Finite Group Databasebeta

← C31×#13050 #21469809

Subgroups — C31×#13050 #21469809

Order 7936.

732 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 118 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×#455C211yesmaximal, characteristic
2C31×#2379C211yesmaximal, characteristic, complemented
3C2×C31×#354C211yesmaximal, characteristic, complemented
4C31×#1790C212yesmaximal
5C31×#1390C212yesmaximal, complemented
6C2×C31×H(5,𝔽2)C2211yescharacteristic
7C2×C31×#137—21no—
8C2×C31×#124C2211yescharacteristic
9C31×#455—22no—
10C31×#455C414yescomplemented
11C2×C31×#104—21no—
12C2×C31×#100C2211yescharacteristic
13C2×C31×#99—22no—
14C2×C31×#99C2212yes—
15C31×#391—21no—
16C31×#388—22no—
17C31×#386—22no—
18C31×#386C2212yes—
19C31×#385—22no—
20C31×#382—21no—
21C31×#354—22no—
22C31×H(5,𝔽2)—22no—
23C31×H(5,𝔽2)—21no—
24C31×H(5,𝔽2)—42no—
25C31×H(5,𝔽2)C2×C412yes—
26C2×#41×C31—21no—
27C2×#41×C31—41no—
28C22×Q8×C31D411yescharacteristic
29C22×D4×C31—22no—
30C22×D4×C31—41no—
31C22×D4×C31C2311yescharacteristic
32C23×C4×C31D412yes—
33C31×#137—24no—
34C31×#137—42no—
35C2×Q16×C31—41no—
36C2×SD16×C31—41no—
37C2×M16×C31—21no—
38C31×#124—22no—
39C31×#124C2×C412yes—
40C31×#124D412yes—
41C2×C4⋊C4×C31—21no—
42C2×C4⋊C4×C31—21no—
43C2×C4⋊C4×C31—21no—
44C2×C22⋊C4×C31—22no—
45C2×C22⋊C4×C31—22no—
46C2×C22⋊C4×C31—22no—
47C2×C22⋊C4×C31—42no—
48C2×C22⋊C4×C31D412yes—
49C2×C42×C31—21no—
50C2×C42×C31D411yescharacteristic
51C31×#104—24no—
52C31×#100—21no—
53C31×#100C2×C412yes—
54C31×#99—22no—
55C31×#99—22no—
56C31×#99—44no—
57C31×#95—22no—
58C31×#95—22no—
59C31×#95—22no—
60C31×#95—42no—
61C24×C31—22no—
62C24×C31C2×D412yes—
63#41×C31—22no—
64#41×C31—22no—
65#41×C31—42no—
66#41×C31—41no—
67#41×C31—44no—
68#41×C31—42no—
69#41×C31—41no—
70C2×Q8×C31—21no—
71C2×Q8×C31—21no—
72C2×Q8×C31—41no—
73C2×Q8×C31—41no—
74C2×Q8×C31C22⋊C412yescomplemented
75C2×D4×C31—21no—
76C2×D4×C31—24no—
77C2×D4×C31—24no—
78C2×D4×C31—41no—
79C2×D4×C31—41no—
80C2×D4×C31—42no—
81C2×D4×C31—42no—
82C2×D4×C31—42no—
83C2×D4×C31—42no—
84C2×D4×C31—44no—
85C2×D4×C31—41no—
86C2×D4×C31—42no—
87C2×D4×C31C22×C411yescharacteristic
88C2×D4×C31C2×D411yescharacteristic
89C2×D4×C31#4112yes—
90C22×C4×C31—22no—
91C22×C4×C31—21no—
92C22×C4×C31—21no—
93C22×C4×C31—22no—
94C22×C4×C31—22no—
95C22×C4×C31—41no—
96C22×C4×C31—42no—
97C22×C4×C31—42no—
98C22×C4×C31—82no—
99C22×C4×C31C2×D411yescharacteristic
100Q16×C31—44no—
101SD16×C31—44no—
102M16×C31—22no—
103M16×C31—41no—
104C2×C8×C31—41no—
105C4⋊C4×C31—81no—
106C4⋊C4×C31—81no—
107C4⋊C4×C31—81no—
108C22⋊C4×C31—24no—
109C22⋊C4×C31—24no—
110C22⋊C4×C31—42no—
111C22⋊C4×C31—42no—
112C22⋊C4×C31—42no—
113C22⋊C4×C31—82no—
114C22⋊C4×C31—84no—
115C42×C31—21no—
116C42×C31—81no—
117C42×C31C22⋊C412yescomplemented
118#13050C3111yesmaximal, characteristic, direct product factor, Sylow
119C23×C31—22no—
120C23×C31—22no—
121C23×C31—24no—
122C23×C31—42no—
123C23×C31—42no—
124C23×C31—42no—
125C23×C31—41no—
126C23×C31—82no—
127C23×C31—82no—
128C23×C31C4×D412yes—
129C23×C31#12011yescharacteristic
130C23×C31#12112yes—
131Q8×C31—22no—
132Q8×C31—42no—
133Q8×C31—41no—
134Q8×C31—81no—
135D4×C31—22no—
136D4×C31—41no—
137D4×C31—41no—
138D4×C31—42no—
139D4×C31—41no—
140D4×C31—44no—
141D4×C31—44no—
142D4×C31—42no—
143D4×C31—42no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—24no—
147C2×C4×C31—22no—
148C2×C4×C31—22no—
149C2×C4×C31—42no—
150C2×C4×C31—42no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—42no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—42no—
158C2×C4×C31—42no—
159C2×C4×C31—42no—
160C2×C4×C31—42no—
161C2×C4×C31—44no—
162C2×C4×C31—42no—
163C2×C4×C31—42no—
164C2×C4×C31—82no—
165C2×C4×C31—82no—
166C2×C4×C31—82no—
167C2×C4×C31—82no—
168C2×C4×C31C2×C22⋊C411yescharacteristic
169C2×C4×C31#12311yescharacteristic
170C8×C31—42no—
171C2×#455C2×C3111yescharacteristic
172#2379C2×C3111yescharacteristic, complemented
173C2×#354C2×C3111yescharacteristic, complemented
174#1790C2×C3112yes—
175#1390C2×C3112yescomplemented
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—22no—
179C22×C31—22no—
180C22×C31—42no—
181C22×C31—42no—
182C22×C31—42no—
