Finite Group Databasebeta

← C31×#10351 #21467110

Subgroups — C31×#10351 #21467110

Order 7936.

500 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 138 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×#2571C211yesmaximal, characteristic, complemented
2C31×#1982C212yesmaximal
3C31×#1957C211yesmaximal, characteristic
4C31×#1296C211yesmaximal, characteristic, complemented
5C31×#1279C212yesmaximal, complemented
6C2×C31×#127C2211yescharacteristic
7C2×C31×C4⋊C8C2212yes—
8C2×C31×#102—22no—
9C2×C31×#102—22no—
10C2×C31×#102—22no—
11C2×C4×C8×C31C2211yescharacteristic
12C31×#400—22no—
13C31×#389—22no—
14C31×#384C2211yescharacteristic
15C31×#382—22no—
16C4×C4⋊C4×C31C2212yes—
17C22×D4×C31—21no—
18C22×D4×C31—22no—
19C22×C8×C31—21no—
20C22×C8×C31—22no—
21C31×#127—22no—
22C31×#127—21no—
23C31×#127Q812yes—
24C2×C4⋊C4×C31—22no—
25C2×C4⋊C4×C31—22no—
26C2×C4⋊C4×C31D412yes—
27C2×C4⋊C4×C31D412yes—
28C2×C4⋊C4×C31D412yes—
29C2×C42×C31—22no—
30C2×C42×C31C2311yescharacteristic
31C31×C4⋊C8—22no—
32C31×C4⋊C8—22no—
33C31×#102—44no—
34C31×#102—44no—
35C31×#102—44no—
36C4×C8×C31—22no—
37C31×#95—22no—
38C31×#95—24no—
39C31×#95—22no—
40C24×C31—41no—
41C2×D4×C31—22no—
42C2×D4×C31—24no—
43C2×D4×C31—42no—
44C2×D4×C31—42no—
45C2×D4×C31—41no—
46C2×D4×C31—42no—
47C2×D4×C31—42no—
48C2×D4×C31—42no—
49C2×D4×C31—42no—
50C2×D4×C31—42no—
51C22×C4×C31—22no—
52C22×C4×C31—22no—
53C22×C4×C31—22no—
54C22×C4×C31C2×D411yescharacteristic
55C22×C4×C31C2×D412yes—
56C2×C8×C31—22no—
57C2×C8×C31—41no—
58C2×C8×C31—41no—
59C2×C8×C31—42no—
60C2×C8×C31—42no—
61C2×C8×C31—42no—
62C4⋊C4×C31—22no—
63C4⋊C4×C31—22no—
64C4⋊C4×C31—22no—
65C4⋊C4×C31—22no—
66C4⋊C4×C31—42no—
67C4⋊C4×C31—42no—
68C4⋊C4×C31—42no—
69C4⋊C4×C31—82no—
70C42×C31—42no—
71C42×C31—42no—
72C42×C31C2×Q811yescharacteristic
73C42×C31#4112yes—
74C42×C31#4111yescharacteristic
75#10351C3111yesmaximal, characteristic, direct product factor, Sylow
76C23×C31—42no—
77C23×C31—82no—
78C23×C31—82no—
79C23×C31—81no—
80C23×C31—81no—
81C23×C31#12011yescharacteristic
82D4×C31—44no—
83D4×C31—44no—
84D4×C31—44no—
85D4×C31—82no—
86D4×C31—82no—
87D4×C31—82no—
88C2×C4×C31—21no—
89C2×C4×C31—21no—
90C2×C4×C31—42no—
91C2×C4×C31—42no—
92C2×C4×C31—42no—
93C2×C4×C31—42no—
94C2×C4×C31—44no—
95C2×C4×C31—42no—
96C2×C4×C31—42no—
97C2×C4×C31—42no—
98C2×C4×C31C22⋊Q811yescharacteristic
99C2×C4×C31C22⋊Q812yes—
100C2×C4×C31#12411yescharacteristic
101C2×C4×C31#12412yes—
102C2×C4×C31#13512yes—
103C2×C4×C31#13512yes—
104C2×C4×C31AGL(1,ℤ/8)12yes—
105C2×C4×C31AGL(1,ℤ/8)12yes—
106C8×C31—42no—
107C8×C31—82no—
108#2571C2×C3111yescharacteristic, complemented
109#1982C2×C3112yes—
110#1957C2×C3111yescharacteristic
111#1296C2×C3111yescharacteristic, complemented
112#1279C2×C3112yescomplemented
113C22×C31—84no—
114C22×C31—82no—
115C22×C31—161no—
116C22×C31—161no—
117C22×C31—162no—
118C22×C31#39311yescharacteristic
119C22×C31#44912yes—
