Finite Group Databasebeta

← C31×#13228 #21469987

Subgroups — C31×#13228 #21469987

Order 7936.

490 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2545C211yesmaximal, characteristic
2C2×C31×#356C212yesmaximal, complemented
3C2×C31×#355C212yesmaximal
4C31×#1801C211yesmaximal, characteristic, complemented
5C4×C31×#101C211yesmaximal, characteristic, complemented
6C2×C31×C4⋊Q8C2211yescharacteristic
7C2×C31×#124C411yescharacteristic
8C2×C31×#124C2211yescharacteristic
9C2×C4×Q8×C31C2211yescharacteristic
10C31×#423—21no—
11C2×C31×#101C2212yes—
12C2×C31×#101C2212yes—
13C4×M16×C31—21no—
14C31×#396—21no—
15C31×#396—21no—
16C31×#391C411yescharacteristic
17C31×#390C411yescharacteristic
18C31×#388—21no—
19C31×#388—21no—
20C4×C4⋊C4×C31C411yescharacteristic
21C31×#356—24no—
22C31×#355—24no—
23C31×#343—21no—
24C31×#343—21no—
25C22×Q8×C31C2×C411yescharacteristic
26C22×Q8×C31C2311yescharacteristic
27C22×D4×C31C2×C411yescharacteristic
28C2×M16×C31—22no—
29C2×M16×C31—22no—
30C31×C4⋊Q8—22no—
31C31×C4⋊Q8—22no—
32C31×#124—22no—
33C31×#124—22no—
34C31×#124—41no—
35C31×#124—41no—
36C4×Q8×C31—41no—
37C4×Q8×C31—41no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31C2×C411yescharacteristic
42C2×C4⋊C4×C31D411yescharacteristic
43C2×C4⋊C4×C31D411yescharacteristic
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C42×C31C2×C411yescharacteristic
51C2×C42×C31C2×C411yescharacteristic
52C2×C42×C31C2×C411yescharacteristic
53C2×C42×C31D411yescharacteristic
54C2×C42×C31D411yescharacteristic
55C31×C4⋊C8—41no—
56C31×C4⋊C8—41no—
57C31×#101—22no—
58C31×#101—22no—
59C31×#101—22no—
60C31×#101D414yes—
61C31×#97—41no—
62C4×C8×C31—41no—
63C31×#95—21no—
64C31×#95—21no—
65C31×#95—21no—
66C24×C31—21no—
67C2×Q8×C31—22no—
68C2×Q8×C31—41no—
69C2×Q8×C31—41no—
70C2×Q8×C31—41no—
71C2×Q8×C31—41no—
72C2×Q8×C31C2×D412yes—
73C2×Q8×C31#4112yes—
74C2×D4×C31—22no—
75C2×D4×C31—41no—
76C2×D4×C31—41no—
77C22×C4×C31—21no—
78C22×C4×C31—21no—
79C22×C4×C31—21no—
80C22×C4×C31C22⋊C411yescharacteristic
81C22×C4×C31C22⋊C411yescharacteristic
82C22×C4×C31C22⋊C411yescharacteristic
83C22×C4×C31C22⋊C411yescharacteristic
84C22×C4×C31C22×C411yescharacteristic
85C22×C4×C31C2×D411yescharacteristic
86C22×C4×C31C2×D411yescharacteristic
87M16×C31—24no—
88M16×C31—42no—
89M16×C31—42no—
90M16×C31—42no—
91C2×C8×C31—42no—
92C2×C8×C31—42no—
93C4⋊C4×C31—41no—
94C4⋊C4×C31—41no—
95C4⋊C4×C31—41no—
96C4⋊C4×C31—42no—
97C4⋊C4×C31—42no—
98C4⋊C4×C31—41no—
99C4⋊C4×C31—41no—
100C22⋊C4×C31—42no—
101C22⋊C4×C31—42no—
102C22⋊C4×C31—42no—
103C22⋊C4×C31—41no—
104C22⋊C4×C31—41no—
105C22⋊C4×C31—42no—
106C22⋊C4×C31—41no—
107C22⋊C4×C31—41no—
108C42×C31—21no—
109C42×C31—21no—
110C42×C31—21no—
111C42×C31—21no—
112C42×C31—21no—
113C42×C31—21no—
114C42×C31—22no—
115C42×C31—22no—
116#13228C3111yesmaximal, characteristic, direct product factor, Sylow
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—41no—
120C23×C31—41no—
121C23×C31—41no—
122C23×C31—41no—
123C23×C31—41no—
124C23×C31C2×C22⋊C411yescharacteristic
125Q8×C31—42no—
126Q8×C31—42no—
127Q8×C31—82no—
128D4×C31—82no—
129C2×C4×C31—21no—
130C2×C4×C31—21no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—22no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
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#9911yescharacteristic
152C2×C4×C31#9911yescharacteristic
153C2×C4×C31C4×D412yes—
154C2×C4×C31#12112yes—
155C2×C4×C31#12411yescharacteristic
156C2×C4×C31#12711yescharacteristic
157C8×C31—44no—
158C8×C31—82no—
159#2545C2×C3111yescharacteristic
160C2×#356C2×C3112yescomplemented
161C2×#355C2×C3112yes—
162#1801C2×C3111yescharacteristic, complemented
163C4×#101C2×C3111yescharacteristic, complemented
164C22×C31—21no—
165C22×C31—21no—
166C22×C31—81no—
167C22×C31—81no—
168C22×C31—81no—
169C22×C31—81no—
