Finite Group Databasebeta

← #15754683

Subgroups — #15754683

Order 3328.

782 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#11719248C211yesmaximal, characteristic
2#13113522C212yesmaximal, complemented
3#13113264C212yesmaximal, complemented
4#13113157C211yesmaximal, characteristic
5C13×#1800C211yesmaximal, characteristic
6C2×C4×D4×C13C2211yescharacteristic
7C2×D4×Dic13—21no—
8C22×C4⋊Dic13C412yes—
9C2×#88178C412yes—
10#11719248—24no—
11#11719248—24no—
12C2×#88144—21no—
13C2×#88132—22no—
14C2×#88123C2211yescharacteristic
15C2×C4×Dic26C2211yescharacteristic
16C13×#423C2212yes—
17C2×C13×#100—21no—
18#11718586C2212yes—
19C13×#343—21no—
20#11718117—22no—
21#11718078—24no—
22#11718076—24no—
23C22×D4×C13D411yescharacteristic
24C23×C4×C13C2×C412yes—
25C23×Dic13—22no—
26C22×Dic26—21no—
27C22×Dic26C2×C411yescharacteristic
28C2×C13×M16—22no—
29C4×D4×C13—22no—
30C4×D4×C13—22no—
31C2×C13×C4⋊C4D411yescharacteristic
32C2×C13×C22⋊C4D412yes—
33C2×C42×C13C2311yescharacteristic
34C2×C22⋊Dic13—22no—
35D4×Dic13—28no—
36C2×C4⋊Dic13—22no—
37C2×C4⋊Dic13—21no—
38C2×C4⋊Dic13—24no—
39C2×C4⋊Dic13—24no—
40C2×C4⋊Dic13C2×C411yescharacteristic
41C2×C4⋊Dic13C2×C412yes—
42#88178—24no—
43#88178—24no—
44C2×#6101—22no—
45C2×#6101—21no—
46C2×C4×Dic13—21no—
47#88144—28no—
48#88132—216no—
49#88123—22no—
50#88123D414yes—
51C4×Dic26—22no—
52C4×Dic26D414yes—
53C13×C4⋊C8—22no—
54C13×C4⋊C8—22no—
55C13×#100—42no—
56C22⋊#1055—22no—
57C4⋊#1055—22no—
58C4×#1055—22no—
59#8395—131nomaximal, Sylow
60C24×C13C22⋊C412yes—
61C2×D4×C13—22no—
62C2×D4×C13—22no—
63C2×D4×C13SD1612yes—
64C2×D4×C13Q1612yes—
65C22×C4×C13—22no—
66C22×C4×C13—22no—
67C22×C4×C13—44no—
68C22×C4×C13C22⋊C412yes—
69C22×C4×C13C22×C411yescharacteristic
70C22×C4×C13C2×D411yescharacteristic
71C22×C4×C13C2×D411yescharacteristic
72C22×Dic13—21no—
73C22×Dic13—21no—
74C22×Dic13—22no—
75C22×Dic13—28no—
76C22×Dic13—84no—
77C2×Dic26—21no—
78C2×Dic26—24no—
79C2×Dic26—42no—
80C2×Dic26—82no—
81C2×Dic26C22⋊C412yes—
82C13×M16—42no—
83C13×M16—42no—
84C2×C8×C13—42no—
85C13×C4⋊C4—22no—
86C13×C22⋊C4—24no—
87C42×C13C2×D412yes—
88C42×C13C2×D412yes—
89C22⋊Dic13—28no—
90C4⋊Dic13—28no—
91C4⋊Dic13—21no—
92C4⋊Dic13—22no—
93C4⋊Dic13—24no—
94C4⋊Dic13C22⋊C412yes—
95C4⋊Dic13C22⋊C414yes—
96#6101—28no—
97#6101—24no—
98C4×Dic13—24no—
99C2×#1055—42no—
100C2×#554—131no—
101#1800D1311yescharacteristic
102#1549—132no—
103#1487—132no—
104#1460—131no—
105C23×C13—22no—
106C23×C13—22no—
107C23×C13—42no—
108C23×C13—42no—
109C23×C13#10314yes—
110C23×C13C2×C22⋊C411yescharacteristic
111D4×C13—24no—
112D4×C13—24no—
113C2×C4×C13—22no—
114C2×C4×C13—22no—
115C2×C4×C13—41no—
116C2×C4×C13—44no—
117C2×C4×C13—44no—
118C2×C4×C13—42no—
119C2×C4×C13C2×C22⋊C411yescharacteristic
120C2×C4×C13C2×C22⋊C411yescharacteristic
121C2×C4×C13#12012yes—
122C2×C4×C13#12012yes—
123C2×C4×C13C2×SD1611yescharacteristic
124C2×C4×C13C2×Q1611yescharacteristic
125C2×C4×C13#13712yes—
126C2×Dic13—22no—
127C2×Dic13—22no—
128C2×Dic13—24no—
129C2×Dic13—42no—
130C2×Dic13—44no—
131C2×Dic13—42no—
132C2×Dic13—88no—
133Dic26—24no—
134Dic26—42no—
135Dic26—84no—
136C8×C13—82no—
137#1055—82no—
138#554—264no—
139#554—264no—
140C2×C4⋊Q8—131no—
141C2×#125—261no—
142C2×C22⋊Q8—132no—
143C2×C22⋊Q8—262no—
144C2×C4×Q8—131no—
145C2×C4×D4—261no—
146C2×C4×D4C2×D1311yescharacteristic
147C22×C4⋊C4—132no—
148#431—132no—
149#423C2×D1312yes—
150C2×#100—21no—
151#343—21no—
152#332—264no—
153Q8⋊C8—264no—
154#323—262no—
155C22×C13—21no—
156C22×C13—21no—
157C22×C13—24no—
158C22×C13—44no—
159C22×C13—44no—
