Finite Group Databasebeta

← C31×#8531 #21465290

Subgroups — C31×#8531 #21465290

Order 7936.

728 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 150 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×#3550C211yesmaximal, characteristic
2C31×#1994C212yesmaximal, complemented
3C31×#1962C211yesmaximal, characteristic
4C31×#1561C212yesmaximal, complemented
5C31×#1560C211yesmaximal, characteristic
6C31×#563C414yescomplemented
7C31×#559—21no—
8Q82×C31—21no—
9C31×#556—22no—
10D4×Q8×C31—22no—
11C31×#542—24no—
12C31×#536—22no—
13C31×#533C2212yes—
14C31×#532C2211yescharacteristic
15C31×#423C2212yes—
16C31×#423C2211yescharacteristic
17C2×C31×#104—22no—
18C43×C31C2211yescharacteristic
19C31×#339—24no—
20C31×#333—21no—
21C31×#332—22no—
22C31×#332—22no—
23C31×#331—21no—
24C2×#41×C31D412yes—
25C22×Q8×C31—22no—
26C2×M16×C31—21no—
27C2×M16×C31—22no—
28C31×C4⋊Q8—24no—
29C31×C4⋊Q8—24no—
30C31×C4⋊Q8D411yescharacteristic
31C31×C4⋊Q8D412yes—
32C31×C4⋊Q8D412yes—
33C31×#127D411yescharacteristic
34C31×#126—28no—
35C31×#125C2×C414yes—
36C31×#124—24no—
37C31×#124—24no—
38C31×#124C2×C412yes—
39C31×#123—24no—
40C31×#123—24no—
41C31×C22⋊Q8—24no—
42C31×C22⋊Q8—22no—
43C31×C22⋊Q8—22no—
44C31×C22⋊Q8—28no—
45C31×#121—22no—
46C4×Q8×C31—24no—
47C4×Q8×C31—22no—
48C4×Q8×C31—22no—
49C4×D4×C31—24no—
50C4×D4×C31—22no—
51C31×#117—22no—
52C31×#117D414yes—
53C2×C42×C31—21no—
54C2×C42×C31—22no—
55C2×C42×C31C2311yescharacteristic
56C31×C4⋊C8—22no—
57C31×C4⋊C8—22no—
58C31×C4⋊C8—22no—
59C31×#104—44no—
60#41×C31—42no—
61#41×C31—42no—
62C2×Q8×C31—24no—
63C2×Q8×C31—22no—
64C2×Q8×C31—22no—
65C2×Q8×C31—82no—
66C2×Q8×C31—82no—
67C2×Q8×C31C22⋊C412yes—
68C2×D4×C31—22no—
69C2×D4×C31—22no—
70C2×D4×C31C22⋊C412yes—
71C22×C4×C31—24no—
72C22×C4×C31—22no—
73C22×C4×C31—41no—
74C22×C4×C31C2×D412yes—
75C22×C4×C31C2×D411yescharacteristic
76M16×C31—42no—
77M16×C31—44no—
78C2×C8×C31—41no—
79C2×C8×C31—42no—
80C4⋊C4×C31—28no—
81C4⋊C4×C31—24no—
82C4⋊C4×C31—24no—
83C4⋊C4×C31—22no—
84C4⋊C4×C31—22no—
85C4⋊C4×C31—22no—
86C4⋊C4×C31C22⋊C414yes—
87C4⋊C4×C31C22⋊C414yes—
88C22⋊C4×C31—24no—
89C22⋊C4×C31—28no—
90C22⋊C4×C31—24no—
91C22⋊C4×C31—24no—
92C42×C31—24no—
93C42×C31—24no—
94C42×C31—22no—
95C42×C31—24no—
96C42×C31—42no—
97C42×C31—41no—
98C42×C31C22×C411yescharacteristic
99C42×C31C2×D412yes—
100C42×C31C2×D411yescharacteristic
101#8531C3111yesmaximal, characteristic, direct product factor, Sylow
102C23×C31—22no—
103C23×C31#12011yescharacteristic
104Q8×C31—42no—
105Q8×C31—42no—
106Q8×C31—42no—
107Q8×C31—42no—
108Q8×C31—42no—
109Q8×C31—84no—
110D4×C31—42no—
111D4×C31—42no—
112D4×C31—42no—
113D4×C31—82no—
114C2×C4×C31—21no—
115C2×C4×C31—22no—
116C2×C4×C31—21no—
117C2×C4×C31—22no—
118C2×C4×C31—24no—
119C2×C4×C31—22no—
120C2×C4×C31—24no—
121C2×C4×C31—42no—
122C2×C4×C31—44no—
123C2×C4×C31—42no—
124C2×C4×C31—42no—
125C2×C4×C31—84no—
126C2×C4×C31C2×C22⋊C412yes—
127C2×C4×C31C2×C22⋊C411yescharacteristic
128C2×C4×C31#12011yescharacteristic
129C2×C4×C31#12012yes—
130C8×C31—81no—
131C8×C31—82no—
132#3550C2×C3111yescharacteristic
133#1994C2×C3112yescomplemented
134#1962C2×C3111yescharacteristic
135#1561C2×C3112yescomplemented
136#1560C2×C3111yescharacteristic
137C22×C31—21no—
138C22×C31—22no—
139C22×C31—42no—
140C22×C31—44no—
141C22×C31#37911yescharacteristic
