Finite Group Databasebeta

← C31×#32762 #21489521

Subgroups — C31×#32762 #21489521

Order 7936.

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

normal subgroups only

This group has 812 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×#2072C214yesmaximal, complemented
3C31×#2016C214yesmaximal, complemented
4C31×#2001C214yesmaximal
5C31×#1817C211yesmaximal, characteristic
6C22×Q16×C31C2214yes—
7C2×C31×C4⋊Q8C2212yes—
8C2×C31×C22⋊Q8C2214yes—
9C22×C42×C31C2211yescharacteristic
10C2×C31×#103C2218yes—
11C2×C31×C22⋊C8C2212yes—
12C31×#391C2214yes—
13C31×#389—22no—
14C31×#389—22no—
15C31×#384C2212yes—
16C31×#380—22no—
17C31×#380—22no—
18C31×#340C2214yes—
19C31×#336C2214yes—
20C22×Q8×C31C2314yes—
21C23×C4×C31D411yescharacteristic
22C23×C4×C31D411yescharacteristic
23C23×C4×C31C2311yescharacteristic
24C2×Q16×C31—44no—
25C2×Q16×C31—48no—
26C22×C8×C31C2314yes—
27C31×C4⋊Q8—44no—
28C31×C22⋊Q8—48no—
29C2×C4⋊C4×C31—24no—
30C2×C4⋊C4×C31—22no—
31C2×C4⋊C4×C31—22no—
32C2×C4⋊C4×C31—22no—
33C2×C4⋊C4×C31—22no—
34C2×C4⋊C4×C31C2314yes—
35C2×C22⋊C4×C31—24no—
36C2×C42×C31—42no—
37C2×C42×C31D412yes—
38C2×C42×C31C2312yes—
39C31×#103—48no—
40C31×C22⋊C8—42no—
41C31×#95—24no—
42C31×#95—24no—
43C24×C31C2×D411yescharacteristic
44C2×Q8×C31—24no—
45C2×Q8×C31—28no—
46C2×Q8×C31—28no—
47C2×Q8×C31—24no—
48C22×C4×C31—24no—
49C22×C4×C31—22no—
50C22×C4×C31—24no—
51C22×C4×C31—24no—
52C22×C4×C31—24no—
53C22×C4×C31—41no—
54C22×C4×C31—41no—
55C22×C4×C31—41no—
56C22×C4×C31Q1614yescomplemented
57C22×C4×C31C2×D411yescharacteristic
58C22×C4×C31C2×D412yes—
59C22×C4×C31C2×D412yes—
60C22×C4×C31C2411yescharacteristic
61Q16×C31—416no—
62C2×C8×C31—24no—
63C2×C8×C31—28no—
64C4⋊C4×C31—28no—
65C4⋊C4×C31—44no—
66C4⋊C4×C31—48no—
67C4⋊C4×C31—44no—
68C4⋊C4×C31—44no—
69C4⋊C4×C31—44no—
70C22⋊C4×C31—48no—
71C42×C31—24no—
72C42×C31—42no—
73#32762C3111yesmaximal, characteristic, direct product factor, Sylow
74C23×C31—21no—
75C23×C31—22no—
76C23×C31—22no—
77C23×C31—21no—
78C23×C31C2×Q1612yes—
79C23×C31C22×D411yescharacteristic
80Q8×C31—48no—
81Q8×C31—48no—
82C2×C4×C31—22no—
83C2×C4×C31—22no—
84C2×C4×C31—22no—
85C2×C4×C31—22no—
86C2×C4×C31—21no—
87C2×C4×C31—22no—
88C2×C4×C31—44no—
89C2×C4×C31—44no—
90C2×C4×C31—44no—
91C2×C4×C31—44no—
92C2×C4×C31—44no—
93C2×C4×C31—44no—
94C2×C4×C31—44no—
95C2×C4×C31—44no—
96C2×C4×C31—44no—
97C2×C4×C31C2×Q1614yes—
98C2×C4×C31#13714yes—
99C2×C4×C31H(5,𝔽2)12yes—
100C2×C4×C31H(5,𝔽2)14yes—
101C8×C31—44no—
102#2857C2×C3112yes—
103#2072C2×C3114yescomplemented
104#2016C2×C3114yescomplemented
105#2001C2×C3114yes—
106#1817C2×C3111yescharacteristic
107C22×C31—22no—
108C22×C31—24no—
109C22×C31—41no—
110C22×C31—42no—
111C22×C31—41no—
112C22×C31#53411yescharacteristic
113C22×C31#53512yes—