160C22×C13—42no—
161C22×C13—42no—
162C22×C13#37911yescharacteristic
163C22×C13C2×#10311yescharacteristic
164C22×C13#41911yescharacteristic
165C4×C13—22no—
166C4×C13—21no—
167C4×C13—42no—
168C4×C13#41312yes—
169C4×C13#44812yes—
170C4×C13#45112yes—
171Dic13—44no—
172Dic13—42no—
173Dic13—42no—
174C22×Q8—131no—
175C22×Q8—261no—
176C22×D4C22⋊D1311yescharacteristic
177C23×C4—262no—
178C23×C4C4×D1312yes—
179C2×M16—22no—
180C4⋊Q8—134no—
181C4⋊Q8—262no—
182#125—268no—
183C22⋊Q8—2616no—
184C22⋊Q8—264no—
185C22⋊Q8—264no—
186C4×Q8—134no—
187C4×Q8—262no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—268no—
191C2×C4⋊C4—131no—
192C2×C4⋊C4—132no—
193C2×C4⋊C4—264no—
194C2×C4⋊C4—264no—
195C2×C4⋊C4—262no—
196C2×C4⋊C4—262no—
197C2×C4⋊C4—261no—
198C2×C4⋊C4—261no—
199C2×C4⋊C4C22⋊D1311yescharacteristic
200C2×C22⋊C4—262no—
201C2×C22⋊C4D5212yes—
202C2×C42—261no—
203C2×C42C22×D1311yescharacteristic
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—262no—
207#100—42no—
208C22⋊C8—262no—
209C4×C8—262no—
210C2×C13—21no—
211C2×C13—21no—
212C2×C13—44no—
213C2×C13#178511yescharacteristic
214C2×C13#178711yescharacteristic
215C2×C13#179511yescharacteristic
216C24#610312yes—
217C2×Q8—132no—
218C2×Q8—261no—
219C2×Q8—264no—
220C2×Q8—522no—
221C2×Q8—1042no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4#610512yes—
225C2×D4#610712yes—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—44no—
229C22×C4—261no—
230C22×C4—261no—
231C22×C4—262no—
232C22×C4—268no—
233C22×C4—1044no—
234C22×C4#610312yes—
235C22×C4C2×C4×D1311yescharacteristic
236C22×C4C2×D5211yescharacteristic
237C22×C4C2×C22⋊D1311yescharacteristic
238M16—42no—
239M16—42no—
240C2×C8—42no—
241C2×C8—522no—
242C4⋊C4—22no—
243C4⋊C4—132no—
244C4⋊C4—134no—
245C4⋊C4—268no—
246C4⋊C4—268no—
247C4⋊C4—261no—
248C4⋊C4—262no—
249C4⋊C4—264no—
250C4⋊C4—264no—
251C22⋊C4—24no—
252C22⋊C4—268no—
253C42—264no—
254C42D4×D1312yes—
255C42D4×D1312yes—
256C13#839511yescharacteristic, complemented, Sylow
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23#8805014yes—
262C23C2×#610311yescharacteristic
263Q8—264no—
264Q8—522no—
265Q8—1044no—
266D4—24no—
267D4—24no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—42no—
274C2×C4—262no—
275C2×C4—262no—
276C2×C4—264no—
277C2×C4—522no—
278C2×C4—524no—
279C2×C4—522no—
280C2×C4—1048no—
281C2×C4C22⋊C4×D1311yescharacteristic
282C2×C4C22⋊C4×D1311yescharacteristic
283C2×C4#8813612yes—
284C2×C4C2×#610511yescharacteristic
285C2×C4#8819112yes—
286C2×C4C2×#610711yescharacteristic
287C2×C4#8820512yes—
288C8—82no—
289C8—1042no—
290C22—21no—
291C22—21no—
292C22—24no—
293C22—44no—
294C22—44no—
295C22—42no—
296C22—42no—
297C22#1171861611yescharacteristic
298C22C2×#8805011yescharacteristic
299C22#1171866511yescharacteristic
300C4—21no—
301C4—22no—
302C4—42no—
303C4—522no—
304C4—522no—
305C4—524no—
306C4#1171840612yes—
307C4#1171870112yes—
308C4#1171871112yes—
309C2—21no—
310C2—21no—
311C2—44no—
312C2#1311476111yescharacteristic
313C2#1311477011yescharacteristic
314C2#1311480711yescharacteristic
subgroupG/H
centerC22#11718616
commutatorC22×C4×C13C22×C4
FrattiniC2×C42C22×D13
FittingC13×#1800C2
socleC22×C13#379
prime radicalC22×D4×C13D4
cosocleC2×C42×C13C23
hypercenter#1800D13
hypocenterC13#8395
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #13113522C2semidirect
3. #13113264C2semidirect
256. C13#8395semidirect

Maximal subgroups

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

subgroupindex
C2×#117192482
#131135222
#131132642
#131131572
C13×#18002
#839513