Finite Group Databasebeta

← C31×#12250 #21469009

Subgroups — C31×#12250 #21469009

Order 7936.

454 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2819C211yesmaximal, characteristic
2C31×#2061C211yesmaximal, characteristic, complemented
3C31×#1599C212yesmaximal, complemented
4C31×#1345C212yesmaximal, complemented
5C31×#1324C211yesmaximal, characteristic, complemented
6C31×#544C2211yescharacteristic
7C2×C31×#124—21no—
8C31×#450—22no—
9C31×#428—21no—
10C31×#394—21no—
11C31×#394C2212yes—
12C31×#393—22no—
13C31×#392—22no—
14C31×#382C2211yescharacteristic
15C31×#343C2211yescharacteristic
16C31×#339—21no—
17C31×#328—22no—
18C31×#324C2212yes—
19C22×Q8×C31—21no—
20C22×D4×C31—21no—
21C22×D4×C31—21no—
22C22×D4×C31D412yes—
23C2×SD16×C31—42no—
24C2×M16×C31—21no—
25C31×C4⋊Q8—41no—
26C31×#125—41no—
27C31×#124—42no—
28C31×#124—42no—
29C31×C22⋊Q8—22no—
30C31×#117—21no—
31C2×C4⋊C4×C31D411yescharacteristic
32C2×C4⋊C4×C31C2311yescharacteristic
33C2×C22⋊C4×C31—22no—
34C2×C22⋊C4×C31—22no—
35C2×C22⋊C4×C31—22no—
36C2×C22⋊C4×C31—22no—
37C2×C42×C31D411yescharacteristic
38C31×#107—41no—
39C31×#106—41no—
40C31×#102—42no—
41C31×C22⋊C8—22no—
42C31×#95—21no—
43C31×#95—22no—
44C31×#95D412yes—
45C24×C31—22no—
46C2×Q8×C31—22no—
47C2×Q8×C31—42no—
48C2×Q8×C31—81no—
49C2×Q8×C31—81no—
50C2×D4×C31—42no—
51C2×D4×C31—42no—
52C2×D4×C31—82no—
53C2×D4×C31—81no—
54C2×D4×C31—81no—
55C2×D4×C31—82no—
56C2×D4×C31—81no—
57C22×C4×C31—21no—
58C22×C4×C31—21no—
59C22×C4×C31—22no—
60C22×C4×C31C2×D411yescharacteristic
61C22×C4×C31C2×D412yes—
62SD16×C31—84no—
63M16×C31—22no—
64M16×C31—41no—
65C2×C8×C31—41no—
66C2×C8×C31—42no—
67C4⋊C4×C31—22no—
68C4⋊C4×C31—21no—
69C4⋊C4×C31—21no—
70C4⋊C4×C31—41no—
71C4⋊C4×C31—42no—
72C4⋊C4×C31—41no—
73C22⋊C4×C31—44no—
74C22⋊C4×C31—41no—
75C22⋊C4×C31—42no—
76C22⋊C4×C31—42no—
77C22⋊C4×C31—82no—
78C22⋊C4×C31—82no—
79C42×C31—21no—
80C42×C31—41no—
81C42×C31SD1612yescomplemented
82#12250C3111yesmaximal, 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—82no—
92Q8×C31—84no—
93D4×C31—44no—
94D4×C31—82no—
95D4×C31—84no—
96D4×C31—84no—
97C2×C4×C31—21no—
98C2×C4×C31—21no—
99C2×C4×C31—22no—
100C2×C4×C31—22no—
101C2×C4×C31—41no—
102C2×C4×C31—41no—
103C2×C4×C31—42no—
104C2×C4×C31—42no—
105C2×C4×C31—42no—
106C2×C4×C31—42no—
107C2×C4×C31—42no—
108C2×C4×C31—41no—
109C2×C4×C31—41no—
110C2×C4×C31—41no—
111C2×C4×C31C2×SD1611yescharacteristic
112C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
113C8×C31—42no—
114C8×C31—82no—
115#2819C2×C3111yescharacteristic
116#2061C2×C3111yescharacteristic, complemented
117#1599C2×C3112yescomplemented
118#1345C2×C3112yescomplemented
119#1324C2×C3111yescharacteristic, complemented
120C22×C31—21no—
121C22×C31—21no—
122C22×C31—82no—
123C22×C31—82no—
124C22×C31—82no—
125C22×C31—82no—
126C22×C31—82no—
127C22×C31—82no—
128C22×C31—82no—
129C22×C31#45011yescharacteristic
130C22×C31#45611yescharacteristic
131C22×C31B(4,𝔽2)11yescharacteristic
