Finite Group Databasebeta

← C31×#12159 #21468918

Subgroups — C31×#12159 #21468918

Order 7936.

390 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3249C211yesmaximal, characteristic, complemented
2C31×#2187C212yesmaximal
3C31×#2054C211yesmaximal, characteristic
4C31×#1348C212yesmaximal, complemented
5C31×#1322C211yesmaximal, characteristic, complemented
6C22×D8×C31C2211yescharacteristic
7C31×#533—21no—
8C2×C31×SD32—22no—
9C31×#477—22no—
10C31×#467—21no—
11C31×#467C2212yes—
12C31×#462—22no—
13C2×C31×C4⋊C8C2211yescharacteristic
14C2×C31×#102—21no—
15C31×#390—21no—
16C31×#357—22no—
17C31×C22⋊C16C2212yes—
18C31×#340C2211yescharacteristic
19C22×D4×C31—21no—
20C2×Q16×C31—21no—
21C2×Q16×C31D412yes—
22C2×D8×C31—22no—
23C2×D8×C31—42no—
24C22×C8×C31D411yescharacteristic
25C22×C8×C31C2311yescharacteristic
26C31×C4⋊Q8—21no—
27C31×C4⋊Q8—21no—
28C31×#125—41no—
29C31×C22⋊Q8—22no—
30C31×C22⋊Q8—41no—
31C4×Q8×C31—22no—
32C4×Q8×C31—41no—
33C31×#117—41no—
34C2×C4⋊C4×C31—21no—
35C2×C22⋊C4×C31—41no—
36C2×C22⋊C4×C31—41no—
37C2×C42×C31D411yescharacteristic
38C31×SD32—44no—
39C2×C16×C31—22no—
40C31×#107—21no—
41C31×#107D412yes—
42C31×C4⋊C8—22no—
43C31×#103—22no—
44C31×#103—22no—
45C31×#102—42no—
46C24×C31—41no—
47C2×Q8×C31—22no—
48C2×Q8×C31—41no—
49C2×D4×C31—22no—
50C2×D4×C31—41no—
51C2×D4×C31—41no—
52C2×D4×C31—41no—
53C2×D4×C31—41no—
54C2×D4×C31—41no—
55C22×C4×C31—41no—
56C22×C4×C31D811yescharacteristic
57C22×C4×C31D811yescharacteristic
58C22×C4×C31C2×D411yescharacteristic
59Q16×C31—42no—
60Q16×C31—81no—
61D8×C31—24no—
62D8×C31—42no—
63C2×C8×C31—21no—
64C2×C8×C31—41no—
65C2×C8×C31—41no—
66C2×C8×C31C2×D412yes—
67C4⋊C4×C31—22no—
68C4⋊C4×C31—41no—
69C4⋊C4×C31—41no—
70C4⋊C4×C31—42no—
71C4⋊C4×C31—41no—
72C4⋊C4×C31—41no—
73C4⋊C4×C31—41no—
74C4⋊C4×C31—42no—
75C22⋊C4×C31—42no—
76C22⋊C4×C31—82no—
77C22⋊C4×C31—82no—
78C42×C31—21no—
79C42×C31—21no—
80C42×C31—42no—
81C16×C31—42no—
82#12159C3111yesmaximal, characteristic, direct product factor, Sylow
83C23×C31—42no—
84C23×C31—81no—
85C23×C31—81no—
86C23×C31—81no—
87C23×C31—81no—
88C23×C31—81no—
89C23×C31—81no—
90C23×C31C2×D811yescharacteristic
91Q8×C31—42no—
92Q8×C31—42no—
93Q8×C31—82no—
94D4×C31—42no—
95D4×C31—42no—
96D4×C31—81no—
97D4×C31—81no—
98C2×C4×C31—21no—
99C2×C4×C31—21no—
100C2×C4×C31—21no—
101C2×C4×C31—21no—
102C2×C4×C31—21no—
103C2×C4×C31—41no—
104C2×C4×C31—42no—
105C2×C4×C31—42no—
106C2×C4×C31—81no—
107C2×C4×C31—81no—
108C2×C4×C31—81no—
109C2×C4×C31SD3212yescomplemented
110C2×C4×C31#12011yescharacteristic
111C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
112C8×C31—22no—
113C8×C31—41no—
114#3249C2×C3111yescharacteristic, complemented
115#2187C2×C3112yes—
116#2054C2×C3111yescharacteristic
117#1348C2×C3112yescomplemented
118#1322C2×C3111yescharacteristic, complemented
119C22×C31—21no—
120C22×C31—21no—
121C22×C31—82no—
122C22×C31—82no—
123C22×C31—82no—
124C22×C31—161no—
125C22×C31—161no—
126C22×C31—161no—
127C22×C31—161no—
128C22×C31#44711yescharacteristic
129C22×C31C2×SD3211yescharacteristic
130C22×C31#51011yescharacteristic
131C4×C31—21no—
132C4×C31—22no—
133C4×C31—41no—
134C4×C31—82no—
135C4×C31—82no—
136C4×C31—161no—
