Finite Group Databasebeta

← C31×#10727 #21467486

Subgroups — C31×#10727 #21467486

Order 7936.

1678 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊C4×C31C211yesmaximal, characteristic
2C2×C31×#407C212yesmaximal
3C2×C31×#343C214yesmaximal, complemented
4C24×C4×C31C412yes—
5C24×C4×C31C2211yescharacteristic
6C22×M16×C31C2214yes—
7C22×C4⋊C4×C31—24no—
8C22×C4⋊C4×C31—28no—
9C22×C4⋊C4×C31C412yes—
10C22×C4⋊C4×C31C2212yes—
11C31×#407—24no—
12C31×#407C418yescomplemented
13C2×C31×C22⋊C8—24no—
14C31×#343—28no—
15C25×C31C2×C411yescharacteristic
16C23×C4×C31—24no—
17C23×C4×C31—24no—
18C23×C4×C31—24no—
19C23×C4×C31—28no—
20C23×C4×C31—48no—
21C23×C4×C31C2×C411yescharacteristic
22C23×C4×C31C2×C414yes—
23C23×C4×C31C2×C414yes—
24C23×C4×C31D412yes—
25C23×C4×C31D414yes—
26C23×C4×C31Q812yes—
27C23×C4×C31C2311yescharacteristic
28C2×M16×C31—24no—
29C2×M16×C31—28no—
30C2×M16×C31—28no—
31C2×M16×C31C2×C418yescomplemented
32C22×C8×C31—24no—
33C2×C4⋊C4×C31—232no—
34C2×C4⋊C4×C31—28no—
35C2×C4⋊C4×C31—24no—
36C2×C4⋊C4×C31—24no—
37C2×C4⋊C4×C31—24no—
38C2×C4⋊C4×C31D418yes—
39C31×C22⋊C8—216no—
40C24×C31—24no—
41C24×C31—24no—
42C24×C31—22no—
43C24×C31—22no—
44C24×C31C4212yes—
45C24×C31C22⋊C412yes—
46C24×C31C4⋊C412yes—
47C24×C31C22×C411yescharacteristic
48C22×C4×C31—22no—
49C22×C4×C31—22no—
50C22×C4×C31—216no—
51C22×C4×C31—28no—
52C22×C4×C31—28no—
53C22×C4×C31—28no—
54C22×C4×C31—28no—
55C22×C4×C31—216no—
56C22×C4×C31—24no—
57C22×C4×C31—24no—
58C22×C4×C31—432no—
59C22×C4×C31—48no—
60C22×C4×C31—48no—
61C22×C4×C31—48no—
62C22×C4×C31C4212yes—
63C22×C4×C31C22⋊C412yes—
64C22×C4×C31C22⋊C418yes—
65C22×C4×C31C22⋊C418yes—
66C22×C4×C31C4⋊C412yes—
67C22×C4×C31C4⋊C418yes—
68C22×C4×C31C22×C412yes—
69C22×C4×C31C2×D411yescharacteristic
70C22×C4×C31C2×D412yes—
71C22×C4×C31C2×D414yes—
72C22×C4×C31C2×Q811yescharacteristic
73C22×C4×C31#4114yes—
74M16×C31—216no—
75M16×C31—48no—
76C2×C8×C31—28no—
77C2×C8×C31—44no—
78C2×C8×C31—44no—
79C4⋊C4×C31—216no—
80C4⋊C4×C31—216no—
81#10727C3111yesmaximal, characteristic, direct product factor, Sylow
82C23×C31—28no—
83C23×C31—216no—
84C23×C31—28no—
85C23×C31—28no—
86C23×C31—28no—
87C23×C31—24no—
88C23×C31—22no—
89C23×C31—24no—
90C23×C31—22no—
91C23×C31—22no—
92C23×C31—22no—
93C23×C31—22no—
94C23×C31—22no—
95C23×C31#9514yes—
96C23×C31#10014yes—
97C23×C31#10114yes—
98C23×C31C2×C4211yescharacteristic
99C23×C31C2×C22⋊C412yes—
100C23×C31C2×C22⋊C411yescharacteristic
101C23×C31C2×C4⋊C411yescharacteristic
102C23×C31#11712yes—
103C2×C4×C31—24no—
104C2×C4×C31—22no—
105C2×C4×C31—24no—
106C2×C4×C31—22no—
107C2×C4×C31—216no—
108C2×C4×C31—216no—
109C2×C4×C31—28no—
110C2×C4×C31—432no—
111C2×C4×C31—48no—
112C2×C4×C31—48no—
113C2×C4×C31—48no—
114C2×C4×C31#9514yes—
115C2×C4×C31C2×C22⋊C412yes—
116C2×C4×C31C2×C4⋊C412yes—
117C2×C4×C31C4×D418yes—
118C2×C4×C31#12014yes—
119C2×C4×C31#12114yes—
120C2×C4×C31C22⋊Q814yes—
121C2×C4×C31#12314yes—
122C8×C31—48no—
123C23×C4⋊C4C2×C3111yescharacteristic
124C2×#407C2×C3112yes—
125C2×#343C2×C3114yescomplemented
126C22×C31—216no—
127C22×C31—216no—
128C22×C31—24no—
129C22×C31—24no—
130C22×C31—24no—
131C22×C31—24no—
132C22×C31—24no—
133C22×C31—22no—
134C22×C31—22no—
135C22×C31—24no—
136C22×C31—22no—
137C22×C31—24no—
138C22×C31—22no—
139C22×C31#34318yescomplemented
140C22×C31C2×#9511yescharacteristic
141C22×C31C4×C22⋊C412yes—
142C22×C31#37911yescharacteristic
143C22×C31#38012yes—
144C22×C31#38111yescharacteristic
145C22×C31C2×#10012yes—
146C22×C31C2×#10112yes—
147C4×C31—24no—
148C4×C31—48no—
149C4×C31#38514yes—
150C4×C31#38614yes—
151C24×C4C4×C3112yes—
152C24×C4C22×C3111yescharacteristic
153C22×M16C22×C3114yes—
154C22×C4⋊C4—24no—
155C22×C4⋊C4—28no—
