Finite Group Databasebeta

← C31×#9976 #21466735

Subgroups — C31×#9976 #21466735

Order 7936.

716 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3189C211yesmaximal, characteristic, complemented
2C31×#3007C212yesmaximal
3C31×#1780C211yesmaximal, characteristic
4C31×#1472C212yesmaximal, complemented
5C31×#1293C211yesmaximal, characteristic, complemented
6C22×#41×C31C2211yescharacteristic
7C2×C31×#137—22no—
8C2×C31×C22⋊Q8C2212yes—
9C31×#484—22no—
10C31×#467—22no—
11C31×#452—24no—
12C31×#451—22no—
13C31×#451—22no—
14C31×#450—22no—
15C31×#417—22no—
16C31×#407C2212yes—
17C2×C31×C22⋊C8—22no—
18C2×C31×C22⋊C8C2211yescharacteristic
19C2×C31×C22⋊C8C2211yescharacteristic
20C31×#324—24no—
21C2×#41×C31—22no—
22C2×#41×C31—22no—
23C2×#41×C31—44no—
24C22×Q8×C31D411yescharacteristic
25C22×Q8×C31D412yes—
26C22×D4×C31—21no—
27C22×D4×C31D411yescharacteristic
28C23×C4×C31—21no—
29C23×C4×C31C2311yescharacteristic
30C31×#137—48no—
31C2×Q16×C31—24no—
32C2×Q16×C31—22no—
33C2×SD16×C31—24no—
34C2×M16×C31—22no—
35C22×C8×C31—21no—
36C22×C8×C31D412yes—
37C31×C22⋊Q8—22no—
38C31×C22⋊Q8—22no—
39C31×C22⋊Q8—22no—
40C31×C22⋊Q8—22no—
41C2×C4⋊C4×C31—22no—
42C2×C4⋊C4×C31D412yes—
43C2×C22⋊C4×C31—22no—
44C31×#107—22no—
45C31×#103—24no—
46C31×#103—24no—
47C31×#102—24no—
48C31×C22⋊C8—22no—
49C31×C22⋊C8—22no—
50C31×C22⋊C8—22no—
51C31×C22⋊C8—44no—
52C31×C22⋊C8D414yes—
53C24×C31—21no—
54C24×C31#4111yescharacteristic
55#41×C31—28no—
56#41×C31—44no—
57#41×C31—44no—
58#41×C31—44no—
59C2×Q8×C31—22no—
60C2×Q8×C31—22no—
61C2×Q8×C31—22no—
62C2×Q8×C31—42no—
63C2×Q8×C31—44no—
64C2×D4×C31—22no—
65C2×D4×C31—42no—
66C2×D4×C31—42no—
67C2×D4×C31—42no—
68C2×D4×C31—42no—
69C22×C4×C31—24no—
70C22×C4×C31—21no—
71C22×C4×C31—22no—
72C22×C4×C31—22no—
73C22×C4×C31—42no—
74C22×C4×C31—42no—
75C22×C4×C31—41no—
76C22×C4×C31—41no—
77C22×C4×C31C2×D412yes—
78C22×C4×C31C2×D411yescharacteristic
79C22×C4×C31C2×D411yescharacteristic
80C22×C4×C31C2×D412yes—
81Q16×C31—48no—
82Q16×C31—82no—
83SD16×C31—48no—
84M16×C31—44no—
85C2×C8×C31—22no—
86C2×C8×C31—24no—
87C2×C8×C31—42no—
88C2×C8×C31—42no—
89C2×C8×C31—41no—
90C4⋊C4×C31—22no—
91C4⋊C4×C31—22no—
92C4⋊C4×C31—42no—
93C4⋊C4×C31—42no—
94C22⋊C4×C31—42no—
95C22⋊C4×C31—42no—
96#9976C3111yesmaximal, characteristic, direct product factor, Sylow
97C23×C31—22no—
98C23×C31—22no—
99C23×C31—41no—
100C23×C31—41no—
101C23×C31—41no—
102C23×C31—42no—
103C23×C31—41no—
104C23×C31—42no—
105C23×C31#12112yes—
106C23×C31#12111yescharacteristic
107Q8×C31—24no—
108Q8×C31—42no—
109Q8×C31—44no—
110Q8×C31—44no—
111D4×C31—24no—
112D4×C31—42no—
113D4×C31—48no—
114C2×C4×C31—22no—
115C2×C4×C31—24no—
116C2×C4×C31—22no—
117C2×C4×C31—41no—
118C2×C4×C31—42no—
119C2×C4×C31—44no—
120C2×C4×C31—41no—
121C2×C4×C31—41no—
122C2×C4×C31—44no—
123C2×C4×C31—44no—
124C2×C4×C31—42no—
125C2×C4×C31—42no—
126C2×C4×C31—82no—
127C2×C4×C31#12012yes—
128C2×C4×C31#12011yescharacteristic
129C2×C4×C31#12711yescharacteristic
130C8×C31—44no—
131C8×C31—42no—
132C8×C31—81no—
133#3189C2×C3111yescharacteristic, complemented
134#3007C2×C3112yes—
135#1780C2×C3111yescharacteristic
136#1472C2×C3112yescomplemented
137#1293C2×C3111yescharacteristic, complemented
138C22×C31—21no—
139C22×C31—21no—
140C22×C31—22no—
