Finite Group Databasebeta

← C31×#10015 #21466774

Subgroups — C31×#10015 #21466774

Order 7936.

696 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3101C212yesmaximal, complemented
2C31×#2397C211yesmaximal, characteristic
3C31×#1995C211yesmaximal, characteristic, complemented
4C31×#1486C212yesmaximal, complemented
5C31×#1274C211yesmaximal, characteristic, complemented
6C31×#533C2211yescharacteristic
7C2×C4×D4×C31—22no—
8C2×C4×D4×C31C2212yes—
9C31×#483—22no—
10C31×#476—22no—
11C31×#461—22no—
12C31×#450—22no—
13C4×SD16×C31—24no—
14C31×#431C2212yes—
15C31×#423C2211yescharacteristic
16C31×#399—22no—
17C31×#392—22no—
18C31×#376C2211yescharacteristic
19C31×#339—22no—
20C31×D4⋊C8—24no—
21C22×D4×C31—22no—
22C22×D4×C31D412yes—
23C23×C4×C31—22no—
24C2×SD16×C31—24no—
25C2×M16×C31—21no—
26C31×C4⋊Q8D412yescomplemented
27C31×#125—21no—
28C31×C22⋊Q8—22no—
29C4×Q8×C31—22no—
30C4×D4×C31—22no—
31C4×D4×C31—22no—
32C4×D4×C31—22no—
33C4×D4×C31—22no—
34C4×D4×C31—44no—
35C4×D4×C31—44no—
36C31×#117—21no—
37C2×C4⋊C4×C31—22no—
38C2×C4⋊C4×C31D412yes—
39C2×C22⋊C4×C31—24no—
40C2×C22⋊C4×C31—22no—
41C2×C42×C31D412yes—
42C2×C42×C31C2311yescharacteristic
43C31×#106—24no—
44C31×C4⋊C8—22no—
45C31×C4⋊C8—22no—
46C31×#103—24no—
47C31×#102—24no—
48C31×C22⋊C8—22no—
49C4×C8×C31D414yescomplemented
50C31×#95—22no—
51C24×C31—22no—
52C2×Q8×C31—22no—
53C2×D4×C31—22no—
54C2×D4×C31—22no—
55C2×D4×C31—44no—
56C2×D4×C31—44no—
57C2×D4×C31—48no—
58C22×C4×C31—22no—
59C22×C4×C31—22no—
60C22×C4×C31—42no—
61C22×C4×C31—42no—
62C22×C4×C31—44no—
63C22×C4×C31—44no—
64C22×C4×C31C2×D412yes—
65C22×C4×C31C2×D411yescharacteristic
66SD16×C31—48no—
67M16×C31—42no—
68C2×C8×C31—24no—
69C2×C8×C31—41no—
70C4⋊C4×C31—22no—
71C4⋊C4×C31—22no—
72C4⋊C4×C31—22no—
73C4⋊C4×C31—22no—
74C4⋊C4×C31—44no—
75C4⋊C4×C31—42no—
76C22⋊C4×C31—48no—
77C22⋊C4×C31—41no—
78C22⋊C4×C31—42no—
79C22⋊C4×C31—42no—
80C42×C31—22no—
81C42×C31—41no—
82C42×C31SD1614yescomplemented
83C42×C31C2×D411yescharacteristic
84C42×C31C2×D412yes—
85C42×C31#4111yescharacteristic
86#10015C3111yesmaximal, characteristic, direct product factor, Sylow
87C23×C31—24no—
88C23×C31—44no—
89C23×C31—42no—
90C23×C31—42no—
91C23×C31—44no—
92C23×C31#12011yescharacteristic
93Q8×C31—44no—
94D4×C31—24no—
95D4×C31—24no—
96D4×C31—42no—
97D4×C31—42no—
98D4×C31—416no—
99C2×C4×C31—24no—
100C2×C4×C31—42no—
101C2×C4×C31—44no—
102C2×C4×C31—42no—
103C2×C4×C31—42no—
104C2×C4×C31—44no—
105C2×C4×C31—84no—
106C2×C4×C31—82no—
107C2×C4×C31—82no—
108C2×C4×C31#12012yes—
109C2×C4×C31#12112yes—
110C2×C4×C31#12111yescharacteristic
111C2×C4×C31#12711yescharacteristic
112C2×C4×C31C2×SD1612yes—
113C2×C4×C31AGL(1,ℤ/8)12yes—
114C8×C31—44no—
115C8×C31—81no—
116#3101C2×C3112yescomplemented
117#2397C2×C3111yescharacteristic
118#1995C2×C3111yescharacteristic, complemented
119#1486C2×C3112yescomplemented
120#1274C2×C3111yescharacteristic, complemented
121C22×C31—22no—
122C22×C31—48no—
123C22×C31—44no—
124C22×C31—84no—
125C22×C31—82no—
126C22×C31—82no—
127C22×C31#39211yescharacteristic
128C22×C31#45012yes—
129C4×C31—22no—
130C4×C31—22no—
131C4×C31—44no—
132C4×C31—82no—
133C4×C31—82no—
134C4×C31#46012yes—
135C4×C31#49211yescharacteristic
