Finite Group Databasebeta

← C131×#8728 #45694612

Subgroups — C131×#8728 #45694612

Order 33536.

514 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 166 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
1#2285×C131C211yesmaximal, characteristic
2#1797×C131C212yesmaximal
3#1296×C131C212yesmaximal, complemented
4#1296×C131C212yesmaximal, complemented
5C2×#117×C131C412yes—
6C22×C42×C131C2211yescharacteristic
7C2×C4⋊C8×C131C2214yes—
8C2×C22⋊C8×C131—22no—
9#384×C131C2211yescharacteristic
10#384×C131C2211yescharacteristic
11#382×C131C412yes—
12#380×C131—22no—
13C4×C4⋊C4×C131—22no—
14#336×C131—22no—
15C23×C4×C131C2×C411yescharacteristic
16C23×C4×C131D412yes—
17C22×C8×C131—24no—
18#117×C131—44no—
19C2×C4⋊C4×C131—22no—
20C2×C4⋊C4×C131—22no—
21C2×C4⋊C4×C131C2×C412yes—
22C2×C4⋊C4×C131C2×C412yes—
23C2×C22⋊C4×C131—22no—
24C2×C42×C131—24no—
25C2×C42×C131—22no—
26C2×C42×C131C2×C411yescharacteristic
27C2×C42×C131D412yes—
28C2×C42×C131C2311yescharacteristic
29C4⋊C8×C131—28no—
30C22⋊C8×C131—44no—
31#95×C131—24no—
32C24×C131C22⋊C411yescharacteristic
33C22×C4×C131—22no—
34C22×C4×C131—22no—
35C22×C4×C131—22no—
36C22×C4×C131—24no—
37C22×C4×C131—24no—
38C22×C4×C131—22no—
39C22×C4×C131—22no—
40C22×C4×C131—22no—
41C22×C4×C131—22no—
42C22×C4×C131C22⋊C412yes—
43C22×C4×C131C22⋊C411yescharacteristic
44C22×C4×C131C22×C411yescharacteristic
45C22×C4×C131C2×D412yes—
46C2×C8×C131—44no—
47C2×C8×C131—44no—
48C2×C8×C131—44no—
49C4⋊C4×C131—42no—
50C4⋊C4×C131—42no—
51C4⋊C4×C131—44no—
52C4⋊C4×C131—42no—
53C4⋊C4×C131—42no—
54C22⋊C4×C131—44no—
55C42×C131—22no—
56C42×C131—24no—
57C42×C131—44no—
58C42×C131#4114yes—
59C23×C131—21no—
60C23×C131—21no—
61C23×C131—22no—
62C23×C131—22no—
63C23×C131#10112yes—
64C23×C131C2×C22⋊C411yescharacteristic
65C2×C4×C131—28no—
66C2×C4×C131—21no—
67C2×C4×C131—21no—
68C2×C4×C131—21no—
69C2×C4×C131—21no—
70C2×C4×C131—22no—
71C2×C4×C131—22no—
72C2×C4×C131—22no—
73C2×C4×C131—44no—
74C2×C4×C131—44no—
75C2×C4×C131—42no—
76C2×C4×C131—42no—
77C2×C4×C131—44no—
78C2×C4×C131—44no—
79C2×C4×C131#10012yes—
80C2×C4×C131#11712yes—
81C2×C4×C131#12314yes—
82C2×C4×C131#13514yes—
83C2×C4×C131AGL(1,ℤ/8)12yes—
84C2×C4×C131#13712yes—
85C8×C131—84no—
86C22×C131—24no—
87C22×C131—22no—
88C22×C131—41no—
89C22×C131—41no—
90C22×C131—42no—
91C22×C131#38111yescharacteristic
92C22×C131C2×#10011yescharacteristic
93C22×C131C2×#10111yescharacteristic
94C22×C131#41612yes—
95C22×C131#41712yes—
96C4×C131—22no—
97C4×C131—22no—
98C4×C131—42no—
99C4×C131—82no—
100C4×C131—82no—
101C4×C131#48214yes—
102C4×C131#48514yes—
103C2×C131—42no—
104C2×C131#154311yescharacteristic
105C2×C131#155212yes—
106C2×C131#156011yescharacteristic
107C2×C131#182111yescharacteristic
108C2×C131#182512yes—
109#8728C13111yesmaximal, characteristic, direct product factor, Sylow
