Finite Group Databasebeta

← C31×#10058 #21466817

Subgroups — C31×#10058 #21466817

Order 7936.

602 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#487C211yesmaximal, characteristic, complemented
2C2×C31×#476C211yesmaximal, characteristic
3C31×#2547C211yesmaximal, characteristic
4C31×#1987C211yesmaximal, characteristic, complemented
5C2×C31×#332C211yesmaximal, characteristic, complemented
6C2×C31×D4⋊C8C211yesmaximal, characteristic
7C31×#1279C211yesmaximal, characteristic, complemented
8C2×C31×C4⋊Q8C2211yescharacteristic
9C2×C31×C4⋊Q8C2211yescharacteristic
10C2×C31×#124—21no—
11C2×C4×D4×C31C2211yescharacteristic
12C31×#487—22no—
13C31×#487—22no—
14C31×#476—22no—
15C31×#476—22no—
16C2×C31×#106—21no—
17C2×C31×C4⋊C8C2211yescharacteristic
18C2×C31×C4⋊C8C2211yescharacteristic
19C2×C31×#103—22no—
20C2×C31×#102—21no—
21C2×C4×C8×C31C2211yescharacteristic
22C31×#397—22no—
23C31×#391—21no—
24C31×#380—22no—
25C4×C4⋊C4×C31C2211yescharacteristic
26C31×#340—22no—
27C31×#332—22no—
28C31×D4⋊C8—22no—
29C22×Q8×C31—21no—
30C22×Q8×C31—21no—
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—
44C31×#124—42no—
45C31×#124—42no—
46C4×D4×C31—21no—
47C4×D4×C31—21no—
48C4×D4×C31—21no—
49C4×D4×C31—21no—
50C2×C4⋊C4×C31—21no—
51C2×C4⋊C4×C31—21no—
52C2×C4⋊C4×C31—21no—
53C2×C4⋊C4×C31D412yes—
54C2×C4⋊C4×C31D411yescharacteristic
55C2×C4⋊C4×C31D412yes—
56C2×C22⋊C4×C31—24no—
57C2×C22⋊C4×C31—21no—
58C2×C42×C31—21no—
59C2×C42×C31C2311yescharacteristic
60C31×#106—24no—
61C31×C4⋊C8—21no—
62C31×C4⋊C8—21no—
63C31×C4⋊C8—21no—
64C31×C4⋊C8D412yes—
65C31×#103—28no—
66C31×#102—24no—
67C4×C8×C31—21no—
68C4×C8×C31D412yes—
69C31×#95—24no—
70C31×#95—21no—
71C24×C31—21no—
72C2×Q8×C31—22no—
73C2×Q8×C31—44no—
74C2×Q8×C31—41no—
75C2×Q8×C31—41no—
76C2×Q8×C31—41no—
77C2×Q8×C31—41no—
78C2×Q8×C31—41no—
79C2×Q8×C31—41no—
80C2×Q8×C31—41no—
81C2×D4×C31—21no—
82C2×D4×C31—21no—
83C2×D4×C31—21no—
84C2×D4×C31—21no—
85C2×D4×C31—21no—
86C2×D4×C31SD1612yes—
87C22×C4×C31—21no—
88C22×C4×C31—22no—
89C22×C4×C31—22no—
90C22×C4×C31—41no—
91C22×C4×C31—41no—
92C22×C4×C31—41no—
93C22×C4×C31—42no—
94C22×C4×C31—42no—
95C22×C4×C31C2×D411yescharacteristic
96C22×C4×C31C2×D411yescharacteristic
97C22×C4×C31C2×D411yescharacteristic
98C2×C8×C31—22no—
99C2×C8×C31—22no—
100C2×C8×C31—42no—
101C2×C8×C31—41no—
102C2×C8×C31—41no—
103C2×C8×C31—41no—
104C2×C8×C31—42no—
105C4⋊C4×C31—22no—
106C4⋊C4×C31—22no—
107C4⋊C4×C31—21no—
108C4⋊C4×C31—22no—
109C4⋊C4×C31—21no—
110C4⋊C4×C31—22no—
111C4⋊C4×C31—22no—
112C4⋊C4×C31—22no—
113C4⋊C4×C31—41no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31—41no—
116C4⋊C4×C31—41no—
117C22⋊C4×C31—48no—
118C22⋊C4×C31—41no—
119C22⋊C4×C31—41no—
120C42×C31—41no—
121C42×C31—41no—
122C42×C31C2×D411yescharacteristic
123C42×C31C2×D411yescharacteristic
124C42×C31C2×D411yescharacteristic
