Finite Group Databasebeta

← C31×#11537 #21468296

Subgroups — C31×#11537 #21468296

Order 7936.

348 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2929C211yesmaximal, characteristic
2C31×#2166C212yesmaximal, complemented
3C31×#1321C212yesmaximal, complemented
4C31×#1320C212yesmaximal, complemented
5C2×C31×#131C411yescharacteristic
6C23×C8×C31C411yescharacteristic
7C31×#525C411yescharacteristic
8C31×#435—22no—
9C8×D4×C31—22no—
10C31×#424C411yescharacteristic
11C31×#408C2211yescharacteristic
12C31×#407C2211yescharacteristic
13C31×#405C2211yescharacteristic
14C31×#364—22no—
15C31×#349C2214yes—
16C31×#347—22no—
17C2×#41×C31C2×C411yescharacteristic
18C23×C4×C31C2×C411yescharacteristic
19C31×#131—42no—
20C2×M16×C31—21no—
21C2×M16×C31C2×C411yescharacteristic
22C2×M16×C31C2311yescharacteristic
23C22×C8×C31—21no—
24C22×C8×C31—21no—
25C22×C8×C31—21no—
26C22×C8×C31—42no—
27C22×C8×C31C2×C411yescharacteristic
28C22×C8×C31C2×C411yescharacteristic
29C31×#123—21no—
30C31×C22⋊Q8—21no—
31C31×#121—21no—
32C31×#120—21no—
33C4×D4×C31—22no—
34C31×#117C2×C411yescharacteristic
35C31×M32—24no—
36C31×C4⋊C8—22no—
37C31×C22⋊C8—22no—
38C31×C22⋊C8—22no—
39C31×#97D412yes—
40C4×C8×C31D412yes—
41C24×C31M1611yescharacteristic
42#41×C31—41no—
43#41×C31—41no—
44C2×Q8×C31M1611yescharacteristic
45C2×D4×C31—22no—
46C2×D4×C31—21no—
47C2×D4×C31M1611yescharacteristic
48C22×C4×C31—21no—
49C22×C4×C31—21no—
50C22×C4×C31—21no—
51C22×C4×C31—41no—
52C22×C4×C31—41no—
53C22×C4×C31M1611yescharacteristic
54C22×C4×C31C22×C411yescharacteristic
55M16×C31—41no—
56M16×C31—42no—
57M16×C31—41no—
58C2×C8×C31—22no—
59C2×C8×C31—21no—
60C2×C8×C31—21no—
61C2×C8×C31—42no—
62C2×C8×C31—42no—
63C2×C8×C31—42no—
64C2×C8×C31—41no—
65C2×C8×C31C22⋊C411yescharacteristic
66C2×C8×C31C22⋊C411yescharacteristic
67C2×C8×C31C2×D412yes—
68C4⋊C4×C31—22no—
69C4⋊C4×C31—21no—
70C22⋊C4×C31—22no—
71C22⋊C4×C31—22no—
72C22⋊C4×C31—21no—
73C42×C31C22⋊C412yes—
74C16×C31—44no—
75#11537C3111yesmaximal, characteristic, direct product factor, Sylow
76C23×C31—21no—
77C23×C31—21no—
78C23×C31—41no—
79C23×C31—42no—
80C23×C31C2×M1611yescharacteristic
81Q8×C31—41no—
82D4×C31—41no—
83D4×C31—44no—
84D4×C31—42no—
85C2×C4×C31—21no—
86C2×C4×C31—21no—
87C2×C4×C31—22no—
88C2×C4×C31—21no—
89C2×C4×C31—21no—
90C2×C4×C31—42no—
91C2×C4×C31—41no—
92C2×C4×C31—41no—
93C2×C4×C31—41no—
94C2×C4×C31—41no—
95C2×C4×C31—41no—
96C2×C4×C31C2×C22⋊C411yescharacteristic
97C2×C4×C31C2×M1611yescharacteristic
98C8×C31—22no—
99C8×C31—21no—
100C8×C31—41no—
101C8×C31—41no—
102C8×C31#10012yes—
103#2929C2×C3111yescharacteristic
104#2166C2×C3112yescomplemented
105#1321C2×C3112yescomplemented
106#1320C2×C3112yescomplemented
107C22×C31—22no—
108C22×C31—21no—
109C22×C31—41no—
110C22×C31—41no—
111C22×C31—42no—
112C22×C31—42no—
113C22×C31—42no—
114C22×C31—41no—
115C22×C31—41no—
116C22×C31#40711yescharacteristic
117C4×C31—21no—
118C4×C31—42no—
119C4×C31—41no—