120C22×C31#44912yes—
121C22×C31#45311yescharacteristic
122C22×C31#45411yescharacteristic
123C4×C31—21no—
124C4×C31—21no—
125C4×C31—82no—
126C4×C31—82no—
127C4×C31—82no—
128C4×C31#47812yes—
129C4×C31#47812yes—
130C4×C31#48812yes—
131C4×C31#49012yes—
132C2×#127C22×C3111yescharacteristic
133C2×C4⋊C8C22×C3112yes—
134C2×#102—22no—
135C2×#102—22no—
136C2×#102—22no—
137C2×C4×C8C22×C3111yescharacteristic
138#400—22no—
139#389—22no—
140#384C22×C3111yescharacteristic
141#382—22no—
142C4×C4⋊C4C22×C3112yes—
143C2×C31—162no—
144C2×C31#162011yescharacteristic
145C2×C31#162411yescharacteristic
146C2×C31#162812yes—
147C2×C31#202512yes—
148C2×C31#202711yescharacteristic
149C22×D4—21no—
150C22×D4—22no—
151C22×C8—21no—
152C22×C8—22no—
153#127—22no—
154#127—21no—
155#127Q8×C3112yes—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4D4×C3112yes—
159C2×C4⋊C4D4×C3112yes—
160C2×C4⋊C4D4×C3112yes—
161C2×C42—22no—
162C2×C42C23×C3111yescharacteristic
163C4⋊C8—22no—
164C4⋊C8—22no—
165#102—44no—
166#102—44no—
167#102—44no—
168C4×C8—22no—
169#95—22no—
170#95—24no—
171#95—22no—
172C31#1035111yescharacteristic, direct product factor, Sylow
173C24—41no—
174C2×D4—22no—
175C2×D4—24no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—41no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4C2×D4×C3111yescharacteristic
188C22×C4C2×D4×C3112yes—
189C2×C8—22no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—82no—
203C42—42no—
204C42—42no—
205C42C2×Q8×C3111yescharacteristic
206C42#41×C3112yes—
207C42#41×C3111yescharacteristic
208C23—42no—
209C23—82no—
210C23—82no—
211C23—81no—
212C23—81no—
213C23C31×#12011yescharacteristic
214D4—44no—
215D4—44no—
216D4—44no—
217D4—82no—
218D4—82no—
219D4—82no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4C31×C22⋊Q811yescharacteristic
231C2×C4C31×C22⋊Q812yes—
232C2×C4C31×#12411yescharacteristic
233C2×C4C31×#12412yes—
234C2×C4C31×#13512yes—
235C2×C4C31×#13512yes—
236C2×C4C31×AGL(1,ℤ/8)12yes—
237C2×C4C31×AGL(1,ℤ/8)12yes—
238C8—42no—
239C8—82no—
240C22—84no—
241C22—82no—
242C22—161no—
243C22—161no—
244C22—162no—
245C22C31×#39311yescharacteristic
246C22C31×#44912yes—
247C22C31×#44912yes—
248C22C31×#45311yescharacteristic
249C22C31×#45411yescharacteristic
250C4—21no—
251C4—21no—
252C4—82no—
253C4—82no—
254C4—82no—
255C4C31×#47812yes—
256C4C31×#47812yes—
257C4C31×#48812yes—
258C4C31×#49012yes—
259C2—162no—
260C2C31×#162011yescharacteristic
261C2C31×#162411yescharacteristic
262C2C31×#162812yes—
263C2C31×#202512yes—
264C2C31×#202711yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2571C2semidirect
4. C31×#1296C2semidirect
5. C31×#1279C2semidirect
75. #10351C31direct (N × H)
108. #2571C2×C31semidirect
111. #1296C2×C31semidirect
112. #1279C2×C31semidirect
172. C31#10351direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25712
C31×#19822
C31×#19572
C31×#12962
C31×#12792
#1035131