Finite Group Databasebeta

← C31×#8540 #21465299

Subgroups — C31×#8540 #21465299

Order 7936.

718 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 142 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×#3491C211yesmaximal, characteristic
2C31×#1822C211yesmaximal, characteristic, complemented
3C31×#1818C211yesmaximal, characteristic, complemented
4C31×#1468C214yesmaximal, complemented
5C31×#552—24no—
6C31×#551—24no—
7C31×#543C411yescharacteristic
8C31×#539—22no—
9C31×#534C411yescharacteristic
10C31×#529—22no—
11C2×C31×#123C2212yes—
12C31×#518C412yes—
13C2×C31×#117C2211yescharacteristic
14C31×#407C2212yes—
15C2×C31×C22⋊C8C2212yes—
16C31×#343—22no—
17C31×#336—22no—
18C31×#324—24no—
19C31×#323—24no—
20C22×D4×C31C2×C412yes—
21C23×C4×C31C2311yescharacteristic
22C2×M16×C31—22no—
23C22×C8×C31—22no—
24C31×#126—24no—
25C31×#126—24no—
26C31×#125—22no—
27C31×#125—24no—
28C31×#124—22no—
29C31×#123—24no—
30C31×#123—22no—
31C31×#123—22no—
32C31×#123—22no—
33C31×#123—24no—
34C31×C22⋊Q8—24no—
35C31×C22⋊Q8—22no—
36C31×C22⋊Q8—24no—
37C31×#121—22no—
38C31×#120—22no—
39C4×Q8×C31—22no—
40C4×D4×C31—22no—
41C4×D4×C31—24no—
42C4×D4×C31—24no—
43C31×#117—24no—
44C31×#117—22no—
45C31×#117—24no—
46C2×C4⋊C4×C31D412yes—
47C2×C4⋊C4×C31D414yes—
48C2×C22⋊C4×C31—22no—
49C2×C22⋊C4×C31C2×C411yescharacteristic
50C2×C22⋊C4×C31C2×C411yescharacteristic
51C2×C22⋊C4×C31C2×C412yes—
52C2×C22⋊C4×C31D414yes—
53C2×C42×C31D412yes—
54C31×C22⋊C8—24no—
55C31×C22⋊C8—24no—
56C24×C31C22⋊C412yes—
57C24×C31C22×C411yescharacteristic
58C2×Q8×C31—22no—
59C2×D4×C31—22no—
60C2×D4×C31—22no—
61C2×D4×C31—22no—
62C2×D4×C31—44no—
63C22×C4×C31—24no—
64C22×C4×C31—42no—
65C22×C4×C31C22⋊C412yes—
66C22×C4×C31C22⋊C412yes—
67C22×C4×C31C22⋊C414yes—
68C22×C4×C31C22⋊C412yes—
69C22×C4×C31C2×D411yescharacteristic
70C22×C4×C31C2×D411yescharacteristic
71C22×C4×C31C2×D414yes—
72M16×C31—44no—
73C2×C8×C31—44no—
74C2×C8×C31—42no—
75C2×C8×C31—42no—
76C4⋊C4×C31—24no—
77C4⋊C4×C31—24no—
78C4⋊C4×C31—22no—
79C4⋊C4×C31—22no—
80C4⋊C4×C31—24no—
81C4⋊C4×C31—24no—
82C4⋊C4×C31—24no—
83C22⋊C4×C31—24no—
84C22⋊C4×C31—24no—
85C22⋊C4×C31—22no—
86C22⋊C4×C31—22no—
87C22⋊C4×C31—22no—
88C22⋊C4×C31—22no—
89C22⋊C4×C31—24no—
90C22⋊C4×C31—24no—
91C42×C31—24no—
92C42×C31—22no—
93C42×C31—22no—
94#8540C3111yesmaximal, characteristic, direct product factor, Sylow
95C23×C31—22no—
96C23×C31—22no—
97C23×C31—42no—
98C23×C31—42no—
99C23×C31—44no—
100C23×C31C2×C22⋊C412yes—
101C23×C31C2×C22⋊C411yescharacteristic
102Q8×C31—82no—
103D4×C31—44no—
104D4×C31—44no—
105D4×C31—82no—
106C2×C4×C31—22no—
107C2×C4×C31—22no—
108C2×C4×C31—22no—
109C2×C4×C31—22no—
110C2×C4×C31—22no—
111C2×C4×C31—24no—
112C2×C4×C31—44no—
113C2×C4×C31—42no—
114C2×C4×C31—44no—
115C2×C4×C31—42no—
116C2×C4×C31—42no—
117C2×C4×C31—84no—
118C2×C4×C31#12012yes—
119C2×C4×C31#12012yes—
120C8×C31—82no—
121C8×C31—82no—
122#3491C2×C3111yescharacteristic
123#1822C2×C3111yescharacteristic, complemented
124#1818C2×C3111yescharacteristic, complemented
125#1468C2×C3114yescomplemented
126C22×C31—22no—
