Finite Group Databasebeta

← C31×#12136 #21468895

Subgroups — C31×#12136 #21468895

Order 7936.

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

normal subgroups only

This group has 456 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×#2824C211yesmaximal, characteristic, complemented
2C31×#2044C211yesmaximal, characteristic, complemented
3C31×#1601C211yesmaximal, characteristic, complemented
4C31×#1595C211yesmaximal, characteristic
5C31×#1351C211yesmaximal, characteristic, complemented
6C31×#1343C211yesmaximal, characteristic, complemented
7C31×#1326C211yesmaximal, characteristic, complemented
8C31×#534C2211yescharacteristic
9C31×#448—21no—
10C31×#447—21no—
11C2×C31×#99—21no—
12C31×#400—21no—
13C31×#397—21no—
14C31×#394—21no—
15C31×#394—21no—
16C31×#393—21no—
17C31×#393C2211yescharacteristic
18C31×#392C2211yescharacteristic
19C31×#386—21no—
20C31×#386—21no—
21C31×#385C2211yescharacteristic
22C31×#342C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×D4×C31—21no—
29C22×D4×C31—21no—
30C22×D4×C31D411yescharacteristic
31C23×C4×C31D411yescharacteristic
32C2×SD16×C31—41no—
33C2×D8×C31—41no—
34C31×#123—41no—
35C31×#121—21no—
36C31×#121—21no—
37C31×#120—41no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—41no—
48C2×C22⋊C4×C31D411yescharacteristic
49C2×C22⋊C4×C31C2311yescharacteristic
50C31×#103—41no—
51C31×#102—41no—
52C31×#99—42no—
53C31×C22⋊C8—21no—
54C31×C22⋊C8—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—41no—
61C31×#95D411yescharacteristic
62C31×#95D411yescharacteristic
63C24×C31—21no—
64C24×C31—21no—
65C24×C31C2×D411yescharacteristic
66C2×Q8×C31—41no—
67C2×Q8×C31—81no—
68C2×D4×C31—21no—
69C2×D4×C31—21no—
70C2×D4×C31—41no—
71C2×D4×C31—41no—
72C2×D4×C31—41no—
73C2×D4×C31—41no—
74C2×D4×C31—41no—
75C2×D4×C31—41no—
76C2×D4×C31—41no—
77C2×D4×C31—41no—
78C2×D4×C31—41no—
79C2×D4×C31—81no—
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—21no—
87C22×C4×C31—21no—
88C22×C4×C31—41no—
89C22×C4×C31—41no—
90C22×C4×C31—81no—
91C22×C4×C31C2×D411yescharacteristic
92C22×C4×C31C2×D411yescharacteristic
93SD16×C31—82no—
94D8×C31—82no—
95C2×C8×C31—41no—
96C2×C8×C31—41no—
97C4⋊C4×C31—21no—
98C4⋊C4×C31—21no—
99C4⋊C4×C31—81no—
100C4⋊C4×C31—81no—
101C22⋊C4×C31—21no—
102C22⋊C4×C31—21no—
103C22⋊C4×C31—41no—
104C22⋊C4×C31—41no—
105C22⋊C4×C31—41no—
106C22⋊C4×C31—41no—
107C22⋊C4×C31—41no—
108C22⋊C4×C31—81no—
109C22⋊C4×C31—81no—
110C22⋊C4×C31—81no—
111C22⋊C4×C31—81no—
112C22⋊C4×C31—81no—
113C22⋊C4×C31—81no—
114C22⋊C4×C31—82no—
115#12136C3111yesmaximal, characteristic, direct product factor, Sylow
116C23×C31—21no—
117C23×C31—21no—
118C23×C31—41no—
119C23×C31—41no—
120C23×C31—41no—
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—81no—
131C23×C31—81no—
132C23×C31#12011yescharacteristic
133C23×C31#13511yescharacteristic
134C23×C31AGL(1,ℤ/8)11yescharacteristic
135Q8×C31—42no—
136Q8×C31—81no—
137D4×C31—42no—
138D4×C31—81no—
139D4×C31—81no—
140D4×C31—81no—
141D4×C31—81no—
142D4×C31—81no—
143D4×C31—82no—
144D4×C31—82no—
145D4×C31—81no—
146D4×C31—81no—
147C2×C4×C31—21no—
148C2×C4×C31—21no—
149C2×C4×C31—41no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
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—42no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—41no—
169C2×C4×C31—41no—
170C2×C4×C31—81no—
171C2×C4×C31—81no—
172C2×C4×C31—81no—
173C2×C4×C31—81no—
174C8×C31—81no—
175C8×C31—81no—
176#2824C2×C3111yescharacteristic, complemented
177#2044C2×C3111yescharacteristic, complemented
178#1601C2×C3111yescharacteristic, complemented
179#1595C2×C3111yescharacteristic
180#1351C2×C3111yescharacteristic, complemented
181#1343C2×C3111yescharacteristic, complemented
182#1326C2×C3111yescharacteristic, complemented
183C22×C31—21no—
184C22×C31—21no—
185C22×C31—21no—
186C22×C31—21no—
187C22×C31—41no—
188C22×C31—41no—
189C22×C31—41no—
190C22×C31—41no—
