Finite Group Databasebeta

← C31×#10270 #21467029

Subgroups — C31×#10270 #21467029

Order 7936.

594 conjugacy classes of subgroups.

The subgroup lattice has 400 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
3C2×C31×#461C211yesmaximal, characteristic
4C31×#1954C211yesmaximal, characteristic
5C2×C31×#332C211yesmaximal, characteristic, complemented
6C2×C31×#332C211yesmaximal, characteristic, complemented
7C31×#1277C211yesmaximal, characteristic, complemented
8C22×SD16×C31—21no—
9C2×C31×C4⋊Q8C2211yescharacteristic
10C2×C31×C4⋊Q8C2211yescharacteristic
11C2×C4×D4×C31C2211yescharacteristic
12C31×#487—22no—
13C31×#487—22no—
14C31×#474—22no—
15C31×#474—22no—
16C31×#461—22no—
17C31×#461—22no—
18C2×C31×#107—21no—
19C2×C31×C4⋊C8C2211yescharacteristic
20C2×C31×C4⋊C8C2211yescharacteristic
21C2×C31×C4⋊C8C2211yescharacteristic
22C2×C31×#103—22no—
23C2×C31×#103—21no—
24C2×C31×#102—21no—
25C2×C31×C22⋊C8—22no—
26C2×C4×C8×C31C2211yescharacteristic
27C31×#332—22no—
28C31×#332—22no—
29C22×Q8×C31—21no—
30C22×Q8×C31—21no—
31C22×D4×C31D411yescharacteristic
32C23×C4×C31—21no—
33C2×SD16×C31—24no—
34C2×SD16×C31—42no—
35C2×SD16×C31—42no—
36C22×C8×C31—21no—
37C22×C8×C31—21no—
38C22×C8×C31—21no—
39C22×C8×C31D412yes—
40C31×C4⋊Q8—21no—
41C31×C4⋊Q8—21no—
42C31×C4⋊Q8—21no—
43C31×C4⋊Q8—21no—
44C31×C4⋊Q8—21no—
45C31×C4⋊Q8—21no—
46C31×C4⋊Q8D412yes—
47C31×C4⋊Q8D412yes—
48C4×D4×C31—21no—
49C4×D4×C31—21no—
50C4×D4×C31—21no—
51C4×D4×C31—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C22⋊C4×C31—21no—
58C2×C42×C31C2311yescharacteristic
59C31×#107—24no—
60C31×C4⋊C8—21no—
61C31×C4⋊C8—21no—
62C31×C4⋊C8—21no—
63C31×C4⋊C8—21no—
64C31×C4⋊C8D412yes—
65C31×C4⋊C8D412yes—
66C31×#103—24no—
67C31×#103—28no—
68C31×#102—24no—
69C31×C22⋊C8—44no—
70C4×C8×C31—21no—
71C4×C8×C31—21no—
72C24×C31—21no—
73C2×Q8×C31—22no—
74C2×Q8×C31—22no—
75C2×Q8×C31—41no—
76C2×Q8×C31—41no—
77C2×Q8×C31—41no—
78C2×Q8×C31—42no—
79C2×Q8×C31—41no—
80C2×Q8×C31—41no—
81C2×Q8×C31—42no—
82C2×Q8×C31—41no—
83C2×D4×C31—21no—
84C2×D4×C31—21no—
85C2×D4×C31—21no—
86C2×D4×C31—21no—
87C2×D4×C31—21no—
88C2×D4×C31SD1612yes—
89C22×C4×C31—21no—
90C22×C4×C31—41no—
91C22×C4×C31—41no—
92C22×C4×C31—41no—
93C22×C4×C31—41no—
94C22×C4×C31—42no—
95C22×C4×C31—42no—
96C22×C4×C31C2×D411yescharacteristic
97C22×C4×C31C2×D411yescharacteristic
98C22×C4×C31#4111yescharacteristic
99SD16×C31—48no—
100C2×C8×C31—22no—
101C2×C8×C31—22no—
102C2×C8×C31—22no—
103C2×C8×C31—22no—
104C2×C8×C31—22no—
105C2×C8×C31—41no—
106C2×C8×C31—41no—
107C2×C8×C31—41no—
108C2×C8×C31—41no—
109C2×C8×C31—41no—
110C2×C8×C31—42no—
111C4⋊C4×C31—22no—
112C4⋊C4×C31—22no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—22no—
116C4⋊C4×C31—22no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C4⋊C4×C31—41no—
120C4⋊C4×C31—41no—
121C22⋊C4×C31—41no—
122C22⋊C4×C31—41no—
123C42×C31C2×D411yescharacteristic
124C42×C31C2×D411yescharacteristic
