Finite Group Databasebeta

← C31×#10212 #21466971

Subgroups — C31×#10212 #21466971

Order 7936.

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

normal subgroups only

This group has 536 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×#3253C211yesmaximal, characteristic, complemented
2C31×#3214C212yesmaximal, complemented
3C31×#1963C211yesmaximal, characteristic, complemented
4C31×#1505C212yesmaximal, complemented
5C31×#1307C211yesmaximal, characteristic, complemented
6C31×#532C2211yescharacteristic
7C31×#532C2212yes—
8C31×#496—22no—
9C31×#489—22no—
10C31×#476—22no—
11C31×#469—22no—
12C31×#461—22no—
13C31×#459—22no—
14C31×#459—22no—
15C31×#449—24no—
16C31×#432C2212yes—
17C31×#423C2211yescharacteristic
18C2×C31×#104—22no—
19C31×#376C2211yescharacteristic
20C31×#327—24no—
21C2×#41×C31—21no—
22C2×#41×C31—22no—
23C2×SD16×C31—24no—
24C2×D8×C31—24no—
25C2×D8×C31—22no—
26C2×M16×C31D412yes—
27C31×C4⋊Q8D412yes—
28C31×C4⋊Q8D411yescharacteristic
29C31×#127D412yes—
30C31×#127D411yescharacteristic
31C31×#124—22no—
32C31×#124—21no—
33C31×#121—22no—
34C31×#121—22no—
35C31×#121—22no—
36C4×D4×C31—22no—
37C4×D4×C31—22no—
38C4×D4×C31—22no—
39C2×C42×C31—21no—
40C2×C42×C31C2311yescharacteristic
41C31×#106—22no—
42C31×C4⋊C8—22no—
43C31×C4⋊C8—22no—
44C31×#104—44no—
45C31×#103—24no—
46C31×#102—24no—
47C31×#102—24no—
48C31×C22⋊C8D414yes—
49C31×#97—22no—
50C31×#95—22no—
51#41×C31—42no—
52#41×C31—42no—
53#41×C31—84no—
54C2×Q8×C31—22no—
55C2×Q8×C31—21no—
56C2×D4×C31—22no—
57C2×D4×C31—22no—
58C2×D4×C31—22no—
59C2×D4×C31—21no—
60C2×D4×C31—22no—
61C2×D4×C31—41no—
62C2×D4×C31—42no—
63C22×C4×C31—41no—
64C22×C4×C31—41no—
65C22×C4×C31—42no—
66C22×C4×C31C2×D411yescharacteristic
67C22×C4×C31C2×D412yes—
68SD16×C31—84no—
69D8×C31—84no—
70D8×C31—82no—
71M16×C31—42no—
72C2×C8×C31—24no—
73C2×C8×C31—22no—
74C4⋊C4×C31—22no—
75C4⋊C4×C31—22no—
76C4⋊C4×C31—22no—
77C22⋊C4×C31—42no—
78C22⋊C4×C31—42no—
79C22⋊C4×C31—42no—
80C42×C31—22no—
81C42×C31—41no—
82C42×C31C2×D411yescharacteristic
83C42×C31C2×D412yes—
84C42×C31#4111yescharacteristic
85#10212C3111yesmaximal, characteristic, direct product factor, Sylow
86C23×C31—41no—
87C23×C31—42no—
88C23×C31#12711yescharacteristic
89Q8×C31—44no—
90Q8×C31—42no—
91D4×C31—42no—
92D4×C31—44no—
93D4×C31—42no—
94D4×C31—42no—
95D4×C31—42no—
96D4×C31—42no—
97D4×C31—82no—
98D4×C31—82no—
99D4×C31—84no—
100C2×C4×C31—21no—
101C2×C4×C31—21no—
102C2×C4×C31—42no—
103C2×C4×C31—42no—
104C2×C4×C31—41no—
105C2×C4×C31—42no—
106C2×C4×C31—84no—
107C2×C4×C31—82no—
108C2×C4×C31—81no—
109C2×C4×C31—81no—
110C2×C4×C31#12012yes—
111C2×C4×C31#12011yescharacteristic
112C2×C4×C31#12111yescharacteristic
113C2×C4×C31#12112yes—
114C8×C31—44no—
115C8×C31—42no—
116#3253C2×C3111yescharacteristic, complemented
117#3214C2×C3112yescomplemented
118#1963C2×C3111yescharacteristic, complemented
119#1505C2×C3112yescomplemented
120#1307C2×C3111yescharacteristic, complemented
121C22×C31—21no—
122C22×C31—22no—
123C22×C31—81no—
124C22×C31—82no—
125C22×C31—82no—
126C22×C31—82no—
127C22×C31—82no—
