Finite Group Databasebeta

← C131×#7817 #45693701

Subgroups — C131×#7817 #45693701

Order 33536.

410 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 106 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#2734×C131C211yesmaximal, characteristic
2#2096×C131C211yesmaximal, characteristic
3#2084×C131C212yesmaximal
4#1272×C131C211yesmaximal, characteristic
5#1271×C131C212yesmaximal
6C2×C4⋊Q8×C131—21no—
7C2×#106×C131—22no—
8C2×#106×C131—21no—
9C2×#103×C131—21no—
10#398×C131—21no—
11#395×C131—22no—
12#391×C131C2211yescharacteristic
13#389×C131C2212yes—
14#387×C131—22no—
15#384×C131C2211yescharacteristic
16#340×C131—22no—
17#336×C131C2211yescharacteristic
18#336×C131C2212yes—
19C22×Q8×C131—21no—
20C22×Q8×C131D411yescharacteristic
21C22×C8×C131—21no—
22C22×C8×C131—22no—
23C4⋊Q8×C131—42no—
24C4⋊Q8×C131—42no—
25C2×C4⋊C4×C131—22no—
26C2×C4⋊C4×C131—22no—
27C2×C4⋊C4×C131—21no—
28C2×C4⋊C4×C131—22no—
29C2×C4⋊C4×C131—21no—
30C2×C4⋊C4×C131D412yes—
31C2×C4⋊C4×C131D412yes—
32C2×C4⋊C4×C131D411yescharacteristic
33C2×C42×C131—21no—
34C2×C42×C131C2311yescharacteristic
35#106×C131—42no—
36#106×C131—44no—
37#103×C131—42no—
38#95×C131—22no—
39#95×C131—21no—
40#95×C131—22no—
41#95×C131—22no—
42C2×Q8×C131—21no—
43C2×Q8×C131—42no—
44C2×Q8×C131—41no—
45C2×Q8×C131—41no—
46C2×Q8×C131—81no—
47C2×Q8×C131—81no—
48C2×Q8×C131SD1612yes—
49C22×C4×C131—22no—
50C22×C4×C131—21no—
51C22×C4×C131—21no—
52C22×C4×C131—22no—
53C22×C4×C131—41no—
54C22×C4×C131C2×D411yescharacteristic
55C22×C4×C131C2×D412yes—
56C2×C8×C131—22no—
57C2×C8×C131—24no—
58C2×C8×C131—42no—
59C2×C8×C131—41no—
60C2×C8×C131—42no—
61C2×C8×C131—41no—
62C4⋊C4×C131—22no—
63C4⋊C4×C131—41no—
64C4⋊C4×C131—42no—
65C4⋊C4×C131—44no—
66C4⋊C4×C131—41no—
67C4⋊C4×C131—41no—
68C4⋊C4×C131—42no—
69C4⋊C4×C131—42no—
70C4⋊C4×C131—42no—
71C4⋊C4×C131—42no—
72C4⋊C4×C131—81no—
73C4⋊C4×C131SD1614yes—
74C42×C131—41no—
75C42×C131—42no—
76C23×C131#12011yescharacteristic
77Q8×C131—42no—
78Q8×C131—81no—
79Q8×C131—84no—
80C2×C4×C131—21no—
81C2×C4×C131—22no—
82C2×C4×C131—22no—
83C2×C4×C131—21no—
84C2×C4×C131—24no—
85C2×C4×C131—22no—
86C2×C4×C131—42no—
87C2×C4×C131—41no—
88C2×C4×C131—42no—
89C2×C4×C131—41no—
90C2×C4×C131—42no—
91C2×C4×C131—41no—
92C2×C4×C131—42no—
93C2×C4×C131—42no—
94C2×C4×C131—42no—
95C2×C4×C131—82no—
96C2×C4×C131—81no—
97C2×C4×C131C2×SD1612yes—
98C2×C4×C131C2×SD1611yescharacteristic
99C2×C4×C131AGL(1,ℤ/8)11yescharacteristic
100C2×C4×C131#13712yes—
101C8×C131—44no—
102C8×C131—42no—
103C22×C131#44812yes—
104C22×C131#45011yescharacteristic
105C22×C131#45311yescharacteristic
106C22×C131#45612yes—
107C22×C131#45811yescharacteristic
108C4×C131—22no—
109C4×C131—22no—
110C4×C131—21no—
111C4×C131—21no—
112C4×C131—42no—
113C4×C131—44no—
114C4×C131—82no—
115C4×C131—81no—
116C4×C131—161no—
117C2×C131#161112yes—
