Finite Group Databasebeta

← C31×#10072 #21466831

Subgroups — C31×#10072 #21466831

Order 7936.

598 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#487C211yesmaximal, characteristic, complemented
2C2×C31×#474C211yesmaximal, characteristic
3C31×#2563C211yesmaximal, characteristic
4C31×#1981C211yesmaximal, characteristic, complemented
5C2×C31×#333C211yesmaximal, characteristic, complemented
6C2×C31×D4⋊C8C211yesmaximal, characteristic
7C31×#1279C211yesmaximal, characteristic, complemented
8C2×C31×C4⋊Q8C2211yescharacteristic
9C2×C31×C4⋊Q8C2211yescharacteristic
10C2×C4×D4×C31C2211yescharacteristic
11C31×#487—22no—
12C31×#487—22no—
13C31×#474—22no—
14C31×#474—22no—
15C2×C31×#107—21no—
16C2×C31×C4⋊C8C2211yescharacteristic
17C2×C31×C4⋊C8C2211yescharacteristic
18C2×C31×#103—22no—
19C2×C31×#103—22no—
20C2×C31×#102—21no—
21C2×C4×C8×C31C2211yescharacteristic
22C31×#393—22no—
23C31×#390—21no—
24C31×#382—21no—
25C31×#380—22no—
26C4×C4⋊C4×C31C2211yescharacteristic
27C31×#333—22no—
28C31×D4⋊C8—22no—
29C22×Q8×C31—21no—
30C22×Q8×C31D412yes—
31C22×D4×C31D411yescharacteristic
32C23×C4×C31—21no—
33C22×C8×C31—21no—
34C22×C8×C31—21no—
35C22×C8×C31—21no—
36C31×C4⋊Q8—21no—
37C31×C4⋊Q8—21no—
38C31×C4⋊Q8—21no—
39C31×C4⋊Q8—21no—
40C31×C4⋊Q8—21no—
41C31×C4⋊Q8—21no—
42C31×C4⋊Q8—21no—
43C31×C4⋊Q8D412yes—
44C4×D4×C31—21no—
45C4×D4×C31—21no—
46C4×D4×C31—21no—
47C4×D4×C31—21no—
48C2×C4⋊C4×C31—21no—
49C2×C4⋊C4×C31—21no—
50C2×C4⋊C4×C31—21no—
51C2×C4⋊C4×C31—21no—
52C2×C4⋊C4×C31D412yes—
53C2×C4⋊C4×C31D411yescharacteristic
54C2×C22⋊C4×C31—21no—
55C2×C22⋊C4×C31—24no—
56C2×C42×C31—21no—
57C2×C42×C31C2311yescharacteristic
58C31×#107—24no—
59C31×C4⋊C8—21no—
60C31×C4⋊C8—21no—
61C31×C4⋊C8—21no—
62C31×C4⋊C8D412yes—
63C31×#103—28no—
64C31×#103—44no—
65C31×#102—24no—
66C4×C8×C31—21no—
67C4×C8×C31D412yes—
68C31×#95—24no—
69C31×#95—21no—
70C24×C31—21no—
71C2×Q8×C31—22no—
72C2×Q8×C31—22no—
73C2×Q8×C31—22no—
74C2×Q8×C31—44no—
75C2×Q8×C31—41no—
76C2×Q8×C31—42no—
77C2×Q8×C31—42no—
78C2×D4×C31—21no—
79C2×D4×C31—21no—
80C2×D4×C31—21no—
81C2×D4×C31—21no—
82C2×D4×C31—21no—
83C2×D4×C31Q1612yes—
84C22×C4×C31—21no—
85C22×C4×C31—22no—
86C22×C4×C31—22no—
87C22×C4×C31—41no—
88C22×C4×C31—41no—
89C22×C4×C31—41no—
90C22×C4×C31—42no—
91C22×C4×C31—42no—
92C22×C4×C31C2×D411yescharacteristic
93C22×C4×C31C2×D411yescharacteristic
94C22×C4×C31C2×D411yescharacteristic
95C2×C8×C31—22no—
96C2×C8×C31—22no—
97C2×C8×C31—42no—
98C2×C8×C31—41no—
99C2×C8×C31—41no—
100C2×C8×C31—41no—
101C2×C8×C31—42no—
102C4⋊C4×C31—22no—
103C4⋊C4×C31—22no—
104C4⋊C4×C31—21no—
105C4⋊C4×C31—21no—
106C4⋊C4×C31—22no—
107C4⋊C4×C31—22no—
108C4⋊C4×C31—41no—
109C4⋊C4×C31—41no—
110C4⋊C4×C31—41no—
111C4⋊C4×C31—41no—
112C4⋊C4×C31—41no—
113C4⋊C4×C31—41no—
114C22⋊C4×C31—41no—
115C22⋊C4×C31—41no—
116C22⋊C4×C31—84no—
117C42×C31—41no—
118C42×C31—41no—
