Finite Group Databasebeta

← C31×#11817 #21468576

Subgroups — C31×#11817 #21468576

Order 7936.

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

normal subgroups only

This group has 386 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×#2627C211yesmaximal, characteristic
2C31×#1821C211yesmaximal, characteristic
3C31×#1463C211yesmaximal, characteristic
4C31×#1354C212yesmaximal, complemented
5C31×#1353C212yesmaximal, complemented
6C2×C31×C22⋊Q8C411yescharacteristic
7C22×C4⋊C4×C31C2211yescharacteristic
8C31×#407—21no—
9C31×#407—21no—
10C31×#400C411yescharacteristic
11C31×#400C2211yescharacteristic
12C31×#398—21no—
13C31×#398—21no—
14C31×#395C411yescharacteristic
15C31×#395C2211yescharacteristic
16C31×#385—21no—
17C31×#381—21no—
18C31×#380C411yescharacteristic
19C31×#343—21no—
20C31×#343C2212yes—
21C31×#324—21no—
22C31×#324C2212yes—
23C22×Q8×C31C2×C411yescharacteristic
24C23×C4×C31C2×C411yescharacteristic
25C23×C4×C31D411yescharacteristic
26C23×C4×C31D411yescharacteristic
27C2×M16×C31—22no—
28C2×M16×C31—41no—
29C31×C22⋊Q8—41no—
30C31×C22⋊Q8—41no—
31C2×C4⋊C4×C31—21no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31—41no—
36C2×C4⋊C4×C31—41no—
37C2×C4⋊C4×C31C2×C411yescharacteristic
38C2×C4⋊C4×C31C2×C411yescharacteristic
39C2×C4⋊C4×C31D411yescharacteristic
40C2×C4⋊C4×C31D411yescharacteristic
41C2×C4⋊C4×C31C2311yescharacteristic
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C31×C22⋊C8—22no—
45C31×C22⋊C8—41no—
46C31×C22⋊C8—41no—
47C31×C22⋊C8—41no—
48C31×#95—21no—
49C31×#95—21no—
50C31×#95—21no—
51C31×#95—21no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95C2×C411yescharacteristic
55C31×#95C2×C411yescharacteristic
56C24×C31C22⋊C411yescharacteristic
57C2×Q8×C31—41no—
58C2×Q8×C31—81no—
59C22×C4×C31—21no—
60C22×C4×C31—21no—
61C22×C4×C31—21no—
62C22×C4×C31—21no—
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—22no—
66C22×C4×C31—21no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—41no—
71C22×C4×C31—81no—
72C22×C4×C31—81no—
73C22×C4×C31C22⋊C411yescharacteristic
74C22×C4×C31C22⋊C411yescharacteristic
75C22×C4×C31C22⋊C411yescharacteristic
76C22×C4×C31C22×C411yescharacteristic
77C22×C4×C31C2×D411yescharacteristic
78C22×C4×C31C2×D411yescharacteristic
79M16×C31—42no—
80M16×C31—42no—
81M16×C31—82no—
82C2×C8×C31—42no—
83C2×C8×C31—42no—
84C4⋊C4×C31—41no—
85C4⋊C4×C31—41no—
86C4⋊C4×C31—41no—
87C4⋊C4×C31—41no—
88C4⋊C4×C31—42no—
89C4⋊C4×C31—41no—
90C4⋊C4×C31—81no—
91C4⋊C4×C31—81no—
92C4⋊C4×C31—81no—
93C22⋊C4×C31—42no—
94C22⋊C4×C31—41no—
95C22⋊C4×C31—41no—
96#11817C3111yesmaximal, characteristic, direct product factor, Sylow
97C23×C31—21no—
98C23×C31—21no—
99C23×C31—41no—
100C23×C31—41no—
101C23×C31#9911yescharacteristic
102C23×C31#9911yescharacteristic
103C23×C31C2×C22⋊C411yescharacteristic
104Q8×C31—82no—
105C2×C4×C31—21no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
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—42no—
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—42no—
128C2×C4×C31—41no—
129C2×C4×C31—41no—
130C2×C4×C31—41no—
131C2×C4×C31—41no—
132C2×C4×C31—81no—
133C2×C4×C31—81no—
134C2×C4×C31—81no—
135C2×C4×C31#13712yes—
136C8×C31—82no—
137C8×C31—82no—
138#2627C2×C3111yescharacteristic
139#1821C2×C3111yescharacteristic
140#1463C2×C3111yescharacteristic
