Finite Group Databasebeta

← #15837774

Subgroups — #15837774

Order 3328.

910 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 207 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#13125781C212yesmaximal
2#13120041C211yesmaximal, characteristic
3#13113810C218yesmaximal
4#13113335C212yesmaximal
5#13113160C211yesmaximal, characteristic
6C13×#3502C211yesmaximal, characteristic
7Q82×C13—22no—
8C13×#557—22no—
9C13×#557—24no—
10C13×#544—21no—
11C13×#544C2214yes—
12C13×#533—22no—
13C13×#533C2212yes—
14C13×#519—21no—
15C2×C4×Q8×C13C2211yescharacteristic
16C2×#88123C2211yescharacteristic
17#11718891—21no—
18#11718754—22no—
19#11718753—22no—
20#11718711C2218yes—
21#11718692—24no—
22#11718640C2214yes—
23#11718637—21no—
24C4×C4⋊Dic13C2211yescharacteristic
25#11718589—22no—
26#11718586C2212yes—
27#11718130C2218yes—
28#11718120C2214yes—
29C22×Q8×C13D411yescharacteristic
30C22×Dic26C2312yes—
31C13×C4⋊Q8—22no—
32C13×C4⋊Q8—22no—
33C13×C4⋊Q8—24no—
34C13×C4⋊Q8—28no—
35C13×C4⋊Q8D412yes—
36C13×C4⋊Q8D412yes—
37C13×#125—24no—
38C13×#125—24no—
39C13×#125—28no—
40C13×#125—22no—
41C13×C22⋊Q8—22no—
42C13×C22⋊Q8—24no—
43C13×C22⋊Q8C2314yes—
44C4×Q8×C13—24no—
45C4×Q8×C13—24no—
46C4×Q8×C13—24no—
47C4×Q8×C13—24no—
48C4×Q8×C13—22no—
49C4×Q8×C13—22no—
50C4×Q8×C13—22no—
51C4×Q8×C13D414yes—
52C13×#117—24no—
53C13×#117—22no—
54C2×C13×C4⋊C4D411yescharacteristic
55C2×C13×C4⋊C4C2312yes—
56C2×C42×C13D412yes—
57C2×C42×C13C2311yescharacteristic
58C2×#6107—28no—
59C2×C4⋊Dic13—21no—
60C2×C4⋊Dic13—21no—
61C2×C4⋊Dic13—21no—
62C2×#6101—24no—
63C2×C4×Dic13—21no—
64#88123—22no—
65#88123—22no—
66#88071—24no—
67#88071C2312yes—
68C22⋊#1055C2314yes—
69#88050—28no—
70C4⋊#1055—22no—
71C4×#1055—22no—
72#18404—131nomaximal, Sylow
73C2×Q8×C13—24no—
74C2×Q8×C13—22no—
75C2×Q8×C13—22no—
76C2×Q8×C13—22no—
77C2×Q8×C13Q1614yescomplemented
78C2×Q8×C13C2×D414yes—
79C22×C4×C13C2×D411yescharacteristic
80C22×C4×C13C2×D412yes—
81C22×C4×C13C2×D412yes—
82C22×C4×C13C2×D411yescharacteristic
83C22×C4×C13C2411yescharacteristic
84C22×Dic13—22no—
85C22×Dic13—22no—
86C2×Dic26—24no—
87C2×Dic26—82no—
88C13×C4⋊C4—24no—
89C13×C4⋊C4—24no—
90C13×C4⋊C4—24no—
91C13×C4⋊C4—28no—
92C13×C4⋊C4—22no—
93C13×C4⋊C4—22no—
94C13×C4⋊C4—22no—
95C13×C4⋊C4—22no—
96C13×C4⋊C4—24no—
97C13×C4⋊C4—24no—
98C13×C4⋊C4—24no—
99C13×C4⋊C4C2×D414yes—
100C13×C22⋊C4—24no—
101C42×C13—28no—
102C42×C13—22no—
103C42×C13—22no—
104C42×C13—24no—
105C42×C13C2×D412yes—
106C42×C13C2×D412yes—
107#6107—88no—
108C4⋊Dic13—24no—
109C4⋊Dic13—41no—
110C4⋊Dic13—42no—
111C4⋊Dic13—41no—
112#6101—84no—
113C4×Dic13—42no—
114C2×#1055—24no—
115#3502D1311yescharacteristic
116#3209—132no—
117#2567—131no—
118#1605—138no—
119#1503—132no—
120#1461—131no—
121C23×C13C22×D411yescharacteristic
122Q8×C13—28no—
123Q8×C13—44no—
124Q8×C13—44no—
125Q8×C13—84no—
126C2×C4×C13—22no—
127C2×C4×C13—22no—
128C2×C4×C13—22no—
129C2×C4×C13—24no—
130C2×C4×C13—24no—
131C2×C4×C13—22no—
132C2×C4×C13—41no—
133C2×C4×C13—42no—
134C2×C4×C13#12011yescharacteristic
135C2×C4×C13#12012yes—
136C2×C4×C13#12011yescharacteristic
137C2×C4×C13C2×Q1614yes—
138C2×C4×C13C2×Q1612yes—
139C2×C4×C13#13712yes—
140C2×C4×C13C22×D412yes—
141C2×Dic13—24no—
142C2×Dic13—42no—
143C2×Dic13—42no—
144C2×Dic13—44no—
145C2×Dic13—42no—
146Dic26—48no—
147#1055—44no—
148Q82—22no—
149#557—22no—
150#557—24no—
151#544—21no—
152#544C2×D1314yes—
153#533—22no—
154#533C2×D1312yes—
155C2×C4⋊Q8—131no—
156#519—21no—
157C2×C4×Q8C2×D1311yescharacteristic
158#494—262no—
159#487—262no—
