Finite Group Databasebeta

← C31×#9360 #21466119

Subgroups — C31×#9360 #21466119

Order 7936.

538 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3038C211yesmaximal, characteristic
2C31×#2991C211yesmaximal, characteristic, complemented
3C31×#1965C211yesmaximal, characteristic, complemented
4C31×#1545C211yesmaximal, characteristic, complemented
5C31×#1489C211yesmaximal, characteristic, complemented
6C31×#1309C211yesmaximal, characteristic, complemented
7C31×#1299C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)C412yes—
9C2×C31×#127C2211yescharacteristic
10C4×#41×C31C2211yescharacteristic
11C31×#460—21no—
12C31×#460—21no—
13C31×#459—21no—
14C31×#459—21no—
15C8×Q8×C31—21no—
16C31×#436—22no—
17C31×#436—21no—
18C31×#431—21no—
19C31×#431C2211yescharacteristic
20C31×#423—21no—
21C31×#423C2211yescharacteristic
22C31×#423C2211yescharacteristic
23C31×#423C2211yescharacteristic
24C31×#418C412yes—
25C2×C31×#100—22no—
26C4×M16×C31C2211yescharacteristic
27C31×#331—21no—
28C31×#331—21no—
29C31×Q8⋊C8—21no—
30C31×D4⋊C8—21no—
31C2×#41×C31C2×C411yescharacteristic
32C22×D4×C31—21no—
33C22×D4×C31—21no—
34C22×D4×C31C2×C412yes—
35C31×AGL(1,ℤ/8)—44no—
36C2×SD16×C31—22no—
37C2×D8×C31—22no—
38C2×M16×C31—22no—
39C2×M16×C31—21no—
40C2×M16×C31—21no—
41C2×M16×C31C2×C412yes—
42C2×M16×C31D412yes—
43C31×#127—21no—
44C31×#127—21no—
45C31×#127—21no—
46C31×#127—21no—
47C4×Q8×C31D411yescharacteristic
48C4×Q8×C31D411yescharacteristic
49C4×D4×C31—21no—
50C4×D4×C31—21no—
51C4×D4×C31D411yescharacteristic
52C4×D4×C31D411yescharacteristic
53C31×#117—21no—
54C31×#117C2×C411yescharacteristic
55C2×C42×C31—21no—
56C2×C42×C31C2311yescharacteristic
57C31×C4⋊C8—22no—
58C31×C4⋊C8—21no—
59C31×C4⋊C8—21no—
60C31×C4⋊C8—21no—
61C31×C4⋊C8—21no—
62C31×C4⋊C8—21no—
63C31×C4⋊C8—21no—
64C31×C4⋊C8D411yescharacteristic
65C31×C4⋊C8D411yescharacteristic
66C31×#102—22no—
67C31×#102—22no—
68C31×#100—44no—
69C31×C22⋊C8—22no—
70C31×C22⋊C8—21no—
71C31×#97—21no—
72C4×C8×C31—22no—
73C4×C8×C31—21no—
74C4×C8×C31—21no—
75C24×C31—22no—
76#41×C31—22no—
77#41×C31—41no—
78C2×Q8×C31C22⋊C411yescharacteristic
79C2×D4×C31—21no—
80C2×D4×C31—22no—
81C2×D4×C31—22no—
82C2×D4×C31—42no—
83C2×D4×C31—42no—
84C2×D4×C31—42no—
85C2×D4×C31—42no—
86C2×D4×C31—41no—
87C2×D4×C31—41no—
88C2×D4×C31—41no—
89C2×D4×C31—41no—
90C2×D4×C31—81no—
91C2×D4×C31—81no—
92C2×D4×C31C22⋊C411yescharacteristic
93C22×C4×C31—21no—
94C22×C4×C31—21no—
95C22×C4×C31—21no—
96C22×C4×C31C22×C411yescharacteristic
97C22×C4×C31C2×D411yescharacteristic
98C22×C4×C31#4111yescharacteristic
99SD16×C31—44no—
100D8×C31—44no—
101M16×C31—22no—
102M16×C31—41no—
103M16×C31—42no—
104M16×C31—42no—
105M16×C31—82no—
106M16×C31—81no—
107C2×C8×C31—22no—
108C2×C8×C31—21no—
109C2×C8×C31—22no—
110C2×C8×C31—21no—
111C2×C8×C31—41no—
112C2×C8×C31—41no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31C22⋊C411yescharacteristic
116C4⋊C4×C31C22⋊C411yescharacteristic
117C22⋊C4×C31—21no—
118C22⋊C4×C31—21no—
119C22⋊C4×C31—21no—
120C42×C31—21no—
121C42×C31—21no—
122C42×C31—21no—
123C42×C31C2×D411yescharacteristic
