Finite Group Databasebeta

← C31×#10169 #21466928

Subgroups — C31×#10169 #21466928

Order 7936.

556 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3208C211yesmaximal, characteristic
2C31×#3139C211yesmaximal, characteristic, complemented
3C2×C31×#486C211yesmaximal, characteristic
4C31×#1965C211yesmaximal, characteristic
5C31×#1546C211yesmaximal, characteristic, complemented
6C31×#1493C211yesmaximal, characteristic, complemented
7C31×#1278C211yesmaximal, characteristic, complemented
8C31×#533C2211yescharacteristic
9C2×C31×C4⋊Q8C2211yescharacteristic
10C2×C31×#127C2211yescharacteristic
11C31×#492—21no—
12C31×#491—22no—
13C31×#487—21no—
14C31×#487—21no—
15C31×#486—22no—
16C31×#486—21no—
17C31×#486—22no—
18C31×#460—22no—
19C31×#450—22no—
20C31×#431C2211yescharacteristic
21C31×#423C2211yescharacteristic
22C31×#419—22no—
23C2×C31×#102—22no—
24C2×C31×#100—22no—
25C4×M16×C31C2211yescharacteristic
26C2×C4×C8×C31C2211yescharacteristic
27C31×#332—22no—
28C31×Q8⋊C8—22no—
29C22×Q8×C31—21no—
30C22×D4×C31—21no—
31C22×D4×C31—21no—
32C22×D4×C31D412yes—
33C2×SD16×C31—24no—
34C2×M16×C31—21no—
35C2×M16×C31D412yes—
36C22×C8×C31—21no—
37C31×C4⋊Q8—21no—
38C31×C4⋊Q8—21no—
39C31×C4⋊Q8—21no—
40C31×C4⋊Q8D411yescharacteristic
41C31×C4⋊Q8D412yes—
42C31×C4⋊Q8D411yescharacteristic
43C31×#127—21no—
44C31×#127—21no—
45C31×#127—21no—
46C31×#127—21no—
47C31×#125—21no—
48C31×C22⋊Q8—21no—
49C31×C22⋊Q8—21no—
50C4×Q8×C31—21no—
51C4×Q8×C31—21no—
52C31×#117—21no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C42×C31C2311yescharacteristic
56C31×C4⋊C8—21no—
57C31×C4⋊C8—21no—
58C31×C4⋊C8D412yes—
59C31×#103—24no—
60C31×#103—24no—
61C31×#102—28no—
62C31×#102—24no—
63C31×#100—44no—
64C31×C22⋊C8—21no—
65C31×#97—21no—
66C4×C8×C31—21no—
67C4×C8×C31—21no—
68C4×C8×C31—21no—
69C4×C8×C31D412yes—
70C24×C31—22no—
71C2×Q8×C31—22no—
72C2×Q8×C31—21no—
73C2×Q8×C31—21no—
74C2×Q8×C31—41no—
75C2×Q8×C31—42no—
76C2×Q8×C31—42no—
77C2×D4×C31—22no—
78C2×D4×C31—22no—
79C2×D4×C31—44no—
80C2×D4×C31—41no—
81C2×D4×C31—41no—
82C2×D4×C31—41no—
83C2×D4×C31—41no—
84C2×D4×C31—42no—
85C2×D4×C31—42no—
86C2×D4×C31—82no—
87C22×C4×C31—41no—
88C22×C4×C31C2×D411yescharacteristic
89C22×C4×C31C2×D411yescharacteristic
90C22×C4×C31#4111yescharacteristic
91SD16×C31—84no—
92M16×C31—22no—
93M16×C31—22no—
94M16×C31—42no—
95C2×C8×C31—22no—
96C2×C8×C31—22no—
97C2×C8×C31—22no—
98C2×C8×C31—42no—
99C2×C8×C31—41no—
100C4⋊C4×C31—22no—
101C4⋊C4×C31—21no—
102C4⋊C4×C31—21no—
103C4⋊C4×C31—21no—
104C4⋊C4×C31—22no—
105C4⋊C4×C31—21no—
106C4⋊C4×C31—41no—
107C4⋊C4×C31—41no—
108C4⋊C4×C31—41no—
109C4⋊C4×C31—41no—
110C22⋊C4×C31—41no—
111C42×C31—41no—
112C42×C31C2×D411yescharacteristic
113C42×C31C2×D411yescharacteristic
114C42×C31C2×D411yescharacteristic
115C42×C31C2×D411yescharacteristic
116#10169C3111yesmaximal, characteristic, direct product factor, Sylow
117C23×C31—42no—
118C23×C31—44no—
119C23×C31—82no—
120C23×C31—82no—
121C23×C31#12111yescharacteristic
122Q8×C31—44no—
123Q8×C31—42no—
124Q8×C31—42no—
125Q8×C31—81no—
126Q8×C31—81no—
127D4×C31—44no—
128D4×C31—44no—
129D4×C31—82no—
130D4×C31—82no—
131D4×C31—82no—
132D4×C31—81no—
133D4×C31—81no—
134C2×C4×C31—22no—
135C2×C4×C31—22no—
136C2×C4×C31—41no—
137C2×C4×C31—41no—
138C2×C4×C31—42no—
139C2×C4×C31—82no—
140C2×C4×C31#12011yescharacteristic
141C2×C4×C31#12011yescharacteristic
142C2×C4×C31#12011yescharacteristic
143C2×C4×C31#12111yescharacteristic
144C2×C4×C31#12111yescharacteristic
145C2×C4×C31#12711yescharacteristic
146C2×C4×C31#13512yes—
147C2×C4×C31AGL(1,ℤ/8)12yes—
148C8×C31—42no—
149C8×C31—42no—
150C8×C31—42no—
151C8×C31—81no—
152#3208C2×C3111yescharacteristic
153#3139C2×C3111yescharacteristic, complemented
154C2×#486C2×C3111yescharacteristic
155#1965C2×C3111yescharacteristic
156#1546C2×C3111yescharacteristic, complemented
157#1493C2×C3111yescharacteristic, complemented
158#1278C2×C3111yescharacteristic, complemented