156C22×C4⋊C4C4×C3112yes—
157C22×C4⋊C4C22×C3112yes—
158#407—24no—
159#407C4×C3118yescomplemented
160C2×C22⋊C8—24no—
161#343—28no—
162C2×C31—28no—
163C2×C31—22no—
164C2×C31—22no—
165C2×C31C22⋊#9511yescharacteristic
166C2×C31C2×#34312yes—
167C2×C31#178311yescharacteristic
168C2×C31#178411yescharacteristic
169C2×C31#182112yes—
170C25C2×C4×C3111yescharacteristic
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—28no—
175C23×C4—48no—
176C23×C4C2×C4×C3111yescharacteristic
177C23×C4C2×C4×C3114yes—
178C23×C4C2×C4×C3114yes—
179C23×C4D4×C3112yes—
180C23×C4D4×C3114yes—
181C23×C4Q8×C3112yes—
182C23×C4C23×C3111yescharacteristic
183C2×M16—24no—
184C2×M16—28no—
185C2×M16—28no—
186C2×M16C2×C4×C3118yescomplemented
187C22×C8—24no—
188C2×C4⋊C4—232no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4D4×C3118yes—
194C22⋊C8—216no—
195C31#1072711yescharacteristic, direct product factor, Sylow
196C24—24no—
197C24—24no—
198C24—22no—
199C24—22no—
200C24C42×C3112yes—
201C24C22⋊C4×C3112yes—
202C24C4⋊C4×C3112yes—
203C24C22×C4×C3111yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—216no—
207C22×C4—28no—
208C22×C4—28no—
209C22×C4—28no—
210C22×C4—28no—
211C22×C4—216no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—432no—
215C22×C4—48no—
216C22×C4—48no—
217C22×C4—48no—
218C22×C4C42×C3112yes—
219C22×C4C22⋊C4×C3112yes—
220C22×C4C22⋊C4×C3118yes—
221C22×C4C22⋊C4×C3118yes—
222C22×C4C4⋊C4×C3112yes—
223C22×C4C4⋊C4×C3118yes—
224C22×C4C22×C4×C3112yes—
225C22×C4C2×D4×C3111yescharacteristic
226C22×C4C2×D4×C3112yes—
227C22×C4C2×D4×C3114yes—
228C22×C4C2×Q8×C3111yescharacteristic
229C22×C4#41×C3114yes—
230M16—216no—
231M16—48no—
232C2×C8—28no—
233C2×C8—44no—
234C2×C8—44no—
235C4⋊C4—216no—
236C4⋊C4—216no—
237C23—28no—
238C23—216no—
239C23—28no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—28no—
245C23—28no—
246C23—24no—
247C23—22no—
248C23—24no—
249C23—22no—
250C23C31×#9514yes—
251C23C31×#10014yes—
252C23C31×#10114yes—
253C23C2×C42×C3111yescharacteristic
254C23C2×C22⋊C4×C3112yes—
255C23C2×C22⋊C4×C3111yescharacteristic
256C23C2×C4⋊C4×C3111yescharacteristic
257C23C31×#11712yes—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—216no—
263C2×C4—216no—
264C2×C4—28no—
265C2×C4—432no—
266C2×C4—48no—
267C2×C4—48no—
268C2×C4—48no—
269C2×C4C31×#9514yes—
270C2×C4C2×C22⋊C4×C3112yes—
271C2×C4C2×C4⋊C4×C3112yes—
272C2×C4C4×D4×C3118yes—
273C2×C4C31×#12014yes—
274C2×C4C31×#12114yes—
275C2×C4C31×C22⋊Q814yes—
276C2×C4C31×#12314yes—
277C8—48no—
278C22—216no—
279C22—216no—
280C22—24no—
281C22—24no—
282C22—24no—
283C22—24no—
284C22—24no—
285C22—22no—
286C22—22no—
287C22—24no—
288C22—22no—
289C22—24no—
290C22—22no—
291C22C31×#34318yescomplemented
292C22C2×C31×#9511yescharacteristic
293C22C4×C22⋊C4×C3112yes—
294C22C31×#37911yescharacteristic
295C22C31×#38012yes—
296C22C31×#38111yescharacteristic
297C22C2×C31×#10012yes—
298C22C2×C31×#10112yes—
299C4—24no—
300C4—48no—
301C4C31×#38514yes—
302C4C31×#38614yes—
303C2—28no—
304C2—22no—
305C2—22no—
306C2C31×C22⋊#9511yescharacteristic
307C2C2×C31×#34312yes—
308C2C31×#178311yescharacteristic
309C2C31×#178411yescharacteristic
310C2C31×#182112yes—
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC23C2×C42×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC25×C31C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C31×#343C2semidirect
12. C31×#407C4semidirect
31. C2×M16×C31C2×C4semidirect
81. #10727C31direct (N × H)
125. C2×#343C2×C31semidirect
139. C22×C31#343semidirect
159. #407C4×C31semidirect
186. C2×M16C2×C4×C31semidirect
195. C31#10727direct (N × H)
291. C22C31×#343semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C4×C312
C2×C31×#4072
C2×C31×#3432
#1072731