Finite Group Databasebeta

← C31×#11810 #21468569

Subgroups — C31×#11810 #21468569

Order 7936.

402 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2396C211yesmaximal, characteristic
2C31×#1820C211yesmaximal, characteristic
3C31×#1464C211yesmaximal, characteristic
4C31×#1352C212yesmaximal
5C31×#1351C212yesmaximal
6C2×C4×Q8×C31C411yescharacteristic, complemented
7C2×C31×#117C2211yescharacteristic
8C31×#432—21no—
9C31×#400C411yescharacteristic, complemented
10C31×#399C2211yescharacteristic
11C31×#398—21no—
12C31×#396—21no—
13C31×#395C411yescharacteristic, complemented
14C31×#393C2211yescharacteristic
15C31×#381—21no—
16C4×C4⋊C4×C31—21no—
17C4×C4⋊C4×C31C411yescharacteristic, complemented
18C4×C22⋊C4×C31—21no—
19C31×#342—21no—
20C31×#342C2212yes—
21C31×#324—21no—
22C31×#323C2212yes—
23C22×Q8×C31C2×C411yescharacteristic
24C23×C4×C31D411yescharacteristic
25C4×Q8×C31—41no—
26C4×Q8×C31—41no—
27C31×#117—41no—
28C31×#117—41no—
29C2×C4⋊C4×C31—21no—
30C2×C4⋊C4×C31—21no—
31C2×C4⋊C4×C31—21no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31C2×C411yescharacteristic
34C2×C4⋊C4×C31C2×C411yescharacteristic
35C2×C4⋊C4×C31D411yescharacteristic
36C2×C22⋊C4×C31—21no—
37C2×C22⋊C4×C31—21no—
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31D411yescharacteristic
40C2×C22⋊C4×C31C2311yescharacteristic
41C2×C42×C31—21no—
42C2×C42×C31—21no—
43C2×C42×C31C2×C411yescharacteristic
44C2×C42×C31D411yescharacteristic
45C31×C4⋊C8—41no—
46C31×C22⋊C8—22no—
47C31×#97—41no—
48C31×#95—21no—
49C31×#95—21no—
50C31×#95—21no—
51C31×#95—21no—
52C31×#95—41no—
53C31×#95—41no—
54C31×#95C2×C411yescharacteristic
55C31×#95C2×C411yescharacteristic
56C24×C31C2×D411yescharacteristic
57C2×Q8×C31—42no—
58C2×Q8×C31—41no—
59C22×C4×C31—21no—
60C22×C4×C31—21no—
61C22×C4×C31—22no—
62C22×C4×C31—21no—
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—21no—
66C22×C4×C31—41no—
67C22×C4×C31—42no—
68C22×C4×C31C22⋊C411yescharacteristic
69C22×C4×C31C22⋊C411yescharacteristic
70C22×C4×C31C22⋊C411yescharacteristic
71C22×C4×C31C22⋊C411yescharacteristic
72C22×C4×C31C22×C411yescharacteristic
73C22×C4×C31C2×D411yescharacteristic
74C2×C8×C31—42no—
75C4⋊C4×C31—41no—
76C4⋊C4×C31—41no—
77C4⋊C4×C31—42no—
78C4⋊C4×C31—42no—
79C4⋊C4×C31—42no—
80C4⋊C4×C31—42no—
81C4⋊C4×C31—41no—
82C22⋊C4×C31—41no—
83C22⋊C4×C31—41no—
84C22⋊C4×C31—42no—
85C22⋊C4×C31—42no—
86C22⋊C4×C31—42no—
87C22⋊C4×C31—42no—
88C42×C31—21no—
89C42×C31—21no—
90C42×C31—21no—
91C42×C31—21no—
92C42×C31—42no—
93C42×C31—41no—
94C42×C31—41no—
95#11810C3111yesmaximal, characteristic, direct product factor, Sylow
96C23×C31—21no—
97C23×C31—21no—
98C23×C31—41no—
99C23×C31—41no—
100C23×C31C2×C22⋊C411yescharacteristic
101C23×C31#13712yes—
102Q8×C31—42no—
103Q8×C31—42no—
104C2×C4×C31—21no—
105C2×C4×C31—21no—
106C2×C4×C31—22no—
107C2×C4×C31—22no—
108C2×C4×C31—21no—
109C2×C4×C31—21no—
110C2×C4×C31—21no—
111C2×C4×C31—21no—
112C2×C4×C31—21no—
113C2×C4×C31—21no—
114C2×C4×C31—21no—
115C2×C4×C31—42no—
116C2×C4×C31—42no—
117C2×C4×C31—41no—
118C2×C4×C31—41no—
119C2×C4×C31—41no—
120C2×C4×C31—41no—
121C2×C4×C31—41no—
122C2×C4×C31—41no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
127C2×C4×C31—41no—
128C2×C4×C31—41no—
129C2×C4×C31—41no—
130C2×C4×C31—41no—
131C2×C4×C31—82no—
132C2×C4×C31—82no—
133C2×C4×C31—82no—
134C2×C4×C31#9911yescharacteristic
135C2×C4×C31#9911yescharacteristic
136C8×C31—82no—
137#2396C2×C3111yescharacteristic
138#1820C2×C3111yescharacteristic