110C131#872811yescharacteristic, direct product factor, Sylow
111#2285C2×C13111yescharacteristic
112#1797C2×C13112yes—
113#1296C2×C13112yescomplemented
114#1296C2×C13112yescomplemented
115C2×#117C4×C13112yes—
116C22×C42C22×C13111yescharacteristic
117C2×C4⋊C8C22×C13114yes—
118C2×C22⋊C8—22no—
119#384C22×C13111yescharacteristic
120#384C22×C13111yescharacteristic
121#382C4×C13112yes—
122#380—22no—
123C4×C4⋊C4—22no—
124#336—22no—
125C23×C4C2×C4×C13111yescharacteristic
126C23×C4D4×C13112yes—
127C22×C8—24no—
128#117—44no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4C2×C4×C13112yes—
132C2×C4⋊C4C2×C4×C13112yes—
133C2×C22⋊C4—22no—
134C2×C42—24no—
135C2×C42—22no—
136C2×C42C2×C4×C13111yescharacteristic
137C2×C42D4×C13112yes—
138C2×C42C23×C13111yescharacteristic
139C4⋊C8—28no—
140C22⋊C8—44no—
141#95—24no—
142C24C22⋊C4×C13111yescharacteristic
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4C22⋊C4×C13112yes—
153C22×C4C22⋊C4×C13111yescharacteristic
154C22×C4C22×C4×C13111yescharacteristic
155C22×C4C2×D4×C13112yes—
156C2×C8—44no—
157C2×C8—44no—
158C2×C8—44no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—44no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C22⋊C4—44no—
165C42—22no—
166C42—24no—
167C42—44no—
168C42#41×C13114yes—
169C23—21no—
170C23—22no—
171C23—22no—
172C23—21no—
173C23#101×C13112yes—
174C23C2×C22⋊C4×C13111yescharacteristic
175C2×C4—28no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—44no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4#100×C13112yes—
190C2×C4#117×C13112yes—
191C2×C4#123×C13114yes—
192C2×C4#135×C13114yes—
193C2×C4AGL(1,ℤ/8)×C13112yes—
194C2×C4#137×C13112yes—
195C8—84no—
196C22—24no—
197C22—22no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22#381×C13111yescharacteristic
202C22C2×#100×C13111yescharacteristic
203C22C2×#101×C13111yescharacteristic
204C22#416×C13112yes—
205C22#417×C13112yes—
206C4—22no—
207C4—22no—
208C4—42no—
209C4—82no—
210C4—82no—
211C4#482×C13114yes—
212C4#485×C13114yes—
213C2—42no—
214C2#1543×C13111yescharacteristic
215C2#1552×C13112yes—
216C2#1560×C13111yescharacteristic
217C2#1821×C13111yescharacteristic
218C2#1825×C13112yes—
subgroupG/H
center, socleC23×C131C2×C22⋊C4
commutatorC22×C4C22×C4×C131
Frattini, cosocleC2×C42C23×C131
prime radicalC24×C131C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C131 ◁ C22×C42×C131 ◁ G

Complements

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

NH ≅ G/Ntype
3. #1296×C131C2semidirect
4. #1296×C131C2semidirect
109. #8728C131direct (N × H)
110. C131#8728direct (N × H)
113. #1296C2×C131semidirect
114. #1296C2×C131semidirect

Maximal subgroups

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

subgroupindex
#2285×C1312
#1797×C1312
#1296×C1312
#8728131