Finite Group Databasebeta

← C31×#11901 #21468660

Subgroups — C31×#11901 #21468660

Order 7936.

508 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2542C211yesmaximal, characteristic
2C31×#1799C211yesmaximal, characteristic
3C31×#1459C211yesmaximal, characteristic
4C31×#1352C214yesmaximal, complemented
5C2×C31×C4⋊Q8C411yescharacteristic, complemented
6C2×C31×C22⋊Q8—21no—
7C2×C31×C22⋊Q8—21no—
8C2×C4×D4×C31C2211yescharacteristic
9C2×C31×#99—21no—
10C2×C31×C22⋊C8—21no—
11C31×#399C412yescomplemented
12C31×#399C2212yes—
13C31×#390—21no—
14C31×#384—21no—
15C31×#380—21no—
16C4×C4⋊C4×C31C411yescharacteristic, complemented
17C31×#342C2212yes—
18C31×#324—21no—
19C31×#323C2212yes—
20C22×Q8×C31—21no—
21C22×D4×C31D411yescharacteristic
22C23×C4×C31D411yescharacteristic
23C23×C4×C31D411yescharacteristic
24C22×C8×C31—41no—
25C31×C4⋊Q8—41no—
26C31×C4⋊Q8—41no—
27C31×C22⋊Q8—44no—
28C31×C22⋊Q8—44no—
29C4×D4×C31—41no—
30C4×D4×C31—41no—
31C2×C4⋊C4×C31—21no—
32C2×C4⋊C4×C31—22no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—22no—
35C2×C4⋊C4×C31—41no—
36C2×C4⋊C4×C31C2×C412yes—
37C2×C4⋊C4×C31D411yescharacteristic
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31—22no—
41C2×C22⋊C4×C31C2×C411yescharacteristic
42C2×C22⋊C4×C31C2311yescharacteristic
43C2×C42×C31—21no—
44C2×C42×C31C2×C411yescharacteristic
45C31×#99—42no—
46C31×C22⋊C8—22no—
47C31×C22⋊C8—41no—
48C31×#95—41no—
49C31×#95C2×C412yes—
50C24×C31C22⋊C411yescharacteristic
51C24×C31C2×D411yescharacteristic
52C2×Q8×C31—42no—
53C2×Q8×C31—42no—
54C2×Q8×C31—41no—
55C2×Q8×C31—41no—
56C2×D4×C31—41no—
57C2×D4×C31—41no—
58C2×D4×C31SD1612yes—
59C2×D4×C31Q1612yes—
60C22×C4×C31—22no—
61C22×C4×C31—21no—
62C22×C4×C31—21no—
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—22no—
66C22×C4×C31—42no—
67C22×C4×C31—42no—
68C22×C4×C31—42no—
69C22×C4×C31C22⋊C411yescharacteristic
70C22×C4×C31C22⋊C411yescharacteristic
71C22×C4×C31C22⋊C411yescharacteristic
72C22×C4×C31C22×C411yescharacteristic
73C22×C4×C31C2×D411yescharacteristic
74C2×C8×C31—42no—
75C2×C8×C31—82no—
76C4⋊C4×C31—24no—
77C4⋊C4×C31—41no—
78C4⋊C4×C31—44no—
79C4⋊C4×C31—44no—
80C4⋊C4×C31—42no—
81C4⋊C4×C31—42no—
82C4⋊C4×C31—82no—
83C22⋊C4×C31—41no—
84C22⋊C4×C31—41no—
85C22⋊C4×C31—44no—
86C22⋊C4×C31—44no—
87C22⋊C4×C31—42no—
88C22⋊C4×C31—42no—
89C42×C31—21no—
90C42×C31—21no—
91C42×C31—42no—
92#11901C3111yesmaximal, characteristic, direct product factor, Sylow
93C23×C31—21no—
94C23×C31—21no—
95C23×C31—21no—
96C23×C31—21no—
97C23×C31—41no—
98C23×C31—41no—
99C23×C31—41no—
100C23×C31—41no—
101C23×C31#10312yes—
102C23×C31C2×C22⋊C411yescharacteristic
103C23×C31C2×SD1611yescharacteristic
104C23×C31C2×Q1611yescharacteristic
105Q8×C31—82no—
106Q8×C31—82no—
107D4×C31—42no—
108D4×C31—42no—
109C2×C4×C31—21no—
110C2×C4×C31—21no—
111C2×C4×C31—22no—
112C2×C4×C31—24no—
113C2×C4×C31—21no—
114C2×C4×C31—21no—
115C2×C4×C31—21no—
116C2×C4×C31—42no—
117C2×C4×C31—42no—
118C2×C4×C31—42no—
119C2×C4×C31—42no—
120C2×C4×C31—41no—
121C2×C4×C31—41no—
122C2×C4×C31—41no—
123C2×C4×C31—42no—
124C2×C4×C31—42no—
125C2×C4×C31—42no—
126C2×C4×C31—81no—
127C2×C4×C31—81no—
128C2×C4×C31—82no—
129C2×C4×C31—82no—
130C2×C4×C31—82no—
131C2×C4×C31#9911yescharacteristic
132C2×C4×C31#9911yescharacteristic
133C2×C4×C31#10312yes—
134C8×C31—82no—
135#2542C2×C3111yescharacteristic
136#1799C2×C3111yescharacteristic
137#1459C2×C3111yescharacteristic
138#1352C2×C3114yescomplemented
139C22×C31—21no—
140C22×C31—21no—
141C22×C31—21no—
