Finite Group Databasebeta

← C31×#13584 #21470343

Subgroups — C31×#13584 #21470343

Order 7936.

474 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3151C211yesmaximal, characteristic
2C2×C31×#353C212yesmaximal, complemented
3C31×#1409C212yesmaximal, complemented
4C31×#1408C212yesmaximal, complemented
5C2×C31×AGL(1,ℤ/8)—22no—
6C31×#532C411yescharacteristic, complemented
7C2×C31×#127C2211yescharacteristic
8C31×#496—22no—
9C31×#493C411yescharacteristic, complemented
10C31×#493C2211yescharacteristic
11C31×#492C411yescharacteristic, complemented
12C31×#492C2211yescharacteristic
13C2×C31×#99—22no—
14C4×M16×C31C411yescharacteristic, complemented
15C31×#353—22no—
16C31×#353C2214yes—
17C2×#41×C31—21no—
18C22×D4×C31—41no—
19C22×D4×C31D412yes—
20C31×AGL(1,ℤ/8)—44no—
21C31×AGL(1,ℤ/8)—44no—
22C2×SD16×C31—22no—
23C2×SD16×C31—22no—
24C2×D8×C31—22no—
25C2×D8×C31—22no—
26C2×M16×C31—21no—
27C31×C4⋊Q8C2×C411yescharacteristic, complemented
28C31×#127—41no—
29C31×#127C2×C411yescharacteristic
30C31×#127C2×C411yescharacteristic, complemented
31C31×#127D412yes—
32C31×#127C2311yescharacteristic
33C31×#124—21no—
34C31×#121—41no—
35C4×D4×C31—41no—
36C2×C22⋊C4×C31—42no—
37C2×C42×C31C2×C411yescharacteristic
38C31×#99—24no—
39C31×#99—42no—
40C31×#97—21no—
41C4×C8×C31C2×C411yescharacteristic, complemented
42C4×C8×C31C2×C411yescharacteristic, complemented
43C24×C31—22no—
44#41×C31—42no—
45#41×C31—42no—
46C2×Q8×C31—21no—
47C2×D4×C31—22no—
48C2×D4×C31—21no—
49C2×D4×C31—42no—
50C2×D4×C31—42no—
51C2×D4×C31—41no—
52C2×D4×C31—42no—
53C2×D4×C31—41no—
54C2×D4×C31—41no—
55C2×D4×C31—41no—
56C2×D4×C31—84no—
57C2×D4×C31C22⋊C412yes—
58C2×D4×C31C2×D412yes—
59C22×C4×C31—21no—
60C22×C4×C31—41no—
61C22×C4×C31—82no—
62C22×C4×C31C22⋊C411yescharacteristic
63SD16×C31—44no—
64SD16×C31—44no—
65D8×C31—44no—
66D8×C31—44no—
67M16×C31—24no—
68C2×C8×C31—21no—
69C2×C8×C31—21no—
70C4⋊C4×C31—41no—
71C22⋊C4×C31—42no—
72C22⋊C4×C31—44no—
73C22⋊C4×C31—82no—
74C42×C31—21no—
75C42×C31C22⋊C411yescharacteristic
76C42×C31C22×C411yescharacteristic
77#13584C3111yesmaximal, characteristic, direct product factor, Sylow
78C23×C31—22no—
79C23×C31—22no—
80C23×C31—42no—
81C23×C31—42no—
82C23×C31—41no—
83C23×C31—84no—
84C23×C31#9911yescharacteristic
85Q8×C31—42no—
86D4×C31—42no—
87D4×C31—42no—
88D4×C31—42no—
89D4×C31—82no—
90D4×C31—82no—
91D4×C31—82no—
92D4×C31—82no—
93D4×C31—84no—
94D4×C31—82no—
95C2×C4×C31—22no—
96C2×C4×C31—21no—
97C2×C4×C31—21no—
98C2×C4×C31—42no—
99C2×C4×C31—41no—
100C2×C4×C31—81no—
101C2×C4×C31—81no—
102C2×C4×C31—84no—
103C2×C4×C31—164no—
104C2×C4×C31#9911yescharacteristic
105C2×C4×C31C2×C22⋊C411yescharacteristic
106C8×C31—22no—
107C8×C31—22no—
108#3151C2×C3111yescharacteristic
109C2×#353C2×C3112yescomplemented
110#1409C2×C3112yescomplemented
111#1408C2×C3112yescomplemented
112C22×C31—41no—
113C22×C31—42no—
114C22×C31—42no—
115C22×C31—82no—
116C22×C31—82no—
117C22×C31—84no—
118C22×C31—82no—
119C22×C31—81no—
120C22×C31—162no—
121C22×C31#35312yes—
122C22×C31C2×#9911yescharacteristic
123C4×C31—22no—
124C4×C31—21no—
125C4×C31—21no—
126C4×C31—41no—
127C4×C31—81no—
128C4×C31—164no—
129C2×AGL(1,ℤ/8)—22no—
130#532C4×C3111yescharacteristic, complemented
131C2×#127C22×C3111yescharacteristic
132#496—22no—
133#493C4×C3111yescharacteristic, complemented
134#493C22×C3111yescharacteristic
