Finite Group Databasebeta

← #28302228

Subgroups — #28302228

Order 13568.

1174 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 136 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#20290537C212yesmaximal
2C2×#16811469C212yesmaximal, complemented
3C2×#16811468C212yesmaximal, complemented
4C22×#13133194C211yesmaximal, characteristic
5C2×C22⋊#88324—24no—
6C2×C22⋊#88324C2214yes—
7#16811469—24no—
8#16811468—24no—
9C23×Dic106C411yescharacteristic
10C24×Dic53C2211yescharacteristic
11C2×#13133194—24no—
12C2×#13133194—24no—
13C2×#13133194—24no—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919C411yescharacteristic
16C22×#11719919C2211yescharacteristic
17C22×C22⋊Dic53C411yescharacteristic
18C22×C22⋊C4×C53C411yescharacteristic
19C22×#88324—24no—
20C22⋊#88324—24no—
21C22⋊#88324—24no—
22C22⋊#88324—48no—
23C22×Dic106—22no—
24C22×Dic106—22no—
25C22×Dic106—22no—
26C22×Dic106—44no—
27C23×Dic53—22no—
28C23×Dic53—22no—
29C23×Dic53—44no—
30C23×Dic53C2×C411yescharacteristic
31C23×Dic53C2×C411yescharacteristic
32C23×Dic53C2×C411yescharacteristic
33C23×Dic53D414yes—
34C23×Dic53C2311yescharacteristic
35#13133194—232no—
36C2×C4⋊Dic53—22no—
37C2×C4⋊Dic53—22no—
38C2×C4⋊Dic53D414yes—
39C2×#11719919—22no—
40C2×#11719919—22no—
41C2×#11719919—22no—
42C2×#11719919—22no—
43C2×#11719919—22no—
44C2×#11719919D414yes—
45C2×C22⋊Dic53—22no—
46C2×C22⋊Dic53—22no—
47C2×C22⋊Dic53—22no—
48C25×C53C2×C411yescharacteristic
49C23×C4×C53C2×C411yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53—22no—
52C2×C22⋊C4×C53—22no—
53C2×C22⋊C4×C53—22no—
54C24×C53—24no—
55C24×C53—21no—
56C24×C53—21no—
57C24×C53—24no—
58C24×C53—42no—
59C24×C53C22⋊C412yes—
60C24×C53C22×C411yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—24no—
66C22×C4×C53—42no—
67C22×C4×C53C22⋊C412yes—
68C22×C4×C53C22⋊C412yes—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—24no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—24no—
77C22×Dic53—28no—
78C22×Dic53—28no—
79C22×Dic53—44no—
80C22×Dic53—42no—
81C22×Dic53—48no—
82C22×Dic53—48no—
83C22×Dic53—42no—
84C22×Dic53C22⋊C412yes—
85C22×Dic53C22⋊C412yes—
86C22×Dic53C22⋊C412yes—
87C22×Dic53C2×D412yes—
88C22×Dic53C2×D412yes—
89C22×Dic53C2×D412yes—
90C2×Dic106—28no—
91C2×Dic106—48no—
92C2×Dic106—48no—
93C2×Dic106—44no—
94C2×Dic106—44no—
95C2×#88324—48no—
96C2×#88324—44no—
97C22⋊C4×C53—28no—
98C22⋊Dic53—28no—
99C4⋊Dic53—28no—
100#11719919—28no—
101#11719919—28no—
102C23×C53—22no—
103C23×C53—22no—
104C23×C53—24no—
105C23×C53—24no—
106C23×C53—22no—
107C23×C53—24no—
108C23×C53—24no—
109C23×C53—22no—
110C23×C53—416no—
111C23×C53—42no—
112C23×C53—42no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53#10114yes—
116C23×C53C2×C22⋊C411yescharacteristic
117C23×C53C2×C22⋊C411yescharacteristic
118C23×C53C2×C22⋊C411yescharacteristic
119C23×C53#13114yes—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—24no—
123C2×C4×C53—42no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×Dic53—24no—
129C2×Dic53—24no—
130C2×Dic53—22no—
131C2×Dic53—24no—
132C2×Dic53—22no—
133C2×Dic53—22no—
134C2×Dic53—22no—
135C2×Dic53—24no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—416no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—42no—
144C2×Dic53#9914yes—
145C2×Dic53#12014yes—
146Dic106—416no—
147#88324—84no—
148#8308—531nomaximal, Sylow
149C22×C53—28no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—28no—
155C22×C53—24no—
156C22×C53—48no—
157C22×C53—48no—
158C22×C53—44no—
159C22×C53—42no—
160C22×C53—42no—
161C22×C53#37911yescharacteristic
162C22×C53#40812yes—
163C22×C53#40812yes—
164C22×C53#40812yes—
165C22×C53C2×#9912yes—
166C22×C53C2×#10112yes—
167C4×C53—42no—
168C4×C53—42no—
169Dic53—24no—
170Dic53—24no—
171Dic53—42no—
172Dic53—42no—
