Finite Group Databasebeta

← C31×#32790 #21489549

Subgroups — C31×#32790 #21489549

Order 7936.

716 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 234 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×#2857C212yesmaximal
2C31×#2100C214yesmaximal, complemented
3C31×#2076C214yesmaximal
4C31×#2057C214yesmaximal, complemented
5C31×#1817C211yesmaximal, characteristic
6C2×C31×C4⋊Q8C2212yes—
7C2×C31×C22⋊Q8—22no—
8C22×C42×C31C2211yescharacteristic
9C2×C31×#107C2214yes—
10C2×C31×C22⋊C8C2212yes—
11C31×#391—22no—
12C31×#389—22no—
13C31×#389C2214yes—
14C31×#384C2212yes—
15C31×#380—22no—
16C31×#380C2214yes—
17C31×#340C2218yes—
18C31×#340C2214yes—
19C31×#336C2214yes—
20C22×Q8×C31—22no—
21C23×C4×C31D411yescharacteristic
22C23×C4×C31D411yescharacteristic
23C23×C4×C31C2311yescharacteristic
24C22×C8×C31C2314yes—
25C31×C4⋊Q8—44no—
26C31×C22⋊Q8—48no—
27C2×C4⋊C4×C31—22no—
28C2×C4⋊C4×C31—24no—
29C2×C4⋊C4×C31—22no—
30C2×C4⋊C4×C31—22no—
31C2×C4⋊C4×C31C2314yes—
32C2×C4⋊C4×C31C2314yes—
33C2×C22⋊C4×C31—24no—
34C2×C42×C31—42no—
35C2×C42×C31D412yes—
36C2×C42×C31C2312yes—
37C31×#107—44no—
38C31×C22⋊C8—42no—
39C31×#95—24no—
40C31×#95—24no—
41C24×C31C2×D411yescharacteristic
42C2×Q8×C31—44no—
43C2×Q8×C31—44no—
44C2×Q8×C31—44no—
45C22×C4×C31—24no—
46C22×C4×C31—22no—
47C22×C4×C31—24no—
48C22×C4×C31—24no—
49C22×C4×C31—24no—
50C22×C4×C31—41no—
51C22×C4×C31—41no—
52C22×C4×C31—41no—
53C22×C4×C31Q1614yes—
54C22×C4×C31C2×D411yescharacteristic
55C22×C4×C31C2×D412yes—
56C22×C4×C31C2×D412yes—
57C22×C4×C31C2411yescharacteristic
58C2×C8×C31—24no—
59C2×C8×C31—28no—
60C4⋊C4×C31—28no—
61C4⋊C4×C31—28no—
62C4⋊C4×C31—44no—
63C4⋊C4×C31—44no—
64C4⋊C4×C31—44no—
65C4⋊C4×C31—48no—
66C22⋊C4×C31—48no—
67C42×C31—24no—
68C42×C31—42no—
69#32790C3111yesmaximal, characteristic, direct product factor, Sylow
70C23×C31—21no—
71C23×C31—22no—
72C23×C31—22no—
73C23×C31—21no—
74C23×C31C2×Q1612yes—
75C23×C31C22×D411yescharacteristic
76Q8×C31—88no—
77C2×C4×C31—22no—
78C2×C4×C31—22no—
79C2×C4×C31—22no—
80C2×C4×C31—22no—
81C2×C4×C31—21no—
82C2×C4×C31—22no—
83C2×C4×C31—44no—
84C2×C4×C31—44no—
85C2×C4×C31—44no—
86C2×C4×C31—44no—
87C2×C4×C31—44no—
88C2×C4×C31—44no—
89C2×C4×C31—44no—
90C2×C4×C31—44no—
91C2×C4×C31—44no—
92C2×C4×C31C2×Q1614yes—
93C2×C4×C31AGL(1,ℤ/8)14yes—
94C2×C4×C31H(5,𝔽2)12yes—
95C2×C4×C312−1+414yes—
96C8×C31—44no—
97#2857C2×C3112yes—
98#2100C2×C3114yescomplemented
99#2076C2×C3114yes—
100#2057C2×C3114yescomplemented
101#1817C2×C3111yescharacteristic
102C22×C31—22no—
103C22×C31—24no—
104C22×C31—41no—
105C22×C31—41no—
106C22×C31—42no—
107C22×C31#53411yescharacteristic
108C22×C31#53612yes—
