Finite Group Databasebeta

← C31×#9027 #21465786

Subgroups — C31×#9027 #21465786

Order 7936.

660 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#398C211yesmaximal, characteristic
2C31×#2316C211yesmaximal, characteristic
3C31×#1823C212yesmaximal
4C2×C31×#324C211yesmaximal, characteristic
5C31×#1294C212yesmaximal
6C2×C31×#117—21no—
7C22×C4⋊C4×C31—21no—
8C22×C4⋊C4×C31C2211yescharacteristic
9C22×C4⋊C4×C31C2211yescharacteristic
10C2×C31×C22⋊C8C2212yes—
11C31×#398—22no—
12C31×#398C414yescomplemented
13C31×#389—24no—
14C31×#380—21no—
15C31×#380C2212yes—
16C2×C31×#95—21no—
17C2×C31×#95C2211yescharacteristic
18C31×#340—24no—
19C31×#324—22no—
20C23×C4×C31—21no—
21C23×C4×C31—21no—
22C23×C4×C31D411yescharacteristic
23C23×C4×C31D411yescharacteristic
24C23×C4×C31C2311yescharacteristic
25C22×C8×C31—22no—
26C31×#117—42no—
27C31×#117—42no—
28C2×C4⋊C4×C31—22no—
29C2×C4⋊C4×C31—22no—
30C2×C4⋊C4×C31—22no—
31C2×C4⋊C4×C31—22no—
32C2×C4⋊C4×C31—22no—
33C2×C4⋊C4×C31—24no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31—22no—
36C2×C4⋊C4×C31—42no—
37C2×C4⋊C4×C31—42no—
38C2×C4⋊C4×C31C2×C412yes—
39C2×C4⋊C4×C31C2×C412yes—
40C2×C4⋊C4×C31D412yes—
41C2×C4⋊C4×C31D414yes—
42C2×C22⋊C4×C31—22no—
43C2×C42×C31—22no—
44C31×C22⋊C8—22no—
45C31×C22⋊C8—22no—
46C31×#95—22no—
47C31×#95—24no—
48C31×#95—21no—
49C31×#95—22no—
50C31×#95C2×C412yes—
51C24×C31C2×D411yescharacteristic
52C22×C4×C31—21no—
53C22×C4×C31—21no—
54C22×C4×C31—21no—
55C22×C4×C31—21no—
56C22×C4×C31—24no—
57C22×C4×C31—22no—
58C22×C4×C31—21no—
59C22×C4×C31—21no—
60C22×C4×C31—22no—
61C22×C4×C31—24no—
62C22×C4×C31—42no—
63C22×C4×C31—42no—
64C22×C4×C31—41no—
65C22×C4×C31—41no—
66C22×C4×C31—42no—
67C22×C4×C31—41no—
68C22×C4×C31—41no—
69C22×C4×C31—41no—
70C22×C4×C31—41no—
71C22×C4×C31—41no—
72C22×C4×C31—41no—
73C22×C4×C31C22⋊C412yes—
74C22×C4×C31C22⋊C412yes—
75C22×C4×C31D812yes—
76C22×C4×C31SD1612yes—
77C22×C4×C31C22×C411yescharacteristic
78C22×C4×C31C2×D412yes—
79C22×C4×C31C2×D411yescharacteristic
80C22×C4×C31#4112yes—
81C2×C8×C31—44no—
82C2×C8×C31—42no—
83C4⋊C4×C31—24no—
84C4⋊C4×C31—24no—
85C4⋊C4×C31—24no—
86C4⋊C4×C31—44no—
87C4⋊C4×C31—44no—
88C4⋊C4×C31—48no—
89C4⋊C4×C31—42no—
90C4⋊C4×C31—82no—
91C22⋊C4×C31—42no—
92C22⋊C4×C31—42no—
93C42×C31—44no—
94#9027C3111yesmaximal, characteristic, direct product factor, Sylow
95C23×C31—22no—
96C23×C31—22no—
97C23×C31C2×C22⋊C411yescharacteristic
98C23×C31#12011yescharacteristic
99C23×C31#12311yescharacteristic
100C23×C31C2×D811yescharacteristic
101C23×C31C2×SD1611yescharacteristic
102C23×C31AGL(1,ℤ/8)11yescharacteristic
103C23×C31#13711yescharacteristic
104C2×C4×C31—22no—
105C2×C4×C31—22no—
106C2×C4×C31—22no—
107C2×C4×C31—22no—
108C2×C4×C31—24no—
109C2×C4×C31—22no—
110C2×C4×C31—22no—
111C2×C4×C31—24no—
112C2×C4×C31—22no—
113C2×C4×C31—41no—
114C2×C4×C31—41no—
115C2×C4×C31—42no—
116C2×C4×C31—44no—
117C2×C4×C31—44no—
118C2×C4×C31—42no—
119C2×C4×C31—42no—
120C2×C4×C31—42no—
121C2×C4×C31—42no—
122C2×C4×C31—42no—
123C2×C4×C31—42no—
124C2×C4×C31—42no—
125C2×C4×C31—44no—
126C2×C4×C31—44no—
127C2×C4×C31—44no—
128C2×C4×C31—81no—
129C2×C4×C31—81no—
130C2×C4×C31—81no—
131C2×C4×C31—81no—
132C2×C4×C31—81no—
133C2×C4×C31#10214yes—
134C2×C4×C31C4×D412yes—
135C2×C4×C31#12112yes—
136C8×C31—82no—
137C2×#398C2×C3111yescharacteristic
138#2316C2×C3111yescharacteristic
139#1823C2×C3112yes—
140C2×#324C2×C3111yescharacteristic
