Finite Group Databasebeta

← C31×#10216 #21466975

Subgroups — C31×#10216 #21466975

Order 7936.

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

normal subgroups only

This group has 468 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×#3255C211yesmaximal, characteristic, complemented
2C31×#3235C212yesmaximal
3C31×#1963C211yesmaximal, characteristic
4C31×#1506C212yesmaximal, complemented
5C31×#1307C211yesmaximal, characteristic, complemented
6C31×#533C2212yes—
7C31×#532C2211yescharacteristic
8C31×#501—22no—
9C31×#490—22no—
10C31×#482—24no—
11C31×#476—22no—
12C31×#475—22no—
13C31×#474—22no—
14C31×#470—22no—
15C31×#462—22no—
16C31×#432C2212yes—
17C31×#423C2211yescharacteristic
18C2×C31×#104—22no—
19C31×#376C2211yescharacteristic
20C31×#328—24no—
21C2×#41×C31—21no—
22C2×Q16×C31—22no—
23C2×M16×C31D412yes—
24C31×C4⋊Q8D412yes—
25C31×C4⋊Q8D412yes—
26C31×C4⋊Q8D411yescharacteristic
27C31×#127D411yescharacteristic
28C31×#125—22no—
29C31×#124—21no—
30C31×C22⋊Q8—22no—
31C31×C22⋊Q8—22no—
32C31×#121—22no—
33C4×Q8×C31—22no—
34C4×Q8×C31—22no—
35C4×D4×C31—22no—
36C31×#117—22no—
37C2×C42×C31—21no—
38C2×C42×C31C2311yescharacteristic
39C31×#107—24no—
40C31×#106—22no—
41C31×#106—24no—
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—
53C2×Q8×C31—22no—
54C2×Q8×C31—21no—
55C2×Q8×C31—22no—
56C2×D4×C31—22no—
57C2×D4×C31—21no—
58C2×D4×C31—41no—
59C22×C4×C31—41no—
60C22×C4×C31—41no—
61C22×C4×C31C2×D411yescharacteristic
62C22×C4×C31C2×D412yes—
63Q16×C31—82no—
64M16×C31—42no—
65C2×C8×C31—24no—
66C2×C8×C31—22no—
67C4⋊C4×C31—22no—
68C4⋊C4×C31—22no—
69C4⋊C4×C31—22no—
70C4⋊C4×C31—22no—
71C4⋊C4×C31—22no—
72C4⋊C4×C31—42no—
73C4⋊C4×C31—42no—
74C22⋊C4×C31—42no—
75C22⋊C4×C31—42no—
76C42×C31—22no—
77C42×C31—41no—
78C42×C31—42no—
79C42×C31C2×D411yescharacteristic
80C42×C31C2×D412yes—
81C42×C31#4111yescharacteristic
82#10216C3111yesmaximal, characteristic, direct product factor, Sylow
83C23×C31—41no—
84C23×C31#12711yescharacteristic
85Q8×C31—42no—
86Q8×C31—42no—
87Q8×C31—42no—
88Q8×C31—82no—
89D4×C31—44no—
90D4×C31—42no—
91D4×C31—82no—
92C2×C4×C31—21no—
93C2×C4×C31—21no—
94C2×C4×C31—42no—
95C2×C4×C31—42no—
96C2×C4×C31—42no—
97C2×C4×C31—42no—
98C2×C4×C31—41no—
99C2×C4×C31—82no—
100C2×C4×C31—81no—
101C2×C4×C31—81no—
102C2×C4×C31#12012yes—
103C2×C4×C31#12011yescharacteristic
104C2×C4×C31#12112yes—
105C2×C4×C31#12111yescharacteristic
106C8×C31—44no—
107C8×C31—42no—
108#3255C2×C3111yescharacteristic, complemented
109#3235C2×C3112yes—
110#1963C2×C3111yescharacteristic
111#1506C2×C3112yescomplemented
112#1307C2×C3111yescharacteristic, complemented
113C22×C31—21no—
114C22×C31—22no—
115C22×C31—81no—
116C22×C31—82no—
117C22×C31#39211yescharacteristic
118C4×C31—21no—
119C4×C31—22no—
120C4×C31—22no—
121C4×C31—82no—
122C4×C31—81no—
123C4×C31—82no—
124C4×C31—82no—
125C4×C31#45311yescharacteristic
126C4×C31#45411yescharacteristic
127#533C22×C3112yes—
128#532C22×C3111yescharacteristic
129#501—22no—
130#490—22no—
131#482—24no—
132#476—22no—
133#475—22no—
134#474—22no—
135#470—22no—
136#462—22no—
137#432C22×C3112yes—
138#423C22×C3111yescharacteristic
139C2×#104—22no—
140#376C22×C3111yescharacteristic
141#328—24no—
142C2×C31—41no—
143C2×C31—161no—
144C2×C31#200611yescharacteristic
145C2×C31#200912yes—
146C2×#41—21no—
147C2×Q16—22no—
148C2×M16D4×C3112yes—
149C4⋊Q8D4×C3111yescharacteristic
150C4⋊Q8D4×C3112yes—
151C4⋊Q8D4×C3112yes—
152#127D4×C3111yescharacteristic
153#125—22no—
154#124—21no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157#121—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×D4—22no—
161#117—22no—
162C2×C42—21no—
163C2×C42C23×C3111yescharacteristic
164#107—24no—
165#106—22no—
166#106—24no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169#104—44no—
170#103—24no—
171#102—24no—
172#102—24no—
173C22⋊C8D4×C3114yes—
174#97—22no—
175#95—22no—
176C31#1021611yescharacteristic, direct product factor, Sylow
177#41—42no—
178#41—42no—
179C2×Q8—22no—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×D4—22no—
183C2×D4—21no—
184C2×D4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4C2×D4×C3111yescharacteristic
188C22×C4C2×D4×C3112yes—
189Q16—82no—
190M16—42no—
191C2×C8—24no—
192C2×C8—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C42—22no—
203C42—41no—
204C42—42no—
205C42C2×D4×C3111yescharacteristic
206C42C2×D4×C3112yes—
207C42#41×C3111yescharacteristic
208C23—41no—
209C23C31×#12711yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—82no—
214D4—44no—
215D4—42no—
216D4—82no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—82no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4C31×#12012yes—
228C2×C4C31×#12011yescharacteristic
229C2×C4C31×#12112yes—
230C2×C4C31×#12111yescharacteristic
231C8—44no—
232C8—42no—
233C22—21no—
234C22—22no—
235C22—81no—
236C22—82no—
237C22C31×#39211yescharacteristic
238C4—21no—
239C4—22no—
240C4—22no—
241C4—82no—
242C4—81no—
243C4—82no—
244C4—82no—
245C4C31×#45311yescharacteristic
246C4C31×#45411yescharacteristic
247C2—41no—
248C2—161no—
249C2C31×#200611yescharacteristic
250C2C31×#200912yes—
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
4. C31×#1506C2semidirect
5. C31×#1307C2semidirect
82. #10216C31direct (N × H)
108. #3255C2×C31semidirect
111. #1506C2×C31semidirect
112. #1307C2×C31semidirect
176. C31#10216direct (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×#32352
C31×#19632
C31×#15062
C31×#13072
#1021631