Finite Group Databasebeta

← C31×#19006 #21475765

Subgroups — C31×#19006 #21475765

Order 7936.

938 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 222 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×#562C211yesmaximal, characteristic
2C31×#3215C212yesmaximal, complemented
3C31×#3166C211yesmaximal, characteristic, complemented
4C31×#1661C218yesmaximal, complemented
5C31×#1534C211yesmaximal, characteristic, complemented
6C31×#1527C212yesmaximal, complemented
7C31×#562—22no—
8C31×#562—24no—
9C31×#562C2214yes—
10C31×#533C2211yescharacteristic
11C2×C31×#125C2211yescharacteristic
12C2×C31×#124C2212yes—
13C2×C31×#123—22no—
14C2×C31×C22⋊Q8—22no—
15C2×C31×#121—22no—
16C31×#501C2214yescomplemented
17C31×#499—22no—
18C31×#497C2214yescomplemented
19C31×#489C2214yescomplemented
20C31×#464—24no—
21C31×#451—22no—
22C31×#450—22no—
23C31×#432C2212yes—
24C31×#428—22no—
25C4×M16×C31C2211yescharacteristic
26C31×#330C2214yescomplemented
27C31×#329C2218yescomplemented
28C22×Q8×C31D412yes—
29C22×D4×C31—22no—
30C22×D4×C31D412yes—
31C23×C4×C31—22no—
32C2×Q16×C31—24no—
33C2×SD16×C31—24no—
34C2×M16×C31—21no—
35C31×C4⋊Q8C2312yes—
36C31×#125—22no—
37C31×#125—21no—
38C31×#125D412yes—
39C31×#125C2312yes—
40C31×#124—22no—
41C31×#124—22no—
42C31×#124D414yes—
43C31×#124C2314yes—
44C31×#123—28no—
45C31×#123—28no—
46C31×C22⋊Q8—28no—
47C31×C22⋊Q8—28no—
48C31×C22⋊Q8—22no—
49C31×#121—28no—
50C31×#121—28no—
51C4×Q8×C31—22no—
52C31×#117—21no—
53C2×C4⋊C4×C31—22no—
54C2×C4⋊C4×C31—22no—
55C2×C4⋊C4×C31D412yes—
56C2×C22⋊C4×C31—22no—
57C2×C22⋊C4×C31—22no—
58C2×C22⋊C4×C31—22no—
59C2×C42×C31C2311yescharacteristic
60C31×#107—24no—
61C31×#106—24no—
62C31×C4⋊C8—22no—
63C31×#103—24no—
64C31×#102—24no—
65C31×C22⋊C8—22no—
66C31×#97C2312yes—
67C31×#97C2314yes—
68C4×C8×C31—21no—
69C24×C31—22no—
70C2×Q8×C31—22no—
71C2×Q8×C31—22no—
72C2×Q8×C31—82no—
73C2×Q8×C31C2×D414yes—
74C2×D4×C31—28no—
75C2×D4×C31—22no—
76C2×D4×C31—82no—
77C2×D4×C31—84no—
78C2×D4×C31C2×D414yes—
79C22×C4×C31—22no—
80C22×C4×C31—22no—
81C22×C4×C31—28no—
82C22×C4×C31—84no—
83C22×C4×C31C2×D411yescharacteristic
84C22×C4×C31C2×D412yes—
85Q16×C31—84no—
86SD16×C31—84no—
87M16×C31—24no—
88C2×C8×C31—22no—
89C2×C8×C31—24no—
90C4⋊C4×C31—28no—
91C4⋊C4×C31—24no—
92C4⋊C4×C31—22no—
93C4⋊C4×C31—24no—
94C4⋊C4×C31—22no—
95C4⋊C4×C31—22no—
96C4⋊C4×C31—42no—
97C4⋊C4×C31C2×D414yes—
98C22⋊C4×C31—28no—
99C22⋊C4×C31—24no—
100C22⋊C4×C31—24no—
101C22⋊C4×C31—24no—
102C22⋊C4×C31—24no—
103C22⋊C4×C31—41no—
104C42×C31—41no—
105C42×C31C2×D411yescharacteristic
106C42×C31C2×D412yes—
107C42×C31C2411yescharacteristic
108#19006C3111yesmaximal, characteristic, direct product factor, Sylow
109C23×C31—24no—
110C23×C31—44no—
111C23×C31—44no—
112C23×C31—44no—
113C23×C31#12011yescharacteristic
114Q8×C31—44no—
115Q8×C31—44no—
116Q8×C31—82no—
117D4×C31—44no—
118D4×C31—82no—
119D4×C31—88no—
120C2×C4×C31—22no—
121C2×C4×C31—24no—
122C2×C4×C31—24no—
123C2×C4×C31—42no—
124C2×C4×C31—44no—
125C2×C4×C31—44no—
126C2×C4×C31—42no—
127C2×C4×C31—88no—
128C2×C4×C31#12011yescharacteristic
129C2×C4×C31#12012yes—
130C2×C4×C31C22×D411yescharacteristic
131C2×C4×C31C22×D412yes—
132C8×C31—24no—
133C8×C31—44no—
134C2×#562C2×C3111yescharacteristic
135#3215C2×C3112yescomplemented
136#3166C2×C3111yescharacteristic, complemented
137#1661C2×C3118yescomplemented
138#1534C2×C3111yescharacteristic, complemented
139#1527C2×C3112yescomplemented
140C22×C31—22no—
141C22×C31—44no—
142C22×C31—44no—
143C22×C31—44no—
144C22×C31—88no—
145C22×C31#45612yes—
146C22×C31C2×#12011yescharacteristic
147C4×C31—21no—
148C4×C31—21no—
149C4×C31—22no—
