Finite Group Databasebeta

← C31×#32804 #21489563

Subgroups — C31×#32804 #21489563

Order 7936.

768 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 206 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×#2658C211yesmaximal, characteristic, complemented
2C31×#2644C211yesmaximal, characteristic, complemented
3C31×#2043C213yesmaximal, complemented
4C31×#2042C213yesmaximal, complemented
5C31×#2013C213yesmaximal, complemented
6C31×#2012C213yesmaximal
7C31×#1840C211yesmaximal, characteristic, complemented
8C22×SD16×C31C2213yes—
9C22×D8×C31C2211yescharacteristic
10C2×C31×C4⋊Q8C2213yes—
11C2×C31×#124C2213yes—
12C2×C31×#103C2213yes—
13C2×C31×#103C2213yes—
14C2×C31×#103C2213yes—
15C2×C31×#102C2213yes—
16C31×#399—21no—
17C31×#399—23no—
18C31×#397—23no—
19C31×#396—21no—
20C31×#390C2213yes—
21C31×#389C2213yes—
22C31×#376C2211yescharacteristic
23C31×#336C2213yes—
24C31×#336C2213yes—
25C22×Q8×C31C2313yes—
26C22×D4×C31C2311yescharacteristic
27C22×D4×C31C2311yescharacteristic
28C2×SD16×C31—43no—
29C2×SD16×C31—43no—
30C2×SD16×C31—43no—
31C2×D8×C31—43no—
32C22×C8×C31C2313yes—
33C22×C8×C31C2311yescharacteristic
34C31×C4⋊Q8—43no—
35C31×C4⋊Q8—43no—
36C31×#124—43no—
37C31×#124—43no—
38C2×C4⋊C4×C31—23no—
39C2×C4⋊C4×C31—23no—
40C2×C4⋊C4×C31—23no—
41C2×C4⋊C4×C31C2313yes—
42C2×C22⋊C4×C31—23no—
43C2×C22⋊C4×C31—23no—
44C2×C22⋊C4×C31—23no—
45C2×C22⋊C4×C31—23no—
46C2×C42×C31C2313yes—
47C31×#103—43no—
48C31×#103—43no—
49C31×#103—43no—
50C31×#102—43no—
51C31×#95—23no—
52C31×#95D411yescharacteristic
53C31×#95D413yes—
54C24×C31—21no—
55C24×C31—21no—
56C2×Q8×C31—23no—
57C2×Q8×C31—23no—
58C2×Q8×C31—23no—
59C2×Q8×C31—23no—
60C2×Q8×C31—23no—
61C2×Q8×C31—23no—
62C2×D4×C31—23no—
63C2×D4×C31—23no—
64C2×D4×C31—23no—
65C2×D4×C31—23no—
66C22×C4×C31—23no—
67C22×C4×C31—23no—
68C22×C4×C31C2×D413yes—
69C22×C4×C31C2×D413yes—
70C22×C4×C31C2411yescharacteristic
71SD16×C31—412no—
72D8×C31—44no—
73C2×C8×C31—23no—
74C2×C8×C31—23no—
75C2×C8×C31—23no—
76C2×C8×C31—23no—
77C4⋊C4×C31—23no—
78C4⋊C4×C31—23no—
79C4⋊C4×C31—46no—
80C4⋊C4×C31—46no—
81C4⋊C4×C31—83no—
82C22⋊C4×C31—46no—
83C22⋊C4×C31—46no—
84C22⋊C4×C31—83no—
85C22⋊C4×C31—83no—
86C42×C31—23no—
87C42×C31—23no—
88#32804C3111yesmaximal, characteristic, direct product factor, Sylow
89C23×C31—43no—
90C23×C31—43no—
91C23×C31—43no—
92C23×C31—43no—
93C23×C31—41no—
94C23×C31—41no—
95C23×C31C22×D411yescharacteristic
96Q8×C31—43no—
97Q8×C31—43no—
98Q8×C31—43no—
99Q8×C31—43no—
100D4×C31—41no—
101D4×C31—41no—
102D4×C31—43no—
103D4×C31—43no—
104C2×C4×C31—23no—
105C2×C4×C31—23no—
106C2×C4×C31—23no—
107C2×C4×C31—23no—
108C2×C4×C31—23no—
109C2×C4×C31—23no—
110C2×C4×C31—43no—
111C2×C4×C31—43no—
112C2×C4×C31—43no—
113C2×C4×C31—43no—
114C2×C4×C31—43no—
115C2×C4×C31—43no—
116C2×C4×C31H(5,𝔽2)13yes—
117C2×C4×C31H(5,𝔽2)13yes—
118C8×C31—41no—
119C8×C31—43no—
120#2658C2×C3111yescharacteristic, complemented
121#2644C2×C3111yescharacteristic, complemented
122#2043C2×C3113yescomplemented
123#2042C2×C3113yescomplemented
124#2013C2×C3113yescomplemented
125#2012C2×C3113yes—
126#1840C2×C3111yescharacteristic, complemented
127C22×C31—81no—
128C22×C31—81no—
129C22×C31—83no—
130C22×C31—83no—
131C22×C31—83no—
132C22×C31—83no—
133C22×C31#53513yes—
134C22×C31#57713yes—
135C22×C31#57811yescharacteristic
136C4×C31—43no—