120C4×C31—41no—
121C4×C31#41011yescharacteristic
122C4×C31C2×#10011yescharacteristic
123C2×#131C4×C3111yescharacteristic
124C23×C8C4×C3111yescharacteristic
125#525C4×C3111yescharacteristic
126#435—22no—
127C8×D4—22no—
128#424C4×C3111yescharacteristic
129#408C22×C3111yescharacteristic
130#407C22×C3111yescharacteristic
131#405C22×C3111yescharacteristic
132#364—22no—
133#349C22×C3114yes—
134#347—22no—
135C2×C31—21no—
136C2×C31—42no—
137C2×C31—41no—
138C2×C31—81no—
139C2×C31#146611yescharacteristic
140C2×#41C2×C4×C3111yescharacteristic
141C23×C4C2×C4×C3111yescharacteristic
142#131—42no—
143C2×M16—21no—
144C2×M16C2×C4×C3111yescharacteristic
145C2×M16C23×C3111yescharacteristic
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—42no—
150C22×C8C2×C4×C3111yescharacteristic
151C22×C8C2×C4×C3111yescharacteristic
152#123—21no—
153C22⋊Q8—21no—
154#121—21no—
155#120—21no—
156C4×D4—22no—
157#117C2×C4×C3111yescharacteristic
158M32—24no—
159C4⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162#97D4×C3112yes—
163C4×C8D4×C3112yes—
164C31#1153711yescharacteristic, direct product factor, Sylow
165C24M16×C3111yescharacteristic
166#41—41no—
167#41—41no—
168C2×Q8M16×C3111yescharacteristic
169C2×D4—22no—
170C2×D4—21no—
171C2×D4M16×C3111yescharacteristic
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4M16×C3111yescharacteristic
178C22×C4C22×C4×C3111yescharacteristic
179M16—41no—
180M16—42no—
181M16—41no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—41no—
189C2×C8C22⋊C4×C3111yescharacteristic
190C2×C8C22⋊C4×C3111yescharacteristic
191C2×C8C2×D4×C3112yes—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—21no—
197C42C22⋊C4×C3112yes—
198C16—44no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—42no—
203C23C2×M16×C3111yescharacteristic
204Q8—41no—
205D4—41no—
206D4—44no—
207D4—42no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4C2×C22⋊C4×C3111yescharacteristic
220C2×C4C2×M16×C3111yescharacteristic
221C8—22no—
222C8—21no—
223C8—41no—
224C8—41no—
225C8C31×#10012yes—
226C22—22no—
227C22—21no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22C31×#40711yescharacteristic
236C4—21no—
237C4—42no—
238C4—41no—
239C4—41no—
240C4C31×#41011yescharacteristic
241C4C2×C31×#10011yescharacteristic
242C2—21no—
243C2—42no—
244C2—41no—
245C2—81no—
246C2C31×#146611yescharacteristic
subgroupG/H
centerC4×C31#410
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×M16C23×C31
socleC2×C31#1466
prime radicalC31×#525C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C31 ◁ C2×C8×C31 ◁ C31×#405 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2166C2semidirect
3. C31×#1321C2semidirect
4. C31×#1320C2semidirect
75. #11537C31direct (N × H)
104. #2166C2×C31semidirect
105. #1321C2×C31semidirect
106. #1320C2×C31semidirect
164. C31#11537direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#29292
C31×#21662
C31×#13212
C31×#13202
#1153731