Finite Group Databasebeta

← #16787112

Subgroups — #16787112

Order 3328.

376 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 74 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#13130098C212yesmaximal
2#13129776C212yesmaximal
3#13129767C212yesmaximal
4#13117555C211yesmaximal, characteristic
5C2×#88123C411yescharacteristic
6C2×C4×Dic26C411yescharacteristic
7C2×C4×Dic26C2211yescharacteristic
8#11719130—22no—
9C13×#376C411yescharacteristic
10#11718597C2211yescharacteristic
11#11718590C411yescharacteristic
12#11718590C2211yescharacteristic
13#11718314—22no—
14#11718314—22no—
15#11718286—22no—
16#11718285C2214yes—
17C22×Dic26C2×C411yescharacteristic
18C22×Dic26C2×C411yescharacteristic
19C2×C42×C13C2×C411yescharacteristic
20C2×C42×C13C2×C411yescharacteristic
21C2×C42×C13C2×C411yescharacteristic
22C2×C4⋊Dic13—22no—
23C2×C4⋊Dic13C2×C411yescharacteristic
24C2×C4⋊Dic13C2311yescharacteristic
25C2×#6101—21no—
26C2×#6101D412yes—
27C2×C4×Dic13—21no—
28C2×C4×Dic13D412yes—
29#88123—22no—
30#88123—22no—
31C4×Dic26—42no—
32C4×Dic26—42no—
33C22⋊#1057—24no—
34#88110—42no—
35C4⋊#1057—42no—
36C13×#95—21no—
37C13×#95—21no—
38#88071—22no—
39#88071—22no—
40#88071—22no—
41#11536—131nomaximal, Sylow
42C22×C4×C13—21no—
43C22×C4×C13—21no—
44C22×C4×C13C22⋊C411yescharacteristic
45C22×C4×C13C22⋊C411yescharacteristic
46C22×C4×C13C22×C411yescharacteristic
47C22×Dic13—21no—
48C22×Dic13—21no—
49C22×Dic13C22⋊C412yes—
50C22×Dic13C2×D412yes—
51C2×Dic26—21no—
52C2×Dic26—21no—
53C2×Dic26—21no—
54C2×Dic26—41no—
55C2×Dic26—41no—
56C2×Dic26—41no—
57C2×Dic26—41no—
58C2×Dic26M1612yes—
59C2×#1057—44no—
60C42×C13—22no—
61C42×C13—21no—
62C42×C13—41no—
63C42×C13M1612yes—
64C4⋊Dic13—28no—
65C4⋊Dic13—41no—
66C4⋊Dic13—41no—
67#6101—42no—
68#6101—42no—
69C4×Dic13—22no—
70C4×Dic13—22no—
71C4×Dic13—42no—
72#2327—131no—
73#1467—132no—
74#1324—132no—
75#1319—132no—
76C23×C13C2×C22⋊C411yescharacteristic
77C2×C4×C13—21no—
78C2×C4×C13—21no—
79C2×C4×C13—22no—
80C2×C4×C13—21no—
81C2×C4×C13—41no—
82C2×C4×C13—41no—
83C2×C4×C13—41no—
84C2×C4×C13—41no—
85C2×C4×C13—41no—
86C2×C4×C13—41no—
87C2×C4×C13—41no—
88C2×C4×C13#9912yes—
89C2×C4×C13C2×M1611yescharacteristic
90C2×C4×C13C2×M1611yescharacteristic
91C2×Dic13—22no—
92C2×Dic13—22no—
93C2×Dic13—22no—
94C2×Dic13—22no—
95C2×Dic13—22no—
96C2×Dic13—41no—
97C2×Dic13—41no—
98C2×Dic13—42no—
99C2×Dic13—42no—
100C2×Dic13—41no—
101Dic26—42no—
102Dic26—42no—
103Dic26—42no—
104Dic26—42no—
105#1057—84no—
106C2×C4⋊Q8—131no—
107C2×C4×Q8—131no—
108C2×C4×Q8—131no—
109#432—262no—
110#391—131no—
111#391—131no—
112#387—262no—
113#387—262no—
114#382—131no—
115#376#26111yescharacteristic
116#324—134no—
117#324—262no—
118C22×C13—21no—
119C22×C13—21no—
120C22×C13#40711yescharacteristic