119C42×C31C2×D411yescharacteristic
120C42×C31C2×D411yescharacteristic
121C42×C31C2×D411yescharacteristic
122C42×C31#4111yescharacteristic
123#10072C3111yesmaximal, characteristic, direct product factor, Sylow
124C23×C31—22no—
125C23×C31—41no—
126C23×C31—42no—
127C23×C31—41no—
128C23×C31—42no—
129C23×C31#12011yescharacteristic
130Q8×C31—44no—
131Q8×C31—44no—
132Q8×C31—82no—
133Q8×C31—81no—
134Q8×C31—81no—
135D4×C31—22no—
136D4×C31—22no—
137D4×C31—41no—
138D4×C31—41no—
139C2×C4×C31—22no—
140C2×C4×C31—41no—
141C2×C4×C31—42no—
142C2×C4×C31—42no—
143C2×C4×C31—42no—
144C2×C4×C31—42no—
145C2×C4×C31—42no—
146C2×C4×C31—42no—
147C2×C4×C31—44no—
148C2×C4×C31—82no—
149C2×C4×C31—82no—
150C2×C4×C31—81no—
151C2×C4×C31—81no—
152C2×C4×C31#12011yescharacteristic
153C2×C4×C31#12011yescharacteristic
154C2×C4×C31#12111yescharacteristic
155C2×C4×C31#12111yescharacteristic
156C2×C4×C31#12111yescharacteristic
157C2×C4×C31#12711yescharacteristic
158C2×C4×C31C2×Q1611yescharacteristic
159C2×C4×C31#13511yescharacteristic
160C2×C4×C31#13512yes—
161C2×C4×C31AGL(1,ℤ/8)12yes—
162C2×C4×C31#13711yescharacteristic
163C2×C4×C31#13711yescharacteristic
164C8×C31—42no—
165C8×C31—42no—
166C8×C31—81no—
167C2×#487C2×C3111yescharacteristic, complemented
168C2×#474C2×C3111yescharacteristic
169#2563C2×C3111yescharacteristic
170#1981C2×C3111yescharacteristic, complemented
171C2×#333C2×C3111yescharacteristic, complemented
172C2×D4⋊C8C2×C3111yescharacteristic
173#1279C2×C3111yescharacteristic, complemented
174C22×C31—44no—
175C22×C31—42no—
176C22×C31—82no—
177C22×C31—81no—
178C22×C31—81no—
179C22×C31#39211yescharacteristic
180C22×C31#44912yes—
181C22×C31#45111yescharacteristic
182C22×C31#45211yescharacteristic
183C22×C31#45512yes—
184C4×C31—21no—
185C4×C31—21no—
186C4×C31—82no—
187C4×C31—82no—
188C4×C31—81no—
189C4×C31—82no—
190C4×C31#45111yescharacteristic
191C4×C31#45211yescharacteristic
192C4×C31#46312yes—
193C4×C31#46711yescharacteristic
194C4×C31#47011yescharacteristic
195C4×C31#49511yescharacteristic
196C4×C31#49611yescharacteristic
197C2×C4⋊Q8C22×C3111yescharacteristic
198C2×C4⋊Q8C22×C3111yescharacteristic
199C2×C4×D4C22×C3111yescharacteristic
200#487—22no—
201#487—22no—
202#474—22no—
203#474—22no—
204C2×#107—21no—
205C2×C4⋊C8C22×C3111yescharacteristic
206C2×C4⋊C8C22×C3111yescharacteristic
207C2×#103—22no—
208C2×#103—22no—
209C2×#102—21no—
210C2×C4×C8C22×C3111yescharacteristic
211#393—22no—
212#390—21no—
213#382—21no—
214#380—22no—
215C4×C4⋊C4C22×C3111yescharacteristic
216#333—22no—
217D4⋊C8—22no—
218C2×C31—82no—
219C2×C31#164412yes—
220C2×C31#164612yes—
221C2×C31#200111yescharacteristic
222C2×C31#200411yescharacteristic
223C2×C31#201211yescharacteristic
224C22×Q8—21no—
225C22×Q8D4×C3112yes—
226C22×D4D4×C3111yescharacteristic
227C23×C4—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8D4×C3112yes—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4D4×C3111yescharacteristic
