Finite Group Databasebeta

← C31×#12859 #21469618

Subgroups — C31×#12859 #21469618

Order 7936.

642 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2311C211yesmaximal, characteristic
2C31×#1782C212yesmaximal
3C31×#1393C214yesmaximal, complemented
4C22×#41×C31C2211yescharacteristic
5C2×C31×#117C411yescharacteristic, complemented
6C22×C4⋊C4×C31C411yescharacteristic, complemented
7C2×C31×#99—22no—
8C2×C31×#99C2214yes—
9C31×#391C412yescomplemented
10C31×#390C2212yes—
11C31×#386—22no—
12C31×#381—22no—
13C31×#380—22no—
14C2×#41×C31—24no—
15C2×#41×C31—42no—
16C2×#41×C31—42no—
17C22×Q8×C31C2×C411yescharacteristic
18C22×D4×C31D412yes—
19C22×D4×C31C2311yescharacteristic
20C23×C4×C31—21no—
21C23×C4×C31C2×C411yescharacteristic
22C23×C4×C31D412yes—
23C31×#117—42no—
24C2×C4⋊C4×C31—21no—
25C2×C4⋊C4×C31—21no—
26C2×C4⋊C4×C31—42no—
27C2×C4⋊C4×C31C2×C412yes—
28C2×C22⋊C4×C31—21no—
29C2×C22⋊C4×C31—24no—
30C2×C22⋊C4×C31—24no—
31C2×C42×C31C2×C412yes—
32C31×#99—28no—
33C31×#99—44no—
34C31×#95—24no—
35C31×#95—44no—
36C24×C31C22⋊C411yescharacteristic
37C24×C31C2×D412yes—
38#41×C31—416no—
39C2×Q8×C31—22no—
40C2×Q8×C31—41no—
41C2×Q8×C31—41no—
42C2×D4×C31—24no—
43C2×D4×C31—24no—
44C2×D4×C31—42no—
45C2×D4×C31—41no—
46C2×D4×C31—41no—
47C2×D4×C31#4114yes—
48C22×C4×C31—22no—
49C22×C4×C31—22no—
50C22×C4×C31—22no—
51C22×C4×C31—24no—
52C22×C4×C31—24no—
53C22×C4×C31—22no—
54C22×C4×C31—22no—
55C22×C4×C31—42no—
56C22×C4×C31—42no—
57C22×C4×C31—44no—
58C22×C4×C31—82no—
59C22×C4×C31C22⋊C412yes—
60C22×C4×C31C22⋊C411yescharacteristic
61C22×C4×C31C22×C411yescharacteristic
62C4⋊C4×C31—42no—
63C4⋊C4×C31—42no—
64C4⋊C4×C31—42no—
65C22⋊C4×C31—42no—
66C22⋊C4×C31—44no—
67C22⋊C4×C31—44no—
68C22⋊C4×C31—84no—
69C42×C31—42no—
70#12859C3111yesmaximal, characteristic, direct product factor, Sylow
71C23×C31—21no—
72C23×C31—21no—
73C23×C31—22no—
74C23×C31—22no—
75C23×C31—41no—
76C23×C31—41no—
77C23×C31—42no—
78C23×C31—42no—
79C23×C31#9912yes—
80C23×C31C2×C22⋊C411yescharacteristic
81C23×C31#12314yes—
82Q8×C31—44no—
83D4×C31—44no—
84D4×C31—48no—
85C2×C4×C31—28no—
86C2×C4×C31—21no—
87C2×C4×C31—21no—
88C2×C4×C31—21no—
89C2×C4×C31—21no—
90C2×C4×C31—22no—
91C2×C4×C31—22no—
92C2×C4×C31—22no—
93C2×C4×C31—48no—
94C2×C4×C31—42no—
95C2×C4×C31—44no—
96C2×C4×C31—42no—
97C2×C4×C31—42no—
98C2×C4×C31—42no—
99C2×C4×C31—84no—
100C2×C4×C31—84no—
101C2×C4×C31—84no—
102C2×C4×C31—84no—
103C2×C4×C31#9912yes—
104C2×C4×C31#11712yes—
105#2311C2×C3111yescharacteristic
106#1782C2×C3112yes—
107#1393C2×C3114yescomplemented
108C22×C31—21no—
109C22×C31—21no—
110C22×C31—22no—
111C22×C31—24no—
112C22×C31—42no—
113C22×C31—44no—
114C22×C31—42no—
115C22×C31—41no—
116C22×C31—42no—
117C22×C31—42no—
118C22×C31—41no—
119C22×C31—82no—
120C22×C31#38111yescharacteristic
121C22×C31C2×#9911yescharacteristic
122C22×C31C2×#9911yescharacteristic
123C4×C31—22no—
124C4×C31—22no—
125C4×C31—42no—
126C4×C31—82no—
