Finite Group Databasebeta

← C31×#18531 #21475290

Subgroups — C31×#18531 #21475290

Order 7936.

982 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 230 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
1C2×C31×#559C211yesmaximal, characteristic
2C31×#3138C211yesmaximal, characteristic
3C31×#3113C212yesmaximal, complemented
4C31×#1640C218yesmaximal, complemented
5C31×#1555C211yesmaximal, characteristic
6C31×#1491C212yesmaximal, complemented
7C31×#559—22no—
8C31×#559—24no—
9C31×#559C2214yes—
10C31×#533C2211yescharacteristic
11C2×C31×#127C2211yescharacteristic
12C2×C31×#124—22no—
13C2×C31×#123—22no—
14C2×C4×Q8×C31C2212yes—
15C2×C4×D4×C31—22no—
16C31×#490—22no—
17C31×#486C2214yes—
18C31×#483—22no—
19C31×#475C2214yescomplemented
20C31×#462C2214yescomplemented
21C31×#451—22no—
22C31×#441—24no—
23C31×#432C2212yes—
24C31×#418—22no—
25C4×M16×C31C2211yescharacteristic
26C31×#331C2214yes—
27C31×Q8⋊C8C2218yescomplemented
28C22×Q8×C31D412yes—
29C22×D4×C31—22no—
30C22×D4×C31D412yes—
31C23×C4×C31—22no—
32C2×Q16×C31—24no—
33C2×M16×C31—21no—
34C31×C4⋊Q8C2312yes—
35C31×#127—22no—
36C31×#127D412yes—
37C31×#127C2312yes—
38C31×#125—21no—
39C31×#124—28no—
40C31×#124—28no—
41C31×#123—28no—
42C31×#123—28no—
43C31×C22⋊Q8—22no—
44C4×Q8×C31—22no—
45C4×Q8×C31—22no—
46C4×Q8×C31—22no—
47C4×Q8×C31D414yes—
48C4×Q8×C31C2314yes—
49C4×D4×C31—28no—
50C4×D4×C31—28no—
51C31×#117—21no—
52C2×C4⋊C4×C31—22no—
53C2×C4⋊C4×C31D412yes—
54C2×C22⋊C4×C31—24no—
55C2×C22⋊C4×C31—22no—
56C2×C42×C31—22no—
57C2×C42×C31C2311yescharacteristic
58C31×#106—24no—
59C31×C4⋊C8C2314yes—
60C31×#103—24no—
61C31×#103—24no—
62C31×#102—28no—
63C31×C22⋊C8—22no—
64C31×#97—21no—
65C31×#97—22no—
66C4×C8×C31C2312yes—
67C24×C31—22no—
68C2×Q8×C31—22no—
69C2×Q8×C31—22no—
70C2×Q8×C31—44no—
71C2×Q8×C31C2×D414yes—
72C2×D4×C31—24no—
73C2×D4×C31—22no—
74C2×D4×C31—24no—
75C2×D4×C31—48no—
76C2×D4×C31—44no—
77C2×D4×C31C2×D414yes—
78C22×C4×C31—22no—
79C22×C4×C31—22no—
80C22×C4×C31—28no—
81C22×C4×C31—84no—
82C22×C4×C31C2×D412yes—
83C22×C4×C31C2×D411yescharacteristic
84Q16×C31—48no—
85M16×C31—42no—
86C2×C8×C31—22no—
87C2×C8×C31—24no—
88C4⋊C4×C31—24no—
89C4⋊C4×C31—22no—
90C4⋊C4×C31—24no—
91C4⋊C4×C31—22no—
92C4⋊C4×C31—22no—
93C4⋊C4×C31—42no—
94C4⋊C4×C31C2×D414yes—
95C22⋊C4×C31—216no—
96C22⋊C4×C31—28no—
97C22⋊C4×C31—41no—
98C42×C31—24no—
99C42×C31—24no—
100C42×C31—41no—
101C42×C31C2×D412yes—
102C42×C31C2×D411yescharacteristic
103C42×C31C2411yescharacteristic
104#18531C3111yesmaximal, characteristic, direct product factor, Sylow
105C23×C31—24no—
106C23×C31—48no—
107C23×C31—44no—
108C23×C31#12011yescharacteristic
109Q8×C31—24no—
110Q8×C31—24no—
111Q8×C31—44no—
112Q8×C31—44no—
113D4×C31—44no—
114D4×C31—48no—
115D4×C31—48no—
116D4×C31—82no—
117C2×C4×C31—22no—
118C2×C4×C31—24no—
119C2×C4×C31—24no—
120C2×C4×C31—24no—
121C2×C4×C31—44no—
122C2×C4×C31—44no—
123C2×C4×C31—42no—
124C2×C4×C31—88no—
125C2×C4×C31#12012yes—
126C2×C4×C31#12011yescharacteristic
127C2×C4×C31C22×D411yescharacteristic
128C2×C4×C31C22×D412yes—
129C8×C31—42no—
130C8×C31—44no—
131C2×#559C2×C3111yescharacteristic
132#3138C2×C3111yescharacteristic
133#3113C2×C3112yescomplemented
134#1640C2×C3118yescomplemented
135#1555C2×C3111yescharacteristic
136#1491C2×C3112yescomplemented
137C22×C31—22no—
138C22×C31—48no—
139C22×C31—44no—
140C22×C31—88no—
141C22×C31#45512yes—