248C2×C4⋊C4D4×C3112yes—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—24no—
251C2×C42—21no—
252C2×C42C23×C3111yescharacteristic
253#107—24no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8D4×C3112yes—
258#103—28no—
259#103—44no—
260#102—24no—
261C4×C8—21no—
262C4×C8D4×C3112yes—
263#95—24no—
264#95—21no—
265C31#1007211yescharacteristic, direct product factor, Sylow
266C24—21no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—44no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4Q16×C3112yes—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4C2×D4×C3111yescharacteristic
289C22×C4C2×D4×C3111yescharacteristic
290C22×C4C2×D4×C3111yescharacteristic
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—42no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—84no—
313C42—41no—
314C42—41no—
315C42C2×D4×C3111yescharacteristic
316C42C2×D4×C3111yescharacteristic
317C42C2×D4×C3111yescharacteristic
318C42#41×C3111yescharacteristic
319C23—22no—
320C23—42no—
321C23—41no—
322C23—42no—
323C23—41no—
324C23C31×#12011yescharacteristic
325Q8—44no—
326Q8—44no—
327Q8—82no—
328Q8—81no—
329Q8—81no—
330D4—22no—
331D4—22no—
332D4—41no—
333D4—41no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4C31×#12011yescharacteristic
348C2×C4C31×#12011yescharacteristic
349C2×C4C31×#12111yescharacteristic
350C2×C4C31×#12111yescharacteristic
351C2×C4C31×#12111yescharacteristic
352C2×C4C31×#12711yescharacteristic
353C2×C4C2×Q16×C3111yescharacteristic
354C2×C4C31×#13511yescharacteristic
355C2×C4C31×#13512yes—
356C2×C4C31×AGL(1,ℤ/8)12yes—
357C2×C4C31×#13711yescharacteristic
358C2×C4C31×#13711yescharacteristic
359C8—42no—
360C8—42no—
361C8—81no—
362C22—44no—
363C22—42no—
364C22—82no—
365C22—81no—
366C22—81no—
367C22C31×#39211yescharacteristic
368C22C31×#44912yes—
369C22C31×#45111yescharacteristic
370C22C31×#45211yescharacteristic
371C22C31×#45512yes—
372C4—21no—
373C4—21no—
374C4—82no—
375C4—82no—
376C4—81no—
377C4—82no—
378C4C31×#45111yescharacteristic
379C4C31×#45211yescharacteristic
380C4C31×#46312yes—
381C4C31×#46711yescharacteristic
382C4C31×#47011yescharacteristic
383C4C31×#49511yescharacteristic
384C4C31×#49611yescharacteristic
385C2—82no—
386C2C31×#164412yes—
387C2C31×#164612yes—
388C2C31×#200111yescharacteristic
389C2C31×#200411yescharacteristic
390C2C31×#201211yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#487C2semidirect
4. C31×#1981C2semidirect
5. C2×C31×#333C2semidirect
7. C31×#1279C2semidirect
123. #10072C31direct (N × H)
167. C2×#487C2×C31semidirect
170. #1981C2×C31semidirect
171. C2×#333C2×C31semidirect
173. #1279C2×C31semidirect
265. C31#10072direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4872
C2×C31×#4742
C31×#25632
C31×#19812
C2×C31×#3332
C2×C31×D4⋊C82
C31×#12792
#1007231