Finite Group Databasebeta

← C31×#10302 #21467061

Subgroups — C31×#10302 #21467061

Order 7936.

516 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3255C211yesmaximal, characteristic, complemented
2C31×#3254C212yesmaximal
3C31×#1994C212yesmaximal, complemented
4C31×#1962C211yesmaximal, characteristic
5C31×#1309C211yesmaximal, characteristic, complemented
6C31×#533C2212yes—
7C31×#532C2211yescharacteristic
8C31×#477—22no—
9C31×#475—22no—
10C31×#474—22no—
11C31×#470—22no—
12C31×#468—24no—
13C31×#462—22no—
14C31×#461—22no—
15C31×#460—22no—
16C31×#423C2211yescharacteristic
17C31×#423C2212yes—
18C2×C31×#104—22no—
19C43×C31C2211yescharacteristic
20C31×#339—24no—
21C2×#41×C31—21no—
22C2×Q16×C31—22no—
23C2×SD16×C31—24no—
24C2×M16×C31D412yes—
25C2×M16×C31D414yes—
26C31×C4⋊Q8D412yes—
27C31×C4⋊Q8D412yes—
28C31×C4⋊Q8D411yescharacteristic
29C31×#127D411yescharacteristic
30C31×#125—22no—
31C31×#124—21no—
32C31×C22⋊Q8—22no—
33C31×C22⋊Q8—22no—
34C31×#121—22no—
35C4×Q8×C31—22no—
36C4×Q8×C31—22no—
37C4×D4×C31—22no—
38C31×#117—22no—
39C2×C42×C31—21no—
40C2×C42×C31—22no—
41C2×C42×C31C2311yescharacteristic
42C31×#107—24no—
43C31×#106—22no—
44C31×C4⋊C8—22no—
45C31×C4⋊C8—22no—
46C31×C4⋊C8—22no—
47C31×#104—44no—
48C31×#103—24no—
49C31×#103—24no—
50C31×#102—24no—
51#41×C31—42no—
52#41×C31—42no—
53C2×Q8×C31—22no—
54C2×Q8×C31—22no—
55C2×Q8×C31—21no—
56C2×D4×C31—22no—
57C2×D4×C31—21no—
58C2×D4×C31—41no—
59C22×C4×C31—41no—
60C22×C4×C31—41no—
61C22×C4×C31C2×D412yes—
62C22×C4×C31C2×D411yescharacteristic
63Q16×C31—82no—
64SD16×C31—84no—
65M16×C31—42no—
66M16×C31—44no—
67C2×C8×C31—24no—
68C2×C8×C31—22no—
69C4⋊C4×C31—22no—
70C4⋊C4×C31—22no—
71C4⋊C4×C31—22no—
72C4⋊C4×C31—22no—
73C4⋊C4×C31—22no—
74C4⋊C4×C31—42no—
75C4⋊C4×C31—42no—
76C22⋊C4×C31—42no—
77C22⋊C4×C31—42no—
78C42×C31—22no—
79C42×C31—24no—
80C42×C31—41no—
81C42×C31—42no—
82C42×C31—42no—
83C42×C31C2×D412yes—
84C42×C31C2×D411yescharacteristic
85C42×C31#4111yescharacteristic
86#10302C3111yesmaximal, characteristic, direct product factor, Sylow
87C23×C31—41no—
88C23×C31#12711yescharacteristic
89Q8×C31—42no—
90Q8×C31—42no—
91Q8×C31—42no—
92Q8×C31—42no—
93Q8×C31—42no—
94D4×C31—42no—
95D4×C31—42no—
96D4×C31—42no—
97D4×C31—82no—
98C2×C4×C31—21no—
99C2×C4×C31—21no—
100C2×C4×C31—22no—
101C2×C4×C31—22no—
102C2×C4×C31—42no—
103C2×C4×C31—44no—
104C2×C4×C31—42no—
105C2×C4×C31—42no—
106C2×C4×C31—41no—
107C2×C4×C31—81no—
108C2×C4×C31—81no—
109C2×C4×C31#12011yescharacteristic
110C2×C4×C31#12012yes—
111C2×C4×C31#12112yes—
112C2×C4×C31#12111yescharacteristic
113C8×C31—42no—
114C8×C31—44no—
115#3255C2×C3111yescharacteristic, complemented
116#3254C2×C3112yes—
117#1994C2×C3112yescomplemented
118#1962C2×C3111yescharacteristic
119#1309C2×C3111yescharacteristic, complemented
120C22×C31—21no—
121C22×C31—22no—
122C22×C31—81no—
123C22×C31—82no—
