Finite Group Databasebeta

← C31×#19056 #21475815

Subgroups — C31×#19056 #21475815

Order 7936.

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

normal subgroups only

This group has 992 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
1C31×#3549C211yesmaximal, characteristic
2C31×#3216C212yesmaximal, complemented
3C31×#3135C211yesmaximal, characteristic, complemented
4C31×#1659C218yesmaximal, complemented
5C31×#1536C211yesmaximal, characteristic, complemented
6C31×#1530C212yesmaximal, complemented
7C31×H(3,𝔽4)C2214yes—
8C31×#555—22no—
9C31×#555—21no—
10D42×C31—22no—
11D42×C31—21no—
12C31×#535—22no—
13C31×#535—24no—
14C31×#533C2211yescharacteristic, complemented
15C31×#532C2211yescharacteristic
16C31×#532C2212yes—
17C31×#497C2214yescomplemented
18C31×#491—21no—
19C31×#490—21no—
20C31×#489C2214yescomplemented
21C31×#489C2214yescomplemented
22C31×#462—22no—
23C31×#460—22no—
24C31×#452—24no—
25C31×#432C2212yescomplemented
26C31×#416—22no—
27C2×C31×#97C2211yescharacteristic, complemented
28C31×#329C2218yescomplemented
29C31×#329C2214yescomplemented
30C2×#41×C31D412yes—
31C2×#41×C31D414yes—
32C22×D4×C31—24no—
33C22×D4×C31—22no—
34C2×Q16×C31—24no—
35C2×SD16×C31—24no—
36C22×C8×C31—21no—
37C31×C4⋊Q8D412yes—
38C31×C4⋊Q8D411yescharacteristic
39C31×C4⋊Q8C2312yes—
40C31×#127—24no—
41C31×#127D412yes—
42C31×#127D411yescharacteristic
43C31×#125—21no—
44C31×#124—24no—
45C31×#124—28no—
46C31×#124C2312yes—
47C31×#124C2314yes—
48C31×C22⋊Q8—22no—
49C31×#121—24no—
50C31×#121—24no—
51C31×#121—24no—
52C31×#121—22no—
53C31×#121—22no—
54C31×#121—22no—
55C31×#120—24no—
56C31×#120—24no—
57C31×#120—28no—
58C4×Q8×C31—22no—
59C4×D4×C31—22no—
60C4×D4×C31—22no—
61C4×D4×C31—22no—
62C31×#117—24no—
63C31×#117—22no—
64C31×#117—21no—
65C2×C42×C31C2311yescharacteristic
66C31×C4⋊C8—22no—
67C31×#103—24no—
68C31×#103—24no—
69C31×#102—24no—
70C31×#102—24no—
71C31×C22⋊C8—22no—
72C31×#97—21no—
73C31×#97C2312yes—
74C31×#97C2314yes—
75C24×C31—24no—
76#41×C31—44no—
77#41×C31—84no—
78C2×Q8×C31—22no—
79C2×Q8×C31C2×D412yes—
80C2×Q8×C31C2×D414yes—
81C2×D4×C31—24no—
82C2×D4×C31—28no—
83C2×D4×C31—22no—
84C2×D4×C31—24no—
85C2×D4×C31—22no—
86C2×D4×C31—22no—
87C2×D4×C31—22no—
88C2×D4×C31—88no—
89C2×D4×C31—84no—
90C2×D4×C31C2×D412yes—
91C2×D4×C31C2×D414yes—
92C22×C4×C31—24no—
93C22×C4×C31—22no—
94C22×C4×C31C2×D411yescharacteristic
95C22×C4×C31C2×D412yes—
96Q16×C31—84no—
97SD16×C31—84no—
98C2×C8×C31—22no—
99C2×C8×C31—24no—
100C2×C8×C31—42no—
101C4⋊C4×C31—22no—
102C4⋊C4×C31—22no—
103C4⋊C4×C31—22no—
104C4⋊C4×C31—22no—
105C4⋊C4×C31—22no—
106C4⋊C4×C31—42no—
107C22⋊C4×C31—28no—
108C22⋊C4×C31—24no—
109C22⋊C4×C31—24no—
110C22⋊C4×C31—24no—
111C22⋊C4×C31—24no—
112C22⋊C4×C31—41no—
113C42×C31—24no—
114C42×C31—41no—
115C42×C31C2×D411yescharacteristic
116C42×C31C2×D412yes—
117C42×C31C2411yescharacteristic
118#19056C3111yesmaximal, characteristic, direct product factor, Sylow
119C23×C31—22no—
120C23×C31—24no—
121C23×C31—84no—
122C23×C31—88no—
123C23×C31#12011yescharacteristic
124Q8×C31—44no—
125Q8×C31—42no—
126Q8×C31—42no—
127Q8×C31—42no—
128D4×C31—44no—
129D4×C31—42no—
130D4×C31—44no—
131D4×C31—44no—
132D4×C31—44no—
133D4×C31—82no—
134D4×C31—84no—
135D4×C31—84no—
136D4×C31—88no—
137C2×C4×C31—22no—
138C2×C4×C31—22no—
139C2×C4×C31—24no—
140C2×C4×C31—42no—
141C2×C4×C31—48no—
142C2×C4×C31—44no—
143C2×C4×C31—42no—
144C2×C4×C31#12012yes—
145C2×C4×C31#12011yescharacteristic
146C2×C4×C31C22×D411yescharacteristic
147C2×C4×C31C22×D412yes—
148C8×C31—42no—
149C8×C31—44no—
150#3549C2×C3111yescharacteristic
151#3216C2×C3112yescomplemented
152#3135C2×C3111yescharacteristic, complemented
153#1659C2×C3118yescomplemented
154#1536C2×C3111yescharacteristic, complemented
155#1530C2×C3112yescomplemented
156C22×C31—21no—
157C22×C31—22no—
158C22×C31—42no—
159C22×C31—44no—
160C22×C31—44no—
161C22×C31—44no—
162C22×C31—44no—
163C22×C31—168no—
164C22×C31C2×#12011yescharacteristic
165C4×C31—21no—
166C4×C31—22no—
167C4×C31—22no—
168C4×C31—42no—
169C4×C31—44no—
170C4×C31—82no—