127C22×C31—22no—
128C22×C31—24no—
129C22×C31—42no—
130C22×C31—44no—
131C22×C31—82no—
132C22×C31—162no—
133C22×C31#37911yescharacteristic
134C22×C31#41012yes—
135C4×C31—22no—
136C4×C31—22no—
137C4×C31—42no—
138C4×C31—42no—
139C4×C31—44no—
140C4×C31—42no—
141#552—24no—
142#551—24no—
143#543C4×C3111yescharacteristic
144#539—22no—
145#534C4×C3111yescharacteristic
146#529—22no—
147C2×#123C22×C3112yes—
148#518C4×C3112yes—
149C2×#117C22×C3111yescharacteristic
150#407C22×C3112yes—
151C2×C22⋊C8C22×C3112yes—
152#343—22no—
153#336—22no—
154#324—24no—
155#323—24no—
156C2×C31—22no—
157C2×C31—42no—
158C2×C31—82no—
159C2×C31#178211yescharacteristic
160C2×C31#179112yes—
161C22×D4C2×C4×C3112yes—
162C23×C4C23×C3111yescharacteristic
163C2×M16—22no—
164C22×C8—22no—
165#126—24no—
166#126—24no—
167#125—22no—
168#125—24no—
169#124—22no—
170#123—24no—
171#123—22no—
172#123—22no—
173#123—22no—
174#123—24no—
175C22⋊Q8—24no—
176C22⋊Q8—22no—
177C22⋊Q8—24no—
178#121—22no—
179#120—22no—
180C4×Q8—22no—
181C4×D4—22no—
182C4×D4—24no—
183C4×D4—24no—
184#117—24no—
185#117—22no—
186#117—24no—
187C2×C4⋊C4D4×C3112yes—
188C2×C4⋊C4D4×C3114yes—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C2×C4×C3111yescharacteristic
191C2×C22⋊C4C2×C4×C3111yescharacteristic
192C2×C22⋊C4C2×C4×C3112yes—
193C2×C22⋊C4D4×C3114yes—
194C2×C42D4×C3112yes—
195C22⋊C8—24no—
196C22⋊C8—24no—
197C31#854011yescharacteristic, direct product factor, Sylow
198C24C22⋊C4×C3112yes—
199C24C22×C4×C3111yescharacteristic
200C2×Q8—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—44no—
205C22×C4—24no—
206C22×C4—42no—
207C22×C4C22⋊C4×C3112yes—
208C22×C4C22⋊C4×C3112yes—
209C22×C4C22⋊C4×C3114yes—
210C22×C4C22⋊C4×C3112yes—
211C22×C4C2×D4×C3111yescharacteristic
212C22×C4C2×D4×C3111yescharacteristic
213C22×C4C2×D4×C3114yes—
214M16—44no—
215C2×C8—44no—
216C2×C8—42no—
217C2×C8—42no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C42—24no—
234C42—22no—
235C42—22no—
236C23—22no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23—44no—
241C23C2×C22⋊C4×C3112yes—
242C23C2×C22⋊C4×C3111yescharacteristic
243Q8—82no—
244D4—44no—
245D4—44no—
246D4—82no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—84no—
259C2×C4C31×#12012yes—
260C2×C4C31×#12012yes—
261C8—82no—
262C8—82no—
263C22—22no—
264C22—22no—
265C22—24no—
266C22—42no—
267C22—44no—
268C22—82no—
269C22—162no—
270C22C31×#37911yescharacteristic
271C22C31×#41012yes—
272C4—22no—
273C4—22no—
274C4—42no—
275C4—42no—
276C4—44no—
277C4—42no—
278C2—22no—
279C2—42no—
280C2—82no—
281C2C31×#178211yescharacteristic
282C2C31×#179112yes—
subgroupG/H
center, socleC22×C31#379
commutatorC24C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC31×#534C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1822C2semidirect
3. C31×#1818C2semidirect
4. C31×#1468C2semidirect
94. #8540C31direct (N × H)
123. #1822C2×C31semidirect
124. #1818C2×C31semidirect
125. #1468C2×C31semidirect
197. C31#8540direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34912
C31×#18222
C31×#18182
C31×#14682
#854031