Finite Group Databasebeta

← #15837723

Subgroups — #15837723

Order 3328.

1010 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1010 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#13125984C212yesmaximal, complemented
2#13119789C211yesmaximal, characteristic
3#13113861C218yesmaximal, complemented
4#13113335C212yesmaximal, complemented
5#13113157C211yesmaximal, characteristic
6C13×#3501C211yesmaximal, characteristic
7C13×#554—22no—
8C13×#554—24no—
9D4×Q8×C13—22no—
10C13×#544—21no—
11C13×#543C2214yes—
12C13×#533C2212yes—
13C2×C13×C22⋊Q8—22no—
14C13×#518—21no—
15C2×C4×D4×C13C2211yescharacteristic
16C2×#88178—22no—
17C2×#88123C2211yescharacteristic
18#11718882—21no—
19#11718742—22no—
20#11718739—22no—
21#11718705C2218yes—
22#11718682—24no—
23#11718637—21no—
24#11718610C2214yes—
25C4×C4⋊Dic13C2211yescharacteristic
26#11718586C2212yes—
27#11718130C2218yes—
28#11718117C2214yes—
29C22×Q8×C13—22no—
30C22×D4×C13D411yescharacteristic
31C23×C4×C13D412yes—
32C22×Dic26—21no—
33C13×C4⋊Q8—24no—
34C13×C4⋊Q8D412yes—
35C13×C4⋊Q8D412yes—
36C13×#125—24no—
37C13×#125—22no—
38C13×C22⋊Q8—24no—
39C13×C22⋊Q8—24no—
40C13×C22⋊Q8—28no—
41C13×C22⋊Q8—22no—
42C13×C22⋊Q8—24no—
43C13×C22⋊Q8—28no—
44C13×C22⋊Q8C2314yes—
45C4×Q8×C13—22no—
46C4×Q8×C13D414yes—
47C4×D4×C13—24no—
48C4×D4×C13—24no—
49C4×D4×C13—22no—
50C4×D4×C13—22no—
51C13×#117—22no—
52C2×C13×C4⋊C4—22no—
53C2×C13×C4⋊C4—24no—
54C2×C13×C4⋊C4D411yescharacteristic
55C2×C13×C22⋊C4—24no—
56C2×C13×C22⋊C4C2312yes—
57C2×C42×C13C2311yescharacteristic
58C2×C22⋊Dic13—24no—
59C2×C4⋊Dic13—21no—
60C2×C4⋊Dic13—21no—
61C2×C4⋊Dic13C2312yes—
62#88178—48no—
63C2×#6101—24no—
64C2×C4×Dic13—21no—
65#88123—22no—
66#88123—22no—
67#88075—28no—
68#88071C2312yes—
69C22⋊#1055C2314yes—
70#88047—28no—
71C4⋊#1055—22no—
72C4×#1055—22no—
73#18397—131nomaximal, Sylow
74C24×C13C2×D412yes—
75C2×Q8×C13—24no—
76C2×Q8×C13—22no—
77C2×Q8×C13—84no—
78C2×Q8×C13C2×D414yes—
79C2×D4×C13—22no—
80C2×D4×C13—22no—
81C2×D4×C13Q1614yes—
82C22×C4×C13—24no—
83C22×C4×C13—24no—
84C22×C4×C13—22no—
85C22×C4×C13—22no—
86C22×C4×C13—44no—
87C22×C4×C13C2×D411yescharacteristic
88C22×C4×C13C2×D412yes—
89C22×C4×C13C2×D411yescharacteristic
90C22×C4×C13C2411yescharacteristic
91C22×Dic13—22no—
92C22×Dic13—22no—
93C2×Dic26—44no—
94C2×Dic26—41no—
95C2×Dic26—41no—
96C13×C4⋊C4—24no—
97C13×C4⋊C4—28no—
98C13×C4⋊C4—22no—
99C13×C4⋊C4—24no—
100C13×C4⋊C4—22no—
101C13×C4⋊C4—22no—
102C13×C4⋊C4—22no—
103C13×C4⋊C4—24no—
104C13×C4⋊C4C2×D414yes—
105C13×C22⋊C4—28no—
106C13×C22⋊C4—24no—
107C13×C22⋊C4—24no—