114C22×C31C22×Q1611yescharacteristic
115C22×C31C2×#13712yes—
116C22×C31#57511yescharacteristic
117C4×C31—24no—
118C4×C31—24no—
119C4×C31—41no—
120C4×C31—84no—
121C4×C31—84no—
122C4×C31#56014yes—
123C22×Q16C22×C3114yes—
124C2×C4⋊Q8C22×C3112yes—
125C2×C22⋊Q8C22×C3114yes—
126C22×C42C22×C3111yescharacteristic
127C2×#103C22×C3118yes—
128C2×C22⋊C8C22×C3112yes—
129#391C22×C3114yes—
130#389—22no—
131#389—22no—
132#384C22×C3112yes—
133#380—22no—
134#380—22no—
135#340C22×C3114yes—
136#336C22×C3114yes—
137C2×C31—42no—
138C2×C31#281511yescharacteristic
139C2×C31#318911yescharacteristic
140C2×C31#319511yescharacteristic
141C2×C31#320912yes—
142C2×C31#321612yes—
143C22×Q8C23×C3114yes—
144C23×C4D4×C3111yescharacteristic
145C23×C4D4×C3111yescharacteristic
146C23×C4C23×C3111yescharacteristic
147C2×Q16—44no—
148C2×Q16—48no—
149C22×C8C23×C3114yes—
150C4⋊Q8—44no—
151C22⋊Q8—48no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C23×C3114yes—
158C2×C22⋊C4—24no—
159C2×C42—42no—
160C2×C42D4×C3112yes—
161C2×C42C23×C3112yes—
162#103—48no—
163C22⋊C8—42no—
164#95—24no—
165#95—24no—
166C31#3276211yescharacteristic, direct product factor, Sylow
167C24C2×D4×C3111yescharacteristic
168C2×Q8—24no—
169C2×Q8—28no—
170C2×Q8—28no—
171C2×Q8—24no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4Q16×C3114yescomplemented
181C22×C4C2×D4×C3111yescharacteristic
182C22×C4C2×D4×C3112yes—
183C22×C4C2×D4×C3112yes—
184C22×C4C24×C3111yescharacteristic
185Q16—416no—
186C2×C8—24no—
187C2×C8—28no—
188C4⋊C4—28no—
189C4⋊C4—44no—
190C4⋊C4—48no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C22⋊C4—48no—
195C42—24no—
196C42—42no—
197C23—21no—
198C23—22no—
199C23—22no—
200C23—21no—
201C23C2×Q16×C3112yes—
202C23C22×D4×C3111yescharacteristic
203Q8—48no—
204Q8—48no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4C2×Q16×C3114yes—
221C2×C4C31×#13714yes—
222C2×C4C31×H(5,𝔽2)12yes—
223C2×C4C31×H(5,𝔽2)14yes—
224C8—44no—
225C22—22no—
226C22—24no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22C31×#53411yescharacteristic
231C22C31×#53512yes—
232C22C22×Q16×C3111yescharacteristic
233C22C2×C31×#13712yes—
234C22C31×#57511yescharacteristic
235C4—24no—
236C4—24no—
237C4—41no—
238C4—84no—
239C4—84no—
240C4C31×#56014yes—
241C2—42no—
242C2C31×#281511yescharacteristic
243C2C31×#318911yescharacteristic
244C2C31×#319511yescharacteristic
245C2C31×#320912yes—
246C2C31×#321612yes—
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×#2072C2semidirect
3. C31×#2016C2semidirect
56. C22×C4×C31Q16semidirect
73. #32762C31direct (N × H)
103. #2072C2×C31semidirect
104. #2016C2×C31semidirect
166. C31#32762direct (N × H)
180. C22×C4Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#28572
C31×#20722
C31×#20162
C31×#20012
C31×#18172
#3276231