124C22×C31#39211yescharacteristic
125C4×C31—21no—
126C4×C31—22no—
127C4×C31—44no—
128C4×C31—81no—
129C4×C31—82no—
130C4×C31—82no—
131C4×C31#45311yescharacteristic
132C4×C31#45411yescharacteristic
133C4×C31#45512yes—
134C4×C31#45612yes—
135#533C22×C3112yes—
136#532C22×C3111yescharacteristic
137#477—22no—
138#475—22no—
139#474—22no—
140#470—22no—
141#468—24no—
142#462—22no—
143#461—22no—
144#460—22no—
145#423C22×C3111yescharacteristic
146#423C22×C3112yes—
147C2×#104—22no—
148C43C22×C3111yescharacteristic
149#339—24no—
150C2×C31—41no—
151C2×C31—161no—
152C2×C31#200611yescharacteristic
153C2×C31#200712yes—
154C2×#41—21no—
155C2×Q16—22no—
156C2×SD16—24no—
157C2×M16D4×C3114yes—
158C2×M16D4×C3112yes—
159C4⋊Q8D4×C3111yescharacteristic
160C4⋊Q8D4×C3112yes—
161C4⋊Q8D4×C3112yes—
162#127D4×C3111yescharacteristic
163#125—22no—
164#124—21no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167#121—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×D4—22no—
171#117—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42C23×C3111yescharacteristic
175#107—24no—
176#106—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180#104—44no—
181#103—24no—
182#103—24no—
183#102—24no—
184C31#1030211yescharacteristic, direct product factor, Sylow
185#41—42no—
186#41—42no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—21no—
190C2×D4—22no—
191C2×D4—21no—
192C2×D4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4C2×D4×C3112yes—
196C22×C4C2×D4×C3111yescharacteristic
197Q16—82no—
198SD16—84no—
199M16—42no—
200M16—44no—
201C2×C8—24no—
202C2×C8—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C42—22no—
213C42—24no—
214C42—41no—
215C42—42no—
216C42—42no—
217C42C2×D4×C3112yes—
218C42C2×D4×C3111yescharacteristic
219C42#41×C3111yescharacteristic
220C23—41no—
221C23C31×#12711yescharacteristic
222Q8—42no—
223Q8—42no—
224Q8—42no—
225Q8—42no—
226Q8—42no—
227D4—42no—
228D4—42no—
229D4—42no—
230D4—82no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4C31×#12011yescharacteristic
243C2×C4C31×#12012yes—
244C2×C4C31×#12112yes—
245C2×C4C31×#12111yescharacteristic
246C8—42no—
247C8—44no—
248C22—21no—
249C22—22no—
250C22—81no—
251C22—82no—
252C22C31×#39211yescharacteristic
253C4—21no—
254C4—22no—
255C4—44no—
256C4—81no—
257C4—82no—
258C4—82no—
259C4C31×#45311yescharacteristic
260C4C31×#45411yescharacteristic
261C4C31×#45512yes—
262C4C31×#45612yes—
263C2—41no—
264C2—161no—
265C2C31×#200611yescharacteristic
266C2C31×#200712yes—
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3255C2semidirect
3. C31×#1994C2semidirect
5. C31×#1309C2semidirect
86. #10302C31direct (N × H)
115. #3255C2×C31semidirect
117. #1994C2×C31semidirect
119. #1309C2×C31semidirect
184. C31#10302direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32552
C31×#32542
C31×#19942
C31×#19622
C31×#13092
#1030231