Finite Group Databasebeta

← C31×#12857 #21469616

Subgroups — C31×#12857 #21469616

Order 7936.

670 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2297C211yesmaximal, characteristic
2C31×#1782C212yesmaximal
3C31×#1393C214yesmaximal, complemented
4C22×#41×C31C2211yescharacteristic
5C2×C31×#117C411yescharacteristic, complemented
6C22×C42×C31C411yescharacteristic, complemented
7C2×C31×#99—22no—
8C2×C31×#99C2214yes—
9C31×#391C412yescomplemented
10C31×#390C2212yes—
11C31×#386—22no—
12C31×#381—22no—
13C4×C22⋊C4×C31—22no—
14C2×#41×C31—24no—
15C2×#41×C31—42no—
16C2×#41×C31—42no—
17C22×Q8×C31C2×C411yescharacteristic
18C22×D4×C31D412yes—
19C22×D4×C31C2311yescharacteristic
20C23×C4×C31—21no—
21C23×C4×C31C2×C411yescharacteristic
22C23×C4×C31D412yes—
23C31×#117—42no—
24C2×C4⋊C4×C31C2×C412yes—
25C2×C22⋊C4×C31—21no—
26C2×C22⋊C4×C31—24no—
27C2×C22⋊C4×C31—24no—
28C2×C42×C31—21no—
29C2×C42×C31—21no—
30C2×C42×C31—42no—
31C2×C42×C31C2×C412yes—
32C31×#99—28no—
33C31×#99—44no—
34C31×#95—24no—
35C31×#95—44no—
36C24×C31C22⋊C411yescharacteristic
37C24×C31C2×D412yes—
38#41×C31—416no—
39C2×Q8×C31—22no—
40C2×Q8×C31—41no—
41C2×Q8×C31—41no—
42C2×D4×C31—24no—
43C2×D4×C31—24no—
44C2×D4×C31—42no—
45C2×D4×C31—41no—
46C2×D4×C31—41no—
47C2×D4×C31#4114yes—
48C22×C4×C31—22no—
49C22×C4×C31—22no—
50C22×C4×C31—22no—
51C22×C4×C31—22no—
52C22×C4×C31—24no—
53C22×C4×C31—24no—
54C22×C4×C31—22no—
55C22×C4×C31—42no—
56C22×C4×C31—44no—
57C22×C4×C31—42no—
58C22×C4×C31—44no—
59C22×C4×C31C22⋊C412yes—
60C22×C4×C31C22⋊C411yescharacteristic
61C22×C4×C31C22×C411yescharacteristic
62C4⋊C4×C31—42no—
63C22⋊C4×C31—48no—
64C22⋊C4×C31—42no—
65C22⋊C4×C31—44no—
66C22⋊C4×C31—44no—
67C42×C31—42no—
68C42×C31—42no—
69C42×C31—42no—
70#12857C3111yesmaximal, characteristic, direct product factor, Sylow
71C23×C31—21no—
72C23×C31—21no—
73C23×C31—22no—
74C23×C31—22no—
75C23×C31—42no—
76C23×C31—44no—
77C23×C31#9912yes—
78C23×C31C2×C22⋊C411yescharacteristic
79C23×C31#12314yes—
80Q8×C31—44no—
81D4×C31—44no—
82D4×C31—48no—
83C2×C4×C31—28no—
84C2×C4×C31—21no—
85C2×C4×C31—21no—
86C2×C4×C31—21no—
87C2×C4×C31—21no—
88C2×C4×C31—24no—
89C2×C4×C31—22no—
90C2×C4×C31—22no—
91C2×C4×C31—22no—
92C2×C4×C31—48no—
93C2×C4×C31—42no—
94C2×C4×C31—48no—
95C2×C4×C31—42no—
96C2×C4×C31—44no—
97C2×C4×C31—42no—
98C2×C4×C31—84no—
99C2×C4×C31—88no—
100C2×C4×C31#9912yes—
101C2×C4×C31#11712yes—
102#2297C2×C3111yescharacteristic
103#1782C2×C3112yes—
104#1393C2×C3114yescomplemented
105C22×C31—22no—
106C22×C31—22no—
107C22×C31—24no—
108C22×C31—42no—
109C22×C31—44no—
110C22×C31—42no—
111C22×C31—44no—
112C22×C31—42no—
113C22×C31—82no—
114C22×C31#38111yescharacteristic
115C22×C31C2×#9911yescharacteristic
116C22×C31C2×#9911yescharacteristic
117C4×C31—22no—
