Finite Group Databasebeta

← C31×#12234 #21468993

Subgroups — C31×#12234 #21468993

Order 7936.

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

normal subgroups only

This group has 448 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×#2680C211yesmaximal, characteristic
2C31×#2060C211yesmaximal, characteristic, complemented
3C31×#1605C211yesmaximal, characteristic, complemented
4C31×#1599C211yesmaximal, characteristic
5C31×#1354C211yesmaximal, characteristic, complemented
6C31×#1345C211yesmaximal, characteristic
7C31×#1325C211yesmaximal, characteristic, complemented
8C31×#544C2211yescharacteristic
9C2×C31×#123—21no—
10C2×C31×C22⋊Q8—21no—
11C31×#451—21no—
12C31×#450—21no—
13C31×#428—21no—
14C31×#398—21no—
15C31×#397—21no—
16C31×#395C2211yescharacteristic
17C31×#394C2211yescharacteristic
18C31×#388—21no—
19C31×#385—21no—
20C31×#384C2211yescharacteristic
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×Q16×C31—41no—
31C2×SD16×C31—41no—
32C2×M16×C31—21no—
33C31×C4⋊Q8—41no—
34C31×#125—41no—
35C31×#123—44no—
36C31×C22⋊Q8—21no—
37C31×C22⋊Q8—21no—
38C31×C22⋊Q8—44no—
39C31×#117—21no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31—21no—
42C2×C4⋊C4×C31—21no—
43C2×C4⋊C4×C31—21no—
44C2×C4⋊C4×C31—21no—
45C2×C4⋊C4×C31D411yescharacteristic
46C2×C4⋊C4×C31C2311yescharacteristic
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C22⋊C4×C31—21no—
51C2×C22⋊C4×C31—21no—
52C2×C42×C31D411yescharacteristic
53C31×#107—41no—
54C31×#106—41no—
55C31×#103—41no—
56C31×#102—41no—
57C31×C22⋊C8—21no—
58C31×C22⋊C8—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—42no—
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
82Q16×C31—82no—
83SD16×C31—82no—
84M16×C31—22no—
85M16×C31—41no—
86C2×C8×C31—41no—
87C2×C8×C31—41no—
88C2×C8×C31—41no—
89C4⋊C4×C31—21no—
90C4⋊C4×C31—21no—
91C4⋊C4×C31—21no—
92C4⋊C4×C31—21no—
93C4⋊C4×C31—41no—
94C4⋊C4×C31—42no—
95C4⋊C4×C31—42no—
96C4⋊C4×C31—42no—
97C4⋊C4×C31—42no—
98C4⋊C4×C31—42no—
99C4⋊C4×C31—41no—
100C4⋊C4×C31—41no—
101C4⋊C4×C31—41no—
102C22⋊C4×C31—24no—
103C22⋊C4×C31—42no—
104C22⋊C4×C31—42no—
105C22⋊C4×C31—42no—
106C22⋊C4×C31—41no—
107C22⋊C4×C31—41no—
108C22⋊C4×C31—41no—
109C42×C31—21no—
110C42×C31—21no—
111C42×C31—41no—
112#12234C3111yesmaximal, characteristic, direct product factor, Sylow
113C23×C31—22no—
114C23×C31—41no—
115C23×C31—41no—
116C23×C31—41no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—41no—
120C23×C31#12011yescharacteristic
121Q8×C31—42no—
122Q8×C31—81no—
123Q8×C31—81no—
124Q8×C31—81no—
125D4×C31—42no—
126D4×C31—81no—
127C2×C4×C31—21no—
128C2×C4×C31—21no—
129C2×C4×C31—22no—
130C2×C4×C31—22no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
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—41no—
150C2×C4×C31—84no—
151C2×C4×C31#13511yescharacteristic
152C2×C4×C31#13711yescharacteristic
153C8×C31—42no—
154C8×C31—81no—
155C8×C31—81no—
156#2680C2×C3111yescharacteristic
157#2060C2×C3111yescharacteristic, complemented
158#1605C2×C3111yescharacteristic, complemented
159#1599C2×C3111yescharacteristic
160#1354C2×C3111yescharacteristic, complemented
161#1345C2×C3111yescharacteristic
162#1325C2×C3111yescharacteristic, complemented
163C22×C31—21no—
164C22×C31—21no—
165C22×C31—42no—
166C22×C31—42no—
167C22×C31—42no—
168C22×C31—81no—
169C22×C31—81no—
170C22×C31—81no—
171C22×C31—81no—
