Finite Group Databasebeta

← C31×#13218 #21469977

Subgroups — C31×#13218 #21469977

Order 7936.

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

normal subgroups only

This group has 840 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×#2408C211yesmaximal, characteristic
2C2×C31×AΣL(1,𝔽16)C214yesmaximal, complemented
3C31×#1799C211yesmaximal, characteristic, complemented
4C4×C31×#100C211yesmaximal, characteristic, complemented
5C23×D4×C31C411yescharacteristic, complemented
6C2×C31×#121—21no—
7C2×C31×#120C2212yes—
8C2×C4×D4×C31C2211yescharacteristic
9C2×C31×#100C2212yes—
10C2×C31×#99C2212yes—
11C4×M16×C31—21no—
12C31×#394—21no—
13C31×#392—21no—
14C31×#390—21no—
15C31×#380—21no—
16C31×#380C411yescharacteristic, complemented
17C31×#379C412yescomplemented
18C31×AΣL(1,𝔽16)—28no—
19C31×#343—21no—
20C31×#342—21no—
21C25×C31C2×C412yes—
22C22×D4×C31—22no—
23C22×D4×C31—21no—
24C22×D4×C31—21no—
25C22×D4×C31—41no—
26C22×D4×C31—41no—
27C22×D4×C31—44no—
28C22×D4×C31C2311yescharacteristic
29C23×C4×C31C2×C411yescharacteristic
30C23×C4×C31D411yescharacteristic
31C2×M16×C31—22no—
32C31×#121—44no—
33C31×#120—24no—
34C31×#120—24no—
35C4×D4×C31—41no—
36C4×D4×C31—41no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—41no—
39C2×C4⋊C4×C31D411yescharacteristic
40C2×C22⋊C4×C31—22no—
41C2×C22⋊C4×C31—22no—
42C2×C22⋊C4×C31—22no—
43C2×C22⋊C4×C31—22no—
44C2×C22⋊C4×C31C2×C412yes—
45C2×C22⋊C4×C31C2×C411yescharacteristic
46C2×C22⋊C4×C31D411yescharacteristic
47C2×C42×C31D411yescharacteristic
48C31×#100—22no—
49C31×#100D414yescomplemented
50C31×#99—24no—
51C31×#97—41no—
52C4×C8×C31—41no—
53C31×#95—21no—
54C31×#95—41no—
55C24×C31—22no—
56C24×C31—24no—
57C24×C31—42no—
58C24×C31—44no—
59C24×C31—42no—
60C24×C31—82no—
61C24×C31C22×C411yescharacteristic
62C24×C31C2×D411yescharacteristic
63C2×D4×C31—24no—
64C2×D4×C31—41no—
65C2×D4×C31—41no—
66C2×D4×C31—42no—
67C2×D4×C31—42no—
68C2×D4×C31—44no—
69C2×D4×C31—48no—
70C2×D4×C31—44no—
71C2×D4×C31—44no—
72C2×D4×C31—44no—
73C2×D4×C31—42no—
74C2×D4×C31—42no—
75C2×D4×C31—41no—
76C2×D4×C31—41no—
77C2×D4×C31—82no—
78C2×D4×C31C2×D412yes—
79C2×D4×C31#4112yes—
80C22×C4×C31—21no—
81C22×C4×C31—21no—
82C22×C4×C31—21no—
83C22×C4×C31—21no—
84C22×C4×C31—22no—
85C22×C4×C31—42no—
86C22×C4×C31—42no—
87C22×C4×C31—42no—
88C22×C4×C31C22⋊C411yescharacteristic
89C22×C4×C31C22⋊C411yescharacteristic
90C22×C4×C31C22⋊C411yescharacteristic
91C22×C4×C31C22⋊C411yescharacteristic
92C22×C4×C31C2×D411yescharacteristic
93M16×C31—24no—
94M16×C31—42no—
95C2×C8×C31—42no—
96C4⋊C4×C31—41no—
97C4⋊C4×C31—42no—
98C4⋊C4×C31—82no—
99C22⋊C4×C31—24no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—41no—
102C22⋊C4×C31—44no—
103C22⋊C4×C31—44no—
104C22⋊C4×C31—42no—
105C22⋊C4×C31—42no—
106C22⋊C4×C31—82no—
107C42×C31—21no—
108C42×C31—21no—
109#13218C3111yesmaximal, characteristic, direct product factor, Sylow
110C23×C31—21no—
111C23×C31—21no—
112C23×C31—21no—
113C23×C31—21no—
114C23×C31—24no—
115C23×C31—41no—
