Finite Group Databasebeta

← C31×#10370 #21467129

Subgroups — C31×#10370 #21467129

Order 7936.

454 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2399C211yesmaximal, characteristic, complemented
2C31×#1963C211yesmaximal, characteristic
3C31×#1502C212yesmaximal
4C31×#1303C212yesmaximal, complemented
5C31×#1274C211yesmaximal, characteristic, complemented
6C31×#532C2211yescharacteristic
7C2×C4×Q8×C31—21no—
8C31×#431C2212yes—
9C31×#423C2211yescharacteristic
10C2×C31×#104—21no—
11C2×C31×#104—21no—
12C31×#398—22no—
13C31×#395—22no—
14C4×C4⋊C4×C31—21no—
15C4×C4⋊C4×C31C2212yes—
16C31×#376C2211yescharacteristic
17C31×#327—22no—
18C31×#327—22no—
19C2×#41×C31—21no—
20C22×Q8×C31—21no—
21C2×M16×C31—21no—
22C31×C4⋊Q8Q811yescharacteristic, complemented
23C31×#127Q811yescharacteristic, complemented
24C31×#124—21no—
25C31×#121—22no—
26C4×Q8×C31—42no—
27C4×Q8×C31—42no—
28C4×D4×C31—22no—
29C2×C4⋊C4×C31—22no—
30C2×C4⋊C4×C31—22no—
31C2×C4⋊C4×C31—21no—
32C2×C4⋊C4×C31—22no—
33C2×C42×C31—22no—
34C2×C42×C31D411yescharacteristic
35C2×C42×C31D411yescharacteristic
36C2×C42×C31C2311yescharacteristic
37C31×C4⋊C8—22no—
38C31×C4⋊C8—22no—
39C31×#104—42no—
40C31×#104—42no—
41C31×C22⋊C8—22no—
42C4×C8×C31—22no—
43C31×#95—21no—
44C31×#95—21no—
45C31×#95—22no—
46C31×#95D412yes—
47C31×#95D412yes—
48#41×C31—41no—
49#41×C31—41no—
50#41×C31—41no—
51#41×C31—41no—
52C2×Q8×C31—21no—
53C2×Q8×C31—42no—
54C2×Q8×C31—44no—
55C2×D4×C31—21no—
56C2×D4×C31—22no—
57C2×D4×C31—41no—
58C22×C4×C31—21no—
59C22×C4×C31—21no—
60C22×C4×C31—22no—
61C22×C4×C31—22no—
62C22×C4×C31—41no—
63C22×C4×C31C2×D412yes—
64C22×C4×C31C2×D411yescharacteristic
65M16×C31—41no—
66M16×C31—41no—
67C2×C8×C31—42no—
68C2×C8×C31—41no—
69C4⋊C4×C31—22no—
70C4⋊C4×C31—44no—
71C4⋊C4×C31—44no—
72C4⋊C4×C31—44no—
73C4⋊C4×C31—42no—
74C22⋊C4×C31—42no—
75C42×C31—21no—
76C42×C31—21no—
77C42×C31—42no—
78C42×C31—42no—
79C42×C31D812yes—
80C42×C31SD1612yes—
81C42×C31C2×Q811yescharacteristic
82C42×C31#4111yescharacteristic
83C42×C31#4112yes—
84#10370C3111yesmaximal, characteristic, direct product factor, Sylow
85C23×C31—41no—
86C23×C31#12011yescharacteristic
87Q8×C31—48no—
88Q8×C31—41no—
89Q8×C31—41no—
90D4×C31—41no—
91D4×C31—41no—
92D4×C31—82no—
93D4×C31—81no—
94D4×C31—81no—
95C2×C4×C31—24no—
96C2×C4×C31—22no—
97C2×C4×C31—22no—
98C2×C4×C31—42no—
99C2×C4×C31—42no—
100C2×C4×C31—42no—
101C2×C4×C31—42no—
102C2×C4×C31—42no—
103C2×C4×C31—42no—
104C2×C4×C31—42no—
105C2×C4×C31—41no—
106C2×C4×C31—42no—
107C2×C4×C31—81no—
108C2×C4×C31—81no—
109C2×C4×C31C22⋊Q812yes—
110C2×C4×C31C22⋊Q811yescharacteristic
111C2×C4×C31#12412yes—
112C2×C4×C31#12411yescharacteristic
113C2×C4×C31C2×D811yescharacteristic
114C2×C4×C31C2×SD1611yescharacteristic
115C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
116C2×C4×C31#13711yescharacteristic
117C8×C31—82no—
118C8×C31—81no—
119#2399C2×C3111yescharacteristic, complemented
120#1963C2×C3111yescharacteristic
121#1502C2×C3112yes—
122#1303C2×C3112yescomplemented
123#1274C2×C3111yescharacteristic, complemented
124C22×C31—22no—
125C22×C31—81no—
126C22×C31—82no—
127C22×C31#39311yescharacteristic
128C22×C31#44711yescharacteristic
129C22×C31#44811yescharacteristic
130C4×C31—22no—
131C4×C31—22no—
132C4×C31—42no—
133C4×C31—44no—
134C4×C31—42no—
135C4×C31—82no—
136C4×C31—81no—