136C4×C31#49611yescharacteristic
137#533C22×C3111yescharacteristic
138C2×C4×D4—22no—
139C2×C4×D4C22×C3112yes—
140#483—22no—
141#476—22no—
142#461—22no—
143#450—22no—
144C4×SD16—24no—
145#431C22×C3112yes—
146#423C22×C3111yescharacteristic
147#399—22no—
148#392—22no—
149#376C22×C3111yescharacteristic
150#339—22no—
151D4⋊C8—24no—
152C2×C31—22no—
153C2×C31—84no—
154C2×C31#201012yes—
155C2×C31#201411yescharacteristic
156C22×D4—22no—
157C22×D4D4×C3112yes—
158C23×C4—22no—
159C2×SD16—24no—
160C2×M16—21no—
161C4⋊Q8D4×C3112yescomplemented
162#125—21no—
163C22⋊Q8—22no—
164C4×Q8—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—44no—
170C4×D4—44no—
171#117—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4D4×C3112yes—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—22no—
176C2×C42D4×C3112yes—
177C2×C42C23×C3111yescharacteristic
178#106—24no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181#103—24no—
182#102—24no—
183C22⋊C8—22no—
184C4×C8D4×C3114yescomplemented
185#95—22no—
186C31#1001511yescharacteristic, direct product factor, Sylow
187C24—22no—
188C2×Q8—22no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—48no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4C2×D4×C3112yes—
201C22×C4C2×D4×C3111yescharacteristic
202SD16—48no—
203M16—42no—
204C2×C8—24no—
205C2×C8—41no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—44no—
211C4⋊C4—42no—
212C22⋊C4—48no—
213C22⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C42—22no—
217C42—41no—
218C42SD16×C3114yescomplemented
219C42C2×D4×C3111yescharacteristic
220C42C2×D4×C3112yes—
221C42#41×C3111yescharacteristic
222C23—24no—
223C23—44no—
224C23—42no—
225C23—44no—
226C23—42no—
227C23C31×#12011yescharacteristic
228Q8—44no—
229D4—24no—
230D4—24no—
231D4—42no—
232D4—42no—
233D4—416no—
234C2×C4—24no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—84no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4C31×#12012yes—
244C2×C4C31×#12112yes—
245C2×C4C31×#12111yescharacteristic
246C2×C4C31×#12711yescharacteristic
247C2×C4C2×SD16×C3112yes—
248C2×C4C31×AGL(1,ℤ/8)12yes—
249C8—44no—
250C8—81no—
251C22—22no—
252C22—48no—
253C22—44no—
254C22—84no—
255C22—82no—
256C22—82no—
257C22C31×#39211yescharacteristic
258C22C31×#45012yes—
259C4—22no—
260C4—22no—
261C4—44no—
262C4—82no—
263C4—82no—
264C4C31×#46012yes—
265C4C31×#49211yescharacteristic
266C4C31×#49611yescharacteristic
267C2—22no—
268C2—84no—
269C2C31×#201012yes—
270C2C31×#201411yescharacteristic
subgroupG/H
centerC2×C4×C31#127
commutator, Frattini, cosocleC2×C42C23×C31
socleC22×C31#392
prime radicalC31×#2397C2
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×#3101C2semidirect
3. C31×#1995C2semidirect
4. C31×#1486C2semidirect
5. C31×#1274C2semidirect
26. C31×C4⋊Q8D4semidirect
49. C4×C8×C31D4semidirect
82. C42×C31SD16semidirect
86. #10015C31direct (N × H)
116. #3101C2×C31semidirect
118. #1995C2×C31semidirect
119. #1486C2×C31semidirect
120. #1274C2×C31semidirect
161. C4⋊Q8D4×C31semidirect
184. C4×C8D4×C31semidirect
186. C31#10015direct (N × H)
218. C42SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#31012
C31×#23972
C31×#19952
C31×#14862
C31×#12742
#1001531