Finite Group Databasebeta

← C31×#36691 #21493450

Subgroups — C31×#36691 #21493450

Order 7936.

1474 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2845C212yesmaximal, complemented
2C31×#2844C212yesmaximal, complemented
3C31×#2843C214yesmaximal, complemented
4C31×#2841C218yesmaximal
5C31×#2402C214yesmaximal, complemented
6C31×#2399C214yesmaximal, complemented
7C31×#2396C214yesmaximal
8C31×#2285C211yesmaximal, characteristic
9C31×#2283C212yesmaximal
10C2×C4×Q8×C31C2214yes—
11C2×C31×#117C2212yes—
12C22×C42×C31C2211yescharacteristic
13C31×#400C2218yes—
14C31×#400C2218yes—
15C31×#400C2214yes—
16C31×#399C2214yes—
17C31×#398C2218yes—
18C31×#398C2218yes—
19C31×#398C2218yes—
20C31×#398C2218yes—
21C31×#398C2218yes—
22C31×#396C2218yes—
23C31×#395C2218yes—
24C31×#395C2218yes—
25C31×#395C2214yes—
26C31×#393C2214yes—
27C31×#384C2212yes—
28C31×#383C2214yes—
29C31×#382C2211yescharacteristic
30C31×#382C2214yes—
31C31×#382C2211yescharacteristic
32C31×#381C2214yes—
33C31×#380C2214yes—
34C31×#380C2214yes—
35C4×C4⋊C4×C31C2218yes—
36C4×C4⋊C4×C31C2214yes—
37C4×C4⋊C4×C31C2212yes—
38C4×C4⋊C4×C31C2214yes—
39C4×C4⋊C4×C31C2212yes—
40C4×C22⋊C4×C31C2214yes—
41C31×#376C2214yes—
42C22×Q8×C31C2314yes—
43C23×C4×C31C2311yescharacteristic
44C23×C4×C31C2312yes—
45C4×Q8×C31—48no—
46C31×#117—44no—
47C2×C4⋊C4×C31C2318yes—
48C2×C4⋊C4×C31C2314yes—
49C2×C4⋊C4×C31C2314yes—
50C2×C4⋊C4×C31C2312yes—
51C2×C4⋊C4×C31C2318yes—
52C2×C4⋊C4×C31C2318yes—
53C2×C4⋊C4×C31C2318yes—
54C2×C4⋊C4×C31C2318yes—
55C2×C4⋊C4×C31C2312yes—
56C2×C4⋊C4×C31C2314yes—
57C2×C22⋊C4×C31C2314yes—
58C2×C22⋊C4×C31C2318yes—
59C2×C42×C31Q818yescomplemented
60C2×C42×C31C2314yes—
61C2×C42×C31C2314yes—
62C2×C42×C31C2314yes—
63C2×C42×C31C2311yescharacteristic
64C2×C42×C31C2311yescharacteristic
65C2×C42×C31C2312yes—
66C31×#95C2318yes—
67C31×#95C2314yes—
68C31×#95C2314yes—
69C31×#95C2318yes—
70C31×#95C2318yes—
71C31×#95C2318yes—
72C31×#95C2314yes—
73C31×#95C2318yes—
74C31×#95C2314yes—
75C31×#95C2314yes—
76C31×#95C2314yes—
77C24×C31C2411yescharacteristic
78C2×Q8×C31—44no—
79C2×Q8×C31—48no—
80C22×C4×C31—22no—
81C22×C4×C31—22no—
82C22×C4×C31—44no—
83C22×C4×C31C2×Q814yes—
84C22×C4×C31C2×Q818yes—
85C22×C4×C31C2418yes—
86C22×C4×C31C2414yes—
87C22×C4×C31C2418yes—
88C22×C4×C31C2414yes—
89C22×C4×C31C2411yescharacteristic
90C22×C4×C31C2412yes—
91C22×C4×C31C2412yes—
92C22×C4×C31C2411yescharacteristic
93C4⋊C4×C31—44no—
94C4⋊C4×C31—48no—
95C4⋊C4×C31—44no—
96C4⋊C4×C31—48no—
97C4⋊C4×C31—48no—
98C4⋊C4×C31—48no—
99C4⋊C4×C31—42no—
100C4⋊C4×C31—42no—
101C4⋊C4×C31—48no—
102C4⋊C4×C31—44no—
103C22⋊C4×C31—48no—
104C22⋊C4×C31—44no—
105C42×C31—28no—
106C42×C31—28no—
107C42×C31—28no—
108C42×C31C2×Q814yes—
