Finite Group Databasebeta

← C31×#11847 #21468606

Subgroups — C31×#11847 #21468606

Order 7936.

432 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2598C211yesmaximal, characteristic
2C31×#1822C211yesmaximal, characteristic, complemented
3C31×#1458C211yesmaximal, characteristic, complemented
4C31×#1353C212yesmaximal, complemented
5C31×#1345C212yesmaximal, complemented
6C2×C4×D4×C31C411yescharacteristic
7C2×C31×#117C2211yescharacteristic
8C31×#431—21no—
9C31×#407—21no—
10C31×#400C2211yescharacteristic
11C31×#399C411yescharacteristic
12C31×#394C2211yescharacteristic
13C31×#392C411yescharacteristic
14C31×#388—21no—
15C31×#386—21no—
16C31×#386—21no—
17C31×#385—21no—
18C31×#383C411yescharacteristic
19C31×#343—21no—
20C31×#343C2212yes—
21C31×#324C2212yes—
22C31×#323—21no—
23C22×D4×C31—21no—
24C22×D4×C31C2×C411yescharacteristic
25C23×C4×C31—21no—
26C23×C4×C31D411yescharacteristic
27C2×M16×C31—22no—
28C4×D4×C31—41no—
29C4×D4×C31—41no—
30C31×#117—41no—
31C31×#117—41no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31C2×C411yescharacteristic
34C2×C4⋊C4×C31D411yescharacteristic
35C2×C4⋊C4×C31C2311yescharacteristic
36C2×C22⋊C4×C31—21no—
37C2×C22⋊C4×C31—21no—
38C2×C22⋊C4×C31—21no—
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31C2×C411yescharacteristic
42C2×C22⋊C4×C31D411yescharacteristic
43C2×C42×C31C2×C411yescharacteristic
44C2×C42×C31D411yescharacteristic
45C31×C4⋊C8—41no—
46C31×C22⋊C8—22no—
47C31×C22⋊C8—41no—
48C31×C22⋊C8—41no—
49C4×C8×C31—41no—
50C31×#95—21no—
51C31×#95—21no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95C2×C411yescharacteristic
55C31×#95C2×C411yescharacteristic
56C24×C31—21no—
57C24×C31C22⋊C411yescharacteristic
58C2×D4×C31—41no—
59C2×D4×C31—42no—
60C2×D4×C31—42no—
61C2×D4×C31—42no—
62C2×D4×C31—81no—
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—22no—
66C22×C4×C31—21no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—41no—
70C22×C4×C31—42no—
71C22×C4×C31—81no—
72C22×C4×C31—81no—
73C22×C4×C31C22⋊C411yescharacteristic
74C22×C4×C31C22⋊C411yescharacteristic
75C22×C4×C31C22⋊C411yescharacteristic
76C22×C4×C31C22×C411yescharacteristic
77C22×C4×C31C2×D411yescharacteristic
78C22×C4×C31C2×D411yescharacteristic
79M16×C31—42no—
80M16×C31—42no—
81C2×C8×C31—42no—
82C2×C8×C31—42no—
83C4⋊C4×C31—41no—
84C4⋊C4×C31—41no—
85C4⋊C4×C31—41no—
86C4⋊C4×C31—81no—
87C22⋊C4×C31—41no—
88C22⋊C4×C31—41no—
89C22⋊C4×C31—42no—
90C22⋊C4×C31—41no—
91C22⋊C4×C31—41no—
92C22⋊C4×C31—42no—
93C22⋊C4×C31—81no—
94C22⋊C4×C31—81no—
95C42×C31—21no—
96C42×C31—21no—
97C42×C31—21no—
98C42×C31—21no—
99#11847C3111yesmaximal, characteristic, direct product factor, Sylow
100C23×C31—22no—
101C23×C31—21no—
102C23×C31—21no—
103C23×C31—21no—
104C23×C31—41no—
105C23×C31—41no—
106C23×C31—41no—
107C23×C31—41no—
108C23×C31—41no—
109C23×C31—41no—
110C23×C31—41no—
111C23×C31—41no—
112C23×C31C2×C22⋊C411yescharacteristic
113D4×C31—42no—
114D4×C31—42no—
115D4×C31—84no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—42no—
125C2×C4×C31—42no—
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—84no—
140C2×C4×C31#9911yescharacteristic
141C2×C4×C31#9911yescharacteristic
142C2×C4×C31#13512yes—
143C8×C31—82no—
144C8×C31—82no—
145#2598C2×C3111yescharacteristic
146#1822C2×C3111yescharacteristic, complemented
147#1458C2×C3111yescharacteristic, complemented
148#1353C2×C3112yescomplemented
149#1345C2×C3112yescomplemented
150C22×C31—21no—
151C22×C31—21no—
152C22×C31—42no—
153C22×C31—42no—
154C22×C31—42no—
155C22×C31—42no—
156C22×C31—41no—
157C22×C31—41no—