125C42×C31#4111yescharacteristic
126#10058C3111yesmaximal, characteristic, direct product factor, Sylow
127C23×C31—22no—
128C23×C31—41no—
129C23×C31—42no—
130C23×C31—41no—
131C23×C31—42no—
132C23×C31#12011yescharacteristic
133Q8×C31—44no—
134Q8×C31—82no—
135Q8×C31—81no—
136Q8×C31—81no—
137Q8×C31—81no—
138Q8×C31—81no—
139D4×C31—22no—
140D4×C31—22no—
141D4×C31—41no—
142D4×C31—41no—
143C2×C4×C31—22no—
144C2×C4×C31—41no—
145C2×C4×C31—42no—
146C2×C4×C31—42no—
147C2×C4×C31—42no—
148C2×C4×C31—42no—
149C2×C4×C31—42no—
150C2×C4×C31—42no—
151C2×C4×C31—42no—
152C2×C4×C31—42no—
153C2×C4×C31—82no—
154C2×C4×C31—82no—
155C2×C4×C31—81no—
156C2×C4×C31—81no—
157C2×C4×C31#12011yescharacteristic
158C2×C4×C31#12011yescharacteristic
159C2×C4×C31#12111yescharacteristic
160C2×C4×C31#12111yescharacteristic
161C2×C4×C31#12111yescharacteristic
162C2×C4×C31#12711yescharacteristic
163C2×C4×C31C2×SD1611yescharacteristic
164C2×C4×C31#13511yescharacteristic
165C2×C4×C31#13512yes—
166C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C31#13711yescharacteristic
168C2×C4×C31#13712yes—
169C8×C31—42no—
170C8×C31—42no—
171C8×C31—81no—
172C2×#487C2×C3111yescharacteristic, complemented
173C2×#476C2×C3111yescharacteristic
174#2547C2×C3111yescharacteristic
175#1987C2×C3111yescharacteristic, complemented
176C2×#332C2×C3111yescharacteristic, complemented
177C2×D4⋊C8C2×C3111yescharacteristic
178#1279C2×C3111yescharacteristic, complemented
179C22×C31—44no—
180C22×C31—42no—
181C22×C31—82no—
182C22×C31—81no—
183C22×C31—81no—
184C22×C31#39211yescharacteristic
185C22×C31#45011yescharacteristic
186C22×C31#45212yes—
187C22×C31#45211yescharacteristic
188C22×C31#45412yes—
189C4×C31—21no—
190C4×C31—21no—
191C4×C31—82no—
192C4×C31—82no—
193C4×C31—82no—
194C4×C31—81no—
195C4×C31#44811yescharacteristic
196C4×C31#44911yescharacteristic
197C4×C31#46412yes—
198C4×C31#46511yescharacteristic
199C4×C31#46811yescharacteristic
200C4×C31#49511yescharacteristic
201C4×C31#49711yescharacteristic
202C2×C4⋊Q8C22×C3111yescharacteristic
203C2×C4⋊Q8C22×C3111yescharacteristic
204C2×#124—21no—
205C2×C4×D4C22×C3111yescharacteristic
206#487—22no—
207#487—22no—
208#476—22no—
209#476—22no—
210C2×#106—21no—
211C2×C4⋊C8C22×C3111yescharacteristic
212C2×C4⋊C8C22×C3111yescharacteristic
213C2×#103—22no—
214C2×#102—21no—
215C2×C4×C8C22×C3111yescharacteristic
216#397—22no—
217#391—21no—
218#380—22no—
219C4×C4⋊C4C22×C3111yescharacteristic
220#340—22no—
221#332—22no—
222D4⋊C8—22no—
223C2×C31—82no—
224C2×C31#163512yes—
225C2×C31#163812yes—
226C2×C31#200011yescharacteristic
227C2×C31#200311yescharacteristic
228C2×C31#201511yescharacteristic
229C22×Q8—21no—
230C22×Q8—21no—
231C22×D4D4×C3111yescharacteristic
232C23×C4—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8D4×C3112yes—
244#124—42no—
