Finite Group Databasebeta

← #15755165

Subgroups — #15755165

Order 3328.

682 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 139 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#13115826C212yesmaximal, complemented
2#13113574C211yesmaximal, characteristic, complemented
3#13113431C211yesmaximal, characteristic, complemented
4#13113275C212yesmaximal
5C13×#3509C211yesmaximal, characteristic
6C13×#557—22no—
7C13×#556—22no—
8C13×#552—24no—
9C13×#541—22no—
10C13×#536C412yes—
11C13×#533—22no—
12C13×#533C2211yescharacteristic
13C2×C13×C4⋊Q8C2211yescharacteristic
14C13×#519C412yes—
15C4×C13×#41C2211yescharacteristic
16#11718585C2212yes—
17C4×#6099C2212yes—
18#11718139—22no—
19#11718135—22no—
20Q8⋊#1055—22no—
21D4⋊#1055—22no—
22#11718122—24no—
23C2×C13×#41C2×C411yescharacteristic
24C22×Q8×C13C2×C412yes—
25C13×C4⋊Q8—22no—
26C13×C4⋊Q8—21no—
27C13×C4⋊Q8—21no—
28C13×C4⋊Q8—22no—
29C13×C4⋊Q8D412yes—
30C13×#126—28no—
31C13×#125—24no—
32C13×#125—28no—
33C13×#125D412yes—
34C13×#124—22no—
35C13×#123—24no—
36C13×C22⋊Q8—24no—
37C13×C22⋊Q8—22no—
38C13×C22⋊Q8—24no—
39C4×Q8×C13—24no—
40C4×Q8×C13—24no—
41C4×Q8×C13—24no—
42C4×Q8×C13—22no—
43C4×Q8×C13D412yes—
44C4×D4×C13—22no—
45C4×D4×C13D412yes—
46C13×#117—24no—
47C13×#117—24no—
48C13×#117—21no—
49C13×#117C2×C411yescharacteristic
50C13×#117C2×C412yes—
51C2×C13×C4⋊C4D414yes—
52C2×C42×C13—21no—
53C2×C42×C13C2311yescharacteristic
54C2×#6099—22no—
55C2×#6099—22no—
56C4⋊#1055—22no—
57C4⋊#1055—22no—
58#88043—22no—
59C4×#1055—22no—
60#8474—131nomaximal, Sylow
61C13×#41—42no—
62C2×Q8×C13—22no—
63C2×Q8×C13—22no—
64C2×Q8×C13—22no—
65C2×Q8×C13—44no—
66C2×Q8×C13C22⋊C411yescharacteristic
67C2×D4×C13—21no—
68C2×D4×C13C22⋊C411yescharacteristic
69C22×C4×C13—21no—
70C22×C4×C13—22no—
71C22×C4×C13C22⋊C414yes—
72C22×C4×C13C22×C411yescharacteristic
73C22×C4×C13C2×D412yes—
74C13×C4⋊C4—28no—
75C13×C4⋊C4—28no—
76C13×C4⋊C4—24no—
77C13×C4⋊C4—24no—
78C13×C4⋊C4—22no—
79C13×C4⋊C4—22no—
80C13×C4⋊C4—24no—
81C13×C4⋊C4C22⋊C412yes—
82C13×C4⋊C4C22⋊C414yes—
83C13×C22⋊C4—28no—
84C13×C22⋊C4—21no—
85C13×C22⋊C4—22no—
86C13×C22⋊C4—22no—
87C42×C13—28no—
88C42×C13—21no—
89C42×C13—22no—
90C42×C13—22no—
91C42×C13C2×D411yescharacteristic
92C42×C13C2×D411yescharacteristic
93C42×C13C2×D412yes—
94#6099—24no—
95#6099—42no—
96#6099—44no—
97C2×#1055—42no—
98C2×#1055—42no—
99#3509D1311yescharacteristic
100#1997—132no—
101#1561—131no—
102#1526—131no—
103#1489—132no—
104C23×C13—21no—
105C23×C13C2×C22⋊C411yescharacteristic
106Q8×C13—22no—
107Q8×C13—44no—
108Q8×C13—44no—
109Q8×C13—42no—
110Q8×C13—42no—
111D4×C13—22no—
112D4×C13—42no—
113C2×C4×C13—22no—
114C2×C4×C13—22no—
115C2×C4×C13—22no—
116C2×C4×C13—22no—
117C2×C4×C13—24no—
118C2×C4×C13—24no—
119C2×C4×C13—21no—
120C2×C4×C13—42no—
121C2×C4×C13—42no—
122C2×C4×C13—42no—
123C2×C4×C13—44no—
124C2×C4×C13C2×C22⋊C411yescharacteristic
125C2×C4×C13C2×C22⋊C411yescharacteristic
126C2×C4×C13#12012yes—
127C2×C4×C13#12012yes—
128#1055—44no—
129#1055—82no—
130#557—22no—
131#556—22no—
132#552—24no—
133#541—22no—
134#536Dic1312yes—
135#533—22no—
136#533C2×D1311yescharacteristic
137C2×C4⋊Q8C2×D1311yescharacteristic
138#519Dic1312yes—
139C4×#41C2×D1311yescharacteristic
