Finite Group Databasebeta

← C31×#8472 #21465231

Subgroups — C31×#8472 #21465231

Order 7936.

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

normal subgroups only

This group has 834 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×#3508C211yesmaximal, characteristic
2C31×#1965C212yesmaximal, complemented
3C31×#1561C211yesmaximal, characteristic, complemented
4C31×#1526C211yesmaximal, characteristic, complemented
5C31×#1489C212yesmaximal, complemented
6C31×#559—22no—
7C31×#550—22no—
8C31×#546—24no—
9C31×#540—22no—
10C31×#535C412yes—
11C31×#533C2211yescharacteristic
12C31×#532—22no—
13C2×C31×#127C2211yescharacteristic
14C31×#518C412yes—
15C4×#41×C31C2211yescharacteristic
16C31×#423C2212yes—
17C2×C31×#100—24no—
18C4×M16×C31C2212yes—
19C31×#332—22no—
20C31×#330—22no—
21C31×Q8⋊C8—22no—
22C31×D4⋊C8—22no—
23C2×#41×C31—24no—
24C2×#41×C31C2×C411yescharacteristic
25C22×D4×C31C2×C412yes—
26C22×D4×C31D414yes—
27C2×M16×C31—22no—
28C2×M16×C31—22no—
29C31×C4⋊Q8D412yes—
30C31×#127—22no—
31C31×#127—21no—
32C31×#127—21no—
33C31×#127—22no—
34C31×#125D412yes—
35C31×#124—24no—
36C31×#124—22no—
37C31×#123—28no—
38C31×C22⋊Q8—22no—
39C31×#121—24no—
40C31×#121—28no—
41C31×#121—24no—
42C31×#120—24no—
43C4×Q8×C31—22no—
44C4×Q8×C31D412yes—
45C4×D4×C31—24no—
46C4×D4×C31—22no—
47C4×D4×C31—24no—
48C4×D4×C31—24no—
49C4×D4×C31D412yes—
50C31×#117—21no—
51C31×#117C2×C411yescharacteristic
52C31×#117C2×C412yes—
53C2×C22⋊C4×C31—24no—
54C2×C42×C31—21no—
55C2×C42×C31C2311yescharacteristic
56C31×C4⋊C8—22no—
57C31×C4⋊C8—22no—
58C31×#100—48no—
59C31×#97—22no—
60C4×C8×C31—22no—
61C24×C31C22⋊C414yes—
62#41×C31—42no—
63#41×C31—84no—
64#41×C31—84no—
65C2×Q8×C31—22no—
66C2×Q8×C31C22⋊C411yescharacteristic
67C2×D4×C31—21no—
68C2×D4×C31—28no—
69C2×D4×C31—22no—
70C2×D4×C31—22no—
71C2×D4×C31—24no—
72C2×D4×C31—24no—
73C2×D4×C31—44no—
74C2×D4×C31—44no—
75C2×D4×C31—44no—
76C2×D4×C31C22⋊C411yescharacteristic
77C22×C4×C31—28no—
78C22×C4×C31—21no—
79C22×C4×C31—22no—
80C22×C4×C31C22×C411yescharacteristic
81C22×C4×C31C2×D412yes—
82M16×C31—24no—
83M16×C31—42no—
84M16×C31—44no—
85C2×C8×C31—42no—
86C2×C8×C31—42no—
87C4⋊C4×C31—22no—
88C4⋊C4×C31—22no—
89C4⋊C4×C31—24no—
90C4⋊C4×C31C22⋊C412yes—
91C4⋊C4×C31C22⋊C414yes—
92C22⋊C4×C31—28no—
93C22⋊C4×C31—28no—
94C22⋊C4×C31—21no—
95C22⋊C4×C31—22no—
96C22⋊C4×C31—22no—
97C42×C31—21no—
98C42×C31—22no—
99C42×C31—22no—
100C42×C31C2×D412yes—
101C42×C31C2×D411yescharacteristic
102C42×C31C2×D411yescharacteristic
103#8472C3111yesmaximal, characteristic, direct product factor, Sylow
104C23×C31—24no—
105C23×C31—21no—
106C23×C31—48no—
107C23×C31—44no—
108C23×C31C2×C22⋊C411yescharacteristic
109Q8×C31—22no—
110Q8×C31—42no—
111Q8×C31—42no—
112D4×C31—22no—
113D4×C31—44no—
114D4×C31—42no—
115D4×C31—44no—
116D4×C31—44no—
117D4×C31—44no—
118D4×C31—84no—
119D4×C31—88no—
120C2×C4×C31—22no—
121C2×C4×C31—22no—
122C2×C4×C31—22no—
123C2×C4×C31—21no—
124C2×C4×C31—22no—
125C2×C4×C31—24no—
126C2×C4×C31—42no—
127C2×C4×C31—42no—
128C2×C4×C31—42no—
129C2×C4×C31—88no—
130C2×C4×C31C2×C22⋊C411yescharacteristic
131C2×C4×C31C2×C22⋊C411yescharacteristic
132C2×C4×C31#12012yes—
133C2×C4×C31#12012yes—
134C8×C31—44no—
135C8×C31—82no—
136#3508C2×C3111yescharacteristic
137#1965C2×C3112yescomplemented
138#1561C2×C3111yescharacteristic, complemented
139#1526C2×C3111yescharacteristic, complemented
140#1489C2×C3112yescomplemented
141C22×C31—22no—
142C22×C31—21no—
143C22×C31—42no—
144C22×C31—48no—
145C22×C31—41no—
