Finite Group Databasebeta

← C31×#9020 #21465779

Subgroups — C31×#9020 #21465779

Order 7936.

1224 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 150 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×#392C211yesmaximal, characteristic
2C31×#2312C211yesmaximal, characteristic, complemented
3C31×#1786C212yesmaximal, complemented
4C2×C31×#323C211yesmaximal, characteristic, complemented
5C31×#1294C212yesmaximal, complemented
6C23×D4×C31—21no—
7C23×D4×C31C2211yescharacteristic
8C22×C22⋊C4×C31—21no—
9C22×C22⋊C4×C31C2211yescharacteristic
10C2×C31×#102—24no—
11C2×C31×C22⋊C8C2212yes—
12C31×#392—22no—
13C31×#392C414yescomplemented
14C31×#388—24no—
15C31×#380C2212yes—
16C31×#379—21no—
17C4×C22⋊C4×C31—21no—
18C2×C31×#95C2211yescharacteristic
19C31×#323—22no—
20C25×C31—21no—
21C25×C31D411yescharacteristic
22C22×D4×C31—24no—
23C22×D4×C31—22no—
24C22×D4×C31—21no—
25C22×D4×C31—22no—
26C22×D4×C31—44no—
27C22×D4×C31—44no—
28C22×D4×C31—44no—
29C22×D4×C31—44no—
30C22×D4×C31C2×C412yes—
31C23×C4×C31—21no—
32C23×C4×C31D411yescharacteristic
33C23×C4×C31C2311yescharacteristic
34C22×C8×C31—22no—
35C2×C4⋊C4×C31D414yescomplemented
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—24no—
40C2×C22⋊C4×C31—22no—
41C2×C22⋊C4×C31—22no—
42C2×C22⋊C4×C31—44no—
43C2×C22⋊C4×C31C2×C412yes—
44C2×C22⋊C4×C31D412yes—
45C2×C42×C31—22no—
46C31×#102—48no—
47C31×C22⋊C8—22no—
48C31×C22⋊C8—22no—
49C31×#95—22no—
50C31×#95—21no—
51C31×#95—22no—
52C31×#95C2×C412yes—
53C24×C31—24no—
54C24×C31—22no—
55C24×C31—21no—
56C24×C31—22no—
57C24×C31—21no—
58C24×C31—22no—
59C24×C31—42no—
60C24×C31—42no—
61C24×C31—42no—
62C24×C31—42no—
63C24×C31—42no—
64C24×C31—41no—
65C24×C31—41no—
66C24×C31—42no—
67C24×C31—42no—
68C24×C31C22⋊C412yes—
69C24×C31C2×D411yescharacteristic
70C2×D4×C31—28no—
71C2×D4×C31—24no—
72C2×D4×C31—44no—
73C2×D4×C31—48no—
74C2×D4×C31—44no—
75C2×D4×C31—48no—
76C2×D4×C31—44no—
77C2×D4×C31—48no—
78C2×D4×C31—48no—
79C2×D4×C31—432no—
80C2×D4×C31—42no—
81C22×C4×C31—24no—
82C22×C4×C31—22no—
83C22×C4×C31—21no—
84C22×C4×C31—21no—
85C22×C4×C31—24no—
86C22×C4×C31—42no—
87C22×C4×C31—42no—
88C22×C4×C31—42no—
89C22×C4×C31—42no—
90C22×C4×C31C22⋊C412yes—
91C22×C4×C31C22×C411yescharacteristic
92C22×C4×C31C2×D412yes—
93C22×C4×C31C2×D411yescharacteristic
94C22×C4×C31#4112yes—
95C2×C8×C31—44no—
96C2×C8×C31—42no—
97C4⋊C4×C31—44no—
98C22⋊C4×C31—24no—
99C22⋊C4×C31—24no—
100C22⋊C4×C31—48no—
101C22⋊C4×C31—42no—
102C22⋊C4×C31—42no—
103C22⋊C4×C31—82no—
104C22⋊C4×C31—82no—
105C42×C31—82no—
106#9020C3111yesmaximal, characteristic, direct product factor, Sylow
107C23×C31—22no—
108C23×C31—22no—
109C23×C31—24no—
110C23×C31—22no—
111C23×C31—22no—
112C23×C31—22no—
113C23×C31—42no—
114C23×C31—42no—
115C23×C31—42no—
116C23×C31—48no—
117C23×C31—48no—
118C23×C31—42no—
119C23×C31—42no—
120C23×C31—44no—
121C23×C31—48no—
122C23×C31—42no—
123C23×C31—42no—