142C22×C31—22no—
143C22×C31—21no—
144C22×C31—42no—
145C22×C31—41no—
146C22×C31—41no—
147C22×C31—41no—
148C22×C31—41no—
149C22×C31—41no—
150C22×C31—41no—
151C22×C31—81no—
152C22×C31—81no—
153C22×C31C2×#9911yescharacteristic
154C22×C31C2×#10311yescharacteristic
155C22×C31#42111yescharacteristic
156C4×C31—21no—
157C4×C31—22no—
158C4×C31—44no—
159C4×C31—41no—
160C4×C31—41no—
161C4×C31—41no—
162C4×C31—82no—
163C4×C31—162no—
164C4×C31#35312yes—
165C2×C4⋊Q8C4×C3111yescharacteristic, complemented
166C2×C22⋊Q8—21no—
167C2×C22⋊Q8—21no—
168C2×C4×D4C22×C3111yescharacteristic
169C2×#99—21no—
170C2×C22⋊C8—21no—
171#399C4×C3112yescomplemented
172#399C22×C3112yes—
173#390—21no—
174#384—21no—
175#380—21no—
176C4×C4⋊C4C4×C3111yescharacteristic, complemented
177#342C22×C3112yes—
178#324—21no—
179#323C22×C3112yes—
180C2×C31—21no—
181C2×C31—21no—
182C2×C31—41no—
183C2×C31—42no—
184C2×C31—41no—
185C2×C31#152011yescharacteristic
186C2×C31#212311yescharacteristic
187C2×C31C2×#35311yescharacteristic
188C22×Q8—21no—
189C22×D4D4×C3111yescharacteristic
190C23×C4D4×C3111yescharacteristic
191C23×C4D4×C3111yescharacteristic
192C22×C8—41no—
193C4⋊Q8—41no—
194C4⋊Q8—41no—
195C22⋊Q8—44no—
196C22⋊Q8—44no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4C2×C4×C3112yes—
205C2×C4⋊C4D4×C3111yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4C2×C4×C3111yescharacteristic
210C2×C22⋊C4C23×C3111yescharacteristic
211C2×C42—21no—
212C2×C42C2×C4×C3111yescharacteristic
213#99—42no—
214C22⋊C8—22no—
215C22⋊C8—41no—
216#95—41no—
217#95C2×C4×C3112yes—
218C31#1190111yescharacteristic, direct product factor, Sylow
219C24C22⋊C4×C3111yescharacteristic
220C24C2×D4×C3111yescharacteristic
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4SD16×C3112yes—
228C2×D4Q16×C3112yes—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4C22⋊C4×C3111yescharacteristic
239C22×C4C22⋊C4×C3111yescharacteristic
240C22×C4C22⋊C4×C3111yescharacteristic
241C22×C4C22×C4×C3111yescharacteristic
242C22×C4C2×D4×C3111yescharacteristic
243C2×C8—42no—
244C2×C8—82no—
245C4⋊C4—24no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—82no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C42—21no—
259C42—21no—
260C42—42no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23C31×#10312yes—
270C23C2×C22⋊C4×C3111yescharacteristic
271C23C2×SD16×C3111yescharacteristic
272C23C2×Q16×C3111yescharacteristic
273Q8—82no—
274Q8—82no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4C31×#9911yescharacteristic
300C2×C4C31×#9911yescharacteristic
301C2×C4C31×#10312yes—
302C8—82no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22C2×C31×#9911yescharacteristic
318C22C2×C31×#10311yescharacteristic
319C22C31×#42111yescharacteristic
320C4—21no—
321C4—22no—
322C4—44no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—162no—
328C4C31×#35312yes—
329C2—21no—
330C2—21no—
331C2—41no—
332C2—42no—
333C2—41no—
334C2C31×#152011yescharacteristic
335C2C31×#212311yescharacteristic
336C2C2×C31×#35311yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊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×#1352C2semidirect
5. C2×C31×C4⋊Q8C4semidirect
11. C31×#399C4semidirect
16. C4×C4⋊C4×C31C4semidirect
92. #11901C31direct (N × H)
138. #1352C2×C31semidirect
165. C2×C4⋊Q8C4×C31semidirect
171. #399C4×C31semidirect
176. C4×C4⋊C4C4×C31semidirect
218. C31#11901direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25422
C31×#17992
C31×#14592
C31×#13522
#1190131