Finite Group Databasebeta

← C31×#12270 #21469029

Subgroups — C31×#12270 #21469029

Order 7936.

512 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2555C211yesmaximal, characteristic
2C31×#2027C211yesmaximal, characteristic, complemented
3C31×#1605C211yesmaximal, characteristic, complemented
4C31×#1599C211yesmaximal, characteristic, complemented
5C31×#1354C211yesmaximal, characteristic, complemented
6C31×#1345C211yesmaximal, characteristic, complemented
7C31×#1318C211yesmaximal, characteristic, complemented
8C31×#544C2211yescharacteristic
9C2×C31×#124—21no—
10C2×C31×#121—21no—
11C31×#451—21no—
12C31×#450—21no—
13C31×#418—21no—
14C31×#395C2211yescharacteristic
15C31×#394C2211yescharacteristic
16C31×#392—21no—
17C31×#390C2211yescharacteristic
18C31×#388—21no—
19C31×#386—21no—
20C4×C22⋊C4×C31—21no—
21C31×#343C2211yescharacteristic
22C31×#339—21no—
23C31×#328—21no—
24C31×#328—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×D4×C31—21no—
29C22×D4×C31—21no—
30C22×D4×C31D411yescharacteristic
31C22×D4×C31D411yescharacteristic
32C23×C4×C31—21no—
33C2×Q16×C31—41no—
34C2×SD16×C31—41no—
35C2×M16×C31—21no—
36C31×C4⋊Q8—41no—
37C31×#125—41no—
38C31×#124—44no—
39C31×C22⋊Q8—21no—
40C31×C22⋊Q8—21no—
41C31×#121—44no—
42C31×#117—21no—
43C2×C4⋊C4×C31—21no—
44C2×C4⋊C4×C31C2311yescharacteristic
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C22⋊C4×C31—21no—
51C2×C22⋊C4×C31—21no—
52C2×C42×C31—21no—
53C2×C42×C31D411yescharacteristic
54C31×#103—41no—
55C31×#102—41no—
56C31×#102—41no—
57C31×#102—41no—
58C31×C22⋊C8—21no—
59C31×C22⋊C8—21no—
60C31×#95—21no—
61C31×#95—21no—
62C31×#95D411yescharacteristic
63C31×#95D411yescharacteristic
64C24×C31—21no—
65C24×C31—21no—
66C2×Q8×C31—21no—
67C2×Q8×C31—21no—
68C2×Q8×C31—42no—
69C2×Q8×C31—41no—
70C2×D4×C31—21no—
71C2×D4×C31—21no—
72C2×D4×C31—41no—
73C2×D4×C31—41no—
74C2×D4×C31—41no—
75C2×D4×C31—42no—
76C2×D4×C31—42no—
77C2×D4×C31—42no—
78C2×D4×C31—42no—
79C2×D4×C31—42no—
80C2×D4×C31—81no—
81C2×D4×C31—81no—
82C22×C4×C31—21no—
83C22×C4×C31—21no—
84C22×C4×C31—21no—
85C22×C4×C31—21no—
86C22×C4×C31—42no—
87C22×C4×C31—42no—
88C22×C4×C31—81no—
89C22×C4×C31C2×D411yescharacteristic
90C22×C4×C31C2×D411yescharacteristic
91C22×C4×C31C2×D411yescharacteristic
92Q16×C31—82no—
93SD16×C31—82no—
94M16×C31—41no—
95M16×C31—41no—
96C2×C8×C31—41no—
97C2×C8×C31—41no—
98C2×C8×C31—41no—
99C4⋊C4×C31—21no—
100C4⋊C4×C31—21no—
101C4⋊C4×C31—21no—
102C4⋊C4×C31—21no—
103C4⋊C4×C31—41no—
104C4⋊C4×C31—42no—
105C4⋊C4×C31—41no—
106C4⋊C4×C31—41no—
107C22⋊C4×C31—24no—
108C22⋊C4×C31—42no—
109C22⋊C4×C31—42no—
110C22⋊C4×C31—42no—
111C22⋊C4×C31—42no—
112C22⋊C4×C31—42no—
113C22⋊C4×C31—41no—
114C22⋊C4×C31—41no—
115C22⋊C4×C31—41no—
116C42×C31—42no—
117C42×C31—41no—
118C42×C31—41no—
119#12270C3111yesmaximal, characteristic, direct product factor, Sylow
120C23×C31—22no—
121C23×C31—41no—
122C23×C31—41no—
123C23×C31—41no—
124C23×C31—41no—
125C23×C31—41no—
126C23×C31—41no—
127C23×C31—41no—
128C23×C31—41no—
129C23×C31—41no—
130C23×C31—41no—
131C23×C31—41no—
132C23×C31—41no—
133C23×C31—41no—
134C23×C31#12011yescharacteristic
135Q8×C31—42no—
136Q8×C31—81no—
137Q8×C31—81no—
138Q8×C31—81no—
139D4×C31—42no—
140D4×C31—42no—
141D4×C31—81no—
142D4×C31—81no—
143D4×C31—84no—
144D4×C31—84no—
145C2×C4×C31—21no—
146C2×C4×C31—21no—
147C2×C4×C31—22no—
148C2×C4×C31—22no—
149C2×C4×C31—22no—
150C2×C4×C31—21no—
151C2×C4×C31—21no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—84no—
166C2×C4×C31#13511yescharacteristic
167C2×C4×C31#13711yescharacteristic
168C8×C31—81no—
169C8×C31—81no—
170C8×C31—81no—
171#2555C2×C3111yescharacteristic
172#2027C2×C3111yescharacteristic, complemented
173#1605C2×C3111yescharacteristic, complemented
174#1599C2×C3111yescharacteristic, complemented
175#1354C2×C3111yescharacteristic, complemented
176#1345C2×C3111yescharacteristic, complemented
177#1318C2×C3111yescharacteristic, complemented