171C4×C31#45512yes—
172H(3,𝔽4)C22×C3114yes—
173#555—22no—
174#555—21no—
175D42—22no—
176D42—21no—
177#535—22no—
178#535—24no—
179#533C22×C3111yescharacteristic, complemented
180#532C22×C3111yescharacteristic
181#532C22×C3112yes—
182#497C22×C3114yescomplemented
183#491—21no—
184#490—21no—
185#489C22×C3114yescomplemented
186#489C22×C3114yescomplemented
187#462—22no—
188#460—22no—
189#452—24no—
190#432C22×C3112yescomplemented
191#416—22no—
192C2×#97C22×C3111yescharacteristic, complemented
193#329C22×C3114yescomplemented
194#329C22×C3118yescomplemented
195C2×C31—41no—
196C2×C31—82no—
197C2×C31—84no—
198C2×C31C2×#45511yescharacteristic
199C2×C31#301912yes—
200C2×#41D4×C3112yes—
201C2×#41D4×C3114yes—
202C22×D4—24no—
203C22×D4—22no—
204C2×Q16—24no—
205C2×SD16—24no—
206C22×C8—21no—
207C4⋊Q8D4×C3112yes—
208C4⋊Q8D4×C3111yescharacteristic
209C4⋊Q8C23×C3112yes—
210#127—24no—
211#127D4×C3112yes—
212#127D4×C3111yescharacteristic
213#125—21no—
214#124—24no—
215#124—28no—
216#124C23×C3112yes—
217#124C23×C3114yes—
218C22⋊Q8—22no—
219#121—24no—
220#121—24no—
221#121—24no—
222#121—22no—
223#121—22no—
224#121—22no—
225#120—24no—
226#120—24no—
227#120—28no—
228C4×Q8—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232#117—24no—
233#117—22no—
234#117—21no—
235C2×C42C23×C3111yescharacteristic
236C4⋊C8—22no—
237#103—24no—
238#103—24no—
239#102—24no—
240#102—24no—
241C22⋊C8—22no—
242#97—21no—
243#97C23×C3112yes—
244#97C23×C3114yes—
245C31#1905611yescharacteristic, direct product factor, Sylow
246C24—24no—
247#41—44no—
248#41—84no—
249C2×Q8—22no—
250C2×Q8C2×D4×C3112yes—
251C2×Q8C2×D4×C3114yes—
252C2×D4—24no—
253C2×D4—28no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—88no—
260C2×D4—84no—
261C2×D4C2×D4×C3112yes—
262C2×D4C2×D4×C3114yes—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4C2×D4×C3111yescharacteristic
266C22×C4C2×D4×C3112yes—
267Q16—84no—
268SD16—84no—
269C2×C8—22no—
270C2×C8—24no—
271C2×C8—42no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—42no—
278C22⋊C4—28no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—41no—
284C42—24no—
285C42—41no—
286C42C2×D4×C3111yescharacteristic
287C42C2×D4×C3112yes—
288C42C24×C3111yescharacteristic
289C23—22no—
290C23—24no—
291C23—84no—
292C23—88no—
293C23C31×#12011yescharacteristic
294Q8—44no—
295Q8—42no—
296Q8—42no—
297Q8—42no—
298D4—44no—
299D4—42no—
300D4—44no—
301D4—44no—
302D4—44no—
303D4—82no—
304D4—84no—
305D4—84no—
306D4—88no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—42no—
311C2×C4—48no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4C31×#12012yes—
315C2×C4C31×#12011yescharacteristic
316C2×C4C22×D4×C3111yescharacteristic
317C2×C4C22×D4×C3112yes—
318C8—42no—
319C8—44no—
320C22—21no—
321C22—22no—
322C22—42no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—44no—
327C22—168no—
328C22C2×C31×#12011yescharacteristic
329C4—21no—
330C4—22no—
331C4—22no—
332C4—42no—
333C4—44no—
334C4—82no—
335C4C31×#45512yes—
336C2—41no—
337C2—82no—
338C2—84no—
339C2C2×C31×#45511yescharacteristic
340C2C31×#301912yes—
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3549C2
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×#3216C2semidirect
3. C31×#3135C2semidirect
4. C31×#1659C2semidirect
5. C31×#1536C2semidirect
6. C31×#1530C2semidirect
14. C31×#533C22semidirect
17. C31×#497C22semidirect
20. C31×#489C22semidirect
21. C31×#489C22semidirect
25. C31×#432C22semidirect
27. C2×C31×#97C22semidirect
28. C31×#329C22semidirect
29. C31×#329C22semidirect
118. #19056C31direct (N × H)
151. #3216C2×C31semidirect
152. #3135C2×C31semidirect
153. #1659C2×C31semidirect
154. #1536C2×C31semidirect
155. #1530C2×C31semidirect
179. #533C22×C31semidirect
182. #497C22×C31semidirect
185. #489C22×C31semidirect
186. #489C22×C31semidirect
190. #432C22×C31semidirect
192. C2×#97C22×C31semidirect
193. #329C22×C31semidirect
194. #329C22×C31semidirect
245. C31#19056direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35492
C31×#32162
C31×#31352
C31×#16592
C31×#15362
C31×#15302
#1905631