Finite Group Databasebeta

← #15754663

Subgroups — #15754663

Order 3328.

966 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 966 conjugacy classes of subgroups — too many to draw. Only its 147 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×D4×Dic26C211yesmaximal, characteristic
2#13113518C212yesmaximal, complemented
3#13113264C212yesmaximal, complemented
4#13113157C211yesmaximal, characteristic
5C13×#1800C211yesmaximal, characteristic
6C23×Dic26C412yes—
7C2×C4×D4×C13C2211yescharacteristic
8C2×D4×Dic13—21no—
9C2×#88178C412yes—
10D4×Dic26—24no—
11D4×Dic26—24no—
12C2×#88142—21no—
13C2×#88132—22no—
14C2×#88123C2211yescharacteristic
15C2×C4×Dic26C2211yescharacteristic
16C13×#423C2212yes—
17C2×C13×#100—21no—
18#11718586C2212yes—
19C13×#343—21no—
20#11718117—22no—
21#11718076—24no—
22#11718073—24no—
23C22×D4×C13D411yescharacteristic
24C23×C4×C13C2×C412yes—
25C23×Dic13—22no—
26C22×Dic26—22no—
27C22×Dic26—24no—
28C22×Dic26—24no—
29C22×Dic26—48no—
30C22×Dic26C2×C411yescharacteristic
31C22×Dic26C2×C412yes—
32C2×C13×M16—22no—
33C4×D4×C13—22no—
34C4×D4×C13—22no—
35C2×C13×C4⋊C4D411yescharacteristic
36C2×C13×C22⋊C4D412yes—
37C2×C42×C13C2311yescharacteristic
38C2×C22⋊Dic13—22no—
39D4×Dic13—28no—
40C2×C4⋊Dic13—21no—
41C2×C4⋊Dic13—21no—
42C2×C4⋊Dic13C2×C411yescharacteristic
43#88178—24no—
44#88178—24no—
45C2×#6101—22no—
46C2×#6101—21no—
47C2×C4×Dic13—21no—
48#88142—28no—
49#88132—216no—
50#88123—22no—
51#88123D414yes—
52C4×Dic26—22no—
53C4×Dic26D414yes—
54C13×C4⋊C8—22no—
55C13×C4⋊C8—22no—
56C13×#100—42no—
57C22⋊#1055—22no—
58C4⋊#1055—22no—
59C4×#1055—22no—
60#8391—131nomaximal, Sylow
61C24×C13C22⋊C412yes—
62C2×D4×C13—22no—
63C2×D4×C13—22no—
64C2×D4×C13SD1612yes—
65C2×D4×C13Q1612yes—
66C22×C4×C13—22no—
67C22×C4×C13—22no—
68C22×C4×C13—44no—
69C22×C4×C13C22⋊C412yes—
70C22×C4×C13C22×C411yescharacteristic
71C22×C4×C13C2×D411yescharacteristic
72C22×C4×C13C2×D411yescharacteristic
73C22×Dic13—21no—
74C22×Dic13—28no—
75C22×Dic13—21no—
76C22×Dic13—22no—
77C22×Dic13—48no—
78C2×Dic26—21no—
79C2×Dic26—24no—
80C2×Dic26—28no—
81C2×Dic26—22no—
82C2×Dic26—416no—
83C2×Dic26—48no—
84C2×Dic26—48no—
85C2×Dic26—48no—
86C2×Dic26—44no—
87C2×Dic26C22⋊C412yes—
88C2×Dic26C22⋊C414yes—
89C13×M16—42no—
90C13×M16—42no—
91C2×C8×C13—42no—
92C13×C4⋊C4—22no—
93C13×C22⋊C4—24no—
94C42×C13C2×D412yes—
95C42×C13C2×D412yes—
96C22⋊Dic13—28no—
97C4⋊Dic13—21no—
98C4⋊Dic13—24no—
99C4⋊Dic13—24no—
100C4⋊Dic13C22⋊C412yes—
101#6101—28no—
102#6101—24no—
103C4×Dic13—24no—
104C2×#1055—42no—
105C2×D4×Q8—131no—
106#1800D1311yescharacteristic
107#1548—132no—
108#1487—132no—
109#1460—131no—
110C23×C13—22no—
111C23×C13—22no—
112C23×C13—42no—
113C23×C13—42no—
114C23×C13#10314yes—
115C23×C13C2×C22⋊C411yescharacteristic
116D4×C13—24no—
117D4×C13—24no—
118C2×C4×C13—22no—
119C2×C4×C13—22no—
120C2×C4×C13—41no—
121C2×C4×C13—44no—
122C2×C4×C13—44no—
123C2×C4×C13—42no—
124C2×C4×C13C2×C22⋊C411yescharacteristic
125C2×C4×C13C2×C22⋊C411yescharacteristic
126C2×C4×C13#12012yes—
127C2×C4×C13#12012yes—
128C2×C4×C13C2×SD1611yescharacteristic
129C2×C4×C13C2×Q1611yescharacteristic
130C2×C4×C13#13712yes—
131C2×Dic13—24no—
132C2×Dic13—22no—
133C2×Dic13—22no—
134C2×Dic13—24no—
135C2×Dic13—416no—
136C2×Dic13—44no—
137C2×Dic13—42no—
138Dic26—24no—
139Dic26—22no—
140Dic26—416no—
141Dic26—44no—
142Dic26—44no—
143Dic26#10014yescomplemented
144C8×C13—82no—
145#1055—82no—
146C23×Q8—132no—
147D4×Q8—264no—
148D4×Q8—264no—
149C2×C4⋊Q8—131no—
150C2×C4⋊Q8—261no—
151C2×C22⋊Q8—132no—
152C2×C22⋊Q8—262no—
153C2×C4×Q8—131no—
154C2×C4×D4—261no—
155C2×C4×D4C2×D1311yescharacteristic
156#431—132no—
