Finite Group Databasebeta

← C31×#9451 #21466210

Subgroups — C31×#9451 #21466210

Order 7936.

518 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2973C211yesmaximal, characteristic
2C31×#2547C211yesmaximal, characteristic
3C31×#1558C212yesmaximal, complemented
4C31×#1466C211yesmaximal, characteristic
5C31×#1285C212yesmaximal, complemented
6C2×C31×C4⋊Q8C2211yescharacteristic
7C2×C31×#124C412yes—
8C2×C4×D4×C31C2211yescharacteristic
9C31×#435—24no—
10C31×#432C2212yes—
11C31×#432C2212yes—
12C31×#407—21no—
13C2×C31×C22⋊C8—21no—
14C31×#397—21no—
15C31×#397—21no—
16C31×#391C412yes—
17C31×#380—21no—
18C31×#380—21no—
19C4×C4⋊C4×C31C2211yescharacteristic
20C31×#333—24no—
21C31×#323—21no—
22C31×#323—21no—
23C22×Q8×C31C2×C412yes—
24C22×D4×C31C2×C411yescharacteristic
25C23×C4×C31D411yescharacteristic
26C23×C4×C31D411yescharacteristic
27C2×M16×C31—22no—
28C22×C8×C31—22no—
29C31×C4⋊Q8—22no—
30C31×C4⋊Q8—22no—
31C31×#124—44no—
32C4×D4×C31—22no—
33C4×D4×C31—22no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31C2×C411yescharacteristic
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31D411yescharacteristic
41C2×C22⋊C4×C31D411yescharacteristic
42C2×C42×C31C2×C412yes—
43C2×C42×C31C2311yescharacteristic
44C31×C4⋊C8—22no—
45C31×C4⋊C8D414yes—
46C31×C22⋊C8—22no—
47C31×C22⋊C8—22no—
48C31×C22⋊C8—22no—
49C31×C22⋊C8—22no—
50C31×#97—22no—
51C31×#97—22no—
52C31×#95—22no—
53C31×#95—22no—
54C31×#95—21no—
55C24×C31C22⋊C411yescharacteristic
56C24×C31C22⋊C411yescharacteristic
57C2×Q8×C31—24no—
58C2×Q8×C31—42no—
59C2×Q8×C31—42no—
60C2×D4×C31—22no—
61C2×D4×C31—22no—
62C2×D4×C31—41no—
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—21no—
66C22×C4×C31—21no—
67C22×C4×C31—22no—
68C22×C4×C31—22no—
69C22×C4×C31—42no—
70C22×C4×C31—42no—
71C22×C4×C31C22⋊C411yescharacteristic
72C22×C4×C31C22⋊C411yescharacteristic
73C22×C4×C31C22×C411yescharacteristic
74C22×C4×C31C2×D411yescharacteristic
75C22×C4×C31C2×D411yescharacteristic
76M16×C31—82no—
77C2×C8×C31—24no—
78C2×C8×C31—42no—
79C2×C8×C31—82no—
80C4⋊C4×C31—22no—
81C4⋊C4×C31—42no—
82C4⋊C4×C31—41no—
83C4⋊C4×C31—42no—
84C4⋊C4×C31—41no—
85C22⋊C4×C31—22no—
86C22⋊C4×C31—22no—
87C22⋊C4×C31—44no—
88C22⋊C4×C31—44no—
89C42×C31—42no—
90C42×C31C2×D412yes—
91C42×C31#4112yes—
92#9451C3111yesmaximal, characteristic, direct product factor, Sylow
93C23×C31—21no—
94C23×C31—21no—
95C23×C31—21no—
96C23×C31—21no—
97C23×C31—21no—
98C23×C31—41no—
99C23×C31—41no—
100C23×C31—41no—
101C23×C31—41no—
102C23×C31#10012yes—
103C23×C31C2×C22⋊C411yescharacteristic
104Q8×C31—84no—
105D4×C31—44no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—22no—
109C2×C4×C31—21no—
110C2×C4×C31—21no—
111C2×C4×C31—41no—
112C2×C4×C31—42no—
113C2×C4×C31—42no—
114C2×C4×C31—41no—
115C2×C4×C31—41no—
116C2×C4×C31—42no—
117C2×C4×C31—42no—
118C2×C4×C31—42no—
119C2×C4×C31—42no—
120C2×C4×C31—42no—
121C2×C4×C31—42no—
122C2×C4×C31—81no—
123C2×C4×C31—81no—
124C2×C4×C31C4×D412yes—
125C2×C4×C31#12011yescharacteristic
126C2×C4×C31#12112yes—
127C2×C4×C31#12311yescharacteristic
128C8×C31—44no—
129C8×C31—82no—
130#2973C2×C3111yescharacteristic
131#2547C2×C3111yescharacteristic
132#1558C2×C3112yescomplemented
133#1466C2×C3111yescharacteristic