118C2×C131#161311yescharacteristic
119C2×C131#162311yescharacteristic
120C2×C131#165112yes—
121C2×C131#166011yescharacteristic
122#7817C13111yesmaximal, characteristic, direct product factor, Sylow
123C131#781711yescharacteristic, direct product factor, Sylow
124#2734C2×C13111yescharacteristic
125#2096C2×C13111yescharacteristic
126#2084C2×C13112yes—
127#1272C2×C13111yescharacteristic
128#1271C2×C13112yes—
129C2×C4⋊Q8—21no—
130C2×#106—22no—
131C2×#106—21no—
132C2×#103—21no—
133#398—21no—
134#395—22no—
135#391C22×C13111yescharacteristic
136#389C22×C13112yes—
137#387—22no—
138#384C22×C13111yescharacteristic
139#340—22no—
140#336C22×C13112yes—
141#336C22×C13111yescharacteristic
142C22×Q8—21no—
143C22×Q8D4×C13111yescharacteristic
144C22×C8—22no—
145C22×C8—21no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4D4×C13112yes—
154C2×C4⋊C4D4×C13112yes—
155C2×C4⋊C4D4×C13111yescharacteristic
156C2×C42—21no—
157C2×C42C23×C13111yescharacteristic
158#106—42no—
159#106—44no—
160#103—42no—
161#95—21no—
162#95—22no—
163#95—22no—
164#95—22no—
165C2×Q8—21no—
166C2×Q8—42no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—81no—
170C2×Q8—81no—
171C2×Q8SD16×C13112yes—
172C22×C4—22no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—41no—
177C22×C4C2×D4×C13111yescharacteristic
178C22×C4C2×D4×C13112yes—
179C2×C8—22no—
180C2×C8—24no—
181C2×C8—42no—
182C2×C8—41no—
183C2×C8—42no—
184C2×C8—41no—
185C4⋊C4—22no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—44no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—81no—
196C4⋊C4SD16×C13114yes—
197C42—41no—
198C42—42no—
199C23#120×C13111yescharacteristic
200Q8—42no—
201Q8—81no—
202Q8—84no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—82no—
219C2×C4—81no—
220C2×C4C2×SD16×C13112yes—
221C2×C4C2×SD16×C13111yescharacteristic
222C2×C4AGL(1,ℤ/8)×C13111yescharacteristic
223C2×C4#137×C13112yes—
224C8—44no—
225C8—42no—
226C22#448×C13112yes—
227C22#450×C13111yescharacteristic
228C22#453×C13111yescharacteristic
229C22#456×C13112yes—
230C22#458×C13111yescharacteristic
231C4—22no—
232C4—22no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—44no—
237C4—82no—
238C4—81no—
239C4—161no—
240C2#1611×C13112yes—
241C2#1613×C13111yescharacteristic
242C2#1623×C13111yescharacteristic
243C2#1651×C13112yes—
244C2#1660×C13111yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C131#120
commutator, Frattini, cosocleC2×C42C23×C131
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C131 ◁ C2×C42×C131 ◁ G

Complements

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

NH ≅ G/Ntype
122. #7817C131direct (N × H)
123. C131#7817direct (N × H)

Maximal subgroups

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

subgroupindex
#2734×C1312
#2096×C1312
#2084×C1312
#1272×C1312
#1271×C1312
#7817131