141#1354C2×C3112yescomplemented
142#1353C2×C3112yescomplemented
143C22×C31—21no—
144C22×C31—21no—
145C22×C31—21no—
146C22×C31—21no—
147C22×C31—21no—
148C22×C31—21no—
149C22×C31—21no—
150C22×C31—21no—
151C22×C31—42no—
152C22×C31—42no—
153C22×C31C2×#9911yescharacteristic
154C22×C31#41911yescharacteristic
155C22×C31#42111yescharacteristic
156C4×C31—22no—
157C4×C31—41no—
158C4×C31—41no—
159C4×C31—41no—
160C4×C31—41no—
161C4×C31—41no—
162C4×C31—81no—
163C4×C31—81no—
164C4×C31—81no—
165C4×C31—81no—
166C2×C22⋊Q8C4×C3111yescharacteristic
167C22×C4⋊C4C22×C3111yescharacteristic
168#407—21no—
169#407—21no—
170#400C4×C3111yescharacteristic
171#400C22×C3111yescharacteristic
172#398—21no—
173#398—21no—
174#395C4×C3111yescharacteristic
175#395C22×C3111yescharacteristic
176#385—21no—
177#381—21no—
178#380C4×C3111yescharacteristic
179#343—21no—
180#343C22×C3112yes—
181#324—21no—
182#324C22×C3112yes—
183C2×C31—21no—
184C2×C31—21no—
185C2×C31—41no—
186C2×C31—41no—
187C2×C31#151111yescharacteristic
188C2×C31#212111yescharacteristic
189C2×C31#213311yescharacteristic
190C22×Q8C2×C4×C3111yescharacteristic
191C23×C4C2×C4×C3111yescharacteristic
192C23×C4D4×C3111yescharacteristic
193C23×C4D4×C3111yescharacteristic
194C2×M16—22no—
195C2×M16—41no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4C2×C4×C3111yescharacteristic
205C2×C4⋊C4C2×C4×C3111yescharacteristic
206C2×C4⋊C4D4×C3111yescharacteristic
207C2×C4⋊C4D4×C3111yescharacteristic
208C2×C4⋊C4C23×C3111yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C22⋊C8—22no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95C2×C4×C3111yescharacteristic
222#95C2×C4×C3111yescharacteristic
223C31#1181711yescharacteristic, direct product factor, Sylow
224C24C22⋊C4×C3111yescharacteristic
225C2×Q8—41no—
226C2×Q8—81no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4C22⋊C4×C3111yescharacteristic
242C22×C4C22⋊C4×C3111yescharacteristic
243C22×C4C22⋊C4×C3111yescharacteristic
244C22×C4C22×C4×C3111yescharacteristic
245C22×C4C2×D4×C3111yescharacteristic
246C22×C4C2×D4×C3111yescharacteristic
247M16—42no—
248M16—42no—
249M16—82no—
250C2×C8—42no—
251C2×C8—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—81no—
259C4⋊C4—81no—
260C4⋊C4—81no—
261C22⋊C4—42no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23C31×#9911yescharacteristic
269C23C31×#9911yescharacteristic
270C23C2×C22⋊C4×C3111yescharacteristic
271Q8—82no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
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—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4C31×#13712yes—
303C8—82no—
304C8—82no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×C31×#9911yescharacteristic
316C22C31×#41911yescharacteristic
317C22C31×#42111yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2C31×#151111yescharacteristic
333C2C31×#212111yescharacteristic
334C2C31×#213311yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC24×C31C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C24×C31 ◁ C22×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1354C2semidirect
5. C31×#1353C2semidirect
96. #11817C31direct (N × H)
141. #1354C2×C31semidirect
142. #1353C2×C31semidirect
223. C31#11817direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26272
C31×#18212
C31×#14632
C31×#13542
C31×#13532
#1181731