Finite Group Databasebeta

← C31×#12271 #21469030

Subgroups — C31×#12271 #21469030

Order 7936.

396 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2572C211yesmaximal, characteristic
2C31×#2028C211yesmaximal, characteristic, complemented
3C31×#1617C211yesmaximal, characteristic, complemented
4C31×#1599C211yesmaximal, characteristic
5C31×#1353C211yesmaximal, characteristic, complemented
6C31×#1345C211yesmaximal, characteristic
7C31×#1319C211yesmaximal, characteristic, complemented
8C31×#544C2211yescharacteristic
9C31×#484—21no—
10C31×#450—21no—
11C31×#419—21no—
12C31×#400C2211yescharacteristic
13C31×#394C2211yescharacteristic
14C31×#393—21no—
15C31×#391C2211yescharacteristic
16C31×#388—21no—
17C31×#386—21no—
18C31×#383—21no—
19C31×#381—21no—
20C4×C22⋊C4×C31—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×#117—21no—
37C2×C4⋊C4×C31D411yescharacteristic
38C2×C4⋊C4×C31C2311yescharacteristic
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C42×C31—21no—
45C2×C42×C31D411yescharacteristic
46C31×#107—41no—
47C31×#103—41no—
48C31×#103—41no—
49C31×#103—41no—
50C31×#102—41no—
51C31×C22⋊C8—21no—
52C31×C22⋊C8—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95D411yescharacteristic
61C31×#95D411yescharacteristic
62C24×C31—21no—
63C2×Q8×C31—21no—
64C2×Q8×C31—21no—
65C2×Q8×C31—21no—
66C2×Q8×C31—21no—
67C2×Q8×C31—41no—
68C2×Q8×C31—41no—
69C2×D4×C31—41no—
70C2×D4×C31—42no—
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—42no—
77C22×C4×C31—42no—
78C22×C4×C31—81no—
79C22×C4×C31C2×D411yescharacteristic
80C22×C4×C31C2×D411yescharacteristic
81C22×C4×C31C2×D411yescharacteristic
82SD16×C31—82no—
83M16×C31—41no—
84M16×C31—41no—
85C2×C8×C31—41no—
86C2×C8×C31—41no—
87C2×C8×C31—41no—
88C4⋊C4×C31—21no—
89C4⋊C4×C31—21no—
90C4⋊C4×C31—21no—
91C4⋊C4×C31—21no—
92C4⋊C4×C31—41no—
93C4⋊C4×C31—41no—
94C4⋊C4×C31—41no—
95C4⋊C4×C31—41no—
96C22⋊C4×C31—42no—
97C22⋊C4×C31—42no—
98C22⋊C4×C31—41no—
99C22⋊C4×C31—41no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—81no—
102C22⋊C4×C31—81no—
103C42×C31—42no—
104C42×C31—41no—
105C42×C31—41no—
106#12271C3111yesmaximal, characteristic, direct product factor, Sylow
107C23×C31—22no—
108C23×C31—41no—
109C23×C31—41no—
110C23×C31—41no—
111C23×C31—41no—
112C23×C31—41no—
113C23×C31—41no—
114C23×C31#12011yescharacteristic
115Q8×C31—42no—
116Q8×C31—81no—
117Q8×C31—81no—
118Q8×C31—81no—
119D4×C31—42no—
120D4×C31—81no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—22no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—41no—
127C2×C4×C31—41no—
128C2×C4×C31—41no—
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—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—84no—
145C2×C4×C31#13511yescharacteristic
146C2×C4×C31#13711yescharacteristic
147C8×C31—81no—
148C8×C31—81no—
149C8×C31—81no—
150#2572C2×C3111yescharacteristic
151#2028C2×C3111yescharacteristic, complemented
152#1617C2×C3111yescharacteristic, complemented
153#1599C2×C3111yescharacteristic
154#1353C2×C3111yescharacteristic, complemented
155#1345C2×C3111yescharacteristic
156#1319C2×C3111yescharacteristic, complemented
157C22×C31—21no—
158C22×C31—21no—
159C22×C31—42no—
160C22×C31—42no—
161C22×C31—42no—
162C22×C31—81no—