178C22×C31—21no—
179C22×C31—21no—
180C22×C31—42no—
181C22×C31—42no—
182C22×C31—42no—
183C22×C31—81no—
184C22×C31—81no—
185C22×C31—81no—
186C22×C31—81no—
187C22×C31—81no—
188C22×C31—81no—
189C22×C31—81no—
190C22×C31—81no—
191C22×C31—81no—
192C22×C31—81no—
193C22×C31—81no—
194C22×C31#45211yescharacteristic
195C22×C31#45511yescharacteristic
196C22×C31B(4,𝔽2)11yescharacteristic
197C4×C31—21no—
198C4×C31—21no—
199C4×C31—42no—
200C4×C31—42no—
201C4×C31—41no—
202C4×C31—41no—
203C4×C31—42no—
204C4×C31—81no—
205C4×C31—81no—
206C4×C31—81no—
207#544C22×C3111yescharacteristic
208C2×#124—21no—
209C2×#121—21no—
210#451—21no—
211#450—21no—
212#418—21no—
213#395C22×C3111yescharacteristic
214#394C22×C3111yescharacteristic
215#392—21no—
216#390C22×C3111yescharacteristic
217#388—21no—
218#386—21no—
219C4×C22⋊C4—21no—
220#343C22×C3111yescharacteristic
221#339—21no—
222#328—21no—
223#328—21no—
224#324C22×C3111yescharacteristic
225#324C22×C3111yescharacteristic
226C2×C31—21no—
227C2×C31—21no—
228C2×C31—82no—
229C2×C31—161no—
230C2×C31#160611yescharacteristic
231C2×C31#219611yescharacteristic
232C2×C31#220411yescharacteristic
233C22×Q8D4×C3111yescharacteristic
234C22×D4—21no—
235C22×D4—21no—
236C22×D4D4×C3111yescharacteristic
237C22×D4D4×C3111yescharacteristic
238C23×C4—21no—
239C2×Q16—41no—
240C2×SD16—41no—
241C2×M16—21no—
242C4⋊Q8—41no—
243#125—41no—
244#124—44no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247#121—44no—
248#117—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4C23×C3111yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C42—21no—
259C2×C42D4×C3111yescharacteristic
260#103—41no—
261#102—41no—
262#102—41no—
263#102—41no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266#95—21no—
267#95—21no—
268#95D4×C3111yescharacteristic
269#95D4×C3111yescharacteristic
270C31#1227011yescharacteristic, direct product factor, Sylow
271C24—21no—
272C24—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—42no—
276C2×Q8—41no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—81no—
288C2×D4—81no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—81no—
296C22×C4C2×D4×C3111yescharacteristic
297C22×C4C2×D4×C3111yescharacteristic
298C22×C4C2×D4×C3111yescharacteristic
299Q16—82no—
300SD16—82no—
301M16—41no—
302M16—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C22⋊C4—24no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C42—41no—
324C42—42no—
325C42—41no—
326C23—22no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23C31×#12011yescharacteristic
341Q8—42no—
342Q8—81no—
343Q8—81no—
344Q8—81no—
345D4—42no—
346D4—42no—
347D4—81no—
348D4—81no—
349D4—84no—
350D4—84no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—84no—
372C2×C4C31×#13511yescharacteristic
373C2×C4C31×#13711yescharacteristic
374C8—81no—
375C8—81no—
376C8—81no—
377C22—21no—
378C22—21no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22C31×#45211yescharacteristic
394C22C31×#45511yescharacteristic
395C22C31×B(4,𝔽2)11yescharacteristic
396C4—21no—
397C4—21no—
398C4—42no—
399C4—42no—
400C4—41no—
401C4—41no—
402C4—42no—
403C4—81no—
404C4—81no—
405C4—81no—
406C2—21no—
407C2—21no—
408C2—82no—
409C2—161no—
410C2C31×#160611yescharacteristic
411C2C31×#219611yescharacteristic
412C2C31×#220411yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC31×#2555C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2027C2semidirect
3. C31×#1605C2semidirect
4. C31×#1599C2semidirect
5. C31×#1354C2semidirect
6. C31×#1345C2semidirect
7. C31×#1318C2semidirect
119. #12270C31direct (N × H)
172. #2027C2×C31semidirect
173. #1605C2×C31semidirect
174. #1599C2×C31semidirect
175. #1354C2×C31semidirect
176. #1345C2×C31semidirect
177. #1318C2×C31semidirect
270. C31#12270direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25552
C31×#20272
C31×#16052
C31×#15992
C31×#13542
C31×#13452
C31×#13182
#1227031