Finite Group Databasebeta

← C31×#12210 #21468969

Subgroups — C31×#12210 #21468969

Order 7936.

414 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2728C211yesmaximal, characteristic
2C31×#2028C211yesmaximal, characteristic, complemented
3C31×#1611C211yesmaximal, characteristic, complemented
4C31×#1599C211yesmaximal, characteristic
5C31×#1346C211yesmaximal, characteristic, complemented
6C31×#1345C211yesmaximal, characteristic
7C31×#1319C211yesmaximal, characteristic, complemented
8C31×#544C2211yescharacteristic
9C2×C31×C22⋊Q8—21no—
10C31×#483—21no—
11C31×#450—21no—
12C31×#419—21no—
13C31×#400—21no—
14C31×#398C2211yescharacteristic
15C31×#397—21no—
16C31×#394C2211yescharacteristic
17C31×#391C2211yescharacteristic
18C31×#388—21no—
19C31×#386—21no—
20C31×#385—21no—
21C31×#343C2211yescharacteristic
22C31×#339—21no—
23C31×#328—21no—
24C31×#328—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×D4×C31D411yescharacteristic
29C23×C4×C31—21no—
30C2×SD16×C31—41no—
31C2×M16×C31—21no—
32C31×C4⋊Q8—41no—
33C31×#125—41no—
34C31×C22⋊Q8—21no—
35C31×C22⋊Q8—21no—
36C31×C22⋊Q8—42no—
37C31×C22⋊Q8—42no—
38C31×#117—21no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31—21no—
42C2×C4⋊C4×C31D411yescharacteristic
43C2×C4⋊C4×C31C2311yescharacteristic
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×C42×C31D411yescharacteristic
50C31×#106—41no—
51C31×#103—41no—
52C31×#103—41no—
53C31×#103—41no—
54C31×#102—41no—
55C31×C22⋊C8—21no—
56C31×C22⋊C8—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—21no—
61C31×#95—21no—
62C31×#95D411yescharacteristic
63C31×#95D411yescharacteristic
64C24×C31—21no—
65C2×Q8×C31—21no—
66C2×Q8×C31—21no—
67C2×Q8×C31—21no—
68C2×Q8×C31—41no—
69C2×Q8×C31—41no—
70C2×Q8×C31SD1612yescomplemented
71C2×D4×C31—41no—
72C2×D4×C31—42no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—42no—
79C22×C4×C31—81no—
80C22×C4×C31—81no—
81C22×C4×C31C2×D411yescharacteristic
82C22×C4×C31C2×D411yescharacteristic
83C22×C4×C31C2×D411yescharacteristic
84SD16×C31—82no—
85M16×C31—41no—
86M16×C31—41no—
87C2×C8×C31—41no—
88C2×C8×C31—41no—
89C2×C8×C31—41no—
90C4⋊C4×C31—21no—
91C4⋊C4×C31—21no—
92C4⋊C4×C31—21no—
93C4⋊C4×C31—21no—
94C4⋊C4×C31—41no—
95C4⋊C4×C31—42no—
96C4⋊C4×C31—42no—
97C4⋊C4×C31—41no—
98C4⋊C4×C31—41no—
99C4⋊C4×C31—41no—
100C4⋊C4×C31—41no—
101C4⋊C4×C31—41no—
102C22⋊C4×C31—42no—
103C22⋊C4×C31—42no—
104C22⋊C4×C31—42no—
105C22⋊C4×C31—41no—
106C22⋊C4×C31—41no—
107C22⋊C4×C31—41no—
108C22⋊C4×C31—81no—
109C42×C31—41no—
110C42×C31—41no—
111#12210C3111yesmaximal, characteristic, direct product factor, Sylow
112C23×C31—22no—
113C23×C31—41no—
114C23×C31—41no—
115C23×C31—41no—
116C23×C31—41no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31#12011yescharacteristic
120Q8×C31—42no—
121Q8×C31—81no—
122Q8×C31—81no—
123Q8×C31—81no—
124D4×C31—42no—
125D4×C31—81no—
126C2×C4×C31—21no—
127C2×C4×C31—21no—
128C2×C4×C31—22no—
129C2×C4×C31—21no—
130C2×C4×C31—21no—
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—41no—
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—41no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—84no—
150C2×C4×C31C2×SD1611yescharacteristic
151C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
152C8×C31—81no—
153C8×C31—81no—
154C8×C31—81no—
155#2728C2×C3111yescharacteristic
156#2028C2×C3111yescharacteristic, complemented
157#1611C2×C3111yescharacteristic, complemented
158#1599C2×C3111yescharacteristic
159#1346C2×C3111yescharacteristic, complemented
160#1345C2×C3111yescharacteristic
