Finite Group Databasebeta

← C31×#10521 #21467280

Subgroups — C31×#10521 #21467280

Order 7936.

420 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3246C211yesmaximal, characteristic
2C31×#1994C211yesmaximal, characteristic, complemented
3C31×#1962C211yesmaximal, characteristic
4C31×#1831C211yesmaximal, characteristic, complemented
5C31×#1550C211yesmaximal, characteristic, complemented
6C31×#1547C211yesmaximal, characteristic
7C31×#1292C211yesmaximal, characteristic, complemented
8C31×#533C2211yescharacteristic
9C31×#532C2211yescharacteristic
10C31×#476—21no—
11C31×#475—21no—
12C31×#465—22no—
13C31×#461—21no—
14C31×#460—21no—
15C31×#423C2211yescharacteristic
16C31×#423C2211yescharacteristic
17C2×C31×C4⋊C8C2211yescharacteristic
18C2×C31×C4⋊C8C2211yescharacteristic
19C2×C31×#104—21no—
20C2×C31×#104—21no—
21C43×C31C2211yescharacteristic
22C31×#339—21no—
23C31×#339—21no—
24C31×#336—22no—
25C31×#333—21no—
26C31×#332—21no—
27C31×#332—21no—
28C31×#331—21no—
29C2×#41×C31—21no—
30C2×SD16×C31—22no—
31C2×M16×C31—21no—
32C2×M16×C31—21no—
33C22×C8×C31—21no—
34C22×C8×C31D412yes—
35C31×C4⋊Q8D411yescharacteristic
36C31×C4⋊Q8D411yescharacteristic
37C31×C4⋊Q8D411yescharacteristic
38C31×#127D411yescharacteristic
39C31×#125—21no—
40C31×#124—21no—
41C31×C22⋊Q8—21no—
42C31×C22⋊Q8—21no—
43C31×#121—21no—
44C31×#121—21no—
45C4×Q8×C31—21no—
46C4×Q8×C31—21no—
47C4×D4×C31—21no—
48C4×D4×C31—21no—
49C31×#117—21no—
50C2×C42×C31—21no—
51C2×C42×C31—21no—
52C2×C42×C31D411yescharacteristic
53C2×C42×C31D411yescharacteristic
54C2×C42×C31C2311yescharacteristic
55C31×#106—22no—
56C31×C4⋊C8—21no—
57C31×C4⋊C8—21no—
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×#104—42no—
65C31×#104—42no—
66C31×#103—22no—
67C31×#102—22no—
68#41×C31—41no—
69#41×C31—41no—
70#41×C31—41no—
71#41×C31—41no—
72C2×Q8×C31—21no—
73C2×Q8×C31—21no—
74C2×Q8×C31—21no—
75C2×D4×C31—21no—
76C2×D4×C31—21no—
77C2×D4×C31—21no—
78C2×D4×C31—41no—
79C22×C4×C31—21no—
80C22×C4×C31—21no—
81C22×C4×C31—41no—
82C22×C4×C31C2×D411yescharacteristic
83C22×C4×C31C2×D411yescharacteristic
84C22×C4×C31#4111yescharacteristic
85SD16×C31—82no—
86M16×C31—41no—
87M16×C31—41no—
88M16×C31—41no—
89M16×C31—41no—
90C2×C8×C31—22no—
91C2×C8×C31—42no—
92C2×C8×C31—42no—
93C2×C8×C31—41no—
94C2×C8×C31—41no—
95C2×C8×C31—41no—
96C4⋊C4×C31—21no—
97C4⋊C4×C31—21no—
98C4⋊C4×C31—21no—
99C4⋊C4×C31—21no—
100C4⋊C4×C31—21no—
101C4⋊C4×C31—21no—
102C4⋊C4×C31—41no—
103C4⋊C4×C31—41no—
104C22⋊C4×C31—41no—
105C22⋊C4×C31—41no—
106C22⋊C4×C31—41no—
107C42×C31—21no—
108C42×C31—21no—
109C42×C31—21no—
110C42×C31—41no—
111C42×C31—41no—
112C42×C31—42no—
113C42×C31—41no—
114C42×C31SD1612yescomplemented
115C42×C31C2×D411yescharacteristic
116C42×C31C2×D411yescharacteristic
117C42×C31#4111yescharacteristic
118C42×C31#4111yescharacteristic
119#10521C3111yesmaximal, characteristic, direct product factor, Sylow
120C23×C31—41no—
121C23×C31#12111yescharacteristic
122Q8×C31—41no—
123Q8×C31—41no—
124Q8×C31—41no—
125Q8×C31—41no—
126Q8×C31—81no—
127D4×C31—41no—
128D4×C31—41no—
129D4×C31—41no—
130D4×C31—41no—
131D4×C31—81no—
132D4×C31—81no—
133D4×C31—81no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—22no—
137C2×C4×C31—22no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—41no—
142C2×C4×C31—41no—
143C2×C4×C31—41no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
146C2×C4×C31—81no—
147C2×C4×C31—81no—
148C2×C4×C31#12011yescharacteristic
149C2×C4×C31#12111yescharacteristic
150C2×C4×C31#12111yescharacteristic
151C2×C4×C31#12311yescharacteristic
152C2×C4×C31#12311yescharacteristic
153C2×C4×C31#12411yescharacteristic
154C2×C4×C31C2×SD1611yescharacteristic
155C2×C4×C31#13511yescharacteristic
156C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C31#13711yescharacteristic
158C8×C31—42no—
159C8×C31—81no—
160C8×C31—81no—
161C8×C31—81no—
162#3246C2×C3111yescharacteristic
163#1994C2×C3111yescharacteristic, complemented
164#1962C2×C3111yescharacteristic
165#1831C2×C3111yescharacteristic, complemented
166#1550C2×C3111yescharacteristic, complemented
167#1547C2×C3111yescharacteristic
168#1292C2×C3111yescharacteristic, complemented
169C22×C31—21no—
170C22×C31—21no—
171C22×C31—81no—
172C22×C31—81no—
173C22×C31—81no—
174C22×C31#39411yescharacteristic
