Finite Group Databasebeta

← C31×#11780 #21468539

Subgroups — C31×#11780 #21468539

Order 7936.

464 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2486C211yesmaximal, characteristic
2C4×C31×#100C211yesmaximal, characteristic
3C31×#1465C211yesmaximal, characteristic
4C31×#1346C214yesmaximal, complemented
5C2×C31×#123—21no—
6C2×C4×D4×C31C2211yescharacteristic
7C22×C4⋊C4×C31C411yescharacteristic
8C2×C31×#100—21no—
9C31×#407—21no—
10C4×M16×C31—21no—
11C31×#398C2212yes—
12C31×#397C412yes—
13C31×#389—21no—
14C31×#381—21no—
15C31×#380—21no—
16C31×#380C411yescharacteristic
17C31×#343C2212yes—
18C31×#324C2212yes—
19C31×#323—21no—
20C22×D4×C31D411yescharacteristic
21C23×C4×C31—21no—
22C23×C4×C31C2×C411yescharacteristic
23C23×C4×C31D411yescharacteristic
24C2×M16×C31—22no—
25C2×M16×C31—41no—
26C31×#123—44no—
27C4×D4×C31—41no—
28C4×D4×C31—41no—
29C2×C4⋊C4×C31—21no—
30C2×C4⋊C4×C31—21no—
31C2×C4⋊C4×C31—22no—
32C2×C4⋊C4×C31—41no—
33C2×C4⋊C4×C31—41no—
34C2×C4⋊C4×C31C2×C412yes—
35C2×C4⋊C4×C31C2311yescharacteristic
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—21no—
38C2×C22⋊C4×C31C2×C411yescharacteristic
39C2×C22⋊C4×C31D411yescharacteristic
40C2×C42×C31D411yescharacteristic
41C31×#100—24no—
42C31×C22⋊C8—22no—
43C31×C22⋊C8—41no—
44C31×#97—41no—
45C4×C8×C31—41no—
46C31×#95—22no—
47C31×#95—22no—
48C31×#95C2×C412yes—
49C24×C31C22⋊C411yescharacteristic
50C24×C31C22⋊C411yescharacteristic
51C2×D4×C31—22no—
52C2×D4×C31—22no—
53C2×D4×C31—41no—
54C22×C4×C31—21no—
55C22×C4×C31—21no—
56C22×C4×C31—21no—
57C22×C4×C31—21no—
58C22×C4×C31—22no—
59C22×C4×C31—22no—
60C22×C4×C31—42no—
61C22×C4×C31—42no—
62C22×C4×C31—42no—
63C22×C4×C31—42no—
64C22×C4×C31—42no—
65C22×C4×C31C22⋊C411yescharacteristic
66C22×C4×C31C22⋊C411yescharacteristic
67C22×C4×C31C22×C411yescharacteristic
68C22×C4×C31C2×D411yescharacteristic
69C22×C4×C31C2×D411yescharacteristic
70M16×C31—24no—
71M16×C31—42no—
72M16×C31—82no—
73C2×C8×C31—42no—
74C2×C8×C31—42no—
75C4⋊C4×C31—41no—
76C4⋊C4×C31—42no—
77C4⋊C4×C31—42no—
78C4⋊C4×C31—41no—
79C4⋊C4×C31—41no—
80C4⋊C4×C31—44no—
81C22⋊C4×C31—41no—
82C22⋊C4×C31—41no—
83C22⋊C4×C31—42no—
84C22⋊C4×C31—44no—
85C42×C31—21no—
86C42×C31—21no—
87#11780C3111yesmaximal, characteristic, direct product factor, Sylow
88C23×C31—21no—
89C23×C31—21no—
90C23×C31—21no—
91C23×C31—21no—
92C23×C31—21no—
93C23×C31—41no—
94C23×C31—41no—
95C23×C31—41no—
96C23×C31—41no—
97C23×C31#10412yes—
98C23×C31C2×C22⋊C411yescharacteristic
99D4×C31—44no—
100C2×C4×C31—21no—
101C2×C4×C31—21no—
102C2×C4×C31—22no—
103C2×C4×C31—21no—
104C2×C4×C31—21no—
105C2×C4×C31—21no—
106C2×C4×C31—42no—
107C2×C4×C31—42no—
108C2×C4×C31—42no—
109C2×C4×C31—41no—
110C2×C4×C31—41no—
111C2×C4×C31—41no—
112C2×C4×C31—42no—
113C2×C4×C31—42no—
114C2×C4×C31—42no—
115C2×C4×C31—42no—
116C2×C4×C31—42no—
117C2×C4×C31—42no—
118C2×C4×C31—81no—
119C2×C4×C31—84no—
120C2×C4×C31#9911yescharacteristic
121C2×C4×C31#9911yescharacteristic
122C2×C4×C31AGL(1,ℤ/8)12yes—
123C8×C31—44no—
124C8×C31—82no—
125#2486C2×C3111yescharacteristic
126C4×#100C2×C3111yescharacteristic
127#1465C2×C3111yescharacteristic
128#1346C2×C3114yescomplemented
