Finite Group Databasebeta

← C31×#11992 #21468751

Subgroups — C31×#11992 #21468751

Order 7936.

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

normal subgroups only

This group has 412 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×#3151C211yesmaximal, characteristic
2C31×#2165C212yesmaximal, complemented
3C31×#1359C212yesmaximal, complemented
4C31×#1357C212yesmaximal, complemented
5C2×C31×AGL(1,ℤ/8)—22no—
6C31×#532C411yescharacteristic
7C2×C31×#127C411yescharacteristic
8C31×#496—22no—
9C31×#493C411yescharacteristic
10C31×#493C2211yescharacteristic
11C31×#492C411yescharacteristic
12C31×#492C2211yescharacteristic
13C4×M16×C31C2211yescharacteristic
14C31×#364C2214yescomplemented
15C31×#349—22no—
16C31×#347—22no—
17C2×#41×C31—21no—
18C22×D4×C31—21no—
19C22×D4×C31—41no—
20C31×AGL(1,ℤ/8)—48no—
21C2×SD16×C31—22no—
22C2×SD16×C31—22no—
23C2×D8×C31—22no—
24C2×D8×C31—22no—
25C2×M16×C31D412yes—
26C31×C4⋊Q8C2×C411yescharacteristic
27C31×#127—21no—
28C31×#127—41no—
29C31×#127C2×C411yescharacteristic
30C31×#127C2×C411yescharacteristic
31C31×#127C2×C411yescharacteristic
32C31×#124—21no—
33C31×#121—41no—
34C4×D4×C31—41no—
35C2×C42×C31C2×C411yescharacteristic
36C31×M32—24no—
37C31×#97D412yes—
38C4×C8×C31C2×C411yescharacteristic
39C4×C8×C31C2311yescharacteristic
40C24×C31—41no—
41#41×C31—44no—
42C2×Q8×C31—21no—
43C2×D4×C31—21no—
44C2×D4×C31—21no—
45C2×D4×C31—21no—
46C2×D4×C31—44no—
47C2×D4×C31—42no—
48C2×D4×C31—41no—
49C2×D4×C31—41no—
50C2×D4×C31—41no—
51C2×D4×C31—41no—
52C2×D4×C31—41no—
53C2×D4×C31—84no—
54C22×C4×C31—21no—
55C22×C4×C31—41no—
56C22×C4×C31C22⋊C411yescharacteristic
57SD16×C31—44no—
58SD16×C31—44no—
59D8×C31—44no—
60D8×C31—44no—
61M16×C31—24no—
62C2×C8×C31C22⋊C412yes—
63C2×C8×C31C2×D412yes—
64C4⋊C4×C31—41no—
65C22⋊C4×C31—42no—
66C42×C31—21no—
67C42×C31C22⋊C411yescharacteristic
68C42×C31C22×C411yescharacteristic
69C16×C31—44no—
70#11992C3111yesmaximal, characteristic, direct product factor, Sylow
71C23×C31—41no—
72C23×C31—41no—
73C23×C31—41no—
74C23×C31—82no—
75C23×C31—84no—
76C23×C31#10011yescharacteristic
77Q8×C31—42no—
78D4×C31—42no—
79D4×C31—42no—
80D4×C31—42no—
81D4×C31—82no—
82D4×C31—82no—
83D4×C31—82no—
84D4×C31—82no—
85D4×C31—84no—
86D4×C31—82no—
87C2×C4×C31—21no—
88C2×C4×C31—22no—
89C2×C4×C31—21no—
90C2×C4×C31—42no—
91C2×C4×C31—41no—
92C2×C4×C31—82no—
93C2×C4×C31#10011yescharacteristic
94C2×C4×C31C2×C22⋊C411yescharacteristic
95C8×C31—22no—
96C8×C31AGL(1,ℤ/8)14yes—
97#3151C2×C3111yescharacteristic
98#2165C2×C3112yescomplemented
99#1359C2×C3112yescomplemented
100#1357C2×C3112yescomplemented
101C22×C31—21no—
102C22×C31—41no—
103C22×C31—82no—
104C22×C31—82no—
105C22×C31—82no—
106C22×C31—81no—
107C22×C31—81no—
108C22×C31—81no—
109C22×C31—164no—
110C22×C31C2×#10011yescharacteristic
111C4×C31—22no—
112C4×C31—21no—
113C4×C31—41no—
114C4×C31—81no—
115C4×C31#41812yes—
