Finite Group Databasebeta

← C31×#12808 #21469567

Subgroups — C31×#12808 #21469567

Order 7936.

420 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2932C211yesmaximal, characteristic
2C31×#2126C212yesmaximal
3C31×#1812C212yesmaximal, complemented
4C31×#1391C212yesmaximal, complemented
5C2×C31×#131C411yescharacteristic, complemented
6C2×C31×#131C2211yescharacteristic
7C31×#532C2211yescharacteristic
8C31×#435—22no—
9C31×#435—22no—
10C31×#427—22no—
11C31×#423C411yescharacteristic, complemented
12C31×#410C2214yes—
13C31×#408C411yescharacteristic, complemented
14C31×#408C2211yescharacteristic
15C2×C31×#97C411yescharacteristic, complemented
16C31×#354—22no—
17C31×#353—22no—
18C31×H(3,ℤ/4)—22no—
19C2×#41×C31C2×C411yescharacteristic
20C2×#41×C31C2311yescharacteristic
21C31×#131—42no—
22C31×#131—42no—
23C2×M16×C31—21no—
24C2×M16×C31C2×C411yescharacteristic
25C2×M16×C31C2×C411yescharacteristic
26C2×M16×C31D412yes—
27C22×C8×C31—21no—
28C22×C8×C31C2×C411yescharacteristic
29C22×C8×C31C2×C411yescharacteristic
30C22×C8×C31D412yes—
31C31×C4⋊Q8Q811yescharacteristic
32C31×#127D411yescharacteristic
33C31×#124D411yescharacteristic
34C31×#124Q811yescharacteristic
35C31×#121—22no—
36C4×D4×C31—22no—
37C31×#117—24no—
38C2×C42×C31C2×C411yescharacteristic
39C31×#107—42no—
40C31×#106—42no—
41C31×C4⋊C8—22no—
42C31×#99—24no—
43C31×#99—24no—
44C31×C22⋊C8—22no—
45C31×C22⋊C8—22no—
46C31×#97—21no—
47C31×#97—21no—
48#41×C31—41no—
49#41×C31—41no—
50#41×C31—41no—
51#41×C31—41no—
52C2×Q8×C31C4⋊C411yescharacteristic
53C2×Q8×C31C2×Q811yescharacteristic
54C2×D4×C31—21no—
55C2×D4×C31—22no—
56C2×D4×C31C4⋊C411yescharacteristic
57C2×D4×C31C2×D411yescharacteristic
58C2×D4×C31#4112yes—
59C22×C4×C31—21no—
60C22×C4×C31—21no—
61C22×C4×C31C22×C411yescharacteristic
62C22×C4×C31C2×D412yes—
63M16×C31—41no—
64M16×C31—42no—
65M16×C31—42no—
66M16×C31—41no—
67C2×C8×C31—22no—
68C2×C8×C31—22no—
69C2×C8×C31—22no—
70C2×C8×C31—21no—
71C2×C8×C31—21no—
72C2×C8×C31—42no—
73C2×C8×C31—41no—
74C2×C8×C31C22⋊C412yes—
75C2×C8×C31C22⋊C411yescharacteristic
76C2×C8×C31C22⋊C411yescharacteristic
77C4⋊C4×C31—22no—
78C4⋊C4×C31—44no—
79C22⋊C4×C31—22no—
80C22⋊C4×C31—22no—
81C22⋊C4×C31—44no—
82C42×C31—21no—
83C42×C31—44no—
84C42×C31C4⋊C411yescharacteristic
85C42×C31C4⋊C411yescharacteristic
86#12808C3111yesmaximal, characteristic, direct product factor, Sylow
87C23×C31—21no—
88C23×C31#11711yescharacteristic
89C23×C31#12112yes—
90Q8×C31—41no—
91Q8×C31—41no—
92D4×C31—41no—
93D4×C31—42no—
94D4×C31—41no—
95D4×C31—42no—
96D4×C31—82no—
97C2×C4×C31—22no—
98C2×C4×C31—21no—
99C2×C4×C31—22no—
100C2×C4×C31—21no—
101C2×C4×C31—21no—
102C2×C4×C31—21no—
103C2×C4×C31—21no—
104C2×C4×C31—41no—
105C2×C4×C31—41no—
106C2×C4×C31—81no—
107C2×C4×C31—84no—
108C2×C4×C31C2×C22⋊C411yescharacteristic
109C2×C4×C31C2×C4⋊C411yescharacteristic
110C2×C4×C31C22⋊Q812yes—
111C8×C31—21no—
112C8×C31—22no—
113C8×C31—41no—
114C8×C31—41no—
115C8×C31#9912yes—
116#2932C2×C3111yescharacteristic
117#2126C2×C3112yes—
118#1812C2×C3112yescomplemented
119#1391C2×C3112yescomplemented
120C22×C31—22no—
121C22×C31—21no—
122C22×C31—41no—
123C22×C31—42no—
124C22×C31—42no—
125C22×C31—41no—
126C22×C31#38011yescharacteristic
127C4×C31—21no—
128C4×C31—22no—
129C4×C31—41no—
130C4×C31—41no—
131C4×C31—41no—
132C4×C31—164no—
133C4×C31C2×#9911yescharacteristic
134C4×C31#41011yescharacteristic
135C2×#131C4×C3111yescharacteristic, complemented
136C2×#131C22×C3111yescharacteristic