161#1319C2×C3111yescharacteristic, complemented
162C22×C31—21no—
163C22×C31—21no—
164C22×C31—42no—
165C22×C31—42no—
166C22×C31—42no—
167C22×C31—81no—
168C22×C31—81no—
169C22×C31—81no—
170C22×C31—81no—
171C22×C31#45011yescharacteristic
172C22×C31#45511yescharacteristic
173C22×C31B(4,𝔽2)11yescharacteristic
174C4×C31—21no—
175C4×C31—21no—
176C4×C31—42no—
177C4×C31—41no—
178C4×C31—41no—
179C4×C31—81no—
180C4×C31—81no—
181C4×C31—81no—
182C4×C31—81no—
183C4×C31—81no—
184C4×C31—81no—
185#544C22×C3111yescharacteristic
186C2×C22⋊Q8—21no—
187#483—21no—
188#450—21no—
189#419—21no—
190#400—21no—
191#398C22×C3111yescharacteristic
192#397—21no—
193#394C22×C3111yescharacteristic
194#391C22×C3111yescharacteristic
195#388—21no—
196#386—21no—
197#385—21no—
198#343C22×C3111yescharacteristic
199#339—21no—
200#328—21no—
201#328—21no—
202#324C22×C3111yescharacteristic
203#324C22×C3111yescharacteristic
204C2×C31—21no—
205C2×C31—21no—
206C2×C31—82no—
207C2×C31#160111yescharacteristic
208C2×C31#219711yescharacteristic
209C2×C31#220311yescharacteristic
210C22×Q8D4×C3111yescharacteristic
211C22×D4D4×C3111yescharacteristic
212C23×C4—21no—
213C2×SD16—41no—
214C2×M16—21no—
215C4⋊Q8—41no—
216#125—41no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C22⋊Q8—42no—
220C22⋊Q8—42no—
221#117—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4D4×C3111yescharacteristic
226C2×C4⋊C4C23×C3111yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C42D4×C3111yescharacteristic
233#106—41no—
234#103—41no—
235#103—41no—
236#103—41no—
237#102—41no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95D4×C3111yescharacteristic
246#95D4×C3111yescharacteristic
247C31#1221011yescharacteristic, direct product factor, Sylow
248C24—21no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8SD16×C3112yescomplemented
255C2×D4—41no—
256C2×D4—42no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4C2×D4×C3111yescharacteristic
266C22×C4C2×D4×C3111yescharacteristic
267C22×C4C2×D4×C3111yescharacteristic
268SD16—82no—
269M16—41no—
270M16—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—81no—
293C42—41no—
294C42—41no—
295C23—22no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23C31×#12011yescharacteristic
303Q8—42no—
304Q8—81no—
305Q8—81no—
306Q8—81no—
307D4—42no—
308D4—81no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
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—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—84no—
333C2×C4C2×SD16×C3111yescharacteristic
334C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
335C8—81no—
336C8—81no—
337C8—81no—
338C22—21no—
339C22—21no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22C31×#45011yescharacteristic
348C22C31×#45511yescharacteristic
349C22C31×B(4,𝔽2)11yescharacteristic
350C4—21no—
351C4—21no—
352C4—42no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—81no—
361C2—21no—
362C2—21no—
363C2—82no—
364C2C31×#160111yescharacteristic
365C2C31×#219711yescharacteristic
366C2C31×#220311yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2028C2semidirect
3. C31×#1611C2semidirect
5. C31×#1346C2semidirect
7. C31×#1319C2semidirect
70. C2×Q8×C31SD16semidirect
111. #12210C31direct (N × H)
156. #2028C2×C31semidirect
157. #1611C2×C31semidirect
159. #1346C2×C31semidirect
161. #1319C2×C31semidirect
247. C31#12210direct (N × H)
254. C2×Q8SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#27282
C31×#20282
C31×#16112
C31×#15992
C31×#13462
C31×#13452
C31×#13192
#1221031