Finite Group Databasebeta

← C31×#12264 #21469023

Subgroups — C31×#12264 #21469023

Order 7936.

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

normal subgroups only

This group has 468 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
2C31×#2029C211yesmaximal, characteristic
3C31×#1608C211yesmaximal, characteristic, complemented
4C31×#1602C211yesmaximal, characteristic, complemented
5C31×#1351C211yesmaximal, characteristic, complemented
6C31×#1344C211yesmaximal, characteristic, complemented
7C31×#1316C211yesmaximal, characteristic
8C31×#543C2211yescharacteristic
9C31×#483—21no—
10C31×#451—21no—
11C31×#397—21no—
12C31×#397C2211yescharacteristic
13C31×#393—21no—
14C31×#393C2211yescharacteristic
15C31×#388—21no—
16C31×#388—21no—
17C31×#388—21no—
18C31×#379—21no—
19C31×#379—21no—
20C31×#379C2211yescharacteristic
21C31×#342—21no—
22C31×#342C2211yescharacteristic
23C31×#328—21no—
24C31×#328—21no—
25C31×#323C2211yescharacteristic
26C31×#323C2211yescharacteristic
27C25×C31D411yescharacteristic
28C22×Q8×C31D411yescharacteristic
29C2×Q16×C31—41no—
30C31×C22⋊Q8—21no—
31C31×C22⋊Q8—21no—
32C31×C22⋊Q8—41no—
33C31×C22⋊Q8—41no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31D411yescharacteristic
36C2×C22⋊C4×C31—21no—
37C2×C22⋊C4×C31—21no—
38C2×C22⋊C4×C31—21no—
39C2×C22⋊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—41no—
47C2×C22⋊C4×C31—41no—
48C2×C22⋊C4×C31D411yescharacteristic
49C2×C22⋊C4×C31C2311yescharacteristic
50C2×C42×C31—21no—
51C31×#106—41no—
52C31×#103—41no—
53C31×#103—41no—
54C31×C22⋊C8—21no—
55C31×C22⋊C8—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95D411yescharacteristic
61C31×#95D411yescharacteristic
62C24×C31—21no—
63C24×C31—21no—
64C24×C31—21no—
65C24×C31—41no—
66C24×C31—41no—
67C24×C31—42no—
68C24×C31—81no—
69C24×C31C2×D411yescharacteristic
70C2×Q8×C31—21no—
71C2×Q8×C31—21no—
72C2×Q8×C31—41no—
73C2×Q8×C31—81no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—21no—
79C22×C4×C31—21no—
80C22×C4×C31—21no—
81C22×C4×C31—41no—
82C22×C4×C31C2×D411yescharacteristic
83C22×C4×C31C2×D411yescharacteristic
84Q16×C31—82no—
85C2×C8×C31—41no—
86C2×C8×C31—41no—
87C4⋊C4×C31—21no—
88C4⋊C4×C31—21no—
89C4⋊C4×C31—41no—
90C4⋊C4×C31—41no—
91C4⋊C4×C31—41no—
92C4⋊C4×C31—81no—
93C22⋊C4×C31—41no—
94C22⋊C4×C31—42no—
95C22⋊C4×C31—42no—
96C22⋊C4×C31—41no—
97C22⋊C4×C31—41no—
98C22⋊C4×C31—41no—
99C22⋊C4×C31—41no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—41no—
102C22⋊C4×C31—41no—
103C22⋊C4×C31—81no—
104C22⋊C4×C31—81no—
105C22⋊C4×C31—81no—
106C22⋊C4×C31—81no—
107C22⋊C4×C31—81no—
108C22⋊C4×C31—82no—
109C22⋊C4×C31—82no—
110C42×C31—81no—
111#12264C3111yesmaximal, characteristic, direct product factor, Sylow
112C23×C31—21no—
113C23×C31—21no—
114C23×C31—22no—
115C23×C31—41no—
116C23×C31—41no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—41no—
120C23×C31—42no—
121C23×C31—42no—
122C23×C31—42no—
123C23×C31—41no—
124C23×C31—41no—
125C23×C31—41no—
126C23×C31—81no—
127C23×C31—82no—
128C23×C31—81no—
129C23×C31—82no—
130C23×C31—84no—
131C23×C31—81no—
132C23×C31#12011yescharacteristic
133C23×C31#13511yescharacteristic
134C23×C31#13711yescharacteristic
135Q8×C31—42no—
136Q8×C31—81no—
137Q8×C31—81no—
138Q8×C31—81no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
141C2×C4×C31—41no—
142C2×C4×C31—41no—
143C2×C4×C31—41no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
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—81no—
161C2×C4×C31—81no—
162C2×C4×C31—81no—
163C2×C4×C31—81no—
164C2×C4×C31—81no—
165C8×C31—81no—
166C8×C31—81no—
167#2736C2×C3111yescharacteristic
168#2029C2×C3111yescharacteristic
169#1608C2×C3111yescharacteristic, complemented
170#1602C2×C3111yescharacteristic, complemented
171#1351C2×C3111yescharacteristic, complemented
172#1344C2×C3111yescharacteristic, complemented
173#1316C2×C3111yescharacteristic
174C22×C31—21no—