121C22×C13C2×#9911yescharacteristic
122C22×C13#41311yescharacteristic
123C4×C13—22no—
124C4×C13—21no—
125C4×C13—21no—
126C4×C13—42no—
127C4×C13—41no—
128C4×C13—81no—
129Dic13—42no—
130Dic13—42no—
131Dic13—42no—
132Dic13—81no—
133C22×Q8—131no—
134C22×Q8—131no—
135C4⋊Q8—262no—
136C4⋊Q8—262no—
137C4×Q8—522no—
138C4×Q8—522no—
139C2×C4⋊C4—132no—
140C2×C4⋊C4—131no—
141C2×C4⋊C4—131no—
142C2×C4⋊C4—262no—
143C2×C4⋊C4—261no—
144C2×C42—132no—
145C2×C42—261no—
146C2×C42C2×#26111yescharacteristic
147C2×C42C2×#26111yescharacteristic
148C2×C42C2×#26111yescharacteristic
149C4⋊C8—522no—
150C22⋊C8—264no—
151#97—522no—
152#95—21no—
153#95—21no—
154#95—262no—
155#95—262no—
156#95—262no—
157C2×C13—21no—
158C2×C13—21no—
159C2×C13#146511yescharacteristic
160C2×C13#212911yescharacteristic
161C2×C13#213311yescharacteristic
162C2×Q8—132no—
163C2×Q8—261no—
164C2×Q8—261no—
165C2×Q8—261no—
166C2×Q8—521no—
167C2×Q8—521no—
168C2×Q8—521no—
169C2×Q8—521no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—132no—
173C22×C4—132no—
174C22×C4—261no—
175C22×C4—261no—
176C22×C4C22⋊#26111yescharacteristic
177C22×C4C22⋊#26111yescharacteristic
178C22×C4C22×#26111yescharacteristic
179C2×C8—524no—
180C4⋊C4—268no—
181C4⋊C4—522no—
182C4⋊C4—521no—
183C4⋊C4—521no—
184C4⋊C4—522no—
185C42—22no—
186C42—21no—
187C42—41no—
188C42—262no—
189C42—262no—
190C42—522no—
191C42#611812yes—
192C13#1153611yescharacteristic, complemented, Sylow
193C23C2×C22⋊#26111yescharacteristic
194Q8—522no—
195Q8—522no—
196Q8—522no—
197Q8—522no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—262no—
210C2×C4—262no—
211C2×C4—262no—
212C2×C4—262no—
213C2×C4—262no—
214C2×C4—521no—
215C2×C4—521no—
216C2×C4—522no—
217C2×C4—522no—
218C2×C4—521no—
219C2×C4#8810712yes—
220C2×C4C2×#611811yescharacteristic
221C2×C4D13⋊M1611yescharacteristic
222C8—1044no—
223C22—21no—
224C22—21no—
225C22C2×#8810711yescharacteristic
226C22#1171912711yescharacteristic
227C22#1171913211yescharacteristic
228C4—21no—
229C4—21no—
230C4—22no—
231C4—42no—
232C4—41no—
233C4—81no—
234C4—522no—
235C4—522no—
236C4—522no—
237C4—1041no—
238C2—21no—
239C2—21no—
240C2#1313009311yescharacteristic
241C2#1313112011yescharacteristic
242C2#1313113311yescharacteristic
subgroupG/H
centerC22C2×#88107
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
FittingC13×#376C4
socleC22×C13C2×#99
prime radicalC23×C13C2×C22⋊C4
cosocleC2×C4⋊Dic13C23
hypercenter#376#261
hypocenterC13#11536
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #376

Complements

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

NH ≅ G/Ntype
192. C13#11536semidirect

Maximal subgroups

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

subgroupindex
#131300982
#131297762
#131297672
#131175552
#1153613