137C4×C31—43no—
138C4×C31—83no—
139C4×C31—83no—
140C4×C31#56011yescharacteristic
141C4×C31#56013yes—
142C22×SD16C22×C3113yes—
143C22×D8C22×C3111yescharacteristic
144C2×C4⋊Q8C22×C3113yes—
145C2×#124C22×C3113yes—
146C2×#103C22×C3113yes—
147C2×#103C22×C3113yes—
148C2×#103C22×C3113yes—
149C2×#102C22×C3113yes—
150#399—21no—
151#399—23no—
152#397—23no—
153#396—21no—
154#390C22×C3113yes—
155#389C22×C3113yes—
156#376C22×C3111yescharacteristic
157#336C22×C3113yes—
158#336C22×C3113yes—
159C2×C31—161no—
160C2×C31—161no—
161C2×C31#281811yescharacteristic
162C2×C31#322113yes—
163C2×C31#322513yes—
164C22×Q8C23×C3113yes—
165C22×D4C23×C3111yescharacteristic
166C22×D4C23×C3111yescharacteristic
167C2×SD16—43no—
168C2×SD16—43no—
169C2×SD16—43no—
170C2×D8—43no—
171C22×C8C23×C3111yescharacteristic
172C22×C8C23×C3113yes—
173C4⋊Q8—43no—
174C4⋊Q8—43no—
175#124—43no—
176#124—43no—
177C2×C4⋊C4—23no—
178C2×C4⋊C4—23no—
179C2×C4⋊C4—23no—
180C2×C4⋊C4C23×C3113yes—
181C2×C22⋊C4—23no—
182C2×C22⋊C4—23no—
183C2×C22⋊C4—23no—
184C2×C22⋊C4—23no—
185C2×C42C23×C3113yes—
186#103—43no—
187#103—43no—
188#103—43no—
189#102—43no—
190#95—23no—
191#95D4×C3111yescharacteristic
192#95D4×C3113yes—
193C31#3280411yescharacteristic, direct product factor, Sylow
194C24—21no—
195C24—21no—
196C2×Q8—23no—
197C2×Q8—23no—
198C2×Q8—23no—
199C2×Q8—23no—
200C2×Q8—23no—
201C2×Q8—23no—
202C2×D4—23no—
203C2×D4—23no—
204C2×D4—23no—
205C2×D4—23no—
206C22×C4—23no—
207C22×C4—23no—
208C22×C4C2×D4×C3113yes—
209C22×C4C2×D4×C3113yes—
210C22×C4C24×C3111yescharacteristic
211SD16—412no—
212D8—44no—
213C2×C8—23no—
214C2×C8—23no—
215C2×C8—23no—
216C2×C8—23no—
217C4⋊C4—23no—
218C4⋊C4—23no—
219C4⋊C4—46no—
220C4⋊C4—46no—
221C4⋊C4—83no—
222C22⋊C4—46no—
223C22⋊C4—46no—
224C22⋊C4—83no—
225C22⋊C4—83no—
226C42—23no—
227C42—23no—
228C23—43no—
229C23—43no—
230C23—43no—
231C23—43no—
232C23—41no—
233C23—41no—
234C23C22×D4×C3111yescharacteristic
235Q8—43no—
236Q8—43no—
237Q8—43no—
238Q8—43no—
239D4—41no—
240D4—41no—
241D4—43no—
242D4—43no—
243C2×C4—23no—
244C2×C4—23no—
245C2×C4—23no—
246C2×C4—23no—
247C2×C4—23no—
248C2×C4—23no—
249C2×C4—43no—
250C2×C4—43no—
251C2×C4—43no—
252C2×C4—43no—
253C2×C4—43no—
254C2×C4—43no—
255C2×C4C31×H(5,𝔽2)13yes—
256C2×C4C31×H(5,𝔽2)13yes—
257C8—41no—
258C8—43no—
259C22—83no—
260C22—83no—
261C22—83no—
262C22—83no—
263C22—81no—
264C22—81no—
265C22C31×#53513yes—
266C22C31×#57713yes—
267C22C31×#57811yescharacteristic
268C4—43no—
269C4—43no—
270C4—83no—
271C4—83no—
272C4C31×#56011yescharacteristic
273C4C31×#56013yes—
274C2—161no—
275C2—161no—
276C2C31×#281811yescharacteristic
277C2C31×#322113yes—
278C2C31×#322513yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×D8×C31C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C31×#376 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2658C2semidirect
2. C31×#2644C2semidirect
3. C31×#2043C2semidirect
4. C31×#2042C2semidirect
5. C31×#2013C2semidirect
7. C31×#1840C2semidirect
88. #32804C31direct (N × H)
120. #2658C2×C31semidirect
121. #2644C2×C31semidirect
122. #2043C2×C31semidirect
123. #2042C2×C31semidirect
124. #2013C2×C31semidirect
126. #1840C2×C31semidirect
193. C31#32804direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26582
C31×#26442
C31×#20432
C31×#20422
C31×#20132
C31×#20122
C31×#18402
#3280431