118C4×C31—22no—
119C4×C31—44no—
120C4×C31—42no—
121C4×C31—82no—
122C4×C31—164no—
123C22×#41C22×C3111yescharacteristic
124C2×#117C4×C3111yescharacteristic, complemented
125C22×C42C4×C3111yescharacteristic, complemented
126C2×#99—22no—
127C2×#99C22×C3114yes—
128#391C4×C3112yescomplemented
129#390C22×C3112yes—
130#386—22no—
131#381—22no—
132C4×C22⋊C4—22no—
133C2×C31—21no—
134C2×C31—21no—
135C2×C31—44no—
136C2×C31—42no—
137C2×C31#181911yescharacteristic
138C2×C31#212612yes—
139C2×#41—24no—
140C2×#41—42no—
141C2×#41—42no—
142C22×Q8C2×C4×C3111yescharacteristic
143C22×D4D4×C3112yes—
144C22×D4C23×C3111yescharacteristic
145C23×C4—21no—
146C23×C4C2×C4×C3111yescharacteristic
147C23×C4D4×C3112yes—
148#117—42no—
149C2×C4⋊C4C2×C4×C3112yes—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—42no—
156C2×C42C2×C4×C3112yes—
157#99—28no—
158#99—44no—
159#95—24no—
160#95—44no—
161C31#1285711yescharacteristic, direct product factor, Sylow
162C24C22⋊C4×C3111yescharacteristic
163C24C2×D4×C3112yes—
164#41—416no—
165C2×Q8—22no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×D4—24no—
169C2×D4—24no—
170C2×D4—42no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4#41×C3114yes—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4C22⋊C4×C3112yes—
186C22×C4C22⋊C4×C3111yescharacteristic
187C22×C4C22×C4×C3111yescharacteristic
188C4⋊C4—42no—
189C22⋊C4—48no—
190C22⋊C4—42no—
191C22⋊C4—44no—
192C22⋊C4—44no—
193C42—42no—
194C42—42no—
195C42—42no—
196C23—21no—
197C23—22no—
198C23—21no—
199C23—22no—
200C23—42no—
201C23—44no—
202C23C31×#9912yes—
203C23C2×C22⋊C4×C3111yescharacteristic
204C23C31×#12314yes—
205Q8—44no—
206D4—44no—
207D4—48no—
208C2×C4—28no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—48no—
218C2×C4—42no—
219C2×C4—48no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—84no—
224C2×C4—88no—
225C2×C4C31×#9912yes—
226C2×C4C31×#11712yes—
227C22—22no—
228C22—22no—
229C22—24no—
230C22—42no—
231C22—44no—
232C22—42no—
233C22—44no—
234C22—42no—
235C22—82no—
236C22C31×#38111yescharacteristic
237C22C2×C31×#9911yescharacteristic
238C22C2×C31×#9911yescharacteristic
239C4—22no—
240C4—22no—
241C4—44no—
242C4—42no—
243C4—82no—
244C4—164no—
245C2—21no—
246C2—21no—
247C2—44no—
248C2—42no—
249C2C31×#181911yescharacteristic
250C2C31×#212612yes—
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC22×D4C23×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1393C2semidirect
5. C2×C31×#117C4semidirect
6. C22×C42×C31C4semidirect
9. C31×#391C4semidirect
70. #12857C31direct (N × H)
104. #1393C2×C31semidirect
124. C2×#117C4×C31semidirect
125. C22×C42C4×C31semidirect
128. #391C4×C31semidirect
161. C31#12857direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#22972
C31×#17822
C31×#13932
#1285731