124C23×C31—42no—
125C23×C31—42no—
126C23×C31—42no—
127C23×C31—42no—
128C23×C31—44no—
129C23×C31—42no—
130C23×C31—81no—
131C23×C31—81no—
132C23×C31—84no—
133C23×C31—84no—
134C23×C31—84no—
135C23×C31—82no—
136C23×C31—82no—
137C23×C31—82no—
138C23×C31—82no—
139C23×C31C2×C22⋊C411yescharacteristic
140C23×C31#12011yescharacteristic
141C23×C31#12311yescharacteristic
142C23×C31#13512yes—
143C23×C31AGL(1,ℤ/8)12yes—
144D4×C31—28no—
145D4×C31—48no—
146D4×C31—44no—
147D4×C31—432no—
148C2×C4×C31—24no—
149C2×C4×C31—24no—
150C2×C4×C31—22no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—42no—
154C2×C4×C31—48no—
155C2×C4×C31—42no—
156C2×C4×C31—42no—
157C2×C4×C31—42no—
158C2×C4×C31—42no—
159C2×C4×C31—44no—
160C2×C4×C31—44no—
161C2×C4×C31—44no—
162C2×C4×C31—81no—
163C2×C4×C31C4×D412yes—
164C2×C4×C31#12112yes—
165C8×C31—82no—
166C2×#392C2×C3111yescharacteristic
167#2312C2×C3111yescharacteristic, complemented
168#1786C2×C3112yescomplemented
169C2×#323C2×C3111yescharacteristic, complemented
170#1294C2×C3112yescomplemented
171C22×C31—24no—
172C22×C31—24no—
173C22×C31—21no—
174C22×C31—21no—
175C22×C31—21no—
176C22×C31—21no—
177C22×C31—44no—
178C22×C31—44no—
179C22×C31—44no—
180C22×C31—42no—
181C22×C31—42no—
182C22×C31—44no—
183C22×C31—42no—
184C22×C31—82no—
185C22×C31—82no—
186C22×C31—82no—
187C22×C31—84no—
188C22×C31—88no—
189C22×C31—84no—
190C22×C31—88no—
191C22×C31—82no—
192C22×C31—82no—
193C22×C31#38611yescharacteristic
194C22×C31#41611yescharacteristic
195C22×C31#41811yescharacteristic
196C22×C31#44912yes—
197C22×C31#45312yes—
198C22×C31B(4,𝔽2)12yes—
199C22×C31#48212yes—
200C4×C31—22no—
201C4×C31—22no—
202C4×C31—44no—
203C4×C31—42no—
204C4×C31—84no—
205C23×D4—21no—
206C23×D4C22×C3111yescharacteristic
207C22×C22⋊C4—21no—
208C22×C22⋊C4C22×C3111yescharacteristic
209C2×#102—24no—
210C2×C22⋊C8C22×C3112yes—
211#392—22no—
212#392C4×C3114yescomplemented
213#388—24no—
214#380C22×C3112yes—
215#379—21no—
216C4×C22⋊C4—21no—
217C2×#95C22×C3111yescharacteristic
218#323—22no—
219C2×C31—21no—
220C2×C31—21no—
221C2×C31—22no—
222C2×C31—82no—
223C2×C31—84no—
224C2×C31#160014yes—
225C2×C31#187511yescharacteristic
226C2×C31#188711yescharacteristic
227C2×C31#188911yescharacteristic
228C25—21no—
229C25D4×C3111yescharacteristic
230C22×D4—24no—
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—44no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4—44no—
238C22×D4C2×C4×C3112yes—
239C23×C4—21no—
240C23×C4D4×C3111yescharacteristic
241C23×C4C23×C3111yescharacteristic
242C22×C8—22no—
243C2×C4⋊C4D4×C3114yescomplemented
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4C2×C4×C3112yes—
252C2×C22⋊C4D4×C3112yes—
253C2×C42—22no—
254#102—48no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257#95—22no—
258#95—21no—
259#95—22no—
260#95C2×C4×C3112yes—
261C31#902011yescharacteristic, direct product factor, Sylow
262C24—24no—