116C23×C31—41no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—42no—
120C23×C31—42no—
121C23×C31—42no—
122C23×C31—44no—
123C23×C31—44no—
124C23×C31—42no—
125C23×C31—42no—
126C23×C31—42no—
127C23×C31—84no—
128C23×C31—88no—
129C23×C31—88no—
130C23×C31—84no—
131C23×C31C2×C22⋊C411yescharacteristic
132C23×C31C4×D412yes—
133C23×C31#12411yescharacteristic
134C23×C31#12711yescharacteristic
135D4×C31—42no—
136D4×C31—42no—
137D4×C31—44no—
138D4×C31—48no—
139D4×C31—44no—
140D4×C31—84no—
141D4×C31—84no—
142C2×C4×C31—21no—
143C2×C4×C31—21no—
144C2×C4×C31—22no—
145C2×C4×C31—21no—
146C2×C4×C31—21no—
147C2×C4×C31—21no—
148C2×C4×C31—42no—
149C2×C4×C31—42no—
150C2×C4×C31—42no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—42no—
155C2×C4×C31—42no—
156C2×C4×C31—42no—
157C2×C4×C31—81no—
158C2×C4×C31—82no—
159C2×C4×C31—82no—
160C2×C4×C31—82no—
161C2×C4×C31#9911yescharacteristic
162C2×C4×C31#9911yescharacteristic
163C2×C4×C31#12112yes—
164C8×C31—44no—
165#2408C2×C3111yescharacteristic
166C2×AΣL(1,𝔽16)C2×C3114yescomplemented
167#1799C2×C3111yescharacteristic, complemented
168C4×#100C2×C3111yescharacteristic, complemented
169C22×C31—21no—
170C22×C31—21no—
171C22×C31—21no—
172C22×C31—22no—
173C22×C31—21no—
174C22×C31—41no—
175C22×C31—41no—
176C22×C31—42no—
177C22×C31—42no—
178C22×C31—44no—
179C22×C31—44no—
180C22×C31—41no—
181C22×C31—41no—
182C22×C31—41no—
183C22×C31—41no—
184C22×C31—44no—
185C22×C31—81no—
186C22×C31—82no—
187C22×C31—82no—
188C22×C31—88no—
189C22×C31—88no—
190C22×C31—82no—
191C22×C31—82no—
192C22×C31#39011yescharacteristic
193C22×C31C2×#9911yescharacteristic
194C22×C31#41011yescharacteristic
195C4×C31—21no—
196C4×C31—22no—
197C4×C31—41no—
198C4×C31—41no—
199C4×C31—41no—
200C4×C31—82no—
201C4×C31—162no—
202C4×C31AΣL(1,𝔽16)12yescomplemented
203C23×D4C4×C3111yescharacteristic, complemented
204C2×#121—21no—
205C2×#120C22×C3112yes—
206C2×C4×D4C22×C3111yescharacteristic
207C2×#100C22×C3112yes—
208C2×#99C22×C3112yes—
209C4×M16—21no—
210#394—21no—
211#392—21no—
212#390—21no—
213#380—21no—
214#380C4×C3111yescharacteristic, complemented
215#379C4×C3112yescomplemented
216AΣL(1,𝔽16)—28no—
217#343—21no—
218#342—21no—
219C2×C31—21no—
220C2×C31—21no—
221C2×C31—41no—
222C2×C31—42no—
223C2×C31—41no—
224C2×C31—84no—
225C2×C31#196411yescharacteristic
226C2×C31C2×AΣL(1,𝔽16)11yescharacteristic
227C2×C31#212211yescharacteristic
228C25C2×C4×C3112yes—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—41no—
233C22×D4—41no—
234C22×D4—44no—
235C22×D4C23×C3111yescharacteristic
236C23×C4C2×C4×C3111yescharacteristic
237C23×C4D4×C3111yescharacteristic
238C2×M16—22no—
239#121—44no—
240#120—24no—
241#120—24no—
242C4×D4—41no—
243C4×D4—41no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4D4×C3111yescharacteristic
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4C2×C4×C3112yes—
252C2×C22⋊C4C2×C4×C3111yescharacteristic
253C2×C22⋊C4D4×C3111yescharacteristic
254C2×C42D4×C3111yescharacteristic