125C42×C31C2×D411yescharacteristic
126C42×C31C2×D411yescharacteristic
127#10270C3111yesmaximal, characteristic, direct product factor, Sylow
128C23×C31—22no—
129C23×C31—41no—
130C23×C31—42no—
131C23×C31—41no—
132C23×C31—42no—
133C23×C31#12111yescharacteristic
134Q8×C31—44no—
135Q8×C31—42no—
136Q8×C31—42no—
137Q8×C31—81no—
138Q8×C31—81no—
139Q8×C31—81no—
140Q8×C31—81no—
141D4×C31—22no—
142D4×C31—22no—
143D4×C31—41no—
144D4×C31—41no—
145C2×C4×C31—22no—
146C2×C4×C31—41no—
147C2×C4×C31—42no—
148C2×C4×C31—42no—
149C2×C4×C31—42no—
150C2×C4×C31—82no—
151C2×C4×C31—81no—
152C2×C4×C31—81no—
153C2×C4×C31—82no—
154C2×C4×C31—81no—
155C2×C4×C31—81no—
156C2×C4×C31#12011yescharacteristic
157C2×C4×C31#12011yescharacteristic
158C2×C4×C31#12011yescharacteristic
159C2×C4×C31#12111yescharacteristic
160C2×C4×C31#12111yescharacteristic
161C2×C4×C31#12711yescharacteristic
162C2×C4×C31C2×SD1611yescharacteristic
163C2×C4×C31#13511yescharacteristic
164C2×C4×C31#13512yes—
165C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C31AGL(1,ℤ/8)12yes—
167C2×C4×C31#13711yescharacteristic
168C8×C31—42no—
169C8×C31—42no—
170C8×C31—42no—
171C8×C31—81no—
172C2×#487C2×C3111yescharacteristic, complemented
173C2×#474C2×C3111yescharacteristic
174C2×#461C2×C3111yescharacteristic
175#1954C2×C3111yescharacteristic
176C2×#332C2×C3111yescharacteristic, complemented
177C2×#332C2×C3111yescharacteristic, complemented
178#1277C2×C3111yescharacteristic, complemented
179C22×C31—44no—
180C22×C31—42no—
181C22×C31—82no—
182C22×C31—81no—
183C22×C31—81no—
184C22×C31#39211yescharacteristic
185C22×C31#46312yes—
186C22×C31#46511yescharacteristic
187C22×C31#46811yescharacteristic
188C22×C31#47312yes—
189C4×C31—21no—
190C4×C31—21no—
191C4×C31—82no—
192C4×C31—81no—
193C4×C31—82no—
194C4×C31#44811yescharacteristic
195C4×C31#44911yescharacteristic
196C4×C31#44912yes—
197C4×C31#45011yescharacteristic
198C4×C31#45211yescharacteristic
199C4×C31#49511yescharacteristic
200C4×C31#49611yescharacteristic
201C22×SD16—21no—
202C2×C4⋊Q8C22×C3111yescharacteristic
203C2×C4⋊Q8C22×C3111yescharacteristic
204C2×C4×D4C22×C3111yescharacteristic
205#487—22no—
206#487—22no—
207#474—22no—
208#474—22no—
209#461—22no—
210#461—22no—
211C2×#107—21no—
212C2×C4⋊C8C22×C3111yescharacteristic
213C2×C4⋊C8C22×C3111yescharacteristic
214C2×C4⋊C8C22×C3111yescharacteristic
215C2×#103—22no—
216C2×#103—21no—
217C2×#102—21no—
218C2×C22⋊C8—22no—
219C2×C4×C8C22×C3111yescharacteristic
220#332—22no—
221#332—22no—
222C2×C31—82no—
223C2×C31#167912yes—
224C2×C31#169912yes—
225C2×C31#200011yescharacteristic
226C2×C31#201211yescharacteristic
227C2×C31#202011yescharacteristic
228C22×Q8—21no—
229C22×Q8—21no—
230C22×D4D4×C3111yescharacteristic
231C23×C4—21no—
232C2×SD16—24no—
233C2×SD16—42no—
234C2×SD16—42no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8D4×C3112yes—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8D4×C3112yes—
