Finite Group Databasebeta

← C31×#12188 #21468947

Subgroups — C31×#12188 #21468947

Order 7936.

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

normal subgroups only

This group has 554 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×#2736C211yesmaximal, characteristic, complemented
2C31×#2021C211yesmaximal, characteristic
3C31×#1607C211yesmaximal, characteristic, complemented
4C31×#1601C211yesmaximal, characteristic, complemented
5C31×#1351C211yesmaximal, characteristic, complemented
6C31×#1344C211yesmaximal, characteristic, complemented
7C31×#1316C211yesmaximal, characteristic, complemented
8C31×#534C2211yescharacteristic
9C2×C31×#120—21no—
10C31×#480—21no—
11C31×#448—21no—
12C2×C31×#99—21no—
13C31×#397—21no—
14C31×#397C2211yescharacteristic
15C31×#393—21no—
16C31×#393C2211yescharacteristic
17C31×#388—21no—
18C31×#388—21no—
19C31×#388—21no—
20C31×#379—21no—
21C31×#379C2211yescharacteristic
22C31×#342C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C25×C31D411yescharacteristic
28C22×Q8×C31D411yescharacteristic, complemented
29C22×D4×C31—21no—
30C22×D4×C31—41no—
31C2×SD16×C31—41no—
32C31×#123—41no—
33C31×#121—21no—
34C31×#121—21no—
35C31×#120—42no—
36C31×#120—41no—
37C31×#120—42no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31D411yescharacteristic, complemented
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—21no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—41no—
50C2×C22⋊C4×C31D411yescharacteristic
51C2×C22⋊C4×C31C2311yescharacteristic
52C2×C42×C31—21no—
53C31×#107—41no—
54C31×#103—41no—
55C31×#102—41no—
56C31×#99—42no—
57C31×C22⋊C8—21no—
58C31×C22⋊C8—21no—
59C31×#95—21no—
60C31×#95—21no—
61C31×#95—21no—
62C31×#95—21no—
63C31×#95D411yescharacteristic, complemented
64C31×#95D411yescharacteristic, complemented
65C24×C31—21no—
66C24×C31—21no—
67C24×C31—21no—
68C24×C31—21no—
69C24×C31—41no—
70C24×C31—41no—
71C24×C31—42no—
72C24×C31—81no—
73C24×C31C2×D411yescharacteristic
74C2×Q8×C31—41no—
75C2×Q8×C31—81no—
76C2×D4×C31—21no—
77C2×D4×C31—21no—
78C2×D4×C31—41no—
79C2×D4×C31—41no—
80C2×D4×C31—41no—
81C2×D4×C31—44no—
82C2×D4×C31—41no—
83C2×D4×C31—41no—
84C2×D4×C31—41no—
85C2×D4×C31—82no—
86C2×D4×C31—82no—
87C22×C4×C31—21no—
88C22×C4×C31—21no—
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—21no—
92C22×C4×C31—21no—
93C22×C4×C31—41no—
94C22×C4×C31C2×D411yescharacteristic
95C22×C4×C31C2×D411yescharacteristic
96SD16×C31—82no—
97C2×C8×C31—41no—
98C2×C8×C31—41no—
99C4⋊C4×C31—21no—
100C4⋊C4×C31—21no—
101C4⋊C4×C31—41no—
102C4⋊C4×C31—81no—
103C22⋊C4×C31—22no—
104C22⋊C4×C31—41no—
105C22⋊C4×C31—41no—
106C22⋊C4×C31—41no—
107C22⋊C4×C31—42no—
108C22⋊C4×C31—42no—
109C22⋊C4×C31—44no—
110C22⋊C4×C31—41no—
111C22⋊C4×C31—41no—
112C22⋊C4×C31—41no—
113C22⋊C4×C31—41no—
114C22⋊C4×C31—81no—
115C22⋊C4×C31—81no—
116C22⋊C4×C31—81no—
117C22⋊C4×C31—81no—
118C22⋊C4×C31—81no—
119C42×C31—81no—
120#12188C3111yesmaximal, characteristic, direct product factor, Sylow
121C23×C31—21no—
122C23×C31—21no—
123C23×C31—22no—
124C23×C31—41no—
125C23×C31—41no—
126C23×C31—42no—
127C23×C31—41no—
128C23×C31—41no—
129C23×C31—41no—
130C23×C31—41no—
131C23×C31—41no—
132C23×C31—42no—
133C23×C31—42no—
134C23×C31—41no—
135C23×C31—41no—
136C23×C31—41no—
137C23×C31—41no—
138C23×C31—81no—
139C23×C31—82no—
140C23×C31—81no—
141C23×C31—82no—
142C23×C31—84no—
143C23×C31—81no—
144C23×C31—81no—
145C23×C31—81no—
146C23×C31#12011yescharacteristic
147C23×C31#13511yescharacteristic
148C23×C31AGL(1,ℤ/8)11yescharacteristic
149Q8×C31—42no—
150Q8×C31—81no—
151D4×C31—81no—
152D4×C31—81no—
153D4×C31—81no—
154D4×C31—81no—
155D4×C31—88no—
156D4×C31—81no—
157D4×C31—81no—
158C2×C4×C31—21no—
159C2×C4×C31—21no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—42no—
163C2×C4×C31—41no—
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—41no—
171C2×C4×C31—41no—
172C2×C4×C31—41no—
173C2×C4×C31—41no—
174C2×C4×C31—41no—
175C2×C4×C31—41no—
176C2×C4×C31—41no—
177C2×C4×C31—41no—
178C2×C4×C31—41no—
179C2×C4×C31—41no—
180C2×C4×C31—41no—
181C2×C4×C31—81no—
182C2×C4×C31—81no—
183C8×C31—81no—
184C8×C31—81no—
185#2736C2×C3111yescharacteristic, complemented
186#2021C2×C3111yescharacteristic
187#1607C2×C3111yescharacteristic, complemented
188#1601C2×C3111yescharacteristic, complemented
189#1351C2×C3111yescharacteristic, complemented
190#1344C2×C3111yescharacteristic, complemented
191#1316C2×C3111yescharacteristic, complemented
192C22×C31—21no—
193C22×C31—21no—
194C22×C31—21no—
195C22×C31—21no—
196C22×C31—41no—
197C22×C31—41no—
198C22×C31—42no—
199C22×C31—42no—