157#423C2×D1312yes—
158C2×#100—21no—
159#343—21no—
160#333—264no—
161Q8⋊C8—264no—
162#323—262no—
163C22×C13—21no—
164C22×C13—21no—
165C22×C13—24no—
166C22×C13—44no—
167C22×C13—44no—
168C22×C13—42no—
169C22×C13—42no—
170C22×C13#37911yescharacteristic
171C22×C13C2×#10311yescharacteristic
172C22×C13#41911yescharacteristic
173C4×C13—22no—
174C4×C13—21no—
175C4×C13—42no—
176C4×C13C2×#10012yes—
177C4×C13#44812yes—
178C4×C13#45112yes—
179Dic13—24no—
180Dic13—42no—
181Dic13—44no—
182C22×Q8—131no—
183C22×Q8—132no—
184C22×Q8—264no—
185C22×Q8—264no—
186C22×Q8—262no—
187C22×Q8—528no—
188C22×D4C22⋊D1311yescharacteristic
189C23×C4—262no—
190C23×C4C4×D1312yes—
191C2×M16—22no—
192C4⋊Q8—134no—
193C4⋊Q8—268no—
194C4⋊Q8—262no—
195C22⋊Q8—2616no—
196C22⋊Q8—264no—
197C22⋊Q8—264no—
198C4×Q8—134no—
199C4×Q8—262no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—268no—
203C2×C4⋊C4—131no—
204C2×C4⋊C4—262no—
205C2×C4⋊C4—261no—
206C2×C4⋊C4—261no—
207C2×C4⋊C4—261no—
208C2×C4⋊C4C22⋊D1311yescharacteristic
209C2×C22⋊C4—262no—
210C2×C22⋊C4D5212yes—
211C2×C42—261no—
212C2×C42C22×D1311yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—262no—
216#100—42no—
217C22⋊C8—262no—
218C4×C8—262no—
219C2×C13—21no—
220C2×C13—21no—
221C2×C13—44no—
222C2×C13#178311yescharacteristic
223C2×C13#178711yescharacteristic
224C2×C13#179611yescharacteristic
225C24#610312yes—
226C2×Q8—132no—
227C2×Q8—134no—
228C2×Q8—264no—
229C2×Q8—268no—
230C2×Q8—261no—
231C2×Q8—262no—
232C2×Q8—524no—
233C2×Q8—528no—
234C2×Q8—528no—
235C2×Q8—5216no—
236C2×Q8—528no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4#610512yes—
240C2×D4#610712yes—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—44no—
244C22×C4—261no—
245C22×C4—268no—
246C22×C4—261no—
247C22×C4—262no—
248C22×C4—528no—
249C22×C4#610312yes—
250C22×C4C2×C4×D1311yescharacteristic
251C22×C4C2×D5211yescharacteristic
252C22×C4C2×C22⋊D1311yescharacteristic
253M16—42no—
254M16—42no—
255C2×C8—42no—
256C2×C8—522no—
257C4⋊C4—22no—
258C4⋊C4—132no—
259C4⋊C4—268no—
260C4⋊C4—261no—
261C4⋊C4—264no—
262C4⋊C4—264no—
263C4⋊C4—264no—
264C22⋊C4—24no—
265C22⋊C4—268no—
266C42—264no—
267C42D4×D1312yes—
268C42D4×D1312yes—
269C13#839111yescharacteristic, complemented, Sylow
270C23—22no—
271C23—22no—
272C23—42no—
273C23—42no—
274C23#8805014yes—
275C23C2×#610311yescharacteristic
276Q8—134no—
277Q8—264no—
278Q8—262no—
279Q8—524no—
280Q8—5216no—
281Q8—524no—
282D4—24no—
283D4—24no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—264no—
291C2×C4—262no—
292C2×C4—262no—
293C2×C4—264no—
294C2×C4—5216no—
295C2×C4—522no—
296C2×C4—524no—
297C2×C4C22⋊C4×D1311yescharacteristic
298C2×C4C22⋊C4×D1311yescharacteristic
299C2×C4#8813612yes—
300C2×C4C2×#610511yescharacteristic
301C2×C4#8819112yes—
302C2×C4C2×#610711yescharacteristic
303C2×C4#8820512yes—
304C8—82no—
305C8—1042no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—44no—
310C22—44no—
311C22—42no—
312C22—42no—
313C22#1171861611yescharacteristic
314C22C2×#8805011yescharacteristic
315C22#1171866511yescharacteristic
316C4—21no—
317C4—22no—
318C4—42no—
319C4—264no—
320C4—524no—
321C4—522no—
322C4D13×#10012yes—
323C4#1171870112yes—
324C4#1171871112yes—
325C2—21no—
326C2—21no—
327C2—44no—
328C2#1311475511yescharacteristic
329C2#1311477011yescharacteristic
330C2#1311481211yescharacteristic
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#8391
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. #13113518C2semidirect
3. #13113264C2semidirect
143. Dic26#100semidirect
269. C13#8391semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic262
#131135182
#131132642
#131131572
C13×#18002
#839113