108C42×C13—24no—
109C42×C13C2×D412yes—
110C42×C13C2×D412yes—
111C22⋊Dic13—48no—
112C4⋊Dic13—24no—
113C4⋊Dic13—24no—
114C4⋊Dic13—42no—
115#6101—48no—
116C4×Dic13—42no—
117C2×#1055—24no—
118#3501D1311yescharacteristic
119#3230—132no—
120#2542—131no—
121#1614—138no—
122#1503—132no—
123#1460—131no—
124C23×C13—22no—
125C23×C13—22no—
126C23×C13—42no—
127C23×C13—42no—
128C23×C13C2×Q1614yes—
129C23×C13C22×D411yescharacteristic
130Q8×C13—44no—
131Q8×C13—44no—
132Q8×C13—84no—
133D4×C13—28no—
134C2×C4×C13—22no—
135C2×C4×C13—24no—
136C2×C4×C13—24no—
137C2×C4×C13—22no—
138C2×C4×C13—41no—
139C2×C4×C13—44no—
140C2×C4×C13—42no—
141C2×C4×C13—82no—
142C2×C4×C13—84no—
143C2×C4×C13—84no—
144C2×C4×C13#12011yescharacteristic
145C2×C4×C13#12012yes—
146C2×C4×C13#12011yescharacteristic
147C2×C4×C13C2×Q1612yes—
148C2×C4×C13#13712yes—
149C2×C4×C13C22×D412yes—
150C2×Dic13—24no—
151C2×Dic13—42no—
152C2×Dic13—42no—
153C2×Dic13—44no—
154C2×Dic13—42no—
155Dic26—82no—
156Dic26—82no—
157#1055—44no—
158#554—22no—
159#554—24no—
160D4×Q8—22no—
161#544—21no—
162#543C2×D1314yes—
163#533C2×D1312yes—
164C2×C4⋊Q8—131no—
165C2×C22⋊Q8—22no—
166C2×C22⋊Q8—262no—
167#518—21no—
168C2×C4×D4C2×D1311yescharacteristic
169#500—262no—
170#487—262no—
171#484—138no—
172#477—264no—
173#431—132no—
174#399—134no—
175#390—261no—
176#384—261no—
177C4×C4⋊C4—131no—
178#328—138no—
179#323—134no—
180C22×C13—21no—
181C22×C13—21no—
182C22×C13—24no—
183C22×C13—44no—
184C22×C13—42no—
185C22×C13—42no—
186C22×C13—82no—
187C22×C13C2×#12011yescharacteristic
188C22×C13C22×Q1611yescharacteristic
189C22×C13C2×#13711yescharacteristic
190C4×C13—21no—
191C4×C13—22no—
192C4×C13—42no—
193C4×C13—44no—
194C4×C13—44no—
195C4×C13#45114yes—
196C4×C13#45312yes—
197Dic13—44no—
198Dic13—82no—
199C22×Q8—22no—
200C22×Q8—261no—
201C22×D4C22⋊D1311yescharacteristic
202C23×C4C22⋊D1312yes—
203C4⋊Q8—24no—
204C4⋊Q8—262no—
205C4⋊Q8—262no—
206C4⋊Q8C22⋊D1312yes—
207C4⋊Q8C22⋊D1312yes—
208#125—24no—
209#125—22no—
210C22⋊Q8—24no—
211C22⋊Q8—24no—
212C22⋊Q8—28no—
213C22⋊Q8—22no—
214C22⋊Q8—24no—
215C22⋊Q8—28no—
216C22⋊Q8—528no—
217C22⋊Q8C22×D1314yes—
218C4×Q8—22no—
219C4×Q8C22⋊D1314yes—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—22no—
223C4×D4—22no—
224#117—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—132no—
228C2×C4⋊C4—261no—
229C2×C4⋊C4—264no—
230C2×C4⋊C4—261no—
231C2×C4⋊C4C22⋊D1311yescharacteristic
232C2×C22⋊C4—24no—
233C2×C22⋊C4—264no—
234C2×C22⋊C4C22×D1312yes—
235C2×C42—261no—
236C2×C42C22×D1311yescharacteristic
237#107—268no—
238C4⋊C8—262no—