132C4×C31—21no—
133C4×C31—21no—
134C4×C31—42no—
135C4×C31—42no—
136C4×C31—82no—
137C4×C31—81no—
138C4×C31—81no—
139C4×C31—81no—
140#544C22×C3111yescharacteristic
141C2×#124—21no—
142#450—22no—
143#428—21no—
144#394—21no—
145#394C22×C3112yes—
146#393—22no—
147#392—22no—
148#382C22×C3111yescharacteristic
149#343C22×C3111yescharacteristic
150#339—21no—
151#328—22no—
152#324C22×C3112yes—
153C2×C31—21no—
154C2×C31—21no—
155C2×C31—162no—
156C2×C31#160411yescharacteristic
157C2×C31#219911yescharacteristic
158C2×C31#220211yescharacteristic
159C22×Q8—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4D4×C3112yes—
163C2×SD16—42no—
164C2×M16—21no—
165C4⋊Q8—41no—
166#125—41no—
167#124—42no—
168#124—42no—
169C22⋊Q8—22no—
170#117—21no—
171C2×C4⋊C4D4×C3111yescharacteristic
172C2×C4⋊C4C23×C3111yescharacteristic
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C42D4×C3111yescharacteristic
178#107—41no—
179#106—41no—
180#102—42no—
181C22⋊C8—22no—
182#95—21no—
183#95—22no—
184#95D4×C3112yes—
185C31#1225011yescharacteristic, direct product factor, Sylow
186C24—22no—
187C2×Q8—22no—
188C2×Q8—42no—
189C2×Q8—81no—
190C2×Q8—81no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—82no—
194C2×D4—81no—
195C2×D4—81no—
196C2×D4—82no—
197C2×D4—81no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4C2×D4×C3111yescharacteristic
202C22×C4C2×D4×C3112yes—
203SD16—84no—
204M16—22no—
205M16—41no—
206C2×C8—41no—
207C2×C8—42no—
208C4⋊C4—22no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C22⋊C4—44no—
215C22⋊C4—41no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—82no—
219C22⋊C4—82no—
220C42—21no—
221C42—41no—
222C42SD16×C3112yescomplemented
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23C31×#12011yescharacteristic
231Q8—82no—
232Q8—84no—
233D4—44no—
234D4—82no—
235D4—84no—
236D4—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4C2×SD16×C3111yescharacteristic
252C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
253C8—42no—
254C8—82no—
255C22—21no—
256C22—21no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22C31×#45011yescharacteristic
265C22C31×#45611yescharacteristic
266C22C31×B(4,𝔽2)11yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—82no—
272C4—81no—
273C4—81no—
274C4—81no—
275C2—21no—
276C2—21no—
277C2—162no—
278C2C31×#160411yescharacteristic
279C2C31×#219911yescharacteristic
280C2C31×#220211yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC31×#2819C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2061C2semidirect
3. C31×#1599C2semidirect
4. C31×#1345C2semidirect
5. C31×#1324C2semidirect
81. C42×C31SD16semidirect
82. #12250C31direct (N × H)
116. #2061C2×C31semidirect
117. #1599C2×C31semidirect
118. #1345C2×C31semidirect
119. #1324C2×C31semidirect
185. C31#12250direct (N × H)
222. C42SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#28192
C31×#20612
C31×#15992
C31×#13452
C31×#13242
#1225031