134#1285C2×C3112yescomplemented
135C22×C31—21no—
136C22×C31—21no—
137C22×C31—22no—
138C22×C31—42no—
139C22×C31—42no—
140C22×C31—41no—
141C22×C31—41no—
142C22×C31—41no—
143C22×C31—41no—
144C22×C31—41no—
145C22×C31—81no—
146C22×C31—81no—
147C22×C31#38611yescharacteristic
148C22×C31C2×#10011yescharacteristic
149C22×C31#41311yescharacteristic
150C4×C31—21no—
151C4×C31—22no—
152C4×C31—22no—
153C4×C31—41no—
154C4×C31—41no—
155C4×C31—82no—
156C4×C31—82no—
157C4×C31#45512yes—
158C2×C4⋊Q8C22×C3111yescharacteristic
159C2×#124C4×C3112yes—
160C2×C4×D4C22×C3111yescharacteristic
161#435—24no—
162#432C22×C3112yes—
163#432C22×C3112yes—
164#407—21no—
165C2×C22⋊C8—21no—
166#397—21no—
167#397—21no—
168#391C4×C3112yes—
169#380—21no—
170#380—21no—
171C4×C4⋊C4C22×C3111yescharacteristic
172#333—24no—
173#323—21no—
174#323—21no—
175C2×C31—21no—
176C2×C31—21no—
177C2×C31—42no—
178C2×C31—81no—
179C2×C31#188511yescharacteristic
180C2×C31#190011yescharacteristic
181C2×C31#190911yescharacteristic
182C22×Q8C2×C4×C3112yes—
183C22×D4C2×C4×C3111yescharacteristic
184C23×C4D4×C3111yescharacteristic
185C23×C4D4×C3111yescharacteristic
186C2×M16—22no—
187C22×C8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190#124—44no—
191C4×D4—22no—
192C4×D4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C2×C4×C3111yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4D4×C3111yescharacteristic
200C2×C22⋊C4D4×C3111yescharacteristic
201C2×C42C2×C4×C3112yes—
202C2×C42C23×C3111yescharacteristic
203C4⋊C8—22no—
204C4⋊C8D4×C3114yes—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209#97—22no—
210#97—22no—
211#95—22no—
212#95—22no—
213#95—21no—
214C31#945111yescharacteristic, direct product factor, Sylow
215C24C22⋊C4×C3111yescharacteristic
216C24C22⋊C4×C3111yescharacteristic
217C2×Q8—24no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C22⋊C4×C3111yescharacteristic
232C22×C4C22⋊C4×C3111yescharacteristic
233C22×C4C22×C4×C3111yescharacteristic
234C22×C4C2×D4×C3111yescharacteristic
235C22×C4C2×D4×C3111yescharacteristic
236M16—82no—
237C2×C8—24no—
238C2×C8—42no—
239C2×C8—82no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C42—42no—
250C42C2×D4×C3112yes—
251C42#41×C3112yes—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23C31×#10012yes—
262C23C2×C22⋊C4×C3111yescharacteristic
263Q8—84no—
264D4—44no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4C4×D4×C3112yes—
284C2×C4C31×#12011yescharacteristic
285C2×C4C31×#12112yes—
286C2×C4C31×#12311yescharacteristic
287C8—44no—
288C8—82no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22C31×#38611yescharacteristic
302C22C2×C31×#10011yescharacteristic
303C22C31×#41311yescharacteristic
304C4—21no—
305C4—22no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—82no—
310C4—82no—
311C4C31×#45512yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2C31×#188511yescharacteristic
317C2C31×#190011yescharacteristic
318C2C31×#190911yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1558C2semidirect
5. C31×#1285C2semidirect
92. #9451C31direct (N × H)
132. #1558C2×C31semidirect
134. #1285C2×C31semidirect
214. C31#9451direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#29732
C31×#25472
C31×#15582
C31×#14662
C31×#12852
#945131