245#124—42no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4D4×C3111yescharacteristic
254C2×C4⋊C4D4×C3112yes—
255C2×C4⋊C4D4×C3112yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—21no—
258C2×C42—21no—
259C2×C42C23×C3111yescharacteristic
260#106—24no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8D4×C3112yes—
265#103—28no—
266#102—24no—
267C4×C8—21no—
268C4×C8D4×C3112yes—
269#95—24no—
270#95—21no—
271C31#1005811yescharacteristic, direct product factor, Sylow
272C24—21no—
273C2×Q8—22no—
274C2×Q8—44no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4SD16×C3112yes—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4C2×D4×C3111yescharacteristic
297C22×C4C2×D4×C3111yescharacteristic
298C22×C4C2×D4×C3111yescharacteristic
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C22⋊C4—48no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C42—41no—
322C42—41no—
323C42C2×D4×C3111yescharacteristic
324C42C2×D4×C3111yescharacteristic
325C42C2×D4×C3111yescharacteristic
326C42#41×C3111yescharacteristic
327C23—22no—
328C23—42no—
329C23—41no—
330C23—42no—
331C23—41no—
332C23C31×#12011yescharacteristic
333Q8—44no—
334Q8—82no—
335Q8—81no—
336Q8—81no—
337Q8—81no—
338Q8—81no—
339D4—22no—
340D4—22no—
341D4—41no—
342D4—41no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4C31×#12011yescharacteristic
358C2×C4C31×#12011yescharacteristic
359C2×C4C31×#12111yescharacteristic
360C2×C4C31×#12111yescharacteristic
361C2×C4C31×#12111yescharacteristic
362C2×C4C31×#12711yescharacteristic
363C2×C4C2×SD16×C3111yescharacteristic
364C2×C4C31×#13511yescharacteristic
365C2×C4C31×#13512yes—
366C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
367C2×C4C31×#13711yescharacteristic
368C2×C4C31×#13712yes—
369C8—42no—
370C8—42no—
371C8—81no—
372C22—44no—
373C22—42no—
374C22—82no—
375C22—81no—
376C22—81no—
377C22C31×#39211yescharacteristic
378C22C31×#45011yescharacteristic
379C22C31×#45212yes—
380C22C31×#45211yescharacteristic
381C22C31×#45412yes—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—82no—
386C4—82no—
387C4—81no—
388C4C31×#44811yescharacteristic
389C4C31×#44911yescharacteristic
390C4C31×#46412yes—
391C4C31×#46511yescharacteristic
392C4C31×#46811yescharacteristic
393C4C31×#49511yescharacteristic
394C4C31×#49711yescharacteristic
395C2—82no—
396C2C31×#163512yes—
397C2C31×#163812yes—
398C2C31×#200011yescharacteristic
399C2C31×#200311yescharacteristic
400C2C31×#201511yescharacteristic
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×#1987C2semidirect
5. C2×C31×#332C2semidirect
7. C31×#1279C2semidirect
126. #10058C31direct (N × H)
172. C2×#487C2×C31semidirect
175. #1987C2×C31semidirect
176. C2×#332C2×C31semidirect
178. #1279C2×C31semidirect
271. C31#10058direct (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×#4762
C31×#25472
C31×#19872
C2×C31×#3322
C2×C31×D4⋊C82
C31×#12792
#1005831