Finite Group Databasebeta

← C31×#8323 #21465082

Subgroups — C31×#8323 #21465082

Order 7936.

930 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 214 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
1C2×C31×#519C211yesmaximal, characteristic
2C31×#1479C214yesmaximal, complemented
3C31×#1458C212yesmaximal
4C23×Q8×C31C411yescharacteristic
5C31×#519—28no—
6C2×C4×Q8×C31—22no—
7C2×C4×Q8×C31C412yes—
8C2×C4×Q8×C31C2212yes—
9C2×C31×#117—22no—
10C2×C31×#117C411yescharacteristic
11C2×C31×#117C2211yescharacteristic
12C31×#431C2214yes—
13C31×Q8⋊C8—28no—
14C31×#324—22no—
15C31×#323—22no—
16C22×Q8×C31—24no—
17C22×Q8×C31—24no—
18C22×Q8×C31—21no—
19C22×Q8×C31—22no—
20C22×Q8×C31C814yescomplemented
21C22×Q8×C31C2×C412yes—
22C23×C4×C31—21no—
23C23×C4×C31C2×C411yescharacteristic
24C4×Q8×C31—216no—
25C4×Q8×C31—24no—
26C4×Q8×C31—24no—
27C4×Q8×C31—24no—
28C4×Q8×C31D418yes—
29C31×#117—216no—
30C31×#117—24no—
31C31×#117—24no—
32C2×C4⋊C4×C31—22no—
33C2×C4⋊C4×C31—22no—
34C2×C4⋊C4×C31C2×C412yes—
35C2×C4⋊C4×C31D412yes—
36C2×C22⋊C4×C31—21no—
37C2×C22⋊C4×C31D412yes—
38C2×C42×C31—22no—
39C2×C42×C31—21no—
40C2×C42×C31—22no—
41C2×C42×C31C2×C411yescharacteristic
42C2×C42×C31C2311yescharacteristic
43C31×C4⋊C8—24no—
44C31×C22⋊C8—24no—
45C4×C8×C31—24no—
46C24×C31C22⋊C411yescharacteristic
47C2×Q8×C31—24no—
48C2×Q8×C31—216no—
49C2×Q8×C31—24no—
50C2×Q8×C31—216no—
51C2×Q8×C31—24no—
52C2×Q8×C31—22no—
53C2×Q8×C31—22no—
54C2×Q8×C31—22no—
55C2×Q8×C31C22⋊C414yes—
56C2×Q8×C31C2×C814yes—
57C2×Q8×C31M1614yes—
58C22×C4×C31—24no—
59C22×C4×C31—24no—
60C22×C4×C31—24no—
61C22×C4×C31—22no—
62C22×C4×C31—22no—
63C22×C4×C31—22no—
64C22×C4×C31—22no—
65C22×C4×C31C22⋊C411yescharacteristic
66C22×C4×C31C22⋊C412yes—
67C22×C4×C31C2×C812yes—
68C22×C4×C31M1612yes—
69C22×C4×C31C22×C411yescharacteristic
70C22×C4×C31C2×D412yes—
71C2×C8×C31—44no—
72C4⋊C4×C31—28no—
73C4⋊C4×C31—22no—
74C4⋊C4×C31—28no—
75C4⋊C4×C31—24no—
76C4⋊C4×C31C22⋊C414yes—
77C22⋊C4×C31—24no—
78C22⋊C4×C31—24no—
79C42×C31—28no—
80C42×C31—24no—
81C42×C31—22no—
82C42×C31—28no—
83C42×C31C2×D414yes—
84#8323C3111yesmaximal, characteristic, direct product factor, Sylow
85C23×C31—21no—
86C23×C31—21no—
87C23×C31—42no—
88C23×C31C22⋊C812yes—
89C23×C31C2×C22⋊C411yescharacteristic
90Q8×C31—216no—
91Q8×C31—24no—
92Q8×C31—24no—
93Q8×C31—24no—
94Q8×C31C22⋊C818yes—
95C2×C4×C31—216no—
96C2×C4×C31—24no—
97C2×C4×C31—21no—
98C2×C4×C31—24no—
99C2×C4×C31—24no—
100C2×C4×C31—21no—
101C2×C4×C31—24no—
102C2×C4×C31—24no—
103C2×C4×C31—22no—
104C2×C4×C31—24no—
105C2×C4×C31—42no—
106C2×C4×C31—44no—
107C2×C4×C31—42no—
108C2×C4×C31C22⋊C812yes—
109C2×C4×C31C2×C22⋊C412yes—
110C2×C4×C31#12014yes—
111C2×C4×C31C22×C811yescharacteristic
112C2×C4×C31C2×M1612yes—
113C2×C4×C31C2×M1611yescharacteristic
114C8×C31—84no—
115C2×#519C2×C3111yescharacteristic
116#1479C2×C3114yescomplemented
117#1458C2×C3112yes—
118C22×C31—22no—
119C22×C31—22no—
120C22×C31—42no—
121C22×C31—42no—
122C22×C31—42no—
123C22×C31#37911yescharacteristic
124C22×C31C2×C22⋊C811yescharacteristic
125C22×C31#40711yescharacteristic
126C4×C31—24no—
127C4×C31—28no—
128C4×C31—22no—