109C22×C31C22×Q1611yescharacteristic
110C22×C31C2×AGL(1,ℤ/8)12yes—
111C22×C31#57511yescharacteristic
112C4×C31—24no—
113C4×C31—24no—
114C4×C31—41no—
115C4×C31—84no—
116C4×C31—84no—
117C4×C31#56314yes—
118C2×C4⋊Q8C22×C3112yes—
119C2×C22⋊Q8—22no—
120C22×C42C22×C3111yescharacteristic
121C2×#107C22×C3114yes—
122C2×C22⋊C8C22×C3112yes—
123#391—22no—
124#389—22no—
125#389C22×C3114yes—
126#384C22×C3112yes—
127#380—22no—
128#380C22×C3114yes—
129#340C22×C3118yes—
130#340C22×C3114yes—
131#336C22×C3114yes—
132C2×C31—42no—
133C2×C31#281711yescharacteristic
134C2×C31#318911yescharacteristic
135C2×C31#319211yescharacteristic
136C2×C31#323012yes—
137C2×C31#323512yes—
138C22×Q8—22no—
139C23×C4D4×C3111yescharacteristic
140C23×C4D4×C3111yescharacteristic
141C23×C4C23×C3111yescharacteristic
142C22×C8C23×C3114yes—
143C4⋊Q8—44no—
144C22⋊Q8—48no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4C23×C3114yes—
150C2×C4⋊C4C23×C3114yes—
151C2×C22⋊C4—24no—
152C2×C42—42no—
153C2×C42D4×C3112yes—
154C2×C42C23×C3112yes—
155#107—44no—
156C22⋊C8—42no—
157#95—24no—
158#95—24no—
159C31#3279011yescharacteristic, direct product factor, Sylow
160C24C2×D4×C3111yescharacteristic
161C2×Q8—44no—
162C2×Q8—44no—
163C2×Q8—44no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4Q16×C3114yes—
173C22×C4C2×D4×C3111yescharacteristic
174C22×C4C2×D4×C3112yes—
175C22×C4C2×D4×C3112yes—
176C22×C4C24×C3111yescharacteristic
177C2×C8—24no—
178C2×C8—28no—
179C4⋊C4—28no—
180C4⋊C4—28no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—48no—
185C22⋊C4—48no—
186C42—24no—
187C42—42no—
188C23—21no—
189C23—22no—
190C23—22no—
191C23—21no—
192C23C2×Q16×C3112yes—
193C23C22×D4×C3111yescharacteristic
194Q8—88no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4C2×Q16×C3114yes—
211C2×C4C31×AGL(1,ℤ/8)14yes—
212C2×C4C31×H(5,𝔽2)12yes—
213C2×C4C31×2−1+414yes—
214C8—44no—
215C22—22no—
216C22—24no—
217C22—41no—
218C22—42no—
219C22—41no—
220C22C31×#53411yescharacteristic
221C22C31×#53612yes—
222C22C22×Q16×C3111yescharacteristic
223C22C2×C31×AGL(1,ℤ/8)12yes—
224C22C31×#57511yescharacteristic
225C4—24no—
226C4—24no—
227C4—41no—
228C4—84no—
229C4—84no—
230C4C31×#56314yes—
231C2—42no—
232C2C31×#281711yescharacteristic
233C2C31×#318911yescharacteristic
234C2C31×#319211yescharacteristic
235C2C31×#323012yes—
236C2C31×#323512yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2100C2semidirect
4. C31×#2057C2semidirect
69. #32790C31direct (N × H)
98. #2100C2×C31semidirect
100. #2057C2×C31semidirect
159. C31#32790direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28572
C31×#21002
C31×#20762
C31×#20572
C31×#18172
#3279031