Finite Group Databasebeta

← C31×#12137 #21468896

Subgroups — C31×#12137 #21468896

Order 7936.

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

normal subgroups only

This group has 446 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×#2826C211yesmaximal, characteristic, complemented
2C31×#2044C211yesmaximal, characteristic, complemented
3C31×#1613C211yesmaximal, characteristic, complemented
4C31×#1595C211yesmaximal, characteristic
5C31×#1352C211yesmaximal, characteristic, complemented
6C31×#1343C211yesmaximal, characteristic, complemented
7C31×#1326C211yesmaximal, characteristic, complemented
8C31×#534C2211yescharacteristic
9C31×#481—21no—
10C31×#447—21no—
11C2×C31×#99—21no—
12C31×#400—21no—
13C31×#399C2211yescharacteristic
14C31×#397—21no—
15C31×#396—21no—
16C31×#394—21no—
17C31×#394—21no—
18C31×#392C2211yescharacteristic
19C31×#386—21no—
20C31×#386—21no—
21C31×#385C2211yescharacteristic
22C31×#342C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C22×D4×C31—21no—
28C22×D4×C31—21no—
29C22×D4×C31D411yescharacteristic
30C23×C4×C31D411yescharacteristic
31C2×D8×C31—41no—
32C31×#123—41no—
33C31×#121—21no—
34C31×#121—21no—
35C31×#120—41no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31D411yescharacteristic
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×#106—41no—
51C31×#102—41no—
52C31×#102—41no—
53C31×#99—42no—
54C31×C22⋊C8—21no—
55C31×C22⋊C8—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×D4×C31—21no—
67C2×D4×C31—21no—
68C2×D4×C31—41no—
69C2×D4×C31—41no—
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—81no—
78C2×D4×C31—81no—
79C2×D4×C31—81no—
80C22×C4×C31—21no—
81C22×C4×C31—21no—
82C22×C4×C31—21no—
83C22×C4×C31—21no—
84C22×C4×C31—21no—
85C22×C4×C31—21no—
86C22×C4×C31—41no—
87C22×C4×C31—41no—
88C22×C4×C31—81no—
89C22×C4×C31C2×D411yescharacteristic
90C22×C4×C31C2×D411yescharacteristic
91D8×C31—82no—
92C2×C8×C31—41no—
93C2×C8×C31—41no—
94C4⋊C4×C31—21no—
95C4⋊C4×C31—21no—
96C4⋊C4×C31—41no—
97C4⋊C4×C31—81no—
98C4⋊C4×C31—81no—
99C4⋊C4×C31—81no—
100C22⋊C4×C31—21no—
101C22⋊C4×C31—21no—
102C22⋊C4×C31—41no—
103C22⋊C4×C31—41no—
104C22⋊C4×C31—41no—
105C22⋊C4×C31—41no—
106C22⋊C4×C31—41no—
107C22⋊C4×C31—81no—
108C22⋊C4×C31—81no—
109C22⋊C4×C31—81no—
110C22⋊C4×C31—81no—
111C22⋊C4×C31—81no—
112C22⋊C4×C31—81no—
113C22⋊C4×C31—82no—
114#12137C3111yesmaximal, characteristic, direct product factor, Sylow
115C23×C31—21no—
116C23×C31—21no—
117C23×C31—41no—
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—81no—
130C23×C31—81no—
131C23×C31#12011yescharacteristic
132C23×C31#13511yescharacteristic
133C23×C31AGL(1,ℤ/8)11yescharacteristic
134D4×C31—42no—
135D4×C31—81no—
136D4×C31—81no—
137D4×C31—81no—
138D4×C31—81no—
139D4×C31—81no—
140D4×C31—82no—
141D4×C31—82no—
142D4×C31—81no—
143D4×C31—81no—
144C2×C4×C31—21no—
145C2×C4×C31—21no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
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—42no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—81no—
168C2×C4×C31—81no—
169C2×C4×C31—81no—
170C2×C4×C31—81no—
171C8×C31—81no—
172C8×C31—81no—
173#2826C2×C3111yescharacteristic, complemented
174#2044C2×C3111yescharacteristic, complemented
175#1613C2×C3111yescharacteristic, complemented
176#1595C2×C3111yescharacteristic
177#1352C2×C3111yescharacteristic, complemented
178#1343C2×C3111yescharacteristic, complemented
179#1326C2×C3111yescharacteristic, complemented
180C22×C31—21no—
181C22×C31—21no—
182C22×C31—21no—
183C22×C31—21no—
184C22×C31—41no—
185C22×C31—41no—
186C22×C31—41no—
187C22×C31—41no—
188C22×C31—81no—
189C22×C31—81no—
190C22×C31—81no—
191C22×C31—81no—
192C22×C31—81no—
193C22×C31—82no—
194C22×C31—81no—