129C4×C31—42no—
130C4×C31—44no—
131C4×C31C2×C22⋊C812yes—
132C4×C31#40712yes—
133C23×Q8C4×C3111yescharacteristic
134#519—28no—
135C2×C4×Q8—22no—
136C2×C4×Q8C4×C3112yes—
137C2×C4×Q8C22×C3112yes—
138C2×#117—22no—
139C2×#117C4×C3111yescharacteristic
140C2×#117C22×C3111yescharacteristic
141#431C22×C3114yes—
142Q8⋊C8—28no—
143#324—22no—
144#323—22no—
145C2×C31—21no—
146C2×C31—21no—
147C2×C31—42no—
148C2×C31#177911yescharacteristic
149C2×C31#179612yes—
150C22×Q8—24no—
151C22×Q8—24no—
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8C8×C3114yescomplemented
155C22×Q8C2×C4×C3112yes—
156C23×C4—21no—
157C23×C4C2×C4×C3111yescharacteristic
158C4×Q8—216no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8D4×C3118yes—
163#117—216no—
164#117—24no—
165#117—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C2×C4×C3112yes—
169C2×C4⋊C4D4×C3112yes—
170C2×C22⋊C4—21no—
171C2×C22⋊C4D4×C3112yes—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42C2×C4×C3111yescharacteristic
176C2×C42C23×C3111yescharacteristic
177C4⋊C8—24no—
178C22⋊C8—24no—
179C4×C8—24no—
180C31#832311yescharacteristic, direct product factor, Sylow
181C24C22⋊C4×C3111yescharacteristic
182C2×Q8—24no—
183C2×Q8—216no—
184C2×Q8—24no—
185C2×Q8—216no—
186C2×Q8—24no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8C22⋊C4×C3114yes—
191C2×Q8C2×C8×C3114yes—
192C2×Q8M16×C3114yes—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4C22⋊C4×C3111yescharacteristic
201C22×C4C22⋊C4×C3112yes—
202C22×C4C2×C8×C3112yes—
203C22×C4M16×C3112yes—
204C22×C4C22×C4×C3111yescharacteristic
205C22×C4C2×D4×C3112yes—
206C2×C8—44no—
207C4⋊C4—28no—
208C4⋊C4—22no—
209C4⋊C4—28no—
210C4⋊C4—24no—
211C4⋊C4C22⋊C4×C3114yes—
212C22⋊C4—24no—
213C22⋊C4—24no—
214C42—28no—
215C42—24no—
216C42—22no—
217C42—28no—
218C42C2×D4×C3114yes—
219C23—21no—
220C23—21no—
221C23—42no—
222C23C31×C22⋊C812yes—
223C23C2×C22⋊C4×C3111yescharacteristic
224Q8—216no—
225Q8—24no—
226Q8—24no—
227Q8—24no—
228Q8C31×C22⋊C818yes—
229C2×C4—216no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—21no—
233C2×C4—24no—
234C2×C4—21no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4C31×C22⋊C812yes—
243C2×C4C2×C22⋊C4×C3112yes—
244C2×C4C31×#12014yes—
245C2×C4C22×C8×C3111yescharacteristic
246C2×C4C2×M16×C3112yes—
247C2×C4C2×M16×C3111yescharacteristic
248C8—84no—
249C22—22no—
250C22—22no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22C31×#37911yescharacteristic
255C22C2×C31×C22⋊C811yescharacteristic
256C22C31×#40711yescharacteristic
257C4—24no—
258C4—28no—
259C4—22no—
260C4—44no—
261C4—42no—
262C4C2×C31×C22⋊C812yes—
263C4C31×#40712yes—
264C2—21no—
265C2—21no—
266C2—42no—
267C2C31×#177911yescharacteristic
268C2C31×#179612yes—
subgroupG/H
center, socleC22×C31#379
commutatorC2×C4C22×C8×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC24×C31C22⋊C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1479C2semidirect
20. C22×Q8×C31C8semidirect
84. #8323C31direct (N × H)
116. #1479C2×C31semidirect
154. C22×Q8C8×C31semidirect
180. C31#8323direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5192
C31×#14792
C31×#14582
#832331