200C22×C31—41no—
201C22×C31—41no—
202C22×C31—44no—
203C22×C31—42no—
204C22×C31—81no—
205C22×C31—81no—
206C22×C31—81no—
207C22×C31—82no—
208C22×C31—82no—
209C22×C31—82no—
210C22×C31—82no—
211C22×C31—81no—
212C22×C31—81no—
213C22×C31—81no—
214C22×C31—81no—
215C22×C31—81no—
216C22×C31—161no—
217C22×C31#44911yescharacteristic
218C22×C31#45311yescharacteristic
219C22×C31B(4,𝔽2)11yescharacteristic
220C4×C31—41no—
221C4×C31—41no—
222C4×C31—81no—
223C4×C31—81no—
224C4×C31—81no—
225C4×C31—82no—
226C4×C31—81no—
227C4×C31—81no—
228C4×C31—81no—
229#534C22×C3111yescharacteristic
230C2×#120—21no—
231#480—21no—
232#448—21no—
233C2×#99—21no—
234#397—21no—
235#397C22×C3111yescharacteristic
236#393—21no—
237#393C22×C3111yescharacteristic
238#388—21no—
239#388—21no—
240#388—21no—
241#379—21no—
242#379C22×C3111yescharacteristic
243#342C22×C3111yescharacteristic
244#327—21no—
245#327—21no—
246#323C22×C3111yescharacteristic
247#323C22×C3111yescharacteristic
248C2×C31—21no—
249C2×C31—21no—
250C2×C31—41no—
251C2×C31—41no—
252C2×C31—82no—
253C2×C31—161no—
254C2×C31#160011yescharacteristic
255C2×C31#219711yescharacteristic
256C2×C31#219911yescharacteristic
257C25D4×C3111yescharacteristic
258C22×Q8D4×C3111yescharacteristic, complemented
259C22×D4—21no—
260C22×D4—41no—
261C2×SD16—41no—
262#123—41no—
263#121—21no—
264#121—21no—
265#120—42no—
266#120—41no—
267#120—42no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4D4×C3111yescharacteristic, complemented
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4D4×C3111yescharacteristic
281C2×C22⋊C4C23×C3111yescharacteristic
282C2×C42—21no—
283#107—41no—
284#103—41no—
285#102—41no—
286#99—42no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95D4×C3111yescharacteristic, complemented
294#95D4×C3111yescharacteristic, complemented
295C31#1218811yescharacteristic, direct product factor, Sylow
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—41no—
301C24—41no—
302C24—42no—
303C24—81no—
304C24C2×D4×C3111yescharacteristic
305C2×Q8—41no—
306C2×Q8—81no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—44no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—82no—
317C2×D4—82no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—41no—
325C22×C4C2×D4×C3111yescharacteristic
326C22×C4C2×D4×C3111yescharacteristic
327SD16—82no—
328C2×C8—41no—
329C2×C8—41no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—81no—
334C22⋊C4—22no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—44no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—81no—
346C22⋊C4—81no—
347C22⋊C4—81no—
348C22⋊C4—81no—
349C22⋊C4—81no—
350C42—81no—
351C23—21no—
352C23—21no—
353C23—22no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—81no—
369C23—82no—
370C23—81no—
371C23—82no—
372C23—84no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23C31×#12011yescharacteristic
377C23C31×#13511yescharacteristic
378C23C31×AGL(1,ℤ/8)11yescharacteristic
379Q8—42no—
380Q8—81no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—88no—
386D4—81no—
387D4—81no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—81no—
412C2×C4—81no—
413C8—81no—
414C8—81no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—41no—
420C22—41no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—44no—
426C22—42no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—161no—
440C22C31×#44911yescharacteristic
441C22C31×#45311yescharacteristic
442C22C31×B(4,𝔽2)11yescharacteristic
443C4—41no—
444C4—41no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—82no—
449C4—81no—
450C4—81no—
451C4—81no—
452C2—21no—
453C2—21no—
454C2—41no—
455C2—41no—
456C2—82no—
457C2—161no—
458C2C31×#160011yescharacteristic
459C2C31×#219711yescharacteristic
460C2C31×#219911yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC31×#2021C2
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×#2736C2semidirect
3. C31×#1607C2semidirect
4. C31×#1601C2semidirect
5. C31×#1351C2semidirect
6. C31×#1344C2semidirect
7. C31×#1316C2semidirect
28. C22×Q8×C31D4semidirect
39. C2×C4⋊C4×C31D4semidirect
63. C31×#95D4semidirect
64. C31×#95D4semidirect
120. #12188C31direct (N × H)
185. #2736C2×C31semidirect
187. #1607C2×C31semidirect
188. #1601C2×C31semidirect
189. #1351C2×C31semidirect
190. #1344C2×C31semidirect
191. #1316C2×C31semidirect
258. C22×Q8D4×C31semidirect
269. C2×C4⋊C4D4×C31semidirect
293. #95D4×C31semidirect
294. #95D4×C31semidirect
295. C31#12188direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27362
C31×#20212
C31×#16072
C31×#16012
C31×#13512
C31×#13442
C31×#13162
#1218831