163C22×C31—81no—
164C22×C31—81no—
165C22×C31—81no—
166C22×C31#45211yescharacteristic
167C22×C31#45511yescharacteristic
168C22×C31B(4,𝔽2)11yescharacteristic
169C4×C31—21no—
170C4×C31—21no—
171C4×C31—42no—
172C4×C31—41no—
173C4×C31—41no—
174C4×C31—81no—
175C4×C31—81no—
176C4×C31—81no—
177C4×C31—81no—
178C4×C31—81no—
179C4×C31—81no—
180#544C22×C3111yescharacteristic
181#484—21no—
182#450—21no—
183#419—21no—
184#400C22×C3111yescharacteristic
185#394C22×C3111yescharacteristic
186#393—21no—
187#391C22×C3111yescharacteristic
188#388—21no—
189#386—21no—
190#383—21no—
191#381—21no—
192C4×C22⋊C4—21no—
193#343C22×C3111yescharacteristic
194#339—21no—
195#328—21no—
196#328—21no—
197#324C22×C3111yescharacteristic
198#324C22×C3111yescharacteristic
199C2×C31—21no—
200C2×C31—21no—
201C2×C31—82no—
202C2×C31#160611yescharacteristic
203C2×C31#219711yescharacteristic
204C2×C31#220411yescharacteristic
205C22×Q8D4×C3111yescharacteristic
206C22×D4D4×C3111yescharacteristic
207C23×C4—21no—
208C2×SD16—41no—
209C2×M16—21no—
210C4⋊Q8—41no—
211#125—41no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214#117—21no—
215C2×C4⋊C4D4×C3111yescharacteristic
216C2×C4⋊C4C23×C3111yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42—21no—
223C2×C42D4×C3111yescharacteristic
224#107—41no—
225#103—41no—
226#103—41no—
227#103—41no—
228#102—41no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95D4×C3111yescharacteristic
239#95D4×C3111yescharacteristic
240C31#1227111yescharacteristic, direct product factor, Sylow
241C24—21no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×D4—41no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—81no—
258C22×C4C2×D4×C3111yescharacteristic
259C22×C4C2×D4×C3111yescharacteristic
260C22×C4C2×D4×C3111yescharacteristic
261SD16—82no—
262M16—41no—
263M16—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—81no—
281C22⋊C4—81no—
282C42—41no—
283C42—42no—
284C42—41no—
285C23—22no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23C31×#12011yescharacteristic
293Q8—42no—
294Q8—81no—
295Q8—81no—
296Q8—81no—
297D4—42no—
298D4—81no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
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×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—84no—
323C2×C4C31×#13511yescharacteristic
324C2×C4C31×#13711yescharacteristic
325C8—81no—
326C8—81no—
327C8—81no—
328C22—21no—
329C22—21no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22C31×#45211yescharacteristic
338C22C31×#45511yescharacteristic
339C22C31×B(4,𝔽2)11yescharacteristic
340C4—21no—
341C4—21no—
342C4—42no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C2—21no—
352C2—21no—
353C2—82no—
354C2C31×#160611yescharacteristic
355C2C31×#219711yescharacteristic
356C2C31×#220411yescharacteristic
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×#1617C2semidirect
5. C31×#1353C2semidirect
7. C31×#1319C2semidirect
106. #12271C31direct (N × H)
151. #2028C2×C31semidirect
152. #1617C2×C31semidirect
154. #1353C2×C31semidirect
156. #1319C2×C31semidirect
240. C31#12271direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25722
C31×#20282
C31×#16172
C31×#15992
C31×#13532
C31×#13452
C31×#13192
#1227131