127C4×C31—82no—
128C4×C31—164no—
129C22×#41C22×C3111yescharacteristic
130C2×#117C4×C3111yescharacteristic, complemented
131C22×C4⋊C4C4×C3111yescharacteristic, complemented
132C2×#99—22no—
133C2×#99C22×C3114yes—
134#391C4×C3112yescomplemented
135#390C22×C3112yes—
136#386—22no—
137#381—22no—
138#380—22no—
139C2×C31—21no—
140C2×C31—21no—
141C2×C31—44no—
142C2×C31—42no—
143C2×C31#181911yescharacteristic
144C2×C31#212811yescharacteristic
145C2×C31#212911yescharacteristic
146C2×#41—24no—
147C2×#41—42no—
148C2×#41—42no—
149C22×Q8C2×C4×C3111yescharacteristic
150C22×D4D4×C3112yes—
151C22×D4C23×C3111yescharacteristic
152C23×C4—21no—
153C23×C4C2×C4×C3111yescharacteristic
154C23×C4D4×C3112yes—
155#117—42no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C2×C4×C3112yes—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C42C2×C4×C3112yes—
164#99—28no—
165#99—44no—
166#95—24no—
167#95—44no—
168C31#1285911yescharacteristic, direct product factor, Sylow
169C24C22⋊C4×C3111yescharacteristic
170C24C2×D4×C3112yes—
171#41—416no—
172C2×Q8—22no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×D4—24no—
176C2×D4—24no—
177C2×D4—42no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4#41×C3114yes—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—82no—
192C22×C4C22⋊C4×C3112yes—
193C22×C4C22⋊C4×C3111yescharacteristic
194C22×C4C22×C4×C3111yescharacteristic
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—44no—
200C22⋊C4—44no—
201C22⋊C4—84no—
202C42—42no—
203C23—21no—
204C23—22no—
205C23—21no—
206C23—22no—
207C23—41no—
208C23—41no—
209C23—42no—
210C23—42no—
211C23C31×#9912yes—
212C23C2×C22⋊C4×C3111yescharacteristic
213C23C31×#12314yes—
214Q8—44no—
215D4—44no—
216D4—48no—
217C2×C4—28no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—48no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—84no—
232C2×C4—84no—
233C2×C4—84no—
234C2×C4—84no—
235C2×C4C31×#9912yes—
236C2×C4C31×#11712yes—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—24no—
241C22—42no—
242C22—44no—
243C22—42no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—41no—
248C22—82no—
249C22C31×#38111yescharacteristic
250C22C2×C31×#9911yescharacteristic
251C22C2×C31×#9911yescharacteristic
252C4—22no—
253C4—22no—
254C4—42no—
255C4—82no—
256C4—82no—
257C4—164no—
258C2—21no—
259C2—21no—
260C2—44no—
261C2—42no—
262C2C31×#181911yescharacteristic
263C2C31×#212811yescharacteristic
264C2C31×#212911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC22×D4C23×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1393C2semidirect
5. C2×C31×#117C4semidirect
6. C22×C4⋊C4×C31C4semidirect
9. C31×#391C4semidirect
70. #12859C31direct (N × H)
107. #1393C2×C31semidirect
130. C2×#117C4×C31semidirect
131. C22×C4⋊C4C4×C31semidirect
134. #391C4×C31semidirect
168. C31#12859direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#23112
C31×#17822
C31×#13932
#1285931