183C22×C31—42no—
184C22×C31—42no—
185C22×C31—82no—
186C22×C31—84no—
187C22×C31—81no—
188C22×C31—81no—
189C22×C31—162no—
190C22×C31#38611yescharacteristic
191C22×C31B(4,𝔽2)11yescharacteristic
192C22×C31B(4,𝔽2)11yescharacteristic
193C4×C31—21no—
194C4×C31—21no—
195C4×C31—42no—
196C4×C31—42no—
197C4×C31—42no—
198C4×C31—82no—
199C4×C31—82no—
200C4×C31—162no—
201C2×H(5,𝔽2)C22×C3111yescharacteristic
202C2×#137—21no—
203C2×#124C22×C3111yescharacteristic
204#455—22no—
205#455C4×C3114yescomplemented
206C2×#104—21no—
207C2×#100C22×C3111yescharacteristic
208C2×#99—22no—
209C2×#99C22×C3112yes—
210#391—21no—
211#388—22no—
212#386—22no—
213#386C22×C3112yes—
214#385—22no—
215#382—21no—
216#354—22no—
217C2×C31—21no—
218C2×C31—21no—
219C2×C31—42no—
220C2×C31—42no—
221C2×C31—82no—
222C2×C31#188911yescharacteristic
223C2×C31#219911yescharacteristic
224C2×C31#220111yescharacteristic
225H(5,𝔽2)—22no—
226H(5,𝔽2)—21no—
227H(5,𝔽2)—42no—
228H(5,𝔽2)C2×C4×C3112yes—
229C2×#41—21no—
230C2×#41—41no—
231C22×Q8D4×C3111yescharacteristic
232C22×D4—22no—
233C22×D4—41no—
234C22×D4C23×C3111yescharacteristic
235C23×C4D4×C3112yes—
236#137—24no—
237#137—42no—
238C2×Q16—41no—
239C2×SD16—41no—
240C2×M16—21no—
241#124—22no—
242#124C2×C4×C3112yes—
243#124D4×C3112yes—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4D4×C3112yes—
252C2×C42—21no—
253C2×C42D4×C3111yescharacteristic
254#104—24no—
255#100—21no—
256#100C2×C4×C3112yes—
257#99—22no—
258#99—22no—
259#99—44no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—42no—
264C31#1305011yescharacteristic, direct product factor, Sylow
265C24—22no—
266C24C2×D4×C3112yes—
267#41—22no—
268#41—22no—
269#41—42no—
270#41—41no—
271#41—44no—
272#41—42no—
273#41—41no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8C22⋊C4×C3112yescomplemented
279C2×D4—21no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—44no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4C22×C4×C3111yescharacteristic
292C2×D4C2×D4×C3111yescharacteristic
293C2×D4#41×C3112yes—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—82no—
303C22×C4C2×D4×C3111yescharacteristic
304Q16—44no—
305SD16—44no—
306M16—22no—
307M16—41no—
308C2×C8—41no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C4⋊C4—81no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—82no—
318C22⋊C4—84no—
319C42—21no—
320C42—81no—
321C42C22⋊C4×C3112yescomplemented
322C23—22no—
323C23—22no—
324C23—24no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23—82no—
330C23—82no—
331C23C4×D4×C3112yes—
332C23C31×#12011yescharacteristic
333C23C31×#12112yes—
334Q8—22no—
335Q8—42no—
336Q8—41no—
337Q8—81no—
338D4—22no—
339D4—41no—
340D4—41no—
341D4—42no—
342D4—41no—
343D4—44no—
344D4—44no—
345D4—42no—
346D4—42no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4C2×C22⋊C4×C3111yescharacteristic
372C2×C4C31×#12311yescharacteristic
373C8—42no—
374C22—21no—
375C22—21no—
376C22—22no—
377C22—22no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—82no—
384C22—84no—
385C22—81no—
386C22—81no—
387C22—162no—
388C22C31×#38611yescharacteristic
389C22C31×B(4,𝔽2)11yescharacteristic
390C22C31×B(4,𝔽2)11yescharacteristic
391C4—21no—
392C4—21no—
393C4—42no—
394C4—42no—
395C4—42no—
396C4—82no—
397C4—82no—
398C4—162no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—42no—
403C2—82no—
404C2C31×#188911yescharacteristic
405C2C31×#219911yescharacteristic
406C2C31×#220111yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutatorC2×D4C22×C4×C31
Frattini, cosocleC22×D4C23×C31
prime radicalC2×C31×H(5,𝔽2)C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C22×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2379C2semidirect
3. C2×C31×#354C2semidirect
5. C31×#1390C2semidirect
10. C31×#455C4semidirect
74. C2×Q8×C31C22⋊C4semidirect
117. C42×C31C22⋊C4semidirect
118. #13050C31direct (N × H)
172. #2379C2×C31semidirect
173. C2×#354C2×C31semidirect
175. #1390C2×C31semidirect
205. #455C4×C31semidirect
264. C31#13050direct (N × H)
278. C2×Q8C22⋊C4×C31semidirect
321. C42C22⋊C4×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4552
C31×#23792
C2×C31×#3542
C31×#17902
C31×#13902
#1305031