109C42×C31C2×Q814yes—
110C42×C31C2×Q818yes—
111#36691C3111yesmaximal, characteristic, direct product factor, Sylow
112C23×C31—21no—
113C23×C31—21no—
114C23×C31—24no—
115C23×C31C22×Q812yes—
116C23×C31C2511yescharacteristic
117Q8×C31—416no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—24no—
121C2×C4×C31—24no—
122C2×C4×C31—28no—
123C2×C4×C31—24no—
124C2×C4×C31—24no—
125C2×C4×C31—28no—
126C2×C4×C31—28no—
127C2×C4×C31—28no—
128C2×C4×C31—28no—
129C2×C4×C31—28no—
130C2×C4×C31—28no—
131C2×C4×C31—28no—
132C2×C4×C31—28no—
133C2×C4×C31—48no—
134C2×C4×C31C22×Q812yes—
135C2×C4×C31C22×Q814yes—
136C2×C4×C31C22×Q814yes—
137C2×C4×C31C22×Q812yes—
138C2×C4×C31H(5,𝔽2)12yes—
139C2×C4×C312−1+412yes—
140#2845C2×C3112yescomplemented
141#2844C2×C3112yescomplemented
142#2843C2×C3114yescomplemented
143#2841C2×C3118yes—
144#2402C2×C3114yescomplemented
145#2399C2×C3114yescomplemented
146#2396C2×C3114yes—
147#2285C2×C3111yescharacteristic
148#2283C2×C3112yes—
149C22×C31—22no—
150C22×C31—24no—
151C22×C31—42no—
152C22×C31—42no—
153C22×C31C23×Q811yescharacteristic
154C22×C31C2×H(5,𝔽2)11yescharacteristic
155C22×C31C2×2−1+411yescharacteristic
156C22×C31#58514yes—
157C4×C31—22no—
158C4×C31—24no—
159C4×C31—24no—
160C4×C31—44no—
161C4×C31—44no—
162C4×C31—48no—
163C4×C31—48no—
164C4×C31#54414yes—
165C2×C4×Q8C22×C3114yes—
166C2×#117C22×C3112yes—
167C22×C42C22×C3111yescharacteristic
168#400C22×C3118yes—
169#400C22×C3118yes—
170#400C22×C3114yes—
171#399C22×C3114yes—
172#398C22×C3118yes—
173#398C22×C3118yes—
174#398C22×C3118yes—
175#398C22×C3118yes—
176#398C22×C3118yes—
177#396C22×C3118yes—
178#395C22×C3118yes—
179#395C22×C3118yes—
180#395C22×C3114yes—
181#393C22×C3114yes—
182#384C22×C3112yes—
183#383C22×C3114yes—
184#382C22×C3111yescharacteristic
185#382C22×C3114yes—
186#382C22×C3111yescharacteristic
187#381C22×C3114yes—
188#380C22×C3114yes—
189#380C22×C3114yes—
190C4×C4⋊C4C22×C3118yes—
191C4×C4⋊C4C22×C3114yes—
192C4×C4⋊C4C22×C3112yes—
193C4×C4⋊C4C22×C3114yes—
194C4×C4⋊C4C22×C3112yes—
195C4×C22⋊C4C22×C3114yes—
196#376C22×C3114yes—
197C2×C31—42no—
198C2×C31C2×#54411yescharacteristic
199C2×C31#345812yes—
200C2×C31#352912yes—
201C2×C31#353112yes—
202C22×Q8C23×C3114yes—
203C23×C4C23×C3111yescharacteristic
204C23×C4C23×C3112yes—
205C4×Q8—48no—
206#117—44no—
207C2×C4⋊C4C23×C3112yes—
208C2×C4⋊C4C23×C3112yes—
209C2×C4⋊C4C23×C3114yes—
210C2×C4⋊C4C23×C3118yes—
211C2×C4⋊C4C23×C3114yes—
212C2×C4⋊C4C23×C3114yes—
213C2×C4⋊C4C23×C3118yes—
214C2×C4⋊C4C23×C3118yes—
215C2×C4⋊C4C23×C3118yes—
216C2×C4⋊C4C23×C3118yes—
217C2×C22⋊C4C23×C3114yes—
218C2×C22⋊C4C23×C3118yes—
219C2×C42Q8×C3118yescomplemented
220C2×C42C23×C3114yes—
221C2×C42C23×C3114yes—