175C22×C31#46511yescharacteristic
176C22×C31#46811yescharacteristic
177C4×C31—21no—
178C4×C31—21no—
179C4×C31—21no—
180C4×C31—42no—
181C4×C31—42no—
182C4×C31—81no—
183C4×C31—81no—
184C4×C31—81no—
185C4×C31#45411yescharacteristic
186C4×C31#45511yescharacteristic
187C4×C31#46011yescharacteristic
188C4×C31#46111yescharacteristic
189C4×C31#46311yescharacteristic
190C4×C31#46411yescharacteristic
191#533C22×C3111yescharacteristic
192#532C22×C3111yescharacteristic
193#476—21no—
194#475—21no—
195#465—22no—
196#461—21no—
197#460—21no—
198#423C22×C3111yescharacteristic
199#423C22×C3111yescharacteristic
200C2×C4⋊C8C22×C3111yescharacteristic
201C2×C4⋊C8C22×C3111yescharacteristic
202C2×#104—21no—
203C2×#104—21no—
204C43C22×C3111yescharacteristic
205#339—21no—
206#339—21no—
207#336—22no—
208#333—21no—
209#332—21no—
210#332—21no—
211#331—21no—
212C2×C31—21no—
213C2×C31—21no—
214C2×C31—161no—
215C2×C31#204011yescharacteristic
216C2×C31#205011yescharacteristic
217C2×C31#205511yescharacteristic
218C2×#41—21no—
219C2×SD16—22no—
220C2×M16—21no—
221C2×M16—21no—
222C22×C8—21no—
223C22×C8D4×C3112yes—
224C4⋊Q8D4×C3111yescharacteristic
225C4⋊Q8D4×C3111yescharacteristic
226C4⋊Q8D4×C3111yescharacteristic
227#127D4×C3111yescharacteristic
228#125—21no—
229#124—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232#121—21no—
233#121—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—21no—
237C4×D4—21no—
238#117—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42D4×C3111yescharacteristic
242C2×C42D4×C3111yescharacteristic
243C2×C42C23×C3111yescharacteristic
244#106—22no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253#104—42no—
254#104—42no—
255#103—22no—
256#102—22no—
257C31#1052111yescharacteristic, direct product factor, Sylow
258#41—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—41no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4C2×D4×C3111yescharacteristic
273C22×C4C2×D4×C3111yescharacteristic
274C22×C4#41×C3111yescharacteristic
275SD16—82no—
276M16—41no—
277M16—41no—
278M16—41no—
279M16—41no—
280C2×C8—22no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—41no—
301C42—41no—
302C42—42no—
303C42—41no—
304C42SD16×C3112yescomplemented
305C42C2×D4×C3111yescharacteristic
306C42C2×D4×C3111yescharacteristic
307C42#41×C3111yescharacteristic
308C42#41×C3111yescharacteristic
309C23—41no—
310C23C31×#12111yescharacteristic
311Q8—41no—
312Q8—41no—
313Q8—41no—
314Q8—41no—
315Q8—81no—
316D4—41no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—81no—
321D4—81no—
322D4—81no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4C31×#12011yescharacteristic
338C2×C4C31×#12111yescharacteristic
339C2×C4C31×#12111yescharacteristic
340C2×C4C31×#12311yescharacteristic
341C2×C4C31×#12311yescharacteristic
342C2×C4C31×#12411yescharacteristic
343C2×C4C2×SD16×C3111yescharacteristic
344C2×C4C31×#13511yescharacteristic
345C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
346C2×C4C31×#13711yescharacteristic
347C8—42no—
348C8—81no—
349C8—81no—
350C8—81no—
351C22—21no—
352C22—21no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22C31×#39411yescharacteristic
357C22C31×#46511yescharacteristic
358C22C31×#46811yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—42no—
363C4—42no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4C31×#45411yescharacteristic
368C4C31×#45511yescharacteristic
369C4C31×#46011yescharacteristic
370C4C31×#46111yescharacteristic
371C4C31×#46311yescharacteristic
372C4C31×#46411yescharacteristic
373C2—21no—
374C2—21no—
375C2—161no—
376C2C31×#204011yescharacteristic
377C2C31×#205011yescharacteristic
378C2C31×#205511yescharacteristic
subgroupG/H
center, socleC22×C31#394
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#532C22
LCSG ▷ C2×C42 ▷ 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×#1994C2semidirect
4. C31×#1831C2semidirect
5. C31×#1550C2semidirect
7. C31×#1292C2semidirect
114. C42×C31SD16semidirect
119. #10521C31direct (N × H)
163. #1994C2×C31semidirect
165. #1831C2×C31semidirect
166. #1550C2×C31semidirect
168. #1292C2×C31semidirect
257. C31#10521direct (N × H)
304. C42SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#32462
C31×#19942
C31×#19622
C31×#18312
C31×#15502
C31×#15472
C31×#12922
#1052131