Finite Group Databasebeta

← C31×#7824 #21464583

Subgroups — C31×#7824 #21464583

Order 7936.

418 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2724C211yesmaximal, characteristic
2C31×#2100C212yesmaximal, complemented
3C31×#2094C211yesmaximal, characteristic
4C31×#1271C212yesmaximal, complemented
5C31×#1270C211yesmaximal, characteristic
6C2×C31×#123—21no—
7C2×C31×#106—21no—
8C2×C31×#102—21no—
9C31×#398—21no—
10C31×#397—22no—
11C31×#390C2211yescharacteristic
12C31×#389C2212yes—
13C31×#385—22no—
14C31×#384C2211yescharacteristic
15C31×#340—22no—
16C31×#340—22no—
17C31×#336C2211yescharacteristic
18C31×#336C2212yes—
19C22×D4×C31D411yescharacteristic
20C23×C4×C31—21no—
21C22×C8×C31—21no—
22C22×C8×C31—22no—
23C31×#123—42no—
24C31×#123—42no—
25C2×C4⋊C4×C31—22no—
26C2×C4⋊C4×C31—21no—
27C2×C4⋊C4×C31—22no—
28C2×C4⋊C4×C31—21no—
29C2×C4⋊C4×C31D412yes—
30C2×C4⋊C4×C31D411yescharacteristic
31C2×C4⋊C4×C31D412yes—
32C2×C22⋊C4×C31—21no—
33C2×C22⋊C4×C31—22no—
34C2×C22⋊C4×C31—22no—
35C2×C42×C31C2311yescharacteristic
36C31×#106—42no—
37C31×#102—42no—
38C31×#95—21no—
39C31×#95—22no—
40C31×#95—22no—
41C24×C31—21no—
42C2×D4×C31—21no—
43C2×D4×C31—41no—
44C2×D4×C31—41no—
45C2×D4×C31SD1612yes—
46C22×C4×C31—22no—
47C22×C4×C31—21no—
48C22×C4×C31—22no—
49C22×C4×C31—42no—
50C22×C4×C31—41no—
51C22×C4×C31—81no—
52C22×C4×C31—81no—
53C22×C4×C31C2×D411yescharacteristic
54C22×C4×C31C2×D412yes—
55C2×C8×C31—22no—
56C2×C8×C31—41no—
57C2×C8×C31—41no—
58C2×C8×C31—42no—
59C2×C8×C31—42no—
60C2×C8×C31—42no—
61C4⋊C4×C31—22no—
62C4⋊C4×C31—22no—
63C4⋊C4×C31—41no—
64C4⋊C4×C31—42no—
65C4⋊C4×C31—41no—
66C4⋊C4×C31—41no—
67C4⋊C4×C31—44no—
68C4⋊C4×C31—42no—
69C4⋊C4×C31—42no—
70C4⋊C4×C31—81no—
71C22⋊C4×C31—42no—
72C22⋊C4×C31—44no—
73C22⋊C4×C31—42no—
74C22⋊C4×C31—42no—
75C42×C31—41no—
76#7824C3111yesmaximal, characteristic, direct product factor, Sylow
77C23×C31—22no—
78C23×C31—42no—
79C23×C31—41no—
80C23×C31—42no—
81C23×C31—41no—
82C23×C31#12011yescharacteristic
83D4×C31—42no—
84D4×C31—81no—
85C2×C4×C31—21no—
86C2×C4×C31—22no—
87C2×C4×C31—22no—
88C2×C4×C31—21no—
89C2×C4×C31—42no—
90C2×C4×C31—42no—
91C2×C4×C31—42no—
92C2×C4×C31—42no—
93C2×C4×C31—41no—
94C2×C4×C31—42no—
95C2×C4×C31—42no—
96C2×C4×C31—42no—
97C2×C4×C31—84no—
98C2×C4×C31—81no—
99C2×C4×C31—82no—
100C2×C4×C31C2×SD1611yescharacteristic
101C2×C4×C31#13512yes—
102C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
103C2×C4×C31#13712yes—
104C8×C31—42no—
105C8×C31—82no—
106#2724C2×C3111yescharacteristic
107#2100C2×C3112yescomplemented
108#2094C2×C3111yescharacteristic
109#1271C2×C3112yescomplemented
110#1270C2×C3111yescharacteristic
111C22×C31—42no—
112C22×C31—44no—
113C22×C31—82no—
114C22×C31—81no—
115C22×C31—81no—
116C22×C31#45011yescharacteristic
117C22×C31#45212yes—
118C22×C31#45311yescharacteristic
119C22×C31#45412yes—
120C22×C31#45811yescharacteristic
121C4×C31—21no—
122C4×C31—21no—
123C4×C31—22no—
124C4×C31—22no—
125C4×C31—82no—
126C4×C31—81no—
127C4×C31—82no—
128C4×C31—161no—
129C2×#123—21no—
130C2×#106—21no—
131C2×#102—21no—
132#398—21no—
133#397—22no—
134#390C22×C3111yescharacteristic
135#389C22×C3112yes—
136#385—22no—
137#384C22×C3111yescharacteristic
138#340—22no—
139#340—22no—
140#336C22×C3111yescharacteristic
141#336C22×C3112yes—
142C2×C31—82no—
143C2×C31#161311yescharacteristic
144C2×C31#161512yes—
145C2×C31#163011yescharacteristic
146C2×C31#163512yes—
147C2×C31#165711yescharacteristic
148C22×D4D4×C3111yescharacteristic
149C23×C4—21no—
150C22×C8—21no—
151C22×C8—22no—
152#123—42no—
153#123—42no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4D4×C3112yes—
159C2×C4⋊C4D4×C3112yes—
160C2×C4⋊C4D4×C3111yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C42C23×C3111yescharacteristic
165#106—42no—
166#102—42no—
167#95—22no—
168#95—21no—
169#95—22no—
170C31#782411yescharacteristic, direct product factor, Sylow
171C24—21no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4SD16×C3112yes—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4C2×D4×C3111yescharacteristic
184C22×C4C2×D4×C3112yes—
185C2×C8—22no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—44no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C22⋊C4—42no—
202C22⋊C4—44no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23C31×#12011yescharacteristic
212D4—42no—
213D4—81no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—84no—
227C2×C4—81no—
228C2×C4—82no—
229C2×C4C2×SD16×C3111yescharacteristic
230C2×C4C31×#13512yes—
231C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
232C2×C4C31×#13712yes—
233C8—42no—
234C8—82no—
235C22—42no—
236C22—44no—
237C22—82no—
238C22—81no—
239C22—81no—
240C22C31×#45011yescharacteristic
241C22C31×#45212yes—
242C22C31×#45311yescharacteristic
243C22C31×#45412yes—
244C22C31×#45811yescharacteristic
245C4—21no—
246C4—21no—
247C4—22no—
248C4—22no—
249C4—82no—
250C4—81no—
251C4—82no—
252C4—161no—
253C2—82no—
254C2C31×#161311yescharacteristic
255C2C31×#161512yes—
256C2C31×#163011yescharacteristic
257C2C31×#163512yes—
258C2C31×#165711yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2100C2semidirect
4. C31×#1271C2semidirect
76. #7824C31direct (N × H)
107. #2100C2×C31semidirect
109. #1271C2×C31semidirect
170. C31#7824direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27242
C31×#21002
C31×#20942
C31×#12712
C31×#12702
#782431