135#492C4×C3111yescharacteristic, complemented
136#492C22×C3111yescharacteristic
137C2×#99—22no—
138C4×M16C4×C3111yescharacteristic, complemented
139#353—22no—
140#353C22×C3114yes—
141C2×C31—21no—
142C2×C31—22no—
143C2×C31—82no—
144C2×C31—82no—
145C2×C31—161no—
146C2×C31C2×#35311yescharacteristic
147C2×#41—21no—
148C22×D4—41no—
149C22×D4D4×C3112yes—
150AGL(1,ℤ/8)—44no—
151AGL(1,ℤ/8)—44no—
152C2×SD16—22no—
153C2×SD16—22no—
154C2×D8—22no—
155C2×D8—22no—
156C2×M16—21no—
157C4⋊Q8C2×C4×C3111yescharacteristic, complemented
158#127—41no—
159#127C2×C4×C3111yescharacteristic
160#127C2×C4×C3111yescharacteristic, complemented
161#127D4×C3112yes—
162#127C23×C3111yescharacteristic
163#124—21no—
164#121—41no—
165C4×D4—41no—
166C2×C22⋊C4—42no—
167C2×C42C2×C4×C3111yescharacteristic
168#99—24no—
169#99—42no—
170#97—21no—
171C4×C8C2×C4×C3111yescharacteristic, complemented
172C4×C8C2×C4×C3111yescharacteristic, complemented
173C31#1358411yescharacteristic, direct product factor, Sylow
174C24—22no—
175#41—42no—
176#41—42no—
177C2×Q8—21no—
178C2×D4—22no—
179C2×D4—21no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—41no—
183C2×D4—42no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—84no—
188C2×D4C22⋊C4×C3112yes—
189C2×D4C2×D4×C3112yes—
190C22×C4—21no—
191C22×C4—41no—
192C22×C4—82no—
193C22×C4C22⋊C4×C3111yescharacteristic
194SD16—44no—
195SD16—44no—
196D8—44no—
197D8—44no—
198M16—24no—
199C2×C8—21no—
200C2×C8—21no—
201C4⋊C4—41no—
202C22⋊C4—42no—
203C22⋊C4—44no—
204C22⋊C4—82no—
205C42—21no—
206C42C22⋊C4×C3111yescharacteristic
207C42C22×C4×C3111yescharacteristic
208C23—22no—
209C23—22no—
210C23—42no—
211C23—42no—
212C23—41no—
213C23—84no—
214C23C31×#9911yescharacteristic
215Q8—42no—
216D4—42no—
217D4—42no—
218D4—42no—
219D4—82no—
220D4—82no—
221D4—82no—
222D4—82no—
223D4—84no—
224D4—82no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—84no—
233C2×C4—164no—
234C2×C4C31×#9911yescharacteristic
235C2×C4C2×C22⋊C4×C3111yescharacteristic
236C8—22no—
237C8—22no—
238C22—41no—
239C22—42no—
240C22—42no—
241C22—82no—
242C22—82no—
243C22—84no—
244C22—82no—
245C22—81no—
246C22—162no—
247C22C31×#35312yes—
248C22C2×C31×#9911yescharacteristic
249C4—22no—
250C4—21no—
251C4—21no—
252C4—41no—
253C4—81no—
254C4—164no—
255C2—21no—
256C2—22no—
257C2—82no—
258C2—82no—
259C2—161no—
260C2C2×C31×#35311yescharacteristic
subgroupG/H
center, socleC2×C31C2×#353
commutatorC42C22×C4×C31
Frattini, cosocle#127C23×C31
prime radicalC31×#3151C2
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C23×C31 ◁ C22×C4×C31 ◁ C2×C31×#127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#353C2semidirect
3. C31×#1409C2semidirect
4. C31×#1408C2semidirect
6. C31×#532C4semidirect
9. C31×#493C4semidirect
11. C31×#492C4semidirect
14. C4×M16×C31C4semidirect
27. C31×C4⋊Q8C2×C4semidirect
30. C31×#127C2×C4semidirect
41. C4×C8×C31C2×C4semidirect
42. C4×C8×C31C2×C4semidirect
77. #13584C31direct (N × H)
109. C2×#353C2×C31semidirect
110. #1409C2×C31semidirect
111. #1408C2×C31semidirect
130. #532C4×C31semidirect
133. #493C4×C31semidirect
135. #492C4×C31semidirect
138. C4×M16C4×C31semidirect
157. C4⋊Q8C2×C4×C31semidirect
160. #127C2×C4×C31semidirect
171. C4×C8C2×C4×C31semidirect
172. C4×C8C2×C4×C31semidirect
173. C31#13584direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31512
C2×C31×#3532
C31×#14092
C31×#14082
#1358431