159C22×C31—21no—
160C22×C31—21no—
161C22×C31—82no—
162C22×C31—82no—
163C22×C31—82no—
164C22×C31—164no—
165C22×C31#39211yescharacteristic
166C22×C31#46312yes—
167C4×C31—21no—
168C4×C31—21no—
169C4×C31—81no—
170C4×C31—82no—
171C4×C31—81no—
172C4×C31#44912yes—
173C4×C31#45512yes—
174C4×C31#47112yes—
175C4×C31#49511yescharacteristic
176C4×C31#49611yescharacteristic
177#533C22×C3111yescharacteristic
178C2×C4⋊Q8C22×C3111yescharacteristic
179C2×#127C22×C3111yescharacteristic
180#492—21no—
181#491—22no—
182#487—21no—
183#487—21no—
184#486—22no—
185#486—21no—
186#486—22no—
187#460—22no—
188#450—22no—
189#431C22×C3111yescharacteristic
190#423C22×C3111yescharacteristic
191#419—22no—
192C2×#102—22no—
193C2×#100—22no—
194C4×M16C22×C3111yescharacteristic
195C2×C4×C8C22×C3111yescharacteristic
196#332—22no—
197Q8⋊C8—22no—
198C2×C31—22no—
199C2×C31—162no—
200C2×C31#200411yescharacteristic
201C2×C31#201211yescharacteristic
202C2×C31#201911yescharacteristic
203C22×Q8—21no—
204C22×D4—21no—
205C22×D4—21no—
206C22×D4D4×C3112yes—
207C2×SD16—24no—
208C2×M16—21no—
209C2×M16D4×C3112yes—
210C22×C8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8D4×C3112yes—
215C4⋊Q8D4×C3111yescharacteristic
216C4⋊Q8D4×C3111yescharacteristic
217#127—21no—
218#127—21no—
219#127—21no—
220#127—21no—
221#125—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226#117—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C42C23×C3111yescharacteristic
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8D4×C3112yes—
233#103—24no—
234#103—24no—
235#102—28no—
236#102—24no—
237#100—44no—
238C22⋊C8—21no—
239#97—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8D4×C3112yes—
244C31#1016911yescharacteristic, direct product factor, Sylow
245C24—22no—
246C2×Q8—22no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—41no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—44no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—82no—
262C22×C4—41no—
263C22×C4C2×D4×C3111yescharacteristic
264C22×C4C2×D4×C3111yescharacteristic
265C22×C4#41×C3111yescharacteristic
266SD16—84no—
267M16—22no—
268M16—22no—
269M16—42no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—42no—
274C2×C8—41no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C22⋊C4—41no—
286C42—41no—
287C42C2×D4×C3111yescharacteristic
288C42C2×D4×C3111yescharacteristic
289C42C2×D4×C3111yescharacteristic
290C42C2×D4×C3111yescharacteristic
291C23—42no—
292C23—44no—
293C23—82no—
294C23—82no—
295C23C31×#12111yescharacteristic
296Q8—44no—
297Q8—42no—
298Q8—42no—
299Q8—81no—
300Q8—81no—
301D4—44no—
302D4—44no—
303D4—82no—
304D4—82no—
305D4—82no—
306D4—81no—
307D4—81no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4C31×#12011yescharacteristic
315C2×C4C31×#12011yescharacteristic
316C2×C4C31×#12011yescharacteristic
317C2×C4C31×#12111yescharacteristic
318C2×C4C31×#12111yescharacteristic
319C2×C4C31×#12711yescharacteristic
320C2×C4C31×#13512yes—
321C2×C4C31×AGL(1,ℤ/8)12yes—
322C8—42no—
323C8—42no—
324C8—42no—
325C8—81no—
326C22—21no—
327C22—21no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—164no—
332C22C31×#39211yescharacteristic
333C22C31×#46312yes—
334C4—21no—
335C4—21no—
336C4—81no—
337C4—82no—
338C4—81no—
339C4C31×#44912yes—
340C4C31×#45512yes—
341C4C31×#47112yes—
342C4C31×#49511yescharacteristic
343C4C31×#49611yescharacteristic
344C2—22no—
345C2—162no—
346C2C31×#200411yescharacteristic
347C2C31×#201211yescharacteristic
348C2C31×#201911yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3139C2semidirect
5. C31×#1546C2semidirect
6. C31×#1493C2semidirect
7. C31×#1278C2semidirect
116. #10169C31direct (N × H)
153. #3139C2×C31semidirect
156. #1546C2×C31semidirect
157. #1493C2×C31semidirect
158. #1278C2×C31semidirect
244. C31#10169direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32082
C31×#31392
C2×C31×#4862
C31×#19652
C31×#15462
C31×#14932
C31×#12782
#1016931