141#1294C2×C3112yes—
142C22×C31—22no—
143C22×C31—22no—
144C22×C31—22no—
145C22×C31—22no—
146C22×C31—21no—
147C22×C31—21no—
148C22×C31—21no—
149C22×C31—21no—
150C22×C31#38611yescharacteristic
151C22×C31C2×#10211yescharacteristic
152C22×C31#41711yescharacteristic
153C22×C31#44711yescharacteristic
154C22×C31#44811yescharacteristic
155C22×C31#45612yes—
156C22×C31#45812yes—
157C22×C31#48011yescharacteristic
158C22×C31#48111yescharacteristic
159C4×C31—24no—
160C4×C31—22no—
161C4×C31—22no—
162C4×C31—44no—
163C4×C31—42no—
164C4×C31—84no—
165C4×C31—82no—
166C2×#117—21no—
167C22×C4⋊C4—21no—
168C22×C4⋊C4C22×C3111yescharacteristic
169C22×C4⋊C4C22×C3111yescharacteristic
170C2×C22⋊C8C22×C3112yes—
171#398—22no—
172#398C4×C3114yescomplemented
173#389—24no—
174#380—21no—
175#380C22×C3112yes—
176C2×#95—21no—
177C2×#95C22×C3111yescharacteristic
178#340—24no—
179#324—22no—
180C2×C31—21no—
181C2×C31—21no—
182C2×C31—21no—
183C2×C31—21no—
184C2×C31#161012yes—
185C2×C31#161112yes—
186C2×C31#187411yescharacteristic
187C2×C31#188611yescharacteristic
188C2×C31#189011yescharacteristic
189C23×C4—21no—
190C23×C4—21no—
191C23×C4D4×C3111yescharacteristic
192C23×C4D4×C3111yescharacteristic
193C23×C4C23×C3111yescharacteristic
194C22×C8—22no—
195#117—42no—
196#117—42no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C2×C4×C3112yes—
208C2×C4⋊C4C2×C4×C3112yes—
209C2×C4⋊C4D4×C3112yes—
210C2×C4⋊C4D4×C3114yes—
211C2×C22⋊C4—22no—
212C2×C42—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215#95—22no—
216#95—24no—
217#95—21no—
218#95—22no—
219#95C2×C4×C3112yes—
220C31#902711yescharacteristic, direct product factor, Sylow
221C24C2×D4×C3111yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4C22⋊C4×C3112yes—
244C22×C4C22⋊C4×C3112yes—
245C22×C4D8×C3112yes—
246C22×C4SD16×C3112yes—
247C22×C4C22×C4×C3111yescharacteristic
248C22×C4C2×D4×C3112yes—
249C22×C4C2×D4×C3111yescharacteristic
250C22×C4#41×C3112yes—
251C2×C8—44no—
252C2×C8—42no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—48no—
259C4⋊C4—42no—
260C4⋊C4—82no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C42—44no—
264C23—22no—
265C23—22no—
266C23C2×C22⋊C4×C3111yescharacteristic
267C23C31×#12011yescharacteristic
268C23C31×#12311yescharacteristic
269C23C2×D8×C3111yescharacteristic
270C23C2×SD16×C3111yescharacteristic
271C23C31×AGL(1,ℤ/8)11yescharacteristic
272C23C31×#13711yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4C31×#10214yes—
303C2×C4C4×D4×C3112yes—
304C2×C4C31×#12112yes—
305C8—82no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22C31×#38611yescharacteristic
315C22C2×C31×#10211yescharacteristic
316C22C31×#41711yescharacteristic
317C22C31×#44711yescharacteristic
318C22C31×#44811yescharacteristic
319C22C31×#45612yes—
320C22C31×#45812yes—
321C22C31×#48011yescharacteristic
322C22C31×#48111yescharacteristic
323C4—24no—
324C4—22no—
325C4—22no—
326C4—44no—
327C4—42no—
328C4—84no—
329C4—82no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2—21no—
334C2C31×#161012yes—
335C2C31×#161112yes—
336C2C31×#187411yescharacteristic
337C2C31×#188611yescharacteristic
338C2C31×#189011yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
12. C31×#398C4semidirect
94. #9027C31direct (N × H)
172. #398C4×C31semidirect
220. C31#9027direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3982
C31×#23162
C31×#18232
C2×C31×#3242
C31×#12942
#902731