141C22×C31—22no—
142C22×C31—22no—
143C22×C31—44no—
144C22×C31—42no—
145C22×C31—82no—
146C22×C31—82no—
147C22×C31—81no—
148C22×C31—81no—
149C22×C31#39211yescharacteristic
150C22×C31#47312yes—
151C4×C31—22no—
152C4×C31—21no—
153C4×C31—42no—
154C4×C31—82no—
155C4×C31—82no—
156C4×C31B(4,𝔽2)12yes—
157C22×#41C22×C3111yescharacteristic
158C2×#137—22no—
159C2×C22⋊Q8C22×C3112yes—
160#484—22no—
161#467—22no—
162#452—24no—
163#451—22no—
164#451—22no—
165#450—22no—
166#417—22no—
167#407C22×C3112yes—
168C2×C22⋊C8—22no—
169C2×C22⋊C8C22×C3111yescharacteristic
170C2×C22⋊C8C22×C3111yescharacteristic
171#324—24no—
172C2×C31—22no—
173C2×C31—42no—
174C2×C31—82no—
175C2×C31#201711yescharacteristic
176C2×C31#202011yescharacteristic
177C2×C31#202111yescharacteristic
178C2×#41—22no—
179C2×#41—22no—
180C2×#41—44no—
181C22×Q8D4×C3112yes—
182C22×Q8D4×C3111yescharacteristic
183C22×D4—21no—
184C22×D4D4×C3111yescharacteristic
185C23×C4—21no—
186C23×C4C23×C3111yescharacteristic
187#137—48no—
188C2×Q16—24no—
189C2×Q16—22no—
190C2×SD16—24no—
191C2×M16—22no—
192C22×C8—21no—
193C22×C8D4×C3112yes—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4D4×C3112yes—
200C2×C22⋊C4—22no—
201#107—22no—
202#103—24no—
203#103—24no—
204#102—24no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—44no—
209C22⋊C8D4×C3114yes—
210C31#997611yescharacteristic, direct product factor, Sylow
211C24—21no—
212C24#41×C3111yescharacteristic
213#41—28no—
214#41—44no—
215#41—44no—
216#41—44no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×D4—22no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C22×C4—24no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4C2×D4×C3112yes—
236C22×C4C2×D4×C3111yescharacteristic
237C22×C4C2×D4×C3111yescharacteristic
238C22×C4C2×D4×C3112yes—
239Q16—48no—
240Q16—82no—
241SD16—48no—
242M16—44no—
243C2×C8—22no—
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—41no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C23—22no—
255C23—22no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—42no—
260C23—42no—
261C23—41no—
262C23C31×#12112yes—
263C23C31×#12111yescharacteristic
264Q8—24no—
265Q8—42no—
266Q8—44no—
267Q8—44no—
268D4—24no—
269D4—42no—
270D4—48no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—82no—
284C2×C4C31×#12012yes—
285C2×C4C31×#12011yescharacteristic
286C2×C4C31×#12711yescharacteristic
287C8—44no—
288C8—42no—
289C8—81no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—44no—
296C22—42no—
297C22—82no—
298C22—82no—
299C22—81no—
300C22—81no—
301C22C31×#39211yescharacteristic
302C22C31×#47312yes—
303C4—22no—
304C4—21no—
305C4—42no—
306C4—82no—
307C4—82no—
308C4C31×B(4,𝔽2)12yes—
309C2—22no—
310C2—42no—
311C2—82no—
312C2C31×#201711yescharacteristic
313C2C31×#202011yescharacteristic
314C2C31×#202111yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC22×#41×C31C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3189C2semidirect
4. C31×#1472C2semidirect
5. C31×#1293C2semidirect
96. #9976C31direct (N × H)
133. #3189C2×C31semidirect
136. #1472C2×C31semidirect
137. #1293C2×C31semidirect
210. C31#9976direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31892
C31×#30072
C31×#17802
C31×#14722
C31×#12932
#997631