Finite Group Databasebeta

← C31×#9990 #21466749

Subgroups — C31×#9990 #21466749

Order 7936.

826 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3195C211yesmaximal, characteristic, complemented
2C2×C31×#450C212yesmaximal
3C31×#1783C211yesmaximal, characteristic
4C31×#1472C212yesmaximal, complemented
5C31×#1298C211yesmaximal, characteristic, complemented
6C23×D4×C31C2211yescharacteristic
7C22×SD16×C31—22no—
8C2×C31×C22⋊Q8C2212yes—
9C31×#483—22no—
10C31×#470—22no—
11C31×#451—22no—
12C31×#450—24no—
13C31×#450—24no—
14C2×C31×#102—22no—
15C2×C31×#100—22no—
16C31×#407C2211yescharacteristic
17C31×#407C2211yescharacteristic
18C2×C31×C22⋊C8C2212yes—
19C31×#324—24no—
20C25×C31—21no—
21C22×Q8×C31D412yes—
22C22×D4×C31—22no—
23C22×D4×C31—21no—
24C22×D4×C31—22no—
25C22×D4×C31—44no—
26C22×D4×C31D412yes—
27C23×C4×C31C2311yescharacteristic
28C2×Q16×C31—22no—
29C2×SD16×C31—28no—
30C2×SD16×C31—48no—
31C2×M16×C31—21no—
32C2×M16×C31D412yes—
33C22×C8×C31—22no—
34C31×C22⋊Q8—22no—
35C31×C22⋊Q8—22no—
36C31×C22⋊Q8—22no—
37C31×C22⋊Q8—22no—
38C2×C4⋊C4×C31—22no—
39C2×C4⋊C4×C31D412yes—
40C2×C22⋊C4×C31—22no—
41C31×#106—22no—
42C31×#103—24no—
43C31×#102—28no—
44C31×#100—44no—
45C31×C22⋊C8—22no—
46C31×C22⋊C8—22no—
47C31×C22⋊C8—22no—
48C31×C22⋊C8D414yes—
49C24×C31—22no—
50C24×C31—44no—
51C24×C31—42no—
52C24×C31—84no—
53C24×C31#4111yescharacteristic
54C2×Q8×C31—22no—
55C2×Q8×C31—22no—
56C2×Q8×C31—44no—
57C2×D4×C31—28no—
58C2×D4×C31—24no—
59C2×D4×C31—24no—
60C2×D4×C31—44no—
61C2×D4×C31—42no—
62C2×D4×C31—44no—
63C2×D4×C31—48no—
64C2×D4×C31—41no—
65C2×D4×C31—41no—
66C22×C4×C31—22no—
67C22×C4×C31—22no—
68C22×C4×C31—42no—
69C22×C4×C31C2×D412yes—
70C22×C4×C31C2×D411yescharacteristic
71C22×C4×C31C2×D411yescharacteristic
72C22×C4×C31C2×D412yes—
73Q16×C31—82no—
74SD16×C31—416no—
75M16×C31—42no—
76M16×C31—42no—
77C2×C8×C31—22no—
78C2×C8×C31—24no—
79C2×C8×C31—44no—
80C2×C8×C31—41no—
81C4⋊C4×C31—22no—
82C4⋊C4×C31—22no—
83C4⋊C4×C31—42no—
84C4⋊C4×C31—42no—
85C22⋊C4×C31—42no—
86C22⋊C4×C31—42no—
87#9990C3111yesmaximal, characteristic, direct product factor, Sylow
88C23×C31—24no—
89C23×C31—24no—
90C23×C31—41no—
91C23×C31—48no—
92C23×C31—42no—
93C23×C31—42no—
94C23×C31—84no—
95C23×C31—84no—
96C23×C31—816no—
97C23×C31—84no—
98C23×C31#12111yescharacteristic
99C23×C31#12112yes—
100Q8×C31—44no—
101Q8×C31—44no—
102D4×C31—28no—
103D4×C31—44no—
104D4×C31—44no—
105D4×C31—44no—
106C2×C4×C31—24no—
107C2×C4×C31—22no—
108C2×C4×C31—41no—
109C2×C4×C31—42no—
110C2×C4×C31—42no—
111C2×C4×C31—42no—
112C2×C4×C31—44no—
113C2×C4×C31—82no—
114C2×C4×C31#12012yes—
115C2×C4×C31#12011yescharacteristic
116C2×C4×C31#12711yescharacteristic
117C8×C31—44no—
118C8×C31—42no—
119C8×C31—81no—
120#3195C2×C3111yescharacteristic, complemented
121C2×#450C2×C3112yes—
122#1783C2×C3111yescharacteristic
123#1472C2×C3112yescomplemented
124#1298C2×C3111yescharacteristic, complemented
125C22×C31—22no—
126C22×C31—22no—
127C22×C31—22no—
128C22×C31—21no—
129C22×C31—44no—
130C22×C31—44no—