173C22×C22⋊Q8—531no—
174#1779—532no—
175C2×#324—532no—
176C2×#324—532no—
177C2×C53—22no—
178C2×C53—22no—
179C2×C53—44no—
180C2×C53#147214yes—
181C2×C53#178011yescharacteristic
182C2×C53#178111yescharacteristic
183C2×C53#178411yescharacteristic
184C23×Q8—531no—
185C24×C4—531no—
186C2×C22⋊Q8—1064no—
187C2×C22⋊Q8—1064no—
188C2×C22⋊Q8—1064no—
189C22×C4⋊C4—531no—
190C22×C4⋊C4—531no—
191C22×C4⋊C4—531no—
192C22×C22⋊C4—531no—
193C22×C22⋊C4#615711yescharacteristic
194C2×C22⋊C8—534no—
195C2×C22⋊C8—1064no—
196#324—1064no—
197#324—1064no—
198C53#830811yescharacteristic, complemented, Sylow
199C25C2×#615711yescharacteristic
200C22×Q8—1062no—
201C22×Q8—1062no—
202C22×Q8—1062no—
203C22×Q8—2124no—
204C23×C4—531no—
205C23×C4—531no—
206C23×C4—531no—
207C23×C4—534no—
208C23×C4—531no—
209C23×C4—1062no—
210C23×C4—1062no—
211C23×C4—2124no—
212C23×C4C2×#615711yescharacteristic
213C23×C4C2×#615711yescharacteristic
214C22×C8—1064no—
215C22⋊Q8—10632no—
216C2×C4⋊C4—534no—
217C2×C4⋊C4—534no—
218C2×C4⋊C4—1062no—
219C2×C4⋊C4—1062no—
220C2×C4⋊C4—1062no—
221C2×C4⋊C4—1062no—
222C2×C4⋊C4—1062no—
223C2×C4⋊C4—1062no—
224C2×C4⋊C4—1062no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—1062no—
229C2×C22⋊C4—1062no—
230C2×C22⋊C4—1062no—
231C22⋊C8—1064no—
232C22⋊C8—1064no—
233C22⋊C8—2128no—
234C24—24no—
235C24—21no—
236C24—21no—
237C24—24no—
238C24—42no—
239C24C22⋊#615712yes—
240C24C22×#615711yescharacteristic
241C2×Q8—1068no—
242C2×Q8—2124no—
243C2×Q8—2128no—
244C2×Q8—2124no—
245C2×Q8—2128no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—24no—
251C22×C4—42no—
252C22×C4—532no—
253C22×C4—532no—
254C22×C4—532no—
255C22×C4—532no—
256C22×C4—532no—
257C22×C4—532no—
258C22×C4—1064no—
259C22×C4—1068no—
260C22×C4—1068no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1064no—
268C22×C4—2122no—
269C22×C4—2124no—
270C22×C4—2128no—
271C22×C4—2128no—
272C22×C4—2122no—
273C22×C4C22⋊#615712yes—
274C22×C4C22⋊#615712yes—
275C2×C8—2124no—
276C2×C8—2128no—
277C4⋊C4—1068no—
278C4⋊C4—1068no—
279C4⋊C4—1068no—
280C22⋊C4—28no—
281C22⋊C4—1068no—
282C23—22no—
283C23—22no—
284C23—24no—
285C23—24no—
286C23—22no—
287C23—24no—
288C23—24no—
289C23—22no—
290C23—416no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23#1313327814yes—
296C23C2×C22⋊#615711yescharacteristic
297C23C2×C22⋊#615711yescharacteristic
298C23C2×C22⋊#615711yescharacteristic
299C23#1313330414yes—
300Q8—21216no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—534no—
310C2×C4—534no—
311C2×C4—1064no—
312C2×C4—1064no—
313C2×C4—1062no—
314C2×C4—1064no—
315C2×C4—1062no—
316C2×C4—1062no—
317C2×C4—1062no—
318C2×C4—1064no—
319C2×C4—2122no—
320C2×C4—2122no—
321C2×C4—21216no—
322C2×C4—2122no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4—2122no—
327C8—4244no—
328C22—28no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—28no—
334C22—24no—
335C22—48no—
336C22—48no—
337C22—44no—
338C22—42no—
339C22—42no—
340C22#1681156411yescharacteristic
341C22#1681159612yes—
342C22#1681159612yes—
343C22#1681159712yes—
344C22D53⋊#9912yes—
345C22C2×#1313327812yes—
346C4—42no—
347C4—42no—
348C4—1064no—
349C4—1064no—
350C4—2122no—
351C4—2122no—
352C2—22no—
353C2—22no—
354C2—44no—
355C2#2028999414yes—
356C2#2029053911yescharacteristic
357C2#2029054211yescharacteristic
358C2#2029054911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C22⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC23×Dic53C23
hypercenterC22×C22⋊C4#6157
hypocenterC53#8308
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16811469C2semidirect
3. C2×#16811468C2semidirect
198. C53#8308semidirect

Maximal subgroups

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

subgroupindex
#202905372
C2×#168114692
C2×#168114682
C22×#131331942
#830853