175C22×C31—21no—
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—41no—
179C22×C31—41no—
180C22×C31—42no—
181C22×C31—42no—
182C22×C31—41no—
183C22×C31—41no—
184C22×C31—44no—
185C22×C31—42no—
186C22×C31—81no—
187C22×C31—81no—
188C22×C31—81no—
189C22×C31—82no—
190C22×C31—82no—
191C22×C31—82no—
192C22×C31—81no—
193C22×C31—81no—
194C22×C31#45211yescharacteristic
195C22×C31#45511yescharacteristic
196C22×C31B(4,𝔽2)11yescharacteristic
197C4×C31—41no—
198C4×C31—41no—
199C4×C31—81no—
200C4×C31—81no—
201C4×C31—81no—
202C4×C31—81no—
203C4×C31—81no—
204C4×C31—81no—
205C4×C31—81no—
206C4×C31—161no—
207#543C22×C3111yescharacteristic
208#483—21no—
209#451—21no—
210#397—21no—
211#397C22×C3111yescharacteristic
212#393—21no—
213#393C22×C3111yescharacteristic
214#388—21no—
215#388—21no—
216#388—21no—
217#379—21no—
218#379—21no—
219#379C22×C3111yescharacteristic
220#342—21no—
221#342C22×C3111yescharacteristic
222#328—21no—
223#328—21no—
224#323C22×C3111yescharacteristic
225#323C22×C3111yescharacteristic
226C2×C31—21no—
227C2×C31—21no—
228C2×C31—41no—
229C2×C31—41no—
230C2×C31—82no—
231C2×C31#160611yescharacteristic
232C2×C31#219711yescharacteristic
233C2×C31#219911yescharacteristic
234C25D4×C3111yescharacteristic
235C22×Q8D4×C3111yescharacteristic
236C2×Q16—41no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—41no—
240C22⋊Q8—41no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4D4×C3111yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4D4×C3111yescharacteristic
256C2×C22⋊C4C23×C3111yescharacteristic
257C2×C42—21no—
258#106—41no—
259#103—41no—
260#103—41no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95D4×C3111yescharacteristic
268#95D4×C3111yescharacteristic
269C31#1226411yescharacteristic, direct product factor, Sylow
270C24—21no—
271C24—21no—
272C24—21no—
273C24—41no—
274C24—41no—
275C24—42no—
276C24—81no—
277C24C2×D4×C3111yescharacteristic
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8—81no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4C2×D4×C3111yescharacteristic
291C22×C4C2×D4×C3111yescharacteristic
292Q16—82no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—81no—
301C22⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—81no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C22⋊C4—81no—
315C22⋊C4—81no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C42—81no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—81no—
334C23—82no—
335C23—81no—
336C23—82no—
337C23—84no—
338C23—81no—
339C23C31×#12011yescharacteristic
340C23C31×#13511yescharacteristic
341C23C31×#13711yescharacteristic
342Q8—42no—
343Q8—81no—
344Q8—81no—
345Q8—81no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
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—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C8—81no—
373C8—81no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22—42no—
382C22—41no—
383C22—41no—
384C22—44no—
385C22—42no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—81no—
393C22—81no—
394C22C31×#45211yescharacteristic
395C22C31×#45511yescharacteristic
396C22C31×B(4,𝔽2)11yescharacteristic
397C4—41no—
398C4—41no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4—161no—
407C2—21no—
408C2—21no—
409C2—41no—
410C2—41no—
411C2—82no—
412C2C31×#160611yescharacteristic
413C2C31×#219711yescharacteristic
414C2C31×#219911yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC25×C31D4
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
3. C31×#1608C2semidirect
4. C31×#1602C2semidirect
5. C31×#1351C2semidirect
6. C31×#1344C2semidirect
111. #12264C31direct (N × H)
169. #1608C2×C31semidirect
170. #1602C2×C31semidirect
171. #1351C2×C31semidirect
172. #1344C2×C31semidirect
269. C31#12264direct (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×#20292
C31×#16082
C31×#16022
C31×#13512
C31×#13442
C31×#13162
#1226431