131C22×C31—816no—
132C22×C31—84no—
133C22×C31—84no—
134C22×C31—84no—
135C22×C31#39211yescharacteristic
136C22×C31#47114yes—
137C4×C31—22no—
138C4×C31—21no—
139C4×C31—82no—
140C4×C31—82no—
141C4×C31B(4,𝔽2)12yes—
142C23×D4C22×C3111yescharacteristic
143C22×SD16—22no—
144C2×C22⋊Q8C22×C3112yes—
145#483—22no—
146#470—22no—
147#451—22no—
148#450—24no—
149#450—24no—
150C2×#102—22no—
151C2×#100—22no—
152#407C22×C3111yescharacteristic
153#407C22×C3111yescharacteristic
154C2×C22⋊C8C22×C3112yes—
155#324—24no—
156C2×C31—24no—
157C2×C31—41no—
158C2×C31—84no—
159C2×C31#201912yes—
160C2×C31#202111yescharacteristic
161C25—21no—
162C22×Q8D4×C3112yes—
163C22×D4—22no—
164C22×D4—21no—
165C22×D4—22no—
166C22×D4—44no—
167C22×D4D4×C3112yes—
168C23×C4C23×C3111yescharacteristic
169C2×Q16—22no—
170C2×SD16—28no—
171C2×SD16—48no—
172C2×M16—21no—
173C2×M16D4×C3112yes—
174C22×C8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4D4×C3112yes—
181C2×C22⋊C4—22no—
182#106—22no—
183#103—24no—
184#102—28no—
185#100—44no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8D4×C3114yes—
190C31#999011yescharacteristic, direct product factor, Sylow
191C24—22no—
192C24—44no—
193C24—42no—
194C24—84no—
195C24#41×C3111yescharacteristic
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—44no—
199C2×D4—28no—
200C2×D4—24no—
201C2×D4—24no—
202C2×D4—44no—
203C2×D4—42no—
204C2×D4—44no—
205C2×D4—48no—
206C2×D4—41no—
207C2×D4—41no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4C2×D4×C3112yes—
212C22×C4C2×D4×C3111yescharacteristic
213C22×C4C2×D4×C3111yescharacteristic
214C22×C4C2×D4×C3112yes—
215Q16—82no—
216SD16—416no—
217M16—42no—
218M16—42no—
219C2×C8—22no—
220C2×C8—24no—
221C2×C8—44no—
222C2×C8—41no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C23—24no—
230C23—24no—
231C23—41no—
232C23—48no—
233C23—42no—
234C23—42no—
235C23—84no—
236C23—84no—
237C23—816no—
238C23—84no—
239C23C31×#12111yescharacteristic
240C23C31×#12112yes—
241Q8—44no—
242Q8—44no—
243D4—28no—
244D4—44no—
245D4—44no—
246D4—44no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—82no—
255C2×C4C31×#12012yes—
256C2×C4C31×#12011yescharacteristic
257C2×C4C31×#12711yescharacteristic
258C8—44no—
259C8—42no—
260C8—81no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—21no—
265C22—44no—
266C22—44no—
267C22—816no—
268C22—84no—
269C22—84no—
270C22—84no—
271C22C31×#39211yescharacteristic
272C22C31×#47114yes—
273C4—22no—
274C4—21no—
275C4—82no—
276C4—82no—
277C4C31×B(4,𝔽2)12yes—
278C2—24no—
279C2—41no—
280C2—84no—
281C2C31×#201912yes—
282C2C31×#202111yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC23×D4×C31C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3195C2semidirect
4. C31×#1472C2semidirect
5. C31×#1298C2semidirect
87. #9990C31direct (N × H)
120. #3195C2×C31semidirect
123. #1472C2×C31semidirect
124. #1298C2×C31semidirect
190. C31#9990direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31952
C2×C31×#4502
C31×#17832
C31×#14722
C31×#12982
#999031