195C22×C31—81no—
196C22×C31—81no—
197C22×C31—81no—
198C22×C31—81no—
199C22×C31—81no—
200C22×C31—161no—
201C22×C31#44911yescharacteristic
202C22×C31#45411yescharacteristic
203C22×C31B(4,𝔽2)11yescharacteristic
204C4×C31—41no—
205C4×C31—41no—
206C4×C31—81no—
207C4×C31—81no—
208C4×C31—81no—
209C4×C31—81no—
210C4×C31—81no—
211C4×C31—81no—
212C4×C31—161no—
213#534C22×C3111yescharacteristic
214#481—21no—
215#447—21no—
216C2×#99—21no—
217#400—21no—
218#399C22×C3111yescharacteristic
219#397—21no—
220#396—21no—
221#394—21no—
222#394—21no—
223#392C22×C3111yescharacteristic
224#386—21no—
225#386—21no—
226#385C22×C3111yescharacteristic
227#342C22×C3111yescharacteristic
228#327—21no—
229#327—21no—
230#323C22×C3111yescharacteristic
231#323C22×C3111yescharacteristic
232C2×C31—21no—
233C2×C31—21no—
234C2×C31—41no—
235C2×C31—41no—
236C2×C31—161no—
237C2×C31—161no—
238C2×C31#159711yescharacteristic
239C2×C31#220011yescharacteristic
240C2×C31#220011yescharacteristic
241C22×D4—21no—
242C22×D4—21no—
243C22×D4D4×C3111yescharacteristic
244C23×C4D4×C3111yescharacteristic
245C2×D8—41no—
246#123—41no—
247#121—21no—
248#121—21no—
249#120—41no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4D4×C3111yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4D4×C3111yescharacteristic
263C2×C22⋊C4C23×C3111yescharacteristic
264#106—41no—
265#102—41no—
266#102—41no—
267#99—42no—
268C22⋊C8—21no—
269C22⋊C8—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—41no—
275#95D4×C3111yescharacteristic
276#95D4×C3111yescharacteristic
277C31#1213711yescharacteristic, direct product factor, Sylow
278C24—21no—
279C24—21no—
280C24C2×D4×C3111yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—81no—
293C2×D4—81no—
294C2×D4—81no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—81no—
304C22×C4C2×D4×C3111yescharacteristic
305C22×C4C2×D4×C3111yescharacteristic
306D8—82no—
307C2×C8—41no—
308C2×C8—41no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—81no—
313C4⋊C4—81no—
314C4⋊C4—81no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—81no—
323C22⋊C4—81no—
324C22⋊C4—81no—
325C22⋊C4—81no—
326C22⋊C4—81no—
327C22⋊C4—81no—
328C22⋊C4—82no—
329C23—21no—
330C23—21no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—81no—
344C23—81no—
345C23C31×#12011yescharacteristic
346C23C31×#13511yescharacteristic
347C23C31×AGL(1,ℤ/8)11yescharacteristic
348D4—42no—
349D4—81no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—82no—
355D4—82no—
356D4—81no—
357D4—81no—
358C2×C4—21no—
359C2×C4—21no—
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—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C8—81no—
386C8—81no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—82no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—161no—
408C22C31×#44911yescharacteristic
409C22C31×#45411yescharacteristic
410C22C31×B(4,𝔽2)11yescharacteristic
411C4—41no—
412C4—41no—
413C4—81no—
414C4—81no—
415C4—81no—
416C4—81no—
417C4—81no—
418C4—81no—
419C4—161no—
420C2—21no—
421C2—21no—
422C2—41no—
423C2—41no—
424C2—161no—
425C2—161no—
426C2C31×#159711yescharacteristic
427C2C31×#220011yescharacteristic
428C2C31×#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×#2826C2semidirect
2. C31×#2044C2semidirect
3. C31×#1613C2semidirect
5. C31×#1352C2semidirect
6. C31×#1343C2semidirect
7. C31×#1326C2semidirect
114. #12137C31direct (N × H)
173. #2826C2×C31semidirect
174. #2044C2×C31semidirect
175. #1613C2×C31semidirect
177. #1352C2×C31semidirect
178. #1343C2×C31semidirect
179. #1326C2×C31semidirect
277. C31#12137direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28262
C31×#20442
C31×#16132
C31×#15952
C31×#13522
C31×#13432
C31×#13262
#1213731