263C24—22no—
264C24—21no—
265C24—22no—
266C24—21no—
267C24—22no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—41no—
274C24—41no—
275C24—42no—
276C24—42no—
277C24C22⋊C4×C3112yes—
278C24C2×D4×C3111yescharacteristic
279C2×D4—28no—
280C2×D4—24no—
281C2×D4—44no—
282C2×D4—48no—
283C2×D4—44no—
284C2×D4—48no—
285C2×D4—44no—
286C2×D4—48no—
287C2×D4—48no—
288C2×D4—432no—
289C2×D4—42no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—24no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4C22⋊C4×C3112yes—
300C22×C4C22×C4×C3111yescharacteristic
301C22×C4C2×D4×C3112yes—
302C22×C4C2×D4×C3111yescharacteristic
303C22×C4#41×C3112yes—
304C2×C8—44no—
305C2×C8—42no—
306C4⋊C4—44no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—48no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C42—82no—
315C23—22no—
316C23—22no—
317C23—24no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—48no—
325C23—48no—
326C23—42no—
327C23—42no—
328C23—44no—
329C23—48no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—44no—
336C23—42no—
337C23—42no—
338C23—81no—
339C23—81no—
340C23—84no—
341C23—84no—
342C23—84no—
343C23—82no—
344C23—82no—
345C23—82no—
346C23—82no—
347C23C2×C22⋊C4×C3111yescharacteristic
348C23C31×#12011yescharacteristic
349C23C31×#12311yescharacteristic
350C23C31×#13512yes—
351C23C31×AGL(1,ℤ/8)12yes—
352D4—28no—
353D4—48no—
354D4—44no—
355D4—432no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—48no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—81no—
371C2×C4C4×D4×C3112yes—
372C2×C4C31×#12112yes—
373C8—82no—
374C22—24no—
375C22—24no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—44no—
381C22—44no—
382C22—44no—
383C22—42no—
384C22—42no—
385C22—44no—
386C22—42no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22—84no—
391C22—88no—
392C22—84no—
393C22—88no—
394C22—82no—
395C22—82no—
396C22C31×#38611yescharacteristic
397C22C31×#41611yescharacteristic
398C22C31×#41811yescharacteristic
399C22C31×#44912yes—
400C22C31×#45312yes—
401C22C31×B(4,𝔽2)12yes—
402C22C31×#48212yes—
403C4—22no—
404C4—22no—
405C4—44no—
406C4—42no—
407C4—84no—
408C2—21no—
409C2—21no—
410C2—22no—
411C2—82no—
412C2—84no—
413C2C31×#160014yes—
414C2C31×#187511yescharacteristic
415C2C31×#188711yescharacteristic
416C2C31×#188911yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC2×C31×#392C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2312C2semidirect
3. C31×#1786C2semidirect
4. C2×C31×#323C2semidirect
5. C31×#1294C2semidirect
13. C31×#392C4semidirect
35. C2×C4⋊C4×C31D4semidirect
106. #9020C31direct (N × H)
167. #2312C2×C31semidirect
168. #1786C2×C31semidirect
169. C2×#323C2×C31semidirect
170. #1294C2×C31semidirect
212. #392C4×C31semidirect
243. C2×C4⋊C4D4×C31semidirect
261. C31#9020direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3922
C31×#23122
C31×#17862
C2×C31×#3232
C31×#12942
#902031