128C22×C31#39211yescharacteristic
129C4×C31—21no—
130C4×C31—22no—
131C4×C31—22no—
132C4×C31—82no—
133C4×C31—81no—
134C4×C31—82no—
135C4×C31#45311yescharacteristic
136C4×C31#45411yescharacteristic
137#532C22×C3111yescharacteristic
138#532C22×C3112yes—
139#496—22no—
140#489—22no—
141#476—22no—
142#469—22no—
143#461—22no—
144#459—22no—
145#459—22no—
146#449—24no—
147#432C22×C3112yes—
148#423C22×C3111yescharacteristic
149C2×#104—22no—
150#376C22×C3111yescharacteristic
151#327—24no—
152C2×C31—41no—
153C2×C31—161no—
154C2×C31—162no—
155C2×C31#200611yescharacteristic
156C2×C31#200812yes—
157C2×#41—21no—
158C2×#41—22no—
159C2×SD16—24no—
160C2×D8—24no—
161C2×D8—22no—
162C2×M16D4×C3112yes—
163C4⋊Q8D4×C3112yes—
164C4⋊Q8D4×C3111yescharacteristic
165#127D4×C3112yes—
166#127D4×C3111yescharacteristic
167#124—22no—
168#124—21no—
169#121—22no—
170#121—22no—
171#121—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C2×C42—21no—
176C2×C42C23×C3111yescharacteristic
177#106—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180#104—44no—
181#103—24no—
182#102—24no—
183#102—24no—
184C22⋊C8D4×C3114yes—
185#97—22no—
186#95—22no—
187C31#1021211yescharacteristic, direct product factor, Sylow
188#41—42no—
189#41—42no—
190#41—84no—
191C2×Q8—22no—
192C2×Q8—21no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4—41no—
199C2×D4—42no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4C2×D4×C3111yescharacteristic
204C22×C4C2×D4×C3112yes—
205SD16—84no—
206D8—84no—
207D8—82no—
208M16—42no—
209C2×C8—24no—
210C2×C8—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C42—22no—
218C42—41no—
219C42C2×D4×C3111yescharacteristic
220C42C2×D4×C3112yes—
221C42#41×C3111yescharacteristic
222C23—41no—
223C23—42no—
224C23C31×#12711yescharacteristic
225Q8—44no—
226Q8—42no—
227D4—42no—
228D4—44no—
229D4—42no—
230D4—42no—
231D4—42no—
232D4—42no—
233D4—82no—
234D4—82no—
235D4—84no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—84no—
243C2×C4—82no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4C31×#12012yes—
247C2×C4C31×#12011yescharacteristic
248C2×C4C31×#12111yescharacteristic
249C2×C4C31×#12112yes—
250C8—44no—
251C8—42no—
252C22—21no—
253C22—22no—
254C22—81no—
255C22—82no—
256C22—82no—
257C22—82no—
258C22—82no—
259C22C31×#39211yescharacteristic
260C4—21no—
261C4—22no—
262C4—22no—
263C4—82no—
264C4—81no—
265C4—82no—
266C4C31×#45311yescharacteristic
267C4C31×#45411yescharacteristic
268C2—41no—
269C2—161no—
270C2—162no—
271C2C31×#200611yescharacteristic
272C2C31×#200812yes—
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
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×#3253C2semidirect
2. C31×#3214C2semidirect
3. C31×#1963C2semidirect
4. C31×#1505C2semidirect
5. C31×#1307C2semidirect
85. #10212C31direct (N × H)
116. #3253C2×C31semidirect
117. #3214C2×C31semidirect
118. #1963C2×C31semidirect
119. #1505C2×C31semidirect
120. #1307C2×C31semidirect
187. C31#10212direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32532
C31×#32142
C31×#19632
C31×#15052
C31×#13072
#1021231