Finite Group Databasebeta

← C31×#12268 #21469027

Subgroups — C31×#12268 #21469027

Order 7936.

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

normal subgroups only

This group has 458 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×#2735C211yesmaximal, characteristic
2C31×#2029C211yesmaximal, characteristic
3C31×#1614C211yesmaximal, characteristic, complemented
4C31×#1608C211yesmaximal, characteristic, complemented
5C31×#1352C211yesmaximal, characteristic, complemented
6C31×#1344C211yesmaximal, characteristic, complemented
7C31×#1316C211yesmaximal, characteristic
8C31×#543C2211yescharacteristic
9C31×#484—21no—
10C31×#483—21no—
11C31×#399C2211yescharacteristic
12C31×#397—21no—
13C31×#397C2211yescharacteristic
14C31×#396—21no—
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
28C31×C22⋊Q8—21no—
29C31×C22⋊Q8—21no—
30C31×C22⋊Q8—41no—
31C31×C22⋊Q8—41no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31D411yescharacteristic
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×#107—41no—
52C31×#106—41no—
53C31×#103—41no—
54C31×#103—41no—
55C31×C22⋊C8—21no—
56C31×C22⋊C8—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—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—21no—
79C22×C4×C31—41no—
80C22×C4×C31C2×D411yescharacteristic
81C22×C4×C31C2×D411yescharacteristic
82C2×C8×C31—41no—
83C2×C8×C31—41no—
84C4⋊C4×C31—21no—
85C4⋊C4×C31—21no—
86C4⋊C4×C31—41no—
87C4⋊C4×C31—41no—
88C4⋊C4×C31—41no—
89C4⋊C4×C31—41no—
90C4⋊C4×C31—81no—
91C4⋊C4×C31—81no—
92C22⋊C4×C31—41no—
93C22⋊C4×C31—42no—
94C22⋊C4×C31—42no—
95C22⋊C4×C31—41no—
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—81no—
103C22⋊C4×C31—81no—
104C22⋊C4×C31—81no—
105C22⋊C4×C31—81no—
106C22⋊C4×C31—81no—
107C22⋊C4×C31—82no—
108C22⋊C4×C31—82no—
109C42×C31—81no—
110#12268C3111yesmaximal, characteristic, direct product factor, Sylow
111C23×C31—21no—
112C23×C31—21no—
113C23×C31—22no—
114C23×C31—41no—
115C23×C31—41no—
116C23×C31—41no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—42no—
120C23×C31—42no—
121C23×C31—42no—
122C23×C31—41no—
123C23×C31—41no—
124C23×C31—41no—
125C23×C31—81no—
126C23×C31—82no—
127C23×C31—81no—
128C23×C31—82no—
129C23×C31—84no—
130C23×C31—81no—
131C23×C31#12011yescharacteristic
132C23×C31#13511yescharacteristic
133C23×C31#13711yescharacteristic
134Q8×C31—81no—
135Q8×C31—81no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
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—81no—
158C2×C4×C31—81no—
159C2×C4×C31—81no—
160C2×C4×C31—81no—
161C2×C4×C31—81no—
162C8×C31—81no—
163C8×C31—81no—
164#2735C2×C3111yescharacteristic
165#2029C2×C3111yescharacteristic
166#1614C2×C3111yescharacteristic, complemented
167#1608C2×C3111yescharacteristic, complemented
168#1352C2×C3111yescharacteristic, complemented
169#1344C2×C3111yescharacteristic, complemented
170#1316C2×C3111yescharacteristic
171C22×C31—21no—
172C22×C31—21no—
173C22×C31—21no—
174C22×C31—21no—
175C22×C31—41no—
176C22×C31—41no—
177C22×C31—42no—
178C22×C31—42no—
179C22×C31—41no—
180C22×C31—41no—
181C22×C31—44no—
182C22×C31—42no—
183C22×C31—81no—
184C22×C31—81no—
185C22×C31—81no—
186C22×C31—82no—
187C22×C31—82no—
188C22×C31—82no—
189C22×C31—81no—
190C22×C31—81no—
191C22×C31#45211yescharacteristic
192C22×C31#45511yescharacteristic
193C22×C31B(4,𝔽2)11yescharacteristic
194C4×C31—41no—
195C4×C31—41no—
196C4×C31—81no—
197C4×C31—81no—
198C4×C31—81no—
199C4×C31—81no—
200C4×C31—81no—
201C4×C31—81no—
202C4×C31—81no—
203C4×C31—161no—
204#543C22×C3111yescharacteristic
205#484—21no—
206#483—21no—
207#399C22×C3111yescharacteristic
208#397—21no—
209#397C22×C3111yescharacteristic
210#396—21no—
211#388—21no—
212#388—21no—
213#388—21no—
214#379—21no—
215#379—21no—
216#379C22×C3111yescharacteristic
217#342—21no—
218#342C22×C3111yescharacteristic
219#328—21no—
220#328—21no—
221#323C22×C3111yescharacteristic
222#323C22×C3111yescharacteristic
223C2×C31—21no—
224C2×C31—21no—
225C2×C31—41no—
226C2×C31—41no—
227C2×C31—82no—
228C2×C31#160611yescharacteristic
229C2×C31#219711yescharacteristic
230C2×C31#220011yescharacteristic
231C25D4×C3111yescharacteristic
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C22⋊Q8—41no—
235C22⋊Q8—41no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4D4×C3111yescharacteristic
239C2×C4⋊C4D4×C3111yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
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—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4D4×C3111yescharacteristic
253C2×C22⋊C4C23×C3111yescharacteristic
254C2×C42—21no—
255#107—41no—
256#106—41no—
257#103—41no—
258#103—41no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95D4×C3111yescharacteristic
265#95D4×C3111yescharacteristic
266C31#1226811yescharacteristic, direct product factor, Sylow
267C24—21no—
268C24—21no—
269C24—21no—
270C24—41no—
271C24—41no—
272C24—42no—
273C24—81no—
274C24C2×D4×C3111yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4C2×D4×C3111yescharacteristic
286C22×C4C2×D4×C3111yescharacteristic
287C2×C8—41no—
288C2×C8—41no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—81no—
296C4⋊C4—81no—
297C22⋊C4—41no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C22⋊C4—81no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C42—81no—
315C23—21no—
316C23—21no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—81no—
330C23—82no—
331C23—81no—
332C23—82no—
333C23—84no—
334C23—81no—
335C23C31×#12011yescharacteristic
336C23C31×#13511yescharacteristic
337C23C31×#13711yescharacteristic
338Q8—81no—
339Q8—81no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
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—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C8—81no—
367C8—81no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—44no—
379C22—42no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—82no—
384C22—82no—
385C22—82no—
386C22—81no—
387C22—81no—
388C22C31×#45211yescharacteristic
389C22C31×#45511yescharacteristic
390C22C31×B(4,𝔽2)11yescharacteristic
391C4—41no—
392C4—41no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4—161no—
401C2—21no—
402C2—21no—
403C2—41no—
404C2—41no—
405C2—82no—
406C2C31×#160611yescharacteristic
407C2C31×#219711yescharacteristic
408C2C31×#220011yescharacteristic
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×#1614C2semidirect
4. C31×#1608C2semidirect
5. C31×#1352C2semidirect
6. C31×#1344C2semidirect
110. #12268C31direct (N × H)
166. #1614C2×C31semidirect
167. #1608C2×C31semidirect
168. #1352C2×C31semidirect
169. #1344C2×C31semidirect
266. C31#12268direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27352
C31×#20292
C31×#16142
C31×#16082
C31×#13522
C31×#13442
C31×#13162
#1226831