116C2×AGL(1,ℤ/8)—22no—
117#532C4×C3111yescharacteristic
118C2×#127C4×C3111yescharacteristic
119#496—22no—
120#493C4×C3111yescharacteristic
121#493C22×C3111yescharacteristic
122#492C4×C3111yescharacteristic
123#492C22×C3111yescharacteristic
124C4×M16C22×C3111yescharacteristic
125#364C22×C3114yescomplemented
126#349—22no—
127#347—22no—
128C2×C31—21no—
129C2×C31—41no—
130C2×C31—161no—
131C2×C31—161no—
132C2×C31—161no—
133C2×C31#155511yescharacteristic
134C2×#41—21no—
135C22×D4—21no—
136C22×D4—41no—
137AGL(1,ℤ/8)—48no—
138C2×SD16—22no—
139C2×SD16—22no—
140C2×D8—22no—
141C2×D8—22no—
142C2×M16D4×C3112yes—
143C4⋊Q8C2×C4×C3111yescharacteristic
144#127—21no—
145#127—41no—
146#127C2×C4×C3111yescharacteristic
147#127C2×C4×C3111yescharacteristic
148#127C2×C4×C3111yescharacteristic
149#124—21no—
150#121—41no—
151C4×D4—41no—
152C2×C42C2×C4×C3111yescharacteristic
153M32—24no—
154#97D4×C3112yes—
155C4×C8C2×C4×C3111yescharacteristic
156C4×C8C23×C3111yescharacteristic
157C31#1199211yescharacteristic, direct product factor, Sylow
158C24—41no—
159#41—44no—
160C2×Q8—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—44no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—42no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—84no—
172C22×C4—21no—
173C22×C4—41no—
174C22×C4C22⋊C4×C3111yescharacteristic
175SD16—44no—
176SD16—44no—
177D8—44no—
178D8—44no—
179M16—24no—
180C2×C8C22⋊C4×C3112yes—
181C2×C8C2×D4×C3112yes—
182C4⋊C4—41no—
183C22⋊C4—42no—
184C42—21no—
185C42C22⋊C4×C3111yescharacteristic
186C42C22×C4×C3111yescharacteristic
187C16—44no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—82no—
192C23—84no—
193C23C31×#10011yescharacteristic
194Q8—42no—
195D4—42no—
196D4—42no—
197D4—42no—
198D4—82no—
199D4—82no—
200D4—82no—
201D4—82no—
202D4—84no—
203D4—82no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—82no—
210C2×C4C31×#10011yescharacteristic
211C2×C4C2×C22⋊C4×C3111yescharacteristic
212C8—22no—
213C8C31×AGL(1,ℤ/8)14yes—
214C22—21no—
215C22—41no—
216C22—82no—
217C22—82no—
218C22—82no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22—164no—
223C22C2×C31×#10011yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—81no—
228C4C31×#41812yes—
229C2—21no—
230C2—41no—
231C2—161no—
232C2—161no—
233C2—161no—
234C2C31×#155511yescharacteristic
subgroupG/H
center, socleC2×C31#1555
commutatorC42C22×C4×C31
Frattini, cosocleC4×C8C23×C31
prime radicalC31×#3151C2
LCSG ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C4×M16×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2165C2semidirect
3. C31×#1359C2semidirect
4. C31×#1357C2semidirect
14. C31×#364C22semidirect
70. #11992C31direct (N × H)
98. #2165C2×C31semidirect
99. #1359C2×C31semidirect
100. #1357C2×C31semidirect
125. #364C22×C31semidirect
157. C31#11992direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31512
C31×#21652
C31×#13592
C31×#13572
#1199231