137#532C22×C3111yescharacteristic
138#435—22no—
139#435—22no—
140#427—22no—
141#423C4×C3111yescharacteristic, complemented
142#410C22×C3114yes—
143#408C4×C3111yescharacteristic, complemented
144#408C22×C3111yescharacteristic
145C2×#97C4×C3111yescharacteristic, complemented
146#354—22no—
147#353—22no—
148H(3,ℤ/4)—22no—
149C2×C31—21no—
150C2×C31—42no—
151C2×C31—41no—
152C2×C31—81no—
153C2×C31#179911yescharacteristic
154C2×#41C2×C4×C3111yescharacteristic
155C2×#41C23×C3111yescharacteristic
156#131—42no—
157#131—42no—
158C2×M16—21no—
159C2×M16C2×C4×C3111yescharacteristic
160C2×M16C2×C4×C3111yescharacteristic
161C2×M16D4×C3112yes—
162C22×C8—21no—
163C22×C8C2×C4×C3111yescharacteristic
164C22×C8C2×C4×C3111yescharacteristic
165C22×C8D4×C3112yes—
166C4⋊Q8Q8×C3111yescharacteristic
167#127D4×C3111yescharacteristic
168#124D4×C3111yescharacteristic
169#124Q8×C3111yescharacteristic
170#121—22no—
171C4×D4—22no—
172#117—24no—
173C2×C42C2×C4×C3111yescharacteristic
174#107—42no—
175#106—42no—
176C4⋊C8—22no—
177#99—24no—
178#99—24no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181#97—21no—
182#97—21no—
183C31#1280811yescharacteristic, direct product factor, Sylow
184#41—41no—
185#41—41no—
186#41—41no—
187#41—41no—
188C2×Q8C4⋊C4×C3111yescharacteristic
189C2×Q8C2×Q8×C3111yescharacteristic
190C2×D4—21no—
191C2×D4—22no—
192C2×D4C4⋊C4×C3111yescharacteristic
193C2×D4C2×D4×C3111yescharacteristic
194C2×D4#41×C3112yes—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4C22×C4×C3111yescharacteristic
198C22×C4C2×D4×C3112yes—
199M16—41no—
200M16—42no—
201M16—42no—
202M16—41no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—42no—
209C2×C8—41no—
210C2×C8C22⋊C4×C3112yes—
211C2×C8C22⋊C4×C3111yescharacteristic
212C2×C8C22⋊C4×C3111yescharacteristic
213C4⋊C4—22no—
214C4⋊C4—44no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—44no—
218C42—21no—
219C42—44no—
220C42C4⋊C4×C3111yescharacteristic
221C42C4⋊C4×C3111yescharacteristic
222C23—21no—
223C23C31×#11711yescharacteristic
224C23C31×#12112yes—
225Q8—41no—
226Q8—41no—
227D4—41no—
228D4—42no—
229D4—41no—
230D4—42no—
231D4—82no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—81no—
242C2×C4—84no—
243C2×C4C2×C22⋊C4×C3111yescharacteristic
244C2×C4C2×C4⋊C4×C3111yescharacteristic
245C2×C4C31×C22⋊Q812yes—
246C8—21no—
247C8—22no—
248C8—41no—
249C8—41no—
250C8C31×#9912yes—
251C22—22no—
252C22—21no—
253C22—41no—
254C22—42no—
255C22—42no—
256C22—41no—
257C22C31×#38011yescharacteristic
258C4—21no—
259C4—22no—
260C4—41no—
261C4—41no—
262C4—41no—
263C4—164no—
264C4C2×C31×#9911yescharacteristic
265C4C31×#41011yescharacteristic
266C2—21no—
267C2—42no—
268C2—41no—
269C2—81no—
270C2C31×#179911yescharacteristic
subgroupG/H
centerC4×C31#410
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×#41C23×C31
socleC2×C31#1799
prime radicalC31×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C31 ◁ C2×C8×C31 ◁ C2×C31×#131 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1812C2semidirect
4. C31×#1391C2semidirect
5. C2×C31×#131C4semidirect
11. C31×#423C4semidirect
13. C31×#408C4semidirect
15. C2×C31×#97C4semidirect
86. #12808C31direct (N × H)
118. #1812C2×C31semidirect
119. #1391C2×C31semidirect
135. C2×#131C4×C31semidirect
141. #423C4×C31semidirect
143. #408C4×C31semidirect
145. C2×#97C4×C31semidirect
183. C31#12808direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#29322
C31×#21262
C31×#18122
C31×#13912
#1280831