137C4×C31#45411yescharacteristic
138C4×C31B(4,𝔽2)11yescharacteristic
139C22×D8C22×C3111yescharacteristic
140#533—21no—
141C2×SD32—22no—
142#477—22no—
143#467—21no—
144#467C22×C3112yes—
145#462—22no—
146C2×C4⋊C8C22×C3111yescharacteristic
147C2×#102—21no—
148#390—21no—
149#357—22no—
150C22⋊C16C22×C3112yes—
151#340C22×C3111yescharacteristic
152C2×C31—21no—
153C2×C31—21no—
154C2×C31—162no—
155C2×C31#159811yescharacteristic
156C2×C31#218511yescharacteristic
157C2×C31#219011yescharacteristic
158C22×D4—21no—
159C2×Q16—21no—
160C2×Q16D4×C3112yes—
161C2×D8—22no—
162C2×D8—42no—
163C22×C8D4×C3111yescharacteristic
164C22×C8C23×C3111yescharacteristic
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167#125—41no—
168C22⋊Q8—22no—
169C22⋊Q8—41no—
170C4×Q8—22no—
171C4×Q8—41no—
172#117—41no—
173C2×C4⋊C4—21no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C42D4×C3111yescharacteristic
177SD32—44no—
178C2×C16—22no—
179#107—21no—
180#107D4×C3112yes—
181C4⋊C8—22no—
182#103—22no—
183#103—22no—
184#102—42no—
185C31#1215911yescharacteristic, direct product factor, Sylow
186C24—41no—
187C2×Q8—22no—
188C2×Q8—41no—
189C2×D4—22no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—41no—
195C22×C4—41no—
196C22×C4D8×C3111yescharacteristic
197C22×C4D8×C3111yescharacteristic
198C22×C4C2×D4×C3111yescharacteristic
199Q16—42no—
200Q16—81no—
201D8—24no—
202D8—42no—
203C2×C8—21no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8C2×D4×C3112yes—
207C4⋊C4—22no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—82no—
217C22⋊C4—82no—
218C42—21no—
219C42—21no—
220C42—42no—
221C16—42no—
222C23—42no—
223C23—81no—
224C23—81no—
225C23—81no—
226C23—81no—
227C23—81no—
228C23—81no—
229C23C2×D8×C3111yescharacteristic
230Q8—42no—
231Q8—42no—
232Q8—82no—
233D4—42no—
234D4—42no—
235D4—81no—
236D4—81no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4C31×SD3212yescomplemented
249C2×C4C31×#12011yescharacteristic
250C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
251C8—22no—
252C8—41no—
253C22—21no—
254C22—21no—
255C22—82no—
256C22—82no—
257C22—82no—
258C22—161no—
259C22—161no—
260C22—161no—
261C22—161no—
262C22C31×#44711yescharacteristic
263C22C2×C31×SD3211yescharacteristic
264C22C31×#51011yescharacteristic
265C4—21no—
266C4—22no—
267C4—41no—
268C4—82no—
269C4—82no—
270C4—161no—
271C4C31×#45411yescharacteristic
272C4C31×B(4,𝔽2)11yescharacteristic
273C2—21no—
274C2—21no—
275C2—162no—
276C2C31×#159811yescharacteristic
277C2C31×#218511yescharacteristic
278C2C31×#219011yescharacteristic
subgroupG/H
center, socleC22×C31#447
commutator, Frattini, cosocleC22×C8C23×C31
prime radicalC22×D8×C31C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C31×C4⋊C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3249C2semidirect
4. C31×#1348C2semidirect
5. C31×#1322C2semidirect
82. #12159C31direct (N × H)
109. C2×C4×C31SD32semidirect
114. #3249C2×C31semidirect
117. #1348C2×C31semidirect
118. #1322C2×C31semidirect
185. C31#12159direct (N × H)
248. C2×C4C31×SD32semidirect

Maximal subgroups

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

subgroupindex
C31×#32492
C31×#21872
C31×#20542
C31×#13482
C31×#13222
#1215931