Finite Group Databasebeta

← C31×#12708 #21469467

Subgroups — C31×#12708 #21469467

Order 7936.

1002 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 146 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×#3487C211yesmaximal, characteristic
2C31×#2127C214yesmaximal, complemented
3C31×#1782C212yesmaximal, complemented
4C2×C31×2−1+4C411yescharacteristic, complemented
5C22×#41×C31C2211yescharacteristic
6C31×#550—24no—
7C31×#548—24no—
8C31×#535—22no—
9C31×#535C2212yes—
10C31×#525—22no—
11C31×#525C412yescomplemented
12C31×#519C411yescharacteristic, complemented
13C2×C31×#99—22no—
14C2×C31×#99C2214yes—
15C31×#381—22no—
16C31×#354—24no—
17C31×#353—24no—
18C31×2−1+4—44no—
19C2×#41×C31—22no—
20C2×#41×C31—24no—
21C2×#41×C31—22no—
22C2×#41×C31—44no—
23C2×#41×C31C2×C412yes—
24C22×Q8×C31—22no—
25C22×Q8×C31C2×C411yescharacteristic
26C22×D4×C31D412yes—
27C22×D4×C31C2311yescharacteristic
28C23×C4×C31C2×C411yescharacteristic
29C23×C4×C31D412yes—
30C31×#127—24no—
31C31×#127D414yes—
32C31×#124—24no—
33C31×#124D414yes—
34C31×#123—22no—
35C31×C22⋊Q8—24no—
36C31×C22⋊Q8—22no—
37C31×#121—24no—
38C31×#121—28no—
39C31×#121—22no—
40C31×#121—24no—
41C31×#120—22no—
42C31×#120—24no—
43C4×Q8×C31—24no—
44C4×D4×C31—28no—
45C4×D4×C31—24no—
46C31×#117—22no—
47C31×#117C2×C412yes—
48C2×C22⋊C4×C31—24no—
49C31×#99—28no—
50C31×#99—44no—
51C31×#95—44no—
52C24×C31C22⋊C411yescharacteristic
53C24×C31C2×D412yes—
54#41×C31—44no—
55#41×C31—44no—
56#41×C31—416no—
57#41×C31—48no—
58#41×C31—44no—
59C2×Q8×C31—22no—
60C2×Q8×C31—24no—
61C2×Q8×C31—22no—
62C2×Q8×C31—48no—
63C2×Q8×C31C22⋊C412yes—
64C2×D4×C31—22no—
65C2×D4×C31—22no—
66C2×D4×C31—24no—
67C2×D4×C31—24no—
68C2×D4×C31—28no—
69C2×D4×C31—24no—
70C2×D4×C31—42no—
71C2×D4×C31—42no—
72C2×D4×C31C22⋊C412yes—
73C2×D4×C31C2×D414yes—
74C22×C4×C31—24no—
75C22×C4×C31—24no—
76C22×C4×C31—24no—
77C22×C4×C31—22no—
78C22×C4×C31—22no—
79C22×C4×C31—42no—
80C22×C4×C31—42no—
81C22×C4×C31—44no—
82C22×C4×C31C22⋊C411yescharacteristic
83C22×C4×C31C22⋊C412yes—
84C22×C4×C31C22×C411yescharacteristic
85C4⋊C4×C31—24no—
86C4⋊C4×C31—24no—
87C4⋊C4×C31—22no—
88C22⋊C4×C31—24no—
89C22⋊C4×C31—24no—
90C22⋊C4×C31—22no—
91C22⋊C4×C31—28no—
92C22⋊C4×C31—44no—
93C22⋊C4×C31—44no—
94C42×C31—24no—
95C42×C31C22⋊C414yes—
96#12708C3111yesmaximal, characteristic, direct product factor, Sylow
97C23×C31—22no—
98C23×C31—22no—
99C23×C31—21no—
100C23×C31—22no—
101C23×C31—22no—
102C23×C31—82no—
103C23×C31—81no—
104C23×C31C2×C22⋊C411yescharacteristic
105C23×C31#12014yes—
106Q8×C31—28no—
107Q8×C31—42no—
108Q8×C31—44no—
109D4×C31—44no—
110D4×C31—48no—
111D4×C31—42no—
112D4×C31—44no—
113D4×C31—44no—
114D4×C31—88no—
115D4×C31—84no—
116C2×C4×C31—22no—
117C2×C4×C31—22no—
118C2×C4×C31—22no—
119C2×C4×C31—24no—
120C2×C4×C31—22no—
121C2×C4×C31—21no—
122C2×C4×C31—22no—
123C2×C4×C31—48no—
124C2×C4×C31—44no—
125C2×C4×C31—48no—
126C2×C4×C31—44no—
127C2×C4×C31—84no—
128C2×C4×C31—88no—
129C2×C4×C31#9914yes—
130C2×C4×C31C2×C22⋊C412yes—
131#3487C2×C3111yescharacteristic
132#2127C2×C3114yescomplemented
133#1782C2×C3112yescomplemented
134C22×C31—24no—
135C22×C31—41no—
136C22×C31—44no—
137C22×C31—44no—
138C22×C31—44no—
139C22×C31—42no—
140C22×C31—42no—
141C22×C31—41no—
142C22×C31—82no—
143C22×C31—82no—
144C22×C31#37911yescharacteristic
145C22×C31C2×#9912yes—
146C4×C31—24no—
147C4×C31—22no—