124C42×C31C2×D411yescharacteristic
125C42×C31C2×D411yescharacteristic
126C42×C31#4111yescharacteristic
127#9360C3111yesmaximal, characteristic, direct product factor, Sylow
128C23×C31—21no—
129C23×C31—44no—
130C23×C31—42no—
131C23×C31—82no—
132C23×C31—82no—
133C23×C31C4×D411yescharacteristic
134Q8×C31—21no—
135Q8×C31#10012yescomplemented
136D4×C31—21no—
137D4×C31—21no—
138D4×C31—42no—
139D4×C31—42no—
140D4×C31—42no—
141D4×C31—82no—
142D4×C31—82no—
143D4×C31—82no—
144D4×C31—81no—
145D4×C31—81no—
146D4×C31—81no—
147D4×C31—81no—
148C2×C4×C31—21no—
149C2×C4×C31—21no—
150C2×C4×C31—21no—
151C2×C4×C31—21no—
152C2×C4×C31—21no—
153C2×C4×C31—21no—
154C2×C4×C31—21no—
155C2×C4×C31—21no—
156C2×C4×C31—21no—
157C2×C4×C31—42no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31C2×C22⋊C411yescharacteristic
163C2×C4×C31C4×D411yescharacteristic
164C2×C4×C31#12011yescharacteristic
165C2×C4×C31#12111yescharacteristic
166C2×C4×C31#12111yescharacteristic
167C2×C4×C31#12311yescharacteristic
168C8×C31—42no—
169C8×C31—42no—
170C8×C31—42no—
171C8×C31—41no—
172C8×C31—41no—
173C8×C31—81no—
174#3038C2×C3111yescharacteristic
175#2991C2×C3111yescharacteristic, complemented
176#1965C2×C3111yescharacteristic, complemented
177#1545C2×C3111yescharacteristic, complemented
178#1489C2×C3111yescharacteristic, complemented
179#1309C2×C3111yescharacteristic, complemented
180#1299C2×C3111yescharacteristic, complemented
181C22×C31—21no—
182C22×C31—21no—
183C22×C31—21no—
184C22×C31—41no—
185C22×C31—41no—
186C22×C31—41no—
187C22×C31—82no—
188C22×C31—82no—
189C22×C31—82no—
190C22×C31—164no—
191C22×C31#38611yescharacteristic
192C4×C31—21no—
193C4×C31—21no—
194C4×C31—22no—
195C4×C31—21no—
196C4×C31—41no—
197C4×C31—41no—
198C4×C31C2×#10011yescharacteristic
199C4×C31#41311yescharacteristic
200C4×C31#45311yescharacteristic
201C4×C31#45411yescharacteristic
202C4×C31#47111yescharacteristic
203C4×C31#47211yescharacteristic
204C2×AGL(1,ℤ/8)C4×C3112yes—
205C2×#127C22×C3111yescharacteristic
206C4×#41C22×C3111yescharacteristic
207#460—21no—
208#460—21no—
209#459—21no—
210#459—21no—
211C8×Q8—21no—
212#436—22no—
213#436—21no—
214#431—21no—
215#431C22×C3111yescharacteristic
216#423—21no—
217#423C22×C3111yescharacteristic
218#423C22×C3111yescharacteristic
219#423C22×C3111yescharacteristic
220#418C4×C3112yes—
221C2×#100—22no—
222C4×M16C22×C3111yescharacteristic
223#331—21no—
224#331—21no—
225Q8⋊C8—21no—
226D4⋊C8—21no—
227C2×C31—41no—
228C2×C31—81no—
229C2×C31—162no—
230C2×C31#188311yescharacteristic
231C2×C31#190011yescharacteristic
232C2×C31#190711yescharacteristic
233C2×#41C2×C4×C3111yescharacteristic
234C22×D4—21no—
235C22×D4—21no—
236C22×D4C2×C4×C3112yes—
237AGL(1,ℤ/8)—44no—
238C2×SD16—22no—
239C2×D8—22no—
240C2×M16—22no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16C2×C4×C3112yes—
244C2×M16D4×C3112yes—
245#127—21no—
246#127—21no—
247#127—21no—
248#127—21no—
249C4×Q8D4×C3111yescharacteristic
250C4×Q8D4×C3111yescharacteristic
251C4×D4—21no—
252C4×D4—21no—
253C4×D4D4×C3111yescharacteristic
254C4×D4D4×C3111yescharacteristic
255#117—21no—