160#462—264no—
161#451—138no—
162#431—132no—
163#395—134no—
164#391—261no—
165#389—262no—
166#384—261no—
167C4×C4⋊C4—131no—
168#328—138no—
169#324—134no—
170C22×C13—21no—
171C22×C13—21no—
172C22×C13C2×#12011yescharacteristic
173C22×C13C22×Q1611yescharacteristic
174C22×C13C2×#13711yescharacteristic
175C4×C13—24no—
176C4×C13—21no—
177C4×C13—22no—
178C4×C13—44no—
179C4×C13—42no—
180C4×C13—44no—
181C4×C13#45114yes—
182C4×C13#45612yes—
183Dic13—44no—
184Dic13—82no—
185C22×Q8—132no—
186C22×Q8C22⋊D1311yescharacteristic
187C2×Q16—268no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—24no—
191C4⋊Q8—28no—
192C4⋊Q8—262no—
193C4⋊Q8—262no—
194C4⋊Q8C22⋊D1312yes—
195C4⋊Q8C22⋊D1312yes—
196#125—24no—
197#125—24no—
198#125—28no—
199#125—22no—
200C22⋊Q8—22no—
201C22⋊Q8—24no—
202C22⋊Q8C22×D1314yes—
203C4×Q8—24no—
204C4×Q8—24no—
205C4×Q8—24no—
206C4×Q8—24no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8C22⋊D1314yes—
211#117—24no—
212#117—22no—
213C2×C4⋊C4—261no—
214C2×C4⋊C4—261no—
215C2×C4⋊C4—261no—
216C2×C4⋊C4—264no—
217C2×C4⋊C4C22⋊D1311yescharacteristic
218C2×C4⋊C4C22×D1312yes—
219C2×C42—261no—
220C2×C42C22⋊D1312yes—
221C2×C42C22×D1311yescharacteristic
222C4⋊C8—262no—
223#103—268no—
224C22⋊C8—134no—
225C4×C8—262no—
226#95—132no—
227#95—264no—
228C2×C13—21no—
229C2×C13—21no—
230C2×C13C2×#45111yescharacteristic
231C2×C13C2×#45611yescharacteristic
232C2×C13#302711yescharacteristic
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—264no—
238C2×Q8—1042no—
239C2×Q8#610714yescomplemented
240C2×Q8C2×C22⋊D1314yes—
241C22×C4—262no—
242C22×C4—262no—
243C22×C4C2×C22⋊D1312yes—
244C22×C4C2×C22⋊D1311yescharacteristic
245C22×C4C2×C22⋊D1312yes—
246C22×C4C2×C22⋊D1311yescharacteristic
247C22×C4C23×D1311yescharacteristic
248Q16—1048no—
249C2×C8—264no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—28no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—264no—
262C4⋊C4—521no—
263C4⋊C4—522no—
264C4⋊C4—521no—
265C4⋊C4—1044no—
266C4⋊C4C2×C22⋊D1314yes—
267C22⋊C4—24no—
268C42—28no—
269C42—22no—
270C42—22no—
271C42—24no—
272C42—522no—
273C42C2×C22⋊D1312yes—
274C42C2×C22⋊D1312yes—
275C13#1840411yescharacteristic, complemented, Sylow
276C23C22×C22⋊D1311yescharacteristic
277Q8—28no—
278Q8—44no—
279Q8—44no—
280Q8—84no—
281Q8—528no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—264no—
291C2×C4—522no—
292C2×C4—522no—
293C2×C4—524no—
294C2×C4—522no—
295C2×C4#8819612yes—
296C2×C4C2×#610714yes—
297C2×C4C2×#610712yes—
298C2×C4#8820711yescharacteristic
299C2×C4#8820712yes—
300C2×C4#8820711yescharacteristic
301C2×C4C22×C22⋊D1312yes—
302C8—524no—
303C22—21no—
304C22—21no—
305C22C2×#8819611yescharacteristic
306C22C22×#610711yescharacteristic
307C22C2×#8820711yescharacteristic
308C4—21no—
309C4—22no—
310C4—24no—
311C4—44no—
312C4—42no—
313C4—44no—
314C4—524no—
315C4—1042no—
316C4#1171888914yes—
317C4#1171890712yes—
318C2—21no—
319C2—21no—
320C2C2×#1171888911yescharacteristic
321C2C2×#1171890711yescharacteristic
322C2#1312423811yescharacteristic
subgroupG/H
centerC22C2×#88207
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#3502C2
socleC22×C13C2×#120
prime radicalC23×C13C22×D4
hypercenter#3502D13
hypocenterC13#18404
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
77. C2×Q8×C13Q16semidirect
239. C2×Q8#6107semidirect
275. C13#18404semidirect

Maximal subgroups

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

subgroupindex
#131257812
#131200412
#131138102
#131133352
#131131602
C13×#35022
#1840413