Finite Group Databasebeta

← C31×#8770 #21465529

Subgroups — C31×#8770 #21465529

Order 7936.

426 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2357C211yesmaximal, characteristic
2C31×#1800C211yesmaximal, characteristic
3C31×#1460C211yesmaximal, characteristic
4C31×#1303C212yesmaximal, complemented
5C31×#1286C212yesmaximal, complemented
6C2×C4×D4×C31C2211yescharacteristic
7C31×#431C2212yes—
8C31×#423C2212yes—
9C2×C31×#100—21no—
10C31×#390—21no—
11C31×#388—22no—
12C31×#382C2211yescharacteristic
13C31×#381C412yes—
14C4×C4⋊C4×C31—21no—
15C4×C4⋊C4×C31C2211yescharacteristic
16C4×C22⋊C4×C31C412yes—
17C31×#343—21no—
18C31×#323—22no—
19C22×D4×C31D411yescharacteristic
20C23×C4×C31C2×C412yes—
21C2×M16×C31—22no—
22C4×D4×C31—22no—
23C4×D4×C31—22no—
24C2×C4⋊C4×C31—21no—
25C2×C4⋊C4×C31—21no—
26C2×C4⋊C4×C31D411yescharacteristic
27C2×C22⋊C4×C31—22no—
28C2×C22⋊C4×C31—22no—
29C2×C22⋊C4×C31D412yes—
30C2×C42×C31—21no—
31C2×C42×C31—22no—
32C2×C42×C31C2311yescharacteristic
33C31×C4⋊C8—22no—
34C31×C4⋊C8—22no—
35C31×C4⋊C8—22no—
36C31×#100—42no—
37C31×C22⋊C8—22no—
38C4×C8×C31—22no—
39C31×#95—22no—
40C31×#95—21no—
41C31×#95C2×C411yescharacteristic
42C31×#95C2×C412yes—
43C31×#95C2×C411yescharacteristic
44C24×C31C22⋊C412yes—
45C2×D4×C31—22no—
46C2×D4×C31—22no—
47C2×D4×C31—21no—
48C2×D4×C31—21no—
49C22×C4×C31—21no—
50C22×C4×C31—22no—
51C22×C4×C31—22no—
52C22×C4×C31—22no—
53C22×C4×C31—21no—
54C22×C4×C31—82no—
55C22×C4×C31C22⋊C412yes—
56C22×C4×C31C22×C411yescharacteristic
57C22×C4×C31C2×D411yescharacteristic
58C22×C4×C31C2×D411yescharacteristic
59M16×C31—42no—
60M16×C31—42no—
61C2×C8×C31—42no—
62C2×C8×C31—42no—
63C4⋊C4×C31—22no—
64C4⋊C4×C31—42no—
65C4⋊C4×C31—42no—
66C22⋊C4×C31—24no—
67C22⋊C4×C31—44no—
68C22⋊C4×C31—82no—
69C42×C31—41no—
70C42×C31—41no—
71C42×C31—82no—
72C42×C31#4112yes—
73C42×C31#4112yes—
74#8770C3111yesmaximal, characteristic, direct product factor, Sylow
75C23×C31—22no—
76C23×C31—22no—
77C23×C31—22no—
78C23×C31—42no—
79C23×C31—42no—
80C23×C31C2×C22⋊C411yescharacteristic
81D4×C31—42no—
82D4×C31—42no—
83C2×C4×C31—22no—
84C2×C4×C31—22no—
85C2×C4×C31—41no—
86C2×C4×C31—41no—
87C2×C4×C31—41no—
88C2×C4×C31—41no—
89C2×C4×C31—42no—
90C2×C4×C31—42no—
91C2×C4×C31—42no—
92C2×C4×C31—41no—
93C2×C4×C31—41no—
94C2×C4×C31—42no—
95C2×C4×C31—41no—
96C2×C4×C31—82no—
97C2×C4×C31—82no—
98C2×C4×C31#11711yescharacteristic
99C2×C4×C31#11711yescharacteristic
100C2×C4×C31#12312yes—
101C2×C4×C31#12312yes—
102C2×C4×C31#13512yes—
103C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
104C2×C4×C31#13711yescharacteristic
105C8×C31—82no—
106C8×C31—82no—
107#2357C2×C3111yescharacteristic
108#1800C2×C3111yescharacteristic
109#1460C2×C3111yescharacteristic
110#1303C2×C3112yescomplemented
111#1286C2×C3112yescomplemented
112C22×C31—21no—
113C22×C31—21no—
114C22×C31—42no—
115C22×C31—42no—
116C22×C31—42no—
117C22×C31—82no—
118C22×C31—82no—
119C22×C31#38111yescharacteristic
120C22×C31#41611yescharacteristic
