Finite Group Databasebeta

← C31×#12239 #21468998

Subgroups — C31×#12239 #21468998

Order 7936.

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

normal subgroups only

This group has 378 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×#2785C211yesmaximal, characteristic
2C31×#2061C211yesmaximal, characteristic, complemented
3C31×#1605C211yesmaximal, characteristic, complemented
4C31×#1599C211yesmaximal, characteristic
5C31×#1354C211yesmaximal, characteristic, complemented
6C31×#1345C211yesmaximal, characteristic
7C31×#1324C211yesmaximal, characteristic, complemented
8C31×#544C2211yescharacteristic
9C31×#451—21no—
10C31×#450—21no—
11C31×#428—21no—
12C31×#395C2211yescharacteristic
13C31×#394—21no—
14C31×#394C2211yescharacteristic
15C31×#393—21no—
16C31×#388—21no—
17C31×#388—21no—
18C31×#388—21no—
19C31×#382—21no—
20C31×#382C2211yescharacteristic
21C31×#343C2211yescharacteristic
22C31×#339—21no—
23C31×#328—21no—
24C31×#328—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×D4×C31D411yescharacteristic
29C2×Q16×C31—41no—
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×C31—21no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31D411yescharacteristic
40C2×C4⋊C4×C31C2311yescharacteristic
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C42×C31—21no—
48C2×C42×C31D411yescharacteristic
49C31×#107—41no—
50C31×#106—41no—
51C31×#103—41no—
52C31×#102—41no—
53C31×C22⋊C8—21no—
54C31×C22⋊C8—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—41no—
66C2×Q8×C31—81no—
67C2×D4×C31—41no—
68C2×D4×C31—81no—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31C2×D411yescharacteristic
75C22×C4×C31C2×D411yescharacteristic
76C22×C4×C31C2×D411yescharacteristic
77Q16×C31—82no—
78SD16×C31—82no—
79M16×C31—22no—
80M16×C31—41no—
81C2×C8×C31—41no—
82C2×C8×C31—41no—
83C2×C8×C31—41no—
84C4⋊C4×C31—21no—
85C4⋊C4×C31—21no—
86C4⋊C4×C31—21no—
87C4⋊C4×C31—21no—
88C4⋊C4×C31—41no—
89C4⋊C4×C31—41no—
90C4⋊C4×C31—41no—
91C4⋊C4×C31—41no—
92C4⋊C4×C31—81no—
93C4⋊C4×C31—81no—
94C22⋊C4×C31—42no—
95C22⋊C4×C31—42no—
96C22⋊C4×C31—41no—
97C22⋊C4×C31—41no—
98C22⋊C4×C31—41no—
99C22⋊C4×C31—81no—
100C22⋊C4×C31—81no—
101C22⋊C4×C31—81no—
102C42×C31—21no—
103C42×C31—21no—
104C42×C31—42no—
105C42×C31—41no—
106#12239C3111yesmaximal, characteristic, direct product factor, Sylow
107C23×C31—41no—
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#13511yescharacteristic
145C2×C4×C31#13711yescharacteristic
146C8×C31—42no—
147C8×C31—81no—
148C8×C31—81no—
149#2785C2×C3111yescharacteristic
150#2061C2×C3111yescharacteristic, complemented
151#1605C2×C3111yescharacteristic, complemented
152#1599C2×C3111yescharacteristic
153#1354C2×C3111yescharacteristic, complemented
154#1345C2×C3111yescharacteristic
155#1324C2×C3111yescharacteristic, complemented
156C22×C31—21no—
157C22×C31—21no—
158C22×C31—81no—
159C22×C31—81no—
160C22×C31—81no—
161C22×C31—81no—
162C22×C31—81no—
163C22×C31—81no—
164C22×C31—81no—
165C22×C31#45211yescharacteristic
166C22×C31#45611yescharacteristic
167C22×C31B(4,𝔽2)11yescharacteristic
168C4×C31—21no—
169C4×C31—21no—
170C4×C31—42no—
171C4×C31—41no—
172C4×C31—41no—
173C4×C31—81no—
174C4×C31—81no—
175C4×C31—81no—
176C4×C31—81no—
177C4×C31—81no—
178C4×C31—81no—
179#544C22×C3111yescharacteristic
180#451—21no—
181#450—21no—
182#428—21no—
183#395C22×C3111yescharacteristic
184#394—21no—
185#394C22×C3111yescharacteristic
186#393—21no—
187#388—21no—
188#388—21no—
189#388—21no—
190#382—21no—
191#382C22×C3111yescharacteristic
192#343C22×C3111yescharacteristic
193#339—21no—
194#328—21no—
195#328—21no—
196#324C22×C3111yescharacteristic
197#324C22×C3111yescharacteristic
198C2×C31—21no—
199C2×C31—21no—
200C2×C31—161no—
201C2×C31#160311yescharacteristic
202C2×C31#219911yescharacteristic
203C2×C31#220411yescharacteristic
204C22×Q8D4×C3111yescharacteristic
205C22×D4D4×C3111yescharacteristic
206C2×Q16—41no—
207C2×SD16—41no—
208C2×M16—21no—
209C4⋊Q8—41no—
210#125—41no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213#117—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4D4×C3111yescharacteristic
217C2×C4⋊C4C23×C3111yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C42—21no—
225C2×C42D4×C3111yescharacteristic
226#107—41no—
227#106—41no—
228#103—41no—
229#102—41no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95D4×C3111yescharacteristic
238#95D4×C3111yescharacteristic
239C31#1223911yescharacteristic, direct product factor, Sylow
240C24—21no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—41no—
244C2×Q8—81no—
245C2×D4—41no—
246C2×D4—81no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4C2×D4×C3111yescharacteristic
253C22×C4C2×D4×C3111yescharacteristic
254C22×C4C2×D4×C3111yescharacteristic
255Q16—82no—
256SD16—82no—
257M16—22no—
258M16—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—81no—
271C4⋊C4—81no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—81no—
278C22⋊C4—81no—
279C22⋊C4—81no—
280C42—21no—
281C42—21no—
282C42—42no—
283C42—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C31×#12011yescharacteristic
292Q8—42no—
293Q8—81no—
294Q8—81no—
295Q8—81no—
296D4—42no—
297D4—81no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
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×C4C31×#13511yescharacteristic
322C2×C4C31×#13711yescharacteristic
323C8—42no—
324C8—81no—
325C8—81no—
326C22—21no—
327C22—21no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22C31×#45211yescharacteristic
336C22C31×#45611yescharacteristic
337C22C31×B(4,𝔽2)11yescharacteristic
338C4—21no—
339C4—21no—
340C4—42no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C2—21no—
350C2—21no—
351C2—161no—
352C2C31×#160311yescharacteristic
353C2C31×#219911yescharacteristic
354C2C31×#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×#2061C2semidirect
3. C31×#1605C2semidirect
5. C31×#1354C2semidirect
7. C31×#1324C2semidirect
106. #12239C31direct (N × H)
150. #2061C2×C31semidirect
151. #1605C2×C31semidirect
153. #1354C2×C31semidirect
155. #1324C2×C31semidirect
239. C31#12239direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27852
C31×#20612
C31×#16052
C31×#15992
C31×#13542
C31×#13452
C31×#13242
#1223931