150C4×C31—22no—
151C4×C31—44no—
152C4×C31—44no—
153C4×C31—82no—
154#562—22no—
155#562—24no—
156#562C22×C3114yes—
157#533C22×C3111yescharacteristic
158C2×#125C22×C3111yescharacteristic
159C2×#124C22×C3112yes—
160C2×#123—22no—
161C2×C22⋊Q8—22no—
162C2×#121—22no—
163#501C22×C3114yescomplemented
164#499—22no—
165#497C22×C3114yescomplemented
166#489C22×C3114yescomplemented
167#464—24no—
168#451—22no—
169#450—22no—
170#432C22×C3112yes—
171#428—22no—
172C4×M16C22×C3111yescharacteristic
173#330C22×C3114yescomplemented
174#329C22×C3118yescomplemented
175C2×C31—22no—
176C2×C31—84no—
177C2×C31C2×#45611yescharacteristic
178C2×C31#301912yes—
179C22×Q8D4×C3112yes—
180C22×D4—22no—
181C22×D4D4×C3112yes—
182C23×C4—22no—
183C2×Q16—24no—
184C2×SD16—24no—
185C2×M16—21no—
186C4⋊Q8C23×C3112yes—
187#125—22no—
188#125—21no—
189#125D4×C3112yes—
190#125C23×C3112yes—
191#124—22no—
192#124—22no—
193#124D4×C3114yes—
194#124C23×C3114yes—
195#123—28no—
196#123—28no—
197C22⋊Q8—28no—
198C22⋊Q8—28no—
199C22⋊Q8—22no—
200#121—28no—
201#121—28no—
202C4×Q8—22no—
203#117—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4D4×C3112yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C42C23×C3111yescharacteristic
211#107—24no—
212#106—24no—
213C4⋊C8—22no—
214#103—24no—
215#102—24no—
216C22⋊C8—22no—
217#97C23×C3112yes—
218#97C23×C3114yes—
219C4×C8—21no—
220C31#1900611yescharacteristic, direct product factor, Sylow
221C24—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—82no—
225C2×Q8C2×D4×C3114yes—
226C2×D4—28no—
227C2×D4—22no—
228C2×D4—82no—
229C2×D4—84no—
230C2×D4C2×D4×C3114yes—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—28no—
234C22×C4—84no—
235C22×C4C2×D4×C3111yescharacteristic
236C22×C4C2×D4×C3112yes—
237Q16—84no—
238SD16—84no—
239M16—24no—
240C2×C8—22no—
241C2×C8—24no—
242C4⋊C4—28no—
243C4⋊C4—24no—
244C4⋊C4—22no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4C2×D4×C3114yes—
250C22⋊C4—28no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—41no—
256C42—41no—
257C42C2×D4×C3111yescharacteristic
258C42C2×D4×C3112yes—
259C42C24×C3111yescharacteristic
260C23—24no—
261C23—44no—
262C23—44no—
263C23—44no—
264C23C31×#12011yescharacteristic
265Q8—44no—
266Q8—44no—
267Q8—82no—
268D4—44no—
269D4—82no—
270D4—88no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—88no—
279C2×C4C31×#12011yescharacteristic
280C2×C4C31×#12012yes—
281C2×C4C22×D4×C3111yescharacteristic
282C2×C4C22×D4×C3112yes—
283C8—24no—
284C8—44no—
285C22—22no—
286C22—44no—
287C22—44no—
288C22—44no—
289C22—88no—
290C22C31×#45612yes—
291C22C2×C31×#12011yescharacteristic
292C4—21no—
293C4—21no—
294C4—22no—
295C4—22no—
296C4—44no—
297C4—44no—
298C4—82no—
299C2—22no—
300C2—84no—
301C2C2×C31×#45611yescharacteristic
302C2C31×#301912yes—
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC2×C31×#562C2
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
2. C31×#3215C2semidirect
3. C31×#3166C2semidirect
4. C31×#1661C2semidirect
5. C31×#1534C2semidirect
6. C31×#1527C2semidirect
16. C31×#501C22semidirect
18. C31×#497C22semidirect
19. C31×#489C22semidirect
26. C31×#330C22semidirect
27. C31×#329C22semidirect
108. #19006C31direct (N × H)
135. #3215C2×C31semidirect
136. #3166C2×C31semidirect
137. #1661C2×C31semidirect
138. #1534C2×C31semidirect
139. #1527C2×C31semidirect
163. #501C22×C31semidirect
165. #497C22×C31semidirect
166. #489C22×C31semidirect
173. #330C22×C31semidirect
174. #329C22×C31semidirect
220. C31#19006direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5622
C31×#32152
C31×#31662
C31×#16612
C31×#15342
C31×#15272
#1900631