170C22×C31—81no—
171C22×C31—81no—
172C22×C31—81no—
173C22×C31#39011yescharacteristic
174C22×C31C2×#9911yescharacteristic
175C22×C31#41011yescharacteristic
176C4×C31—21no—
177C4×C31—21no—
178C4×C31—22no—
179C4×C31—21no—
180C4×C31—22no—
181C4×C31—21no—
182C4×C31—42no—
183C4×C31—41no—
184C4×C31—41no—
185C4×C31—41no—
186C4×C31—81no—
187C4×C31—81no—
188C2×C4⋊Q8C22×C3111yescharacteristic
189C2×#124C4×C3111yescharacteristic
190C2×#124C22×C3111yescharacteristic
191C2×C4×Q8C22×C3111yescharacteristic
192#423—21no—
193C2×#101C22×C3112yes—
194C2×#101C22×C3112yes—
195C4×M16—21no—
196#396—21no—
197#396—21no—
198#391C4×C3111yescharacteristic
199#390C4×C3111yescharacteristic
200#388—21no—
201#388—21no—
202C4×C4⋊C4C4×C3111yescharacteristic
203#356—24no—
204#355—24no—
205#343—21no—
206#343—21no—
207C2×C31—21no—
208C2×C31—21no—
209C2×C31—161no—
210C2×C31#196411yescharacteristic
211C2×C31#213211yescharacteristic
212C2×C31#213311yescharacteristic
213C22×Q8C2×C4×C3111yescharacteristic
214C22×Q8C23×C3111yescharacteristic
215C22×D4C2×C4×C3111yescharacteristic
216C2×M16—22no—
217C2×M16—22no—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220#124—22no—
221#124—22no—
222#124—41no—
223#124—41no—
224C4×Q8—41no—
225C4×Q8—41no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C2×C4×C3111yescharacteristic
230C2×C4⋊C4D4×C3111yescharacteristic
231C2×C4⋊C4D4×C3111yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C42C2×C4×C3111yescharacteristic
239C2×C42C2×C4×C3111yescharacteristic
240C2×C42C2×C4×C3111yescharacteristic
241C2×C42D4×C3111yescharacteristic
242C2×C42D4×C3111yescharacteristic
243C4⋊C8—41no—
244C4⋊C8—41no—
245#101—22no—
246#101—22no—
247#101—22no—
248#101D4×C3114yes—
249#97—41no—
250C4×C8—41no—
251#95—21no—
252#95—21no—
253#95—21no—
254C31#1322811yescharacteristic, direct product factor, Sylow
255C24—21no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8C2×D4×C3112yes—
262C2×Q8#41×C3112yes—
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—41no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4C22⋊C4×C3111yescharacteristic
270C22×C4C22⋊C4×C3111yescharacteristic
271C22×C4C22⋊C4×C3111yescharacteristic
272C22×C4C22⋊C4×C3111yescharacteristic
273C22×C4C22×C4×C3111yescharacteristic
274C22×C4C2×D4×C3111yescharacteristic
275C22×C4C2×D4×C3111yescharacteristic
276M16—24no—
277M16—42no—
278M16—42no—
279M16—42no—
280C2×C8—42no—
281C2×C8—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—22no—
304C42—22no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23C2×C22⋊C4×C3111yescharacteristic
313Q8—42no—
314Q8—42no—
315Q8—82no—
316D4—82no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4C31×#9911yescharacteristic
340C2×C4C31×#9911yescharacteristic
341C2×C4C4×D4×C3112yes—
342C2×C4C31×#12112yes—
343C2×C4C31×#12411yescharacteristic
344C2×C4C31×#12711yescharacteristic
345C8—44no—
346C8—82no—
347C22—21no—
348C22—21no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22C31×#39011yescharacteristic
357C22C2×C31×#9911yescharacteristic
358C22C31×#41011yescharacteristic
359C4—21no—
360C4—21no—
361C4—22no—
362C4—21no—
363C4—22no—
364C4—21no—
365C4—42no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—81no—
370C4—81no—
371C2—21no—
372C2—21no—
373C2—161no—
374C2C31×#196411yescharacteristic
375C2C31×#213211yescharacteristic
376C2C31×#213311yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC22×Q8C23×C31
prime radicalC22×D4×C31C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#356C2semidirect
4. C31×#1801C2semidirect
5. C4×C31×#101C2semidirect
116. #13228C31direct (N × H)
160. C2×#356C2×C31semidirect
162. #1801C2×C31semidirect
163. C4×#101C2×C31semidirect
254. C31#13228direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25452
C2×C31×#3562
C2×C31×#3552
C31×#18012
C4×C31×#1012
#1322831