239#103—268no—
240C22⋊C8—134no—
241C4×C8—262no—
242#95—132no—
243C2×C13—21no—
244C2×C13—21no—
245C2×C13—44no—
246C2×C13C2×#45111yescharacteristic
247C2×C13C2×#45311yescharacteristic
248C2×C13#302811yescharacteristic
249C24C2×C22⋊D1312yes—
250C2×Q8—24no—
251C2×Q8—22no—
252C2×Q8—84no—
253C2×Q8—524no—
254C2×Q8—521no—
255C2×Q8—521no—
256C2×Q8C2×C22⋊D1314yes—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4#610714yes—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—44no—
265C22×C4—262no—
266C22×C4—262no—
267C22×C4C2×C22⋊D1311yescharacteristic
268C22×C4C2×C22⋊D1312yes—
269C22×C4C2×C22⋊D1311yescharacteristic
270C22×C4C23×D1311yescharacteristic
271C2×C8—264no—
272C4⋊C4—24no—
273C4⋊C4—28no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—264no—
281C4⋊C4—264no—
282C4⋊C4—528no—
283C4⋊C4—522no—
284C4⋊C4C2×C22⋊D1314yes—
285C22⋊C4—28no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—528no—
289C42—24no—
290C42—522no—
291C42C2×C22⋊D1312yes—
292C42C2×C22⋊D1312yes—
293C13#1839711yescharacteristic, complemented, Sylow
294C23—22no—
295C23—22no—
296C23—42no—
297C23—42no—
298C23C2×#610714yes—
299C23C22×C22⋊D1311yescharacteristic
300Q8—44no—
301Q8—44no—
302Q8—84no—
303Q8—1042no—
304Q8—1042no—
305D4—28no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4—84no—
315C2×C4—84no—
316C2×C4—264no—
317C2×C4—522no—
318C2×C4—522no—
319C2×C4—524no—
320C2×C4—522no—
321C2×C4#8819612yes—
322C2×C4C2×#610712yes—
323C2×C4#8820711yescharacteristic
324C2×C4#8820712yes—
325C2×C4#8820711yescharacteristic
326C2×C4C22×C22⋊D1312yes—
327C8—524no—
328C22—21no—
329C22—21no—
330C22—24no—
331C22—44no—
332C22—42no—
333C22—42no—
334C22—82no—
335C22C2×#8819611yescharacteristic
336C22C22×#610711yescharacteristic
337C22C2×#8820711yescharacteristic
338C4—21no—
339C4—22no—
340C4—42no—
341C4—44no—
342C4—44no—
343C4—524no—
344C4—1042no—
345C4#1171888914yes—
346C4#1171890212yes—
347C2—21no—
348C2—21no—
349C2—44no—
350C2C2×#1171888911yescharacteristic
351C2C2×#1171890211yescharacteristic
352C2#1312424211yescharacteristic
subgroupG/H
centerC22C2×#88207
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#3501C2
socleC22×C13C2×#120
prime radicalC22×D4×C13D4
hypercenter#3501D13
hypocenterC13#18397
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #13125984C2semidirect
3. #13113861C2semidirect
4. #13113335C2semidirect
293. C13#18397semidirect

Maximal subgroups

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

subgroupindex
#131259842
#131197892
#131138612
#131133352
#131131572
C13×#35012
#1839713