142C22×C31C2×#12011yescharacteristic
143C4×C31—22no—
144C4×C31—21no—
145C4×C31—21no—
146C4×C31—44no—
147C4×C31—44no—
148C4×C31—82no—
149C4×C31#57514yes—
150#559—22no—
151#559—24no—
152#559C22×C3114yes—
153#533C22×C3111yescharacteristic
154C2×#127C22×C3111yescharacteristic
155C2×#124—22no—
156C2×#123—22no—
157C2×C4×Q8C22×C3112yes—
158C2×C4×D4—22no—
159#490—22no—
160#486C22×C3114yes—
161#483—22no—
162#475C22×C3114yescomplemented
163#462C22×C3114yescomplemented
164#451—22no—
165#441—24no—
166#432C22×C3112yes—
167#418—22no—
168C4×M16C22×C3111yescharacteristic
169#331C22×C3114yes—
170Q8⋊C8C22×C3118yescomplemented
171C2×C31—22no—
172C2×C31—84no—
173C2×C31#300512yes—
174C2×C31C2×#45511yescharacteristic
175C22×Q8D4×C3112yes—
176C22×D4—22no—
177C22×D4D4×C3112yes—
178C23×C4—22no—
179C2×Q16—24no—
180C2×M16—21no—
181C4⋊Q8C23×C3112yes—
182#127—22no—
183#127D4×C3112yes—
184#127C23×C3112yes—
185#125—21no—
186#124—28no—
187#124—28no—
188#123—28no—
189#123—28no—
190C22⋊Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8D4×C3114yes—
195C4×Q8C23×C3114yes—
196C4×D4—28no—
197C4×D4—28no—
198#117—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4D4×C3112yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C42—22no—
204C2×C42C23×C3111yescharacteristic
205#106—24no—
206C4⋊C8C23×C3114yes—
207#103—24no—
208#103—24no—
209#102—28no—
210C22⋊C8—22no—
211#97—21no—
212#97—22no—
213C4×C8C23×C3112yes—
214C31#1853111yescharacteristic, direct product factor, Sylow
215C24—22no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—44no—
219C2×Q8C2×D4×C3114yes—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—48no—
224C2×D4—44no—
225C2×D4C2×D4×C3114yes—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—28no—
229C22×C4—84no—
230C22×C4C2×D4×C3112yes—
231C22×C4C2×D4×C3111yescharacteristic
232Q16—48no—
233M16—42no—
234C2×C8—22no—
235C2×C8—24no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4C2×D4×C3114yes—
243C22⋊C4—216no—
244C22⋊C4—28no—
245C22⋊C4—41no—
246C42—24no—
247C42—24no—
248C42—41no—
249C42C2×D4×C3112yes—
250C42C2×D4×C3111yescharacteristic
251C42C24×C3111yescharacteristic
252C23—24no—
253C23—48no—
254C23—44no—
255C23C31×#12011yescharacteristic
256Q8—24no—
257Q8—24no—
258Q8—44no—
259Q8—44no—
260D4—44no—
261D4—48no—
262D4—48no—
263D4—82no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—88no—
272C2×C4C31×#12012yes—
273C2×C4C31×#12011yescharacteristic
274C2×C4C22×D4×C3111yescharacteristic
275C2×C4C22×D4×C3112yes—
276C8—42no—
277C8—44no—
278C22—22no—
279C22—48no—
280C22—44no—
281C22—88no—
282C22C31×#45512yes—
283C22C2×C31×#12011yescharacteristic
284C4—22no—
285C4—21no—
286C4—21no—
287C4—44no—
288C4—44no—
289C4—82no—
290C4C31×#57514yes—
291C2—22no—
292C2—84no—
293C2C31×#300512yes—
294C2C2×C31×#45511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC2×C31×#127C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#3113C2semidirect
4. C31×#1640C2semidirect
6. C31×#1491C2semidirect
19. C31×#475C22semidirect
20. C31×#462C22semidirect
27. C31×Q8⋊C8C22semidirect
104. #18531C31direct (N × H)
133. #3113C2×C31semidirect
134. #1640C2×C31semidirect
136. #1491C2×C31semidirect
162. #475C22×C31semidirect
163. #462C22×C31semidirect
170. Q8⋊C8C22×C31semidirect
214. C31#18531direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5592
C31×#31382
C31×#31132
C31×#16402
C31×#15552
C31×#14912
#1853131