121C22×C31#41711yescharacteristic
122C4×C31—22no—
123C4×C31—21no—
124C4×C31—21no—
125C4×C31—42no—
126C4×C31—42no—
127C4×C31—81no—
128C4×C31—82no—
129C4×C31#48212yes—
130C4×C31#48512yes—
131C2×C4×D4C22×C3111yescharacteristic
132#431C22×C3112yes—
133#423C22×C3112yes—
134C2×#100—21no—
135#390—21no—
136#388—22no—
137#382C22×C3111yescharacteristic
138#381C4×C3112yes—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C22×C3111yescharacteristic
141C4×C22⋊C4C4×C3112yes—
142#343—21no—
143#323—22no—
144C2×C31—21no—
145C2×C31—21no—
146C2×C31—82no—
147C2×C31#182511yescharacteristic
148C2×C31#182711yescharacteristic
149C2×C31#182911yescharacteristic
150C22×D4D4×C3111yescharacteristic
151C23×C4C2×C4×C3112yes—
152C2×M16—22no—
153C4×D4—22no—
154C4×D4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4D4×C3111yescharacteristic
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4D4×C3112yes—
161C2×C42—21no—
162C2×C42—22no—
163C2×C42C23×C3111yescharacteristic
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167#100—42no—
168C22⋊C8—22no—
169C4×C8—22no—
170#95—22no—
171#95—21no—
172#95C2×C4×C3111yescharacteristic
173#95C2×C4×C3112yes—
174#95C2×C4×C3111yescharacteristic
175C31#877011yescharacteristic, direct product factor, Sylow
176C24C22⋊C4×C3112yes—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—21no—
180C2×D4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—82no—
187C22×C4C22⋊C4×C3112yes—
188C22×C4C22×C4×C3111yescharacteristic
189C22×C4C2×D4×C3111yescharacteristic
190C22×C4C2×D4×C3111yescharacteristic
191M16—42no—
192M16—42no—
193C2×C8—42no—
194C2×C8—42no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C22⋊C4—24no—
199C22⋊C4—44no—
200C22⋊C4—82no—
201C42—41no—
202C42—41no—
203C42—82no—
204C42#41×C3112yes—
205C42#41×C3112yes—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23C2×C22⋊C4×C3111yescharacteristic
212D4—42no—
213D4—42no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4C31×#11711yescharacteristic
230C2×C4C31×#11711yescharacteristic
231C2×C4C31×#12312yes—
232C2×C4C31×#12312yes—
233C2×C4C31×#13512yes—
234C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
235C2×C4C31×#13711yescharacteristic
236C8—82no—
237C8—82no—
238C22—21no—
239C22—21no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22—82no—
245C22C31×#38111yescharacteristic
246C22C31×#41611yescharacteristic
247C22C31×#41711yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—81no—
254C4—82no—
255C4C31×#48212yes—
256C4C31×#48512yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2C31×#182511yescharacteristic
261C2C31×#182711yescharacteristic
262C2C31×#182911yescharacteristic
subgroupG/H
center, socleC22×C31#381
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1303C2semidirect
5. C31×#1286C2semidirect
74. #8770C31direct (N × H)
110. #1303C2×C31semidirect
111. #1286C2×C31semidirect
175. C31#8770direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#23572
C31×#18002
C31×#14602
C31×#13032
C31×#12862
#877031