172C22×C31#45211yescharacteristic
173C22×C31#45611yescharacteristic
174C22×C31B(4,𝔽2)11yescharacteristic
175C4×C31—21no—
176C4×C31—21no—
177C4×C31—42no—
178C4×C31—41no—
179C4×C31—41no—
180C4×C31—42no—
181C4×C31—81no—
182C4×C31—81no—
183C4×C31—81no—
184C4×C31—81no—
185C4×C31—81no—
186#544C22×C3111yescharacteristic
187C2×#123—21no—
188C2×C22⋊Q8—21no—
189#451—21no—
190#450—21no—
191#428—21no—
192#398—21no—
193#397—21no—
194#395C22×C3111yescharacteristic
195#394C22×C3111yescharacteristic
196#388—21no—
197#385—21no—
198#384C22×C3111yescharacteristic
199#343C22×C3111yescharacteristic
200#339—21no—
201#328—21no—
202#328—21no—
203#324C22×C3111yescharacteristic
204#324C22×C3111yescharacteristic
205C2×C31—21no—
206C2×C31—21no—
207C2×C31—82no—
208C2×C31#160311yescharacteristic
209C2×C31#219611yescharacteristic
210C2×C31#220311yescharacteristic
211C22×Q8D4×C3111yescharacteristic
212C22×D4D4×C3111yescharacteristic
213C23×C4—21no—
214C2×Q16—41no—
215C2×SD16—41no—
216C2×M16—21no—
217C4⋊Q8—41no—
218#125—41no—
219#123—44no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C22⋊Q8—44no—
223#117—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4D4×C3111yescharacteristic
230C2×C4⋊C4C23×C3111yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C42D4×C3111yescharacteristic
237#107—41no—
238#106—41no—
239#103—41no—
240#102—41no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95D4×C3111yescharacteristic
247#95D4×C3111yescharacteristic
248C31#1223411yescharacteristic, direct product factor, Sylow
249C24—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—42no—
253C2×Q8—41no—
254C2×D4—41no—
255C2×D4—42no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—81no—
264C22×C4C2×D4×C3111yescharacteristic
265C22×C4C2×D4×C3111yescharacteristic
266C22×C4C2×D4×C3111yescharacteristic
267Q16—82no—
268SD16—82no—
269M16—22no—
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—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C22⋊C4—24no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C42—21no—
295C42—21no—
296C42—41no—
297C23—22no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23C31×#12011yescharacteristic
305Q8—42no—
306Q8—81no—
307Q8—81no—
308Q8—81no—
309D4—42no—
310D4—81no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
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—41no—
333C2×C4—41no—
334C2×C4—84no—
335C2×C4C31×#13511yescharacteristic
336C2×C4C31×#13711yescharacteristic
337C8—42no—
338C8—81no—
339C8—81no—
340C22—21no—
341C22—21no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22C31×#45211yescharacteristic
350C22C31×#45611yescharacteristic
351C22C31×B(4,𝔽2)11yescharacteristic
352C4—21no—
353C4—21no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—42no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C2—21no—
364C2—21no—
365C2—82no—
366C2C31×#160311yescharacteristic
367C2C31×#219611yescharacteristic
368C2C31×#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×#2060C2semidirect
3. C31×#1605C2semidirect
5. C31×#1354C2semidirect
7. C31×#1325C2semidirect
112. #12234C31direct (N × H)
157. #2060C2×C31semidirect
158. #1605C2×C31semidirect
160. #1354C2×C31semidirect
162. #1325C2×C31semidirect
248. C31#12234direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26802
C31×#20602
C31×#16052
C31×#15992
C31×#13542
C31×#13452
C31×#13252
#1223431