142C4×C31—21no—
143C4×C31—22no—
144C4×C31—44no—
145C4×C31—42no—
146C4×C31—44no—
147C4×C31—44no—
148C4×C31#45311yescharacteristic
149C4×C31#45412yes—
150C4×C31#45511yescharacteristic
151C4×C31#45612yes—
152#563C4×C3114yescomplemented
153#559—21no—
154Q82—21no—
155#556—22no—
156D4×Q8—22no—
157#542—24no—
158#536—22no—
159#533C22×C3112yes—
160#532C22×C3111yescharacteristic
161#423C22×C3112yes—
162#423C22×C3111yescharacteristic
163C2×#104—22no—
164C43C22×C3111yescharacteristic
165#339—24no—
166#333—21no—
167#332—22no—
168#332—22no—
169#331—21no—
170C2×C31—41no—
171C2×C31—82no—
172C2×C31#179211yescharacteristic
173C2×C31#179312yes—
174C2×#41D4×C3112yes—
175C22×Q8—22no—
176C2×M16—22no—
177C2×M16—21no—
178C4⋊Q8—24no—
179C4⋊Q8—24no—
180C4⋊Q8D4×C3111yescharacteristic
181C4⋊Q8D4×C3112yes—
182C4⋊Q8D4×C3112yes—
183#127D4×C3111yescharacteristic
184#126—28no—
185#125C2×C4×C3114yes—
186#124—24no—
187#124—24no—
188#124C2×C4×C3112yes—
189#123—24no—
190#123—24no—
191C22⋊Q8—24no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—28no—
195#121—22no—
196C4×Q8—24no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×D4—24no—
200C4×D4—22no—
201#117—22no—
202#117D4×C3114yes—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42C23×C3111yescharacteristic
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209#104—44no—
210C31#853111yescharacteristic, direct product factor, Sylow
211#41—42no—
212#41—42no—
213C2×Q8—24no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—82no—
217C2×Q8—82no—
218C2×Q8C22⋊C4×C3112yes—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4C22⋊C4×C3112yes—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4C2×D4×C3112yes—
226C22×C4C2×D4×C3111yescharacteristic
227M16—42no—
228M16—44no—
229C2×C8—41no—
230C2×C8—42no—
231C4⋊C4—28no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4C22⋊C4×C3114yes—
238C4⋊C4C22⋊C4×C3114yes—
239C22⋊C4—24no—
240C22⋊C4—28no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C42—24no—
244C42—24no—
245C42—22no—
246C42—24no—
247C42—42no—
248C42—41no—
249C42C22×C4×C3111yescharacteristic
250C42C2×D4×C3112yes—
251C42C2×D4×C3111yescharacteristic
252C23—22no—
253C23C31×#12011yescharacteristic
254Q8—42no—
255Q8—42no—
256Q8—42no—
257Q8—42no—
258Q8—42no—
259Q8—84no—
260D4—42no—
261D4—42no—
262D4—42no—
263D4—82no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—84no—
276C2×C4C2×C22⋊C4×C3112yes—
277C2×C4C2×C22⋊C4×C3111yescharacteristic
278C2×C4C31×#12011yescharacteristic
279C2×C4C31×#12012yes—
280C8—81no—
281C8—82no—
282C22—21no—
283C22—22no—
284C22—42no—
285C22—44no—
286C22C31×#37911yescharacteristic
287C4—21no—
288C4—22no—
289C4—44no—
290C4—42no—
291C4—44no—
292C4—44no—
293C4C31×#45311yescharacteristic
294C4C31×#45412yes—
295C4C31×#45511yescharacteristic
296C4C31×#45612yes—
297C2—41no—
298C2—82no—
299C2C31×#179211yescharacteristic
300C2C31×#179312yes—
subgroupG/H
center, socleC22×C31#379
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1994C2semidirect
4. C31×#1561C2semidirect
6. C31×#563C4semidirect
101. #8531C31direct (N × H)
133. #1994C2×C31semidirect
135. #1561C2×C31semidirect
152. #563C4×C31semidirect
210. C31#8531direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35502
C31×#19942
C31×#19622
C31×#15612
C31×#15602
#853131