129C22×C31—21no—
130C22×C31—21no—
131C22×C31—22no—
132C22×C31—42no—
133C22×C31—42no—
134C22×C31—42no—
135C22×C31—41no—
136C22×C31—41no—
137C22×C31—41no—
138C22×C31—41no—
139C22×C31—41no—
140C22×C31—81no—
141C22×C31C2×#9911yescharacteristic
142C22×C31#41811yescharacteristic
143C22×C31C2×#10411yescharacteristic
144C4×C31—21no—
145C4×C31—22no—
146C4×C31—41no—
147C4×C31—41no—
148C4×C31—41no—
149C4×C31—82no—
150C4×C31—82no—
151C4×C31#35212yes—
152C2×#123—21no—
153C2×C4×D4C22×C3111yescharacteristic
154C22×C4⋊C4C4×C3111yescharacteristic
155C2×#100—21no—
156#407—21no—
157C4×M16—21no—
158#398C22×C3112yes—
159#397C4×C3112yes—
160#389—21no—
161#381—21no—
162#380—21no—
163#380C4×C3111yescharacteristic
164#343C22×C3112yes—
165#324C22×C3112yes—
166#323—21no—
167C2×C31—21no—
168C2×C31—21no—
169C2×C31—42no—
170C2×C31—81no—
171C2×C31#150711yescharacteristic
172C2×C31C2×#35211yescharacteristic
173C2×C31#213511yescharacteristic
174C22×D4D4×C3111yescharacteristic
175C23×C4—21no—
176C23×C4C2×C4×C3111yescharacteristic
177C23×C4D4×C3111yescharacteristic
178C2×M16—22no—
179C2×M16—41no—
180#123—44no—
181C4×D4—41no—
182C4×D4—41no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4C2×C4×C3112yes—
189C2×C4⋊C4C23×C3111yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4C2×C4×C3111yescharacteristic
193C2×C22⋊C4D4×C3111yescharacteristic
194C2×C42D4×C3111yescharacteristic
195#100—24no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198#97—41no—
199C4×C8—41no—
200#95—22no—
201#95—22no—
202#95C2×C4×C3112yes—
203C31#1178011yescharacteristic, direct product factor, Sylow
204C24C22⋊C4×C3111yescharacteristic
205C24C22⋊C4×C3111yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4C22⋊C4×C3111yescharacteristic
221C22×C4C22⋊C4×C3111yescharacteristic
222C22×C4C22×C4×C3111yescharacteristic
223C22×C4C2×D4×C3111yescharacteristic
224C22×C4C2×D4×C3111yescharacteristic
225M16—24no—
226M16—42no—
227M16—82no—
228C2×C8—42no—
229C2×C8—42no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—44no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—44no—
240C42—21no—
241C42—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23C31×#10412yes—
252C23C2×C22⋊C4×C3111yescharacteristic
253D4—44no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—84no—
274C2×C4C31×#9911yescharacteristic
275C2×C4C31×#9911yescharacteristic
276C2×C4C31×AGL(1,ℤ/8)12yes—
277C8—44no—
278C8—82no—
279C22—21no—
280C22—21no—
281C22—22no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—81no—
291C22C2×C31×#9911yescharacteristic
292C22C31×#41811yescharacteristic
293C22C2×C31×#10411yescharacteristic
294C4—21no—
295C4—22no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—82no—
300C4—82no—
301C4C31×#35212yes—
302C2—21no—
303C2—21no—
304C2—42no—
305C2—81no—
306C2C31×#150711yescharacteristic
307C2C2×C31×#35211yescharacteristic
308C2C31×#213511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1346C2semidirect
87. #11780C31direct (N × H)
128. #1346C2×C31semidirect
203. C31#11780direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#24862
C4×C31×#1002
C31×#14652
C31×#13462
#1178031