158C22×C31—41no—
159C22×C31—81no—
160C22×C31—81no—
161C22×C31—81no—
162C22×C31—81no—
163C22×C31—81no—
164C22×C31C2×#9911yescharacteristic
165C22×C31#41611yescharacteristic
166C22×C31#42111yescharacteristic
167C4×C31—22no—
168C4×C31—21no—
169C4×C31—21no—
170C4×C31—41no—
171C4×C31—41no—
172C4×C31—41no—
173C4×C31—41no—
174C4×C31—81no—
175C4×C31—81no—
176C4×C31—81no—
177C2×C4×D4C4×C3111yescharacteristic
178C2×#117C22×C3111yescharacteristic
179#431—21no—
180#407—21no—
181#400C22×C3111yescharacteristic
182#399C4×C3111yescharacteristic
183#394C22×C3111yescharacteristic
184#392C4×C3111yescharacteristic
185#388—21no—
186#386—21no—
187#386—21no—
188#385—21no—
189#383C4×C3111yescharacteristic
190#343—21no—
191#343C22×C3112yes—
192#324C22×C3112yes—
193#323—21no—
194C2×C31—21no—
195C2×C31—21no—
196C2×C31—81no—
197C2×C31—82no—
198C2×C31#151611yescharacteristic
199C2×C31#212011yescharacteristic
200C2×C31#213411yescharacteristic
201C22×D4—21no—
202C22×D4C2×C4×C3111yescharacteristic
203C23×C4—21no—
204C23×C4D4×C3111yescharacteristic
205C2×M16—22no—
206C4×D4—41no—
207C4×D4—41no—
208#117—41no—
209#117—41no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C2×C4×C3111yescharacteristic
212C2×C4⋊C4D4×C3111yescharacteristic
213C2×C4⋊C4C23×C3111yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4C2×C4×C3111yescharacteristic
220C2×C22⋊C4D4×C3111yescharacteristic
221C2×C42C2×C4×C3111yescharacteristic
222C2×C42D4×C3111yescharacteristic
223C4⋊C8—41no—
224C22⋊C8—22no—
225C22⋊C8—41no—
226C22⋊C8—41no—
227C4×C8—41no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95C2×C4×C3111yescharacteristic
233#95C2×C4×C3111yescharacteristic
234C31#1184711yescharacteristic, direct product factor, Sylow
235C24—21no—
236C24C22⋊C4×C3111yescharacteristic
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—81no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4C22⋊C4×C3111yescharacteristic
253C22×C4C22⋊C4×C3111yescharacteristic
254C22×C4C22⋊C4×C3111yescharacteristic
255C22×C4C22×C4×C3111yescharacteristic
256C22×C4C2×D4×C3111yescharacteristic
257C22×C4C2×D4×C3111yescharacteristic
258M16—42no—
259M16—42no—
260C2×C8—42no—
261C2×C8—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—81no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—81no—
273C22⋊C4—81no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—21no—
278C23—22no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23C2×C22⋊C4×C3111yescharacteristic
291D4—42no—
292D4—42no—
293D4—84no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—42no—
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—84no—
318C2×C4C31×#9911yescharacteristic
319C2×C4C31×#9911yescharacteristic
320C2×C4C31×#13512yes—
321C8—82no—
322C8—82no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22C2×C31×#9911yescharacteristic
338C22C31×#41611yescharacteristic
339C22C31×#42111yescharacteristic
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4—81no—
350C2—21no—
351C2—21no—
352C2—81no—
353C2—82no—
354C2C31×#151611yescharacteristic
355C2C31×#212011yescharacteristic
356C2C31×#213411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC31×#392C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1822C2semidirect
3. C31×#1458C2semidirect
4. C31×#1353C2semidirect
5. C31×#1345C2semidirect
99. #11847C31direct (N × H)
146. #1822C2×C31semidirect
147. #1458C2×C31semidirect
148. #1353C2×C31semidirect
149. #1345C2×C31semidirect
234. C31#11847direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25982
C31×#18222
C31×#14582
C31×#13532
C31×#13452
#1184731