255#100—22no—
256#100D4×C3114yescomplemented
257#99—24no—
258#97—41no—
259C4×C8—41no—
260#95—21no—
261#95—41no—
262C31#1321811yescharacteristic, direct product factor, Sylow
263C24—22no—
264C24—24no—
265C24—42no—
266C24—44no—
267C24—42no—
268C24—82no—
269C24C22×C4×C3111yescharacteristic
270C24C2×D4×C3111yescharacteristic
271C2×D4—24no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—44no—
277C2×D4—48no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—82no—
286C2×D4C2×D4×C3112yes—
287C2×D4#41×C3112yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4C22⋊C4×C3111yescharacteristic
297C22×C4C22⋊C4×C3111yescharacteristic
298C22×C4C22⋊C4×C3111yescharacteristic
299C22×C4C22⋊C4×C3111yescharacteristic
300C22×C4C2×D4×C3111yescharacteristic
301M16—24no—
302M16—42no—
303C2×C8—42no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—82no—
307C22⋊C4—24no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—82no—
315C42—21no—
316C42—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—24no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—44no—
330C23—44no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—84no—
335C23—88no—
336C23—88no—
337C23—84no—
338C23C2×C22⋊C4×C3111yescharacteristic
339C23C4×D4×C3112yes—
340C23C31×#12411yescharacteristic
341C23C31×#12711yescharacteristic
342D4—42no—
343D4—42no—
344D4—44no—
345D4—48no—
346D4—44no—
347D4—84no—
348D4—84no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4C31×#9911yescharacteristic
369C2×C4C31×#9911yescharacteristic
370C2×C4C31×#12112yes—
371C8—44no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—42no—
380C22—42no—
381C22—44no—
382C22—44no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—44no—
388C22—81no—
389C22—82no—
390C22—82no—
391C22—88no—
392C22—88no—
393C22—82no—
394C22—82no—
395C22C31×#39011yescharacteristic
396C22C2×C31×#9911yescharacteristic
397C22C31×#41011yescharacteristic
398C4—21no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—82no—
404C4—162no—
405C4C31×AΣL(1,𝔽16)12yescomplemented
406C2—21no—
407C2—21no—
408C2—41no—
409C2—42no—
410C2—41no—
411C2—84no—
412C2C31×#196411yescharacteristic
413C2C2×C31×AΣL(1,𝔽16)11yescharacteristic
414C2C31×#212211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC24C22×C4×C31
Frattini, cosocleC22×D4C23×C31
prime radicalC31×#2408C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×AΣL(1,𝔽16)C2semidirect
3. C31×#1799C2semidirect
4. C4×C31×#100C2semidirect
5. C23×D4×C31C4semidirect
16. C31×#380C4semidirect
17. C31×#379C4semidirect
49. C31×#100D4semidirect
109. #13218C31direct (N × H)
166. C2×AΣL(1,𝔽16)C2×C31semidirect
167. #1799C2×C31semidirect
168. C4×#100C2×C31semidirect
202. C4×C31AΣL(1,𝔽16)semidirect
203. C23×D4C4×C31semidirect
214. #380C4×C31semidirect
215. #379C4×C31semidirect
256. #100D4×C31semidirect
262. C31#13218direct (N × H)
405. C4C31×AΣL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
C31×#24082
C2×C31×AΣL(1,𝔽16)2
C31×#17992
C4×C31×#1002
#1321831