246C4⋊Q8D4×C3112yes—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4D4×C3111yescharacteristic
256C2×C22⋊C4—21no—
257C2×C42C23×C3111yescharacteristic
258#107—24no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8D4×C3112yes—
264C4⋊C8D4×C3112yes—
265#103—24no—
266#103—28no—
267#102—24no—
268C22⋊C8—44no—
269C4×C8—21no—
270C4×C8—21no—
271C31#1027011yescharacteristic, direct product factor, Sylow
272C24—21no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—42no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—42no—
282C2×Q8—41no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4SD16×C3112yes—
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4C2×D4×C3111yescharacteristic
297C22×C4C2×D4×C3111yescharacteristic
298C22×C4#41×C3111yescharacteristic
299SD16—48no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C42C2×D4×C3111yescharacteristic
324C42C2×D4×C3111yescharacteristic
325C42C2×D4×C3111yescharacteristic
326C42C2×D4×C3111yescharacteristic
327C23—22no—
328C23—42no—
329C23—41no—
330C23—42no—
331C23—41no—
332C23C31×#12111yescharacteristic
333Q8—44no—
334Q8—42no—
335Q8—42no—
336Q8—81no—
337Q8—81no—
338Q8—81no—
339Q8—81no—
340D4—22no—
341D4—22no—
342D4—41no—
343D4—41no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—82no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4C31×#12011yescharacteristic
356C2×C4C31×#12011yescharacteristic
357C2×C4C31×#12011yescharacteristic
358C2×C4C31×#12111yescharacteristic
359C2×C4C31×#12111yescharacteristic
360C2×C4C31×#12711yescharacteristic
361C2×C4C2×SD16×C3111yescharacteristic
362C2×C4C31×#13511yescharacteristic
363C2×C4C31×#13512yes—
364C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
365C2×C4C31×AGL(1,ℤ/8)12yes—
366C2×C4C31×#13711yescharacteristic
367C8—42no—
368C8—42no—
369C8—42no—
370C8—81no—
371C22—44no—
372C22—42no—
373C22—82no—
374C22—81no—
375C22—81no—
376C22C31×#39211yescharacteristic
377C22C31×#46312yes—
378C22C31×#46511yescharacteristic
379C22C31×#46811yescharacteristic
380C22C31×#47312yes—
381C4—21no—
382C4—21no—
383C4—82no—
384C4—81no—
385C4—82no—
386C4C31×#44811yescharacteristic
387C4C31×#44911yescharacteristic
388C4C31×#44912yes—
389C4C31×#45011yescharacteristic
390C4C31×#45211yescharacteristic
391C4C31×#49511yescharacteristic
392C4C31×#49611yescharacteristic
393C2—82no—
394C2C31×#167912yes—
395C2C31×#169912yes—
396C2C31×#200011yescharacteristic
397C2C31×#201211yescharacteristic
398C2C31×#202011yescharacteristic
subgroupG/H
center, socleC23×C31#121
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
5. C2×C31×#332C2semidirect
6. C2×C31×#332C2semidirect
7. C31×#1277C2semidirect
127. #10270C31direct (N × H)
172. C2×#487C2×C31semidirect
176. C2×#332C2×C31semidirect
177. C2×#332C2×C31semidirect
178. #1277C2×C31semidirect
271. C31#10270direct (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
C2×C31×#4612
C31×#19542
C2×C31×#3322
C31×#12772
#1027031