146C22×C31—44no—
147C22×C31—164no—
148C22×C31#37911yescharacteristic
149C4×C31—22no—
150C4×C31—21no—
151C4×C31—42no—
152C4×C31—44no—
153C4×C31—41no—
154C4×C31#41312yes—
155C4×C31#45312yes—
156C4×C31#45512yes—
157#559—22no—
158#550—22no—
159#546—24no—
160#540—22no—
161#535C4×C3112yes—
162#533C22×C3111yescharacteristic
163#532—22no—
164C2×#127C22×C3111yescharacteristic
165#518C4×C3112yes—
166C4×#41C22×C3111yescharacteristic
167#423C22×C3112yes—
168C2×#100—24no—
169C4×M16C22×C3112yes—
170#332—22no—
171#330—22no—
172Q8⋊C8—22no—
173D4⋊C8—22no—
174C2×C31—41no—
175C2×C31—81no—
176C2×C31—84no—
177C2×C31#178511yescharacteristic
178C2×C31#179212yes—
179C2×#41—24no—
180C2×#41C2×C4×C3111yescharacteristic
181C22×D4C2×C4×C3112yes—
182C22×D4D4×C3114yes—
183C2×M16—22no—
184C2×M16—22no—
185C4⋊Q8D4×C3112yes—
186#127—22no—
187#127—21no—
188#127—21no—
189#127—22no—
190#125D4×C3112yes—
191#124—24no—
192#124—22no—
193#123—28no—
194C22⋊Q8—22no—
195#121—24no—
196#121—28no—
197#121—24no—
198#120—24no—
199C4×Q8—22no—
200C4×Q8D4×C3112yes—
201C4×D4—24no—
202C4×D4—22no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4D4×C3112yes—
206#117—21no—
207#117C2×C4×C3111yescharacteristic
208#117C2×C4×C3112yes—
209C2×C22⋊C4—24no—
210C2×C42—21no—
211C2×C42C23×C3111yescharacteristic
212C4⋊C8—22no—
213C4⋊C8—22no—
214#100—48no—
215#97—22no—
216C4×C8—22no—
217C31#847211yescharacteristic, direct product factor, Sylow
218C24C22⋊C4×C3114yes—
219#41—42no—
220#41—84no—
221#41—84no—
222C2×Q8—22no—
223C2×Q8C22⋊C4×C3111yescharacteristic
224C2×D4—21no—
225C2×D4—28no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—44no—
233C2×D4C22⋊C4×C3111yescharacteristic
234C22×C4—28no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4C22×C4×C3111yescharacteristic
238C22×C4C2×D4×C3112yes—
239M16—24no—
240M16—42no—
241M16—44no—
242C2×C8—42no—
243C2×C8—42no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—24no—
247C4⋊C4C22⋊C4×C3112yes—
248C4⋊C4C22⋊C4×C3114yes—
249C22⋊C4—28no—
250C22⋊C4—28no—
251C22⋊C4—21no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C42—21no—
255C42—22no—
256C42—22no—
257C42C2×D4×C3112yes—
258C42C2×D4×C3111yescharacteristic
259C42C2×D4×C3111yescharacteristic
260C23—24no—
261C23—21no—
262C23—44no—
263C23—48no—
264C23C2×C22⋊C4×C3111yescharacteristic
265Q8—22no—
266Q8—42no—
267Q8—42no—
268D4—22no—
269D4—44no—
270D4—42no—
271D4—44no—
272D4—44no—
273D4—44no—
274D4—84no—
275D4—88no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—88no—
286C2×C4C2×C22⋊C4×C3111yescharacteristic
287C2×C4C2×C22⋊C4×C3111yescharacteristic
288C2×C4C31×#12012yes—
289C2×C4C31×#12012yes—
290C8—44no—
291C8—82no—
292C22—21no—
293C22—22no—
294C22—41no—
295C22—44no—
296C22—42no—
297C22—48no—
298C22—164no—
299C22C31×#37911yescharacteristic
300C4—21no—
301C4—22no—
302C4—41no—
303C4—42no—
304C4—44no—
305C4C31×#41312yes—
306C4C31×#45312yes—
307C4C31×#45512yes—
308C2—41no—
309C2—84no—
310C2—81no—
311C2C31×#178511yescharacteristic
312C2C31×#179212yes—
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3508C2
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
2. C31×#1965C2semidirect
3. C31×#1561C2semidirect
4. C31×#1526C2semidirect
5. C31×#1489C2semidirect
103. #8472C31direct (N × H)
137. #1965C2×C31semidirect
138. #1561C2×C31semidirect
139. #1526C2×C31semidirect
140. #1489C2×C31semidirect
217. C31#8472direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35082
C31×#19652
C31×#15612
C31×#15262
C31×#14892
#847231