139#1464C2×C3111yescharacteristic
140#1352C2×C3112yes—
141#1351C2×C3112yes—
142C22×C31—21no—
143C22×C31—21no—
144C22×C31—22no—
145C22×C31—42no—
146C22×C31—41no—
147C22×C31—41no—
148C22×C31—81no—
149C22×C31C2×#9911yescharacteristic
150C22×C31#41911yescharacteristic
151C22×C31#42111yescharacteristic
152C4×C31—21no—
153C4×C31—21no—
154C4×C31—42no—
155C4×C31—41no—
156C4×C31—42no—
157C4×C31—41no—
158C4×C31—41no—
159C4×C31—41no—
160C4×C31—41no—
161C4×C31—81no—
162C4×C31—81no—
163C4×C31—162no—
164C2×C4×Q8C4×C3111yescharacteristic, complemented
165C2×#117C22×C3111yescharacteristic
166#432—21no—
167#400C4×C3111yescharacteristic, complemented
168#399C22×C3111yescharacteristic
169#398—21no—
170#396—21no—
171#395C4×C3111yescharacteristic, complemented
172#393C22×C3111yescharacteristic
173#381—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4C4×C3111yescharacteristic, complemented
176C4×C22⋊C4—21no—
177#342—21no—
178#342C22×C3112yes—
179#324—21no—
180#323C22×C3112yes—
181C2×C31—21no—
182C2×C31—21no—
183C2×C31—42no—
184C2×C31#151111yescharacteristic
185C2×C31#212011yescharacteristic
186C2×C31#212611yescharacteristic
187C22×Q8C2×C4×C3111yescharacteristic
188C23×C4D4×C3111yescharacteristic
189C4×Q8—41no—
190C4×Q8—41no—
191#117—41no—
192#117—41no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×C4×C3111yescharacteristic
198C2×C4⋊C4C2×C4×C3111yescharacteristic
199C2×C4⋊C4D4×C3111yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4D4×C3111yescharacteristic
204C2×C22⋊C4C23×C3111yescharacteristic
205C2×C42—21no—
206C2×C42—21no—
207C2×C42C2×C4×C3111yescharacteristic
208C2×C42D4×C3111yescharacteristic
209C4⋊C8—41no—
210C22⋊C8—22no—
211#97—41no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—41no—
217#95—41no—
218#95C2×C4×C3111yescharacteristic
219#95C2×C4×C3111yescharacteristic
220C31#1181011yescharacteristic, direct product factor, Sylow
221C24C2×D4×C3111yescharacteristic
222C2×Q8—42no—
223C2×Q8—41no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4C22⋊C4×C3111yescharacteristic
234C22×C4C22⋊C4×C3111yescharacteristic
235C22×C4C22⋊C4×C3111yescharacteristic
236C22×C4C22⋊C4×C3111yescharacteristic
237C22×C4C22×C4×C3111yescharacteristic
238C22×C4C2×D4×C3111yescharacteristic
239C2×C8—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—42no—
258C42—41no—
259C42—41no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×C22⋊C4×C3111yescharacteristic
265C23C31×#13712yes—
266Q8—42no—
267Q8—42no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4C31×#9911yescharacteristic
299C2×C4C31×#9911yescharacteristic
300C8—82no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22C2×C31×#9911yescharacteristic
309C22C31×#41911yescharacteristic
310C22C31×#42111yescharacteristic
311C4—21no—
312C4—21no—
313C4—42no—
314C4—42no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—81no—
321C4—81no—
322C4—162no—
323C2—21no—
324C2—21no—
325C2—42no—
326C2C31×#151111yescharacteristic
327C2C31×#212011yescharacteristic
328C2C31×#212611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×C4×Q8×C31C4semidirect
9. C31×#400C4semidirect
13. C31×#395C4semidirect
17. C4×C4⋊C4×C31C4semidirect
95. #11810C31direct (N × H)
164. C2×C4×Q8C4×C31semidirect
167. #400C4×C31semidirect
171. #395C4×C31semidirect
175. C4×C4⋊C4C4×C31semidirect
220. C31#11810direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#23962
C31×#18202
C31×#14642
C31×#13522
C31×#13512
#1181031