacteristic
292C22×C4C2×D4×C3111yescharacteristic
293M16—48no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—48no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—22no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42—41no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23C31×#9912yes—
331C23C31×#10212yes—
332C23C31×#10412yes—
333C23C2×C22⋊C4×C3111yescharacteristic
334D4—82no—
335D4—84no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—44no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4C31×#10212yes—
364C2×C4C31×#10412yes—
365C2×C4C31×#13112yes—
366C2×C4C2×D8×C3111yescharacteristic
367C2×C4C2×SD16×C3111yescharacteristic
368C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
369C2×C4C31×#13711yescharacteristic
370C8—44no—
371C8—44no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C31×#32714yes—
386C22C31×#40811yescharacteristic
387C22C2×C31×#9911yescharacteristic
388C22C31×#41311yescharacteristic
389C22C2×C31×#10211yescharacteristic
390C22C31×#41711yescharacteristic
391C22C2×C31×#10411yescharacteristic
392C22C2×C31×#10411yescharacteristic
393C4—22no—
394C4—21no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—22no—
399C4—21no—
400C4—41no—
401C4—81no—
402C4—81no—
403C4—81no—
404C2—24no—
405C2—161no—
406C2C31×#146811yescharacteristic
407C2C31×#148311yescharacteristic
408C2C31×#149211yescharacteristic
409C2C2×C31×#32711yescharacteristic
410C2C31×#150411yescharacteristic
411C2C31×#152411yescharacteristic
412C2C31×#154711yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#121C4
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
2. C31×#1728C2semidirect
3. C2×C31×#336C2semidirect
4. C31×#1271C2semidirect
5. C31×#1270C2semidirect
112. #7678C31direct (N × H)
167. #1728C2×C31semidirect
168. C2×#336C2×C31semidirect
169. #1271C2×C31semidirect
170. #1270C2×C31semidirect
264. C31#7678direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28542
C31×#17282
C2×C31×#3362
C31×#12712
C31×#12702
#767831