Finite Group Databasebeta

← C31×#7810 #21464569

Subgroups — C31×#7810 #21464569

Order 7936.

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

normal subgroups only

This group has 478 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×#2733C211yesmaximal, characteristic
2C31×#2096C211yesmaximal, characteristic, complemented
3C31×#2071C212yesmaximal, complemented
4C31×#1272C211yesmaximal, characteristic, complemented
5C31×#1270C212yesmaximal, complemented
6C2×C31×#124—21no—
7C2×C31×#106—21no—
8C2×C31×#103—21no—
9C2×C31×#102—22no—
10C31×#394—21no—
11C31×#392—22no—
12C31×#391C2211yescharacteristic
13C31×#390C2212yes—
14C31×#388—22no—
15C31×#384C2211yescharacteristic
16C31×#340—22no—
17C31×#336C2212yes—
18C31×#336C2211yescharacteristic
19C22×Q8×C31D411yescharacteristic
20C22×D4×C31—21no—
21C22×D4×C31—21no—
22C22×D4×C31D412yes—
23C22×C8×C31—21no—
24C22×C8×C31—22no—
25C31×#124—42no—
26C31×#124—42no—
27C2×C4⋊C4×C31—22no—
28C2×C4⋊C4×C31—21no—
29C2×C4⋊C4×C31D411yescharacteristic
30C2×C4⋊C4×C31D412yes—
31C2×C22⋊C4×C31—22no—
32C2×C22⋊C4×C31—22no—
33C2×C22⋊C4×C31—22no—
34C2×C22⋊C4×C31—22no—
35C2×C42×C31—21no—
36C2×C42×C31C2311yescharacteristic
37C31×#106—42no—
38C31×#103—42no—
39C31×#102—44no—
40C31×#95—21no—
41C31×#95—22no—
42C24×C31—22no—
43C2×Q8×C31—21no—
44C2×Q8×C31—41no—
45C2×Q8×C31—41no—
46C2×Q8×C31SD1612yes—
47C2×D4×C31—22no—
48C2×D4×C31—22no—
49C2×D4×C31—42no—
50C2×D4×C31—42no—
51C2×D4×C31—42no—
52C2×D4×C31—81no—
53C2×D4×C31—81no—
54C2×D4×C31—81no—
55C2×D4×C31—82no—
56C22×C4×C31—21no—
57C22×C4×C31—21no—
58C22×C4×C31—22no—
59C22×C4×C31—41no—
60C22×C4×C31C2×D411yescharacteristic
61C22×C4×C31C2×D412yes—
62C2×C8×C31—22no—
63C2×C8×C31—41no—
64C2×C8×C31—41no—
65C2×C8×C31—42no—
66C2×C8×C31—42no—
67C2×C8×C31—42no—
68C4⋊C4×C31—41no—
69C4⋊C4×C31—42no—
70C4⋊C4×C31—41no—
71C4⋊C4×C31—41no—
72C4⋊C4×C31—42no—
73C4⋊C4×C31—42no—
74C22⋊C4×C31—44no—
75C22⋊C4×C31—42no—
76C22⋊C4×C31—42no—
77C22⋊C4×C31—42no—
78C22⋊C4×C31—42no—
79C22⋊C4×C31—82no—
80C42×C31—41no—
81C42×C31—42no—
82#7810C3111yesmaximal, characteristic, direct product factor, Sylow
83C23×C31—42no—
84C23×C31—42no—
85C23×C31—42no—
86C23×C31—42no—
87C23×C31—42no—
88C23×C31—42no—
89C23×C31—42no—
90C23×C31#12011yescharacteristic
91Q8×C31—42no—
92Q8×C31—81no—
93D4×C31—44no—
94D4×C31—82no—
95D4×C31—84no—
96D4×C31—84no—
97C2×C4×C31—21no—
98C2×C4×C31—22no—
99C2×C4×C31—22no—
100C2×C4×C31—21no—
101C2×C4×C31—22no—
102C2×C4×C31—41no—
103C2×C4×C31—41no—
104C2×C4×C31—42no—
105C2×C4×C31—42no—
106C2×C4×C31—41no—
107C2×C4×C31—42no—
108C2×C4×C31—42no—
109C2×C4×C31—82no—
110C2×C4×C31—81no—
111C2×C4×C31C2×SD1611yescharacteristic
112C2×C4×C31#13512yes—
113C2×C4×C31AGL(1,ℤ/8)12yes—
114C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
115C8×C31—42no—
116C8×C31—82no—
117#2733C2×C3111yescharacteristic
118#2096C2×C3111yescharacteristic, complemented
119#2071C2×C3112yescomplemented
120#1272C2×C3111yescharacteristic, complemented
121#1270C2×C3112yescomplemented
122C22×C31—82no—
123C22×C31—82no—
124C22×C31—82no—