140#423—132no—
141C4×M16—132no—
142#343—264no—
143#332—262no—
144#330—262no—
145Q8⋊C8—262no—
146D4⋊C8—262no—
147C22×C13—22no—
148C22×C13—21no—
149C22×C13—42no—
150C22×C13—41no—
151C22×C13#37911yescharacteristic
152C4×C13—22no—
153C4×C13—21no—
154C4×C13—42no—
155C4×C13—44no—
156C4×C13—41no—
157C4×C13—44no—
158C4×C13#41312yes—
159C4×C13#45412yes—
160C4×C13#45612yes—
161C2×#41C2×Dic1311yescharacteristic
162C22×Q8C2×Dic1312yes—
163C2×M16—262no—
164C2×M16—262no—
165C4⋊Q8—22no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—22no—
169C4⋊Q8C22⋊D1312yes—
170#126—28no—
171#125—24no—
172#125—28no—
173#125C22⋊D1312yes—
174#124—22no—
175#123—24no—
176C22⋊Q8—24no—
177C22⋊Q8—22no—
178C22⋊Q8—24no—
179C4×Q8—24no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—22no—
183C4×Q8C22⋊D1312yes—
184C4×D4—22no—
185C4×D4C22⋊D1312yes—
186#117—24no—
187#117—24no—
188#117—21no—
189#117C2×Dic1311yescharacteristic
190#117C2×Dic1312yes—
191C2×C4⋊C4C22⋊D1314yes—
192C2×C42—21no—
193C2×C42C22×D1311yescharacteristic
194C4⋊C8—262no—
195C4⋊C8—262no—
196#97—262no—
197C4×C8—262no—
198C2×C13—41no—
199C2×C13—81no—
200C2×C13#178511yescharacteristic
201C2×C13#179312yes—
202#41—42no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—44no—
207C2×Q8C22⋊Dic1311yescharacteristic
208C2×D4—21no—
209C2×D4C22⋊Dic1311yescharacteristic
210C22×C4—21no—
211C22×C4—22no—
212C22×C4C22⋊Dic1314yes—
213C22×C4C22×Dic1311yescharacteristic
214C22×C4C2×C22⋊D1312yes—
215M16—264no—
216M16—522no—
217M16—524no—
218C2×C8—522no—
219C2×C8—522no—
220C4⋊C4—28no—
221C4⋊C4—28no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—24no—
227C4⋊C4C22⋊Dic1312yes—
228C4⋊C4C22⋊Dic1314yes—
229C22⋊C4—28no—
230C22⋊C4—21no—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C42—28no—
234C42—21no—
235C42—22no—
236C42—22no—
237C42C2×C22⋊D1312yes—
238C42C2×C22⋊D1311yescharacteristic
239C42C2×C22⋊D1311yescharacteristic
240C13#847411yescharacteristic, complemented, Sylow
241C23—21no—
242C23C2×C22⋊Dic1311yescharacteristic
243Q8—22no—
244Q8—44no—
245Q8—44no—
246Q8—42no—
247Q8—42no—
248D4—22no—
249D4—42no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4C2×C22⋊Dic1311yescharacteristic
262C2×C4C2×C22⋊Dic1311yescharacteristic
263C2×C4#8820712yes—
264C2×C4#8820712yes—
265C8—524no—
266C8—1042no—
267C22—21no—
268C22—22no—
269C22—41no—
270C22—42no—
271C22#1171890811yescharacteristic
272C4—21no—
273C4—22no—
274C4—41no—
275C4—44no—
276C4—42no—
277C4—44no—
278C4#1171889812yes—
279C4#1171890612yes—
280C4#1171890712yes—
281C2—41no—
282C2—81no—
283C2#1311476411yescharacteristic
284C2#1311479512yes—
subgroupG/H
centerC22#11718908
commutatorC22×C4×C13C22×C4
FrattiniC2×C42C22×D13
FittingC13×#3509C2
socleC22×C13#379
prime radicalC2×C13×#41C2×C4
cosocleC2×C42×C13C23
hypercenter#3509D13
hypocenterC13#8474
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3509

Complements

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

NH ≅ G/Ntype
1. #13115826C2semidirect
2. #13113574C2semidirect
3. #13113431C2semidirect
240. C13#8474semidirect

Maximal subgroups

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

subgroupindex
#131158262
#131135742
#131134312
#131132752
C13×#35092
#847413