222C2×C42C23×C3114yes—
223C2×C42C23×C3112yes—
224C2×C42C23×C3111yescharacteristic
225C2×C42C23×C3111yescharacteristic
226#95C23×C3118yes—
227#95C23×C3118yes—
228#95C23×C3118yes—
229#95C23×C3114yes—
230#95C23×C3114yes—
231#95C23×C3114yes—
232#95C23×C3114yes—
233#95C23×C3118yes—
234#95C23×C3114yes—
235#95C23×C3118yes—
236#95C23×C3114yes—
237C31#3669111yescharacteristic, direct product factor, Sylow
238C24C24×C3111yescharacteristic
239C2×Q8—44no—
240C2×Q8—48no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—44no—
244C22×C4C2×Q8×C3114yes—
245C22×C4C2×Q8×C3118yes—
246C22×C4C24×C3118yes—
247C22×C4C24×C3114yes—
248C22×C4C24×C3118yes—
249C22×C4C24×C3114yes—
250C22×C4C24×C3111yescharacteristic
251C22×C4C24×C3112yes—
252C22×C4C24×C3112yes—
253C22×C4C24×C3111yescharacteristic
254C4⋊C4—44no—
255C4⋊C4—48no—
256C4⋊C4—44no—
257C4⋊C4—48no—
258C4⋊C4—48no—
259C4⋊C4—48no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—48no—
263C4⋊C4—44no—
264C22⋊C4—48no—
265C22⋊C4—44no—
266C42—28no—
267C42—28no—
268C42—28no—
269C42C2×Q8×C3114yes—
270C42C2×Q8×C3114yes—
271C42C2×Q8×C3118yes—
272C23—21no—
273C23—21no—
274C23—24no—
275C23C22×Q8×C3112yes—
276C23C25×C3111yescharacteristic
277Q8—416no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—28no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—28no—
286C2×C4—28no—
287C2×C4—28no—
288C2×C4—28no—
289C2×C4—28no—
290C2×C4—28no—
291C2×C4—28no—
292C2×C4—28no—
293C2×C4—48no—
294C2×C4C22×Q8×C3112yes—
295C2×C4C22×Q8×C3114yes—
296C2×C4C22×Q8×C3114yes—
297C2×C4C22×Q8×C3112yes—
298C2×C4C31×H(5,𝔽2)12yes—
299C2×C4C31×2−1+412yes—
300C22—22no—
301C22—24no—
302C22—42no—
303C22—42no—
304C22C23×Q8×C3111yescharacteristic
305C22C2×C31×H(5,𝔽2)11yescharacteristic
306C22C2×C31×2−1+411yescharacteristic
307C22C31×#58514yes—
308C4—22no—
309C4—24no—
310C4—24no—
311C4—44no—
312C4—48no—
313C4—44no—
314C4—48no—
315C4C31×#54414yes—
316C2—42no—
317C2C2×C31×#54411yescharacteristic
318C2C31×#345812yes—
319C2C31×#352912yes—
320C2C31×#353112yes—
subgroupG/H
centerC22×C4×C31C24
commutator, Frattini, cosocleC23C25×C31
socleC23×C31C25
prime radicalC24×C31C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2845C2semidirect
2. C31×#2844C2semidirect
3. C31×#2843C2semidirect
5. C31×#2402C2semidirect
6. C31×#2399C2semidirect
59. C2×C42×C31Q8semidirect
111. #36691C31direct (N × H)
140. #2845C2×C31semidirect
141. #2844C2×C31semidirect
142. #2843C2×C31semidirect
144. #2402C2×C31semidirect
145. #2399C2×C31semidirect
219. C2×C42Q8×C31semidirect
237. C31#36691direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28452
C31×#28442
C31×#28432
C31×#28412
C31×#24022
C31×#23992
C31×#23962
C31×#22852
C31×#22832
#3669131