125C22×C31—82no—
126C22×C31—82no—
127C22×C31—82no—
128C22×C31—82no—
129C22×C31#44912yes—
130C22×C31#45011yescharacteristic
131C22×C31#45311yescharacteristic
132C22×C31#45512yes—
133C22×C31#45811yescharacteristic
134C4×C31—22no—
135C4×C31—22no—
136C4×C31—21no—
137C4×C31—21no—
138C4×C31—42no—
139C4×C31—82no—
140C4×C31—81no—
141C4×C31—161no—
142C2×#124—21no—
143C2×#106—21no—
144C2×#103—21no—
145C2×#102—22no—
146#394—21no—
147#392—22no—
148#391C22×C3111yescharacteristic
149#390C22×C3112yes—
150#388—22no—
151#384C22×C3111yescharacteristic
152#340—22no—
153#336C22×C3112yes—
154#336C22×C3111yescharacteristic
155C2×C31—162no—
156C2×C31#161212yes—
157C2×C31#161311yescharacteristic
158C2×C31#162411yescharacteristic
159C2×C31#164212yes—
160C2×C31#165911yescharacteristic
161C22×Q8D4×C3111yescharacteristic
162C22×D4—21no—
163C22×D4—21no—
164C22×D4D4×C3112yes—
165C22×C8—21no—
166C22×C8—22no—
167#124—42no—
168#124—42no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4D4×C3112yes—
172C2×C4⋊C4D4×C3111yescharacteristic
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C42—21no—
178C2×C42C23×C3111yescharacteristic
179#106—42no—
180#103—42no—
181#102—44no—
182#95—21no—
183#95—22no—
184C31#781011yescharacteristic, direct product factor, Sylow
185C24—22no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8SD16×C3112yes—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—81no—
196C2×D4—81no—
197C2×D4—81no—
198C2×D4—82no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—41no—
203C22×C4C2×D4×C3111yescharacteristic
204C22×C4C2×D4×C3112yes—
205C2×C8—22no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C22⋊C4—44no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—82no—
223C42—41no—
224C42—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23C31×#12011yescharacteristic
233Q8—42no—
234Q8—81no—
235D4—44no—
236D4—82no—
237D4—84no—
238D4—84no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—82no—
252C2×C4—81no—
253C2×C4C2×SD16×C3111yescharacteristic
254C2×C4C31×#13512yes—
255C2×C4C31×AGL(1,ℤ/8)12yes—
256C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
257C8—42no—
258C8—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—82no—
266C22C31×#44912yes—
267C22C31×#45011yescharacteristic
268C22C31×#45311yescharacteristic
269C22C31×#45512yes—
270C22C31×#45811yescharacteristic
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—82no—
277C4—81no—
278C4—161no—
279C2—162no—
280C2C31×#161212yes—
281C2C31×#161311yescharacteristic
282C2C31×#162411yescharacteristic
283C2C31×#164212yes—
284C2C31×#165911yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#2733C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2096C2semidirect
3. C31×#2071C2semidirect
4. C31×#1272C2semidirect
5. C31×#1270C2semidirect
82. #7810C31direct (N × H)
118. #2096C2×C31semidirect
119. #2071C2×C31semidirect
120. #1272C2×C31semidirect
121. #1270C2×C31semidirect
184. C31#7810direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27332
C31×#20962
C31×#20712
C31×#12722
C31×#12702
#781031