191C22×C31—81no—
192C22×C31—81no—
193C22×C31—81no—
194C22×C31—81no—
195C22×C31—81no—
196C22×C31—82no—
197C22×C31—81no—
198C22×C31—81no—
199C22×C31—81no—
200C22×C31—81no—
201C22×C31—81no—
202C22×C31—81no—
203C22×C31—161no—
204C22×C31#44911yescharacteristic
205C22×C31#45411yescharacteristic
206C22×C31B(4,𝔽2)11yescharacteristic
207C4×C31—41no—
208C4×C31—41no—
209C4×C31—81no—
210C4×C31—81no—
211C4×C31—81no—
212C4×C31—81no—
213C4×C31—81no—
214C4×C31—81no—
215C4×C31—161no—
216#534C22×C3111yescharacteristic
217#448—21no—
218#447—21no—
219C2×#99—21no—
220#400—21no—
221#397—21no—
222#394—21no—
223#394—21no—
224#393—21no—
225#393C22×C3111yescharacteristic
226#392C22×C3111yescharacteristic
227#386—21no—
228#386—21no—
229#385C22×C3111yescharacteristic
230#342C22×C3111yescharacteristic
231#327—21no—
232#327—21no—
233#323C22×C3111yescharacteristic
234#323C22×C3111yescharacteristic
235C2×C31—21no—
236C2×C31—21no—
237C2×C31—41no—
238C2×C31—41no—
239C2×C31—161no—
240C2×C31—161no—
241C2×C31#159711yescharacteristic
242C2×C31#219911yescharacteristic
243C2×C31#220011yescharacteristic
244C22×Q8D4×C3111yescharacteristic
245C22×D4—21no—
246C22×D4—21no—
247C22×D4D4×C3111yescharacteristic
248C23×C4D4×C3111yescharacteristic
249C2×SD16—41no—
250C2×D8—41no—
251#123—41no—
252#121—21no—
253#121—21no—
254#120—41no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4D4×C3111yescharacteristic
266C2×C22⋊C4C23×C3111yescharacteristic
267#103—41no—
268#102—41no—
269#99—42no—
270C22⋊C8—21no—
271C22⋊C8—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—41no—
278#95D4×C3111yescharacteristic
279#95D4×C3111yescharacteristic
280C31#1213611yescharacteristic, direct product factor, Sylow
281C24—21no—
282C24—21no—
283C24C2×D4×C3111yescharacteristic
284C2×Q8—41no—
285C2×Q8—81no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—81no—
298C2×D4—81no—
299C2×D4—81no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—81no—
309C22×C4C2×D4×C3111yescharacteristic
310C22×C4C2×D4×C3111yescharacteristic
311SD16—82no—
312D8—82no—
313C2×C8—41no—
314C2×C8—41no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—81no—
318C4⋊C4—81no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—81no—
327C22⋊C4—81no—
328C22⋊C4—81no—
329C22⋊C4—81no—
330C22⋊C4—81no—
331C22⋊C4—81no—
332C22⋊C4—82no—
333C23—21no—
334C23—21no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—81no—
348C23—81no—
349C23C31×#12011yescharacteristic
350C23C31×#13511yescharacteristic
351C23C31×AGL(1,ℤ/8)11yescharacteristic
352Q8—42no—
353Q8—81no—
354D4—42no—
355D4—81no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—82no—
361D4—82no—
362D4—81no—
363D4—81no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C8—81no—
392C8—81no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—82no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—161no—
414C22C31×#44911yescharacteristic
415C22C31×#45411yescharacteristic
416C22C31×B(4,𝔽2)11yescharacteristic
417C4—41no—
418C4—41no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4—161no—
426C2—21no—
427C2—21no—
428C2—41no—
429C2—41no—
430C2—161no—
431C2—161no—
432C2C31×#159711yescharacteristic
433C2C31×#219911yescharacteristic
434C2C31×#220011yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC31×#1595C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C22⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2824C2semidirect
2. C31×#2044C2semidirect
3. C31×#1601C2semidirect
5. C31×#1351C2semidirect
6. C31×#1343C2semidirect
7. C31×#1326C2semidirect
115. #12136C31direct (N × H)
176. #2824C2×C31semidirect
177. #2044C2×C31semidirect
178. #1601C2×C31semidirect
180. #1351C2×C31semidirect
181. #1343C2×C31semidirect
182. #1326C2×C31semidirect
280. C31#12136direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28242
C31×#20442
C31×#16012
C31×#15952
C31×#13512
C31×#13432
C31×#13262
#1213631