148C4×C31—44no—
149C4×C31—41no—
150C4×C31—42no—
151C4×C31—164no—
152C2×2−1+4C4×C3111yescharacteristic, complemented
153C22×#41C22×C3111yescharacteristic
154#550—24no—
155#548—24no—
156#535—22no—
157#535C22×C3112yes—
158#525—22no—
159#525C4×C3112yescomplemented
160#519C4×C3111yescharacteristic, complemented
161C2×#99—22no—
162C2×#99C22×C3114yes—
163#381—22no—
164#354—24no—
165#353—24no—
166C2×C31—21no—
167C2×C31—22no—
168C2×C31—44no—
169C2×C31—82no—
170C2×C31—81no—
171C2×C31#178111yescharacteristic
1722−1+4—44no—
173C2×#41—22no—
174C2×#41—24no—
175C2×#41—22no—
176C2×#41—44no—
177C2×#41C2×C4×C3112yes—
178C22×Q8—22no—
179C22×Q8C2×C4×C3111yescharacteristic
180C22×D4D4×C3112yes—
181C22×D4C23×C3111yescharacteristic
182C23×C4C2×C4×C3111yescharacteristic
183C23×C4D4×C3112yes—
184#127—24no—
185#127D4×C3114yes—
186#124—24no—
187#124D4×C3114yes—
188#123—22no—
189C22⋊Q8—24no—
190C22⋊Q8—22no—
191#121—24no—
192#121—28no—
193#121—22no—
194#121—24no—
195#120—22no—
196#120—24no—
197C4×Q8—24no—
198C4×D4—28no—
199C4×D4—24no—
200#117—22no—
201#117C2×C4×C3112yes—
202C2×C22⋊C4—24no—
203#99—28no—
204#99—44no—
205#95—44no—
206C31#1270811yescharacteristic, direct product factor, Sylow
207C24C22⋊C4×C3111yescharacteristic
208C24C2×D4×C3112yes—
209#41—44no—
210#41—44no—
211#41—416no—
212#41—48no—
213#41—44no—
214C2×Q8—22no—
215C2×Q8—24no—
216C2×Q8—22no—
217C2×Q8—48no—
218C2×Q8C22⋊C4×C3112yes—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—28no—
224C2×D4—24no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4C22⋊C4×C3112yes—
228C2×D4C2×D4×C3114yes—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4C22⋊C4×C3111yescharacteristic
238C22×C4C22⋊C4×C3112yes—
239C22×C4C22×C4×C3111yescharacteristic
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—22no—
246C22⋊C4—28no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C42—24no—
250C42C22⋊C4×C3114yes—
251C23—22no—
252C23—22no—
253C23—21no—
254C23—22no—
255C23—22no—
256C23—82no—
257C23—81no—
258C23C2×C22⋊C4×C3111yescharacteristic
259C23C31×#12014yes—
260Q8—28no—
261Q8—42no—
262Q8—44no—
263D4—44no—
264D4—48no—
265D4—42no—
266D4—44no—
267D4—44no—
268D4—88no—
269D4—84no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—48no—
278C2×C4—44no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—84no—
282C2×C4—88no—
283C2×C4C31×#9914yes—
284C2×C4C2×C22⋊C4×C3112yes—
285C22—24no—
286C22—41no—
287C22—44no—
288C22—44no—
289C22—44no—
290C22—42no—
291C22—42no—
292C22—41no—
293C22—82no—
294C22—82no—
295C22C31×#37911yescharacteristic
296C22C2×C31×#9912yes—
297C4—24no—
298C4—22no—
299C4—44no—
300C4—41no—
301C4—42no—
302C4—164no—
303C2—21no—
304C2—22no—
305C2—44no—
306C2—81no—
307C2—82no—
308C2C31×#178111yescharacteristic
subgroupG/H
center, socleC2×C31#1781
commutatorC22×C4C22×C4×C31
Frattini, cosocleC22×D4C23×C31
prime radicalC31×#3487C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C23×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2127C2semidirect
3. C31×#1782C2semidirect
4. C2×C31×2−1+4C4semidirect
11. C31×#525C4semidirect
12. C31×#519C4semidirect
96. #12708C31direct (N × H)
132. #2127C2×C31semidirect
133. #1782C2×C31semidirect
152. C2×2−1+4C4×C31semidirect
159. #525C4×C31semidirect
160. #519C4×C31semidirect
206. C31#12708direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34872
C31×#21272
C31×#17822
#1270831