137C4×C31#47411yescharacteristic
138C4×C31#47611yescharacteristic
139C4×C31#48611yescharacteristic
140C4×C31#48911yescharacteristic
141#532C22×C3111yescharacteristic
142C2×C4×Q8—21no—
143#431C22×C3112yes—
144#423C22×C3111yescharacteristic
145C2×#104—21no—
146C2×#104—21no—
147#398—22no—
148#395—22no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4C22×C3112yes—
151#376C22×C3111yescharacteristic
152#327—22no—
153#327—22no—
154C2×C31—21no—
155C2×C31—21no—
156C2×C31—161no—
157C2×C31#202311yescharacteristic
158C2×C31#203112yes—
159C2×#41—21no—
160C22×Q8—21no—
161C2×M16—21no—
162C4⋊Q8Q8×C3111yescharacteristic, complemented
163#127Q8×C3111yescharacteristic, complemented
164#124—21no—
165#121—22no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×D4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C42—22no—
174C2×C42D4×C3111yescharacteristic
175C2×C42D4×C3111yescharacteristic
176C2×C42C23×C3111yescharacteristic
177C4⋊C8—22no—
178C4⋊C8—22no—
179#104—42no—
180#104—42no—
181C22⋊C8—22no—
182C4×C8—22no—
183#95—21no—
184#95—21no—
185#95—22no—
186#95D4×C3112yes—
187#95D4×C3112yes—
188C31#1037011yescharacteristic, direct product factor, Sylow
189#41—41no—
190#41—41no—
191#41—41no—
192#41—41no—
193C2×Q8—21no—
194C2×Q8—42no—
195C2×Q8—44no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4—41no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—41no—
204C22×C4C2×D4×C3112yes—
205C22×C4C2×D4×C3111yescharacteristic
206M16—41no—
207M16—41no—
208C2×C8—42no—
209C2×C8—41no—
210C4⋊C4—22no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—42no—
215C22⋊C4—42no—
216C42—21no—
217C42—21no—
218C42—42no—
219C42—42no—
220C42D8×C3112yes—
221C42SD16×C3112yes—
222C42C2×Q8×C3111yescharacteristic
223C42#41×C3112yes—
224C42#41×C3111yescharacteristic
225C23—41no—
226C23C31×#12011yescharacteristic
227Q8—48no—
228Q8—41no—
229Q8—41no—
230D4—41no—
231D4—41no—
232D4—82no—
233D4—81no—
234D4—81no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4C31×C22⋊Q812yes—
250C2×C4C31×C22⋊Q811yescharacteristic
251C2×C4C31×#12412yes—
252C2×C4C31×#12411yescharacteristic
253C2×C4C2×D8×C3111yescharacteristic
254C2×C4C2×SD16×C3111yescharacteristic
255C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
256C2×C4C31×#13711yescharacteristic
257C8—82no—
258C8—81no—
259C22—22no—
260C22—81no—
261C22—82no—
262C22C31×#39311yescharacteristic
263C22C31×#44711yescharacteristic
264C22C31×#44811yescharacteristic
265C4—22no—
266C4—22no—
267C4—44no—
268C4—42no—
269C4—42no—
270C4—82no—
271C4—81no—
272C4C31×#47411yescharacteristic
273C4C31×#47611yescharacteristic
274C4C31×#48611yescharacteristic
275C4C31×#48911yescharacteristic
276C2—21no—
277C2—21no—
278C2—161no—
279C2C31×#202311yescharacteristic
280C2C31×#203112yes—
subgroupG/H
centerC2×C4×C31#124
commutator, Frattini, cosocleC2×C42C23×C31
socleC22×C31#393
prime radicalC31×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2399C2semidirect
4. C31×#1303C2semidirect
5. C31×#1274C2semidirect
22. C31×C4⋊Q8Q8semidirect
23. C31×#127Q8semidirect
84. #10370C31direct (N × H)
119. #2399C2×C31semidirect
122. #1303C2×C31semidirect
123. #1274C2×C31semidirect
162. C4⋊Q8Q8×C31semidirect
163. #127Q8×C31semidirect
188. C31#10370direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#23992
C31×#19632
C31×#15022
C31×#13032
C31×#12742
#1037031