256#117C2×C4×C3111yescharacteristic
257C2×C42—21no—
258C2×C42C23×C3111yescharacteristic
259C4⋊C8—22no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8D4×C3111yescharacteristic
267C4⋊C8D4×C3111yescharacteristic
268#102—22no—
269#102—22no—
270#100—44no—
271C22⋊C8—22no—
272C22⋊C8—21no—
273#97—21no—
274C4×C8—22no—
275C4×C8—21no—
276C4×C8—21no—
277C31#936011yescharacteristic, direct product factor, Sylow
278C24—22no—
279#41—22no—
280#41—41no—
281C2×Q8C22⋊C4×C3111yescharacteristic
282C2×D4—21no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—81no—
294C2×D4—81no—
295C2×D4C22⋊C4×C3111yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4C22×C4×C3111yescharacteristic
300C22×C4C2×D4×C3111yescharacteristic
301C22×C4#41×C3111yescharacteristic
302SD16—44no—
303D8—44no—
304M16—22no—
305M16—41no—
306M16—42no—
307M16—42no—
308M16—82no—
309M16—81no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—21no—
314C2×C8—41no—
315C2×C8—41no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4C22⋊C4×C3111yescharacteristic
319C4⋊C4C22⋊C4×C3111yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42C2×D4×C3111yescharacteristic
327C42C2×D4×C3111yescharacteristic
328C42C2×D4×C3111yescharacteristic
329C42#41×C3111yescharacteristic
330C23—21no—
331C23—44no—
332C23—42no—
333C23—82no—
334C23—82no—
335C23C4×D4×C3111yescharacteristic
336Q8—21no—
337Q8C31×#10012yescomplemented
338D4—21no—
339D4—21no—
340D4—42no—
341D4—42no—
342D4—42no—
343D4—82no—
344D4—82no—
345D4—82no—
346D4—81no—
347D4—81no—
348D4—81no—
349D4—81no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4C2×C22⋊C4×C3111yescharacteristic
365C2×C4C4×D4×C3111yescharacteristic
366C2×C4C31×#12011yescharacteristic
367C2×C4C31×#12111yescharacteristic
368C2×C4C31×#12111yescharacteristic
369C2×C4C31×#12311yescharacteristic
370C8—42no—
371C8—42no—
372C8—42no—
373C8—41no—
374C8—41no—
375C8—81no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22—164no—
386C22C31×#38611yescharacteristic
387C4—21no—
388C4—21no—
389C4—22no—
390C4—21no—
391C4—41no—
392C4—41no—
393C4C2×C31×#10011yescharacteristic
394C4C31×#41311yescharacteristic
395C4C31×#45311yescharacteristic
396C4C31×#45411yescharacteristic
397C4C31×#47111yescharacteristic
398C4C31×#47211yescharacteristic
399C2—41no—
400C2—81no—
401C2—162no—
402C2C31×#188311yescharacteristic
403C2C31×#190011yescharacteristic
404C2C31×#190711yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3038C2
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
2. C31×#2991C2semidirect
3. C31×#1965C2semidirect
4. C31×#1545C2semidirect
5. C31×#1489C2semidirect
6. C31×#1309C2semidirect
7. C31×#1299C2semidirect
127. #9360C31direct (N × H)
135. Q8×C31#100semidirect
175. #2991C2×C31semidirect
176. #1965C2×C31semidirect
177. #1545C2×C31semidirect
178. #1489C2×C31semidirect
179. #1309C2×C31semidirect
180. #1299C2×C31semidirect
277. C31#9360direct (N × H)
337. Q8C31×#100semidirect

Maximal subgroups

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

subgroupindex
C31×#30382
C31×#29912
C31×#19652
C31×#15452
C31×#14892
C31×#13092
C31×#12992
#936031