Finite Group Databasebeta

← #27288495

Subgroups — #27288495

Order 13568.

604 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 119 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#20280829C211yesmaximal, characteristic
2#20274903C212yesmaximal
3#20272594C212yesmaximal
4#20272558C211yesmaximal, characteristic
5C53×#1444C211yesmaximal, characteristic
6C2×#13133196—21no—
7C2×#13133150—22no—
8C2×C22⋊#88322C2212yes—
9#16810555—22no—
10#16810544—22no—
11#16810483—21no—
12#16810483C2212yes—
13#16810385—22no—
14C22×C4⋊C4×C53—21no—
15C22×C4⋊C4×C53C2211yescharacteristic
16C2×#95×C53—21no—
17C2×#95×C53C2211yescharacteristic
18C2×#95×C53C2211yescharacteristic
19C22×Dic106—21no—
20C22×#88322—22no—
21#13133196—42no—
22#13133196—42no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53D412yes—
27C2×C4⋊Dic53D412yes—
28C2×#11719919—22no—
29C2×C22⋊Dic53—22no—
30#13133150—44no—
31C22⋊#88322—22no—
32C22⋊#88322Q814yescomplemented
33#13133124—22no—
34#13133124—22no—
35C23×C4×C53—21no—
36C23×C4×C53—21no—
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—42no—
42C2×C4⋊C4×C53—42no—
43C2×C4⋊C4×C53—42no—
44C2×C4⋊C4×C53—42no—
45#95×C53—22no—
46#95×C53—21no—
47#95×C53—42no—
48#95×C53Q812yescomplemented
49C24×C53C2×D411yescharacteristic
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—22no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—22no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53D812yes—
75C22×C4×C53Q1612yes—
76C22×C4×C53C2×D412yes—
77C22×C4×C53C2×Q812yes—
78C22×C4×C53C2×Q811yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×Dic53—22no—
81C22×Dic53—22no—
82C2×Dic106—42no—
83C2×Dic106—81no—
84C2×Dic106—81no—
85C4⋊C4×C53—24no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—48no—
88C22⋊Dic53—42no—
89C22⋊Dic53—42no—
90C4⋊Dic53—42no—
91C4⋊Dic53—42no—
92C4⋊Dic53—42no—
93C4⋊Dic53—81no—
94C4⋊Dic53—81no—
95#11719919—44no—
96C2×#88322—24no—
97C2×#88322—42no—
98C2×#88322—42no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53#12011yescharacteristic
104C23×C53C22⋊Q811yescharacteristic
105C23×C53C4⋊Q811yescharacteristic
106C23×C53C2×D811yescharacteristic
107C23×C53C2×Q1611yescharacteristic
108C23×C53AGL(1,ℤ/8)11yescharacteristic
109C23×C53#13711yescharacteristic
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—24no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53—81no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53C22⋊Q812yes—
140C2×C4×C53C4⋊Q812yes—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—42no—
144C2×Dic53—42no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147Dic106—84no—
148#88322—44no—
149#10567—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53#39511yescharacteristic
163C22×C53#44711yescharacteristic
164C22×C53#45111yescharacteristic
165C22×C53#45311yescharacteristic
166C22×C53#45611yescharacteristic
167C22×C53B(4,𝔽2)11yescharacteristic
168C22×C53#45811yescharacteristic
169C22×C53#47411yescharacteristic
170C22×C53#47711yescharacteristic
171C22×C53#50011yescharacteristic
172C22×C53#50111yescharacteristic
173C4×C53—22no—
174C4×C53—22no—
175C4×C53—44no—
176C4×C53—42no—
177C4×C53—42no—
178C4×C53—82no—
179Dic53—82no—
180Dic53—82no—
181#2659—531no—
182#1852—532no—
183#1444D5311yescharacteristic
184#1294—532no—
185#1280—531no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—21no—
190C2×C53#159511yescharacteristic
191C2×C53#160511yescharacteristic
192C2×C53#161011yescharacteristic
193C2×C53#161411yescharacteristic
194C2×C53#205711yescharacteristic
195C2×C53#206011yescharacteristic
196C2×C53#206211yescharacteristic
197C2×C22⋊Q8—1061no—
198C22×C4⋊C4—21no—
199C22×C4⋊C4C2×D5311yescharacteristic
200C2×#107—1062no—
201C2×C22⋊C8—532no—
202#398—1062no—
203#395—1062no—
204#380—532no—
205#380—1061no—
206C2×#95—21no—
207C2×#95C2×D5311yescharacteristic
208C2×#95C2×D5311yescharacteristic
209#340—1062no—
210C53#1056711yescharacteristic, complemented, Sylow
211C22×Q8—1061no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4C22⋊D5311yescharacteristic
215C23×C4C22⋊D5311yescharacteristic
216C23×C4C22×D5311yescharacteristic
217C22×C8—1062no—
218C22⋊Q8—2122no—
219C22⋊Q8—2122no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—532no—
226C2×C4⋊C4—532no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4—1062no—
230C2×C4⋊C4—1061no—
231C2×C22⋊C4—1062no—
232#107—2124no—
233C22⋊C8—534no—
234C22⋊C8—1062no—
235#95—21no—
236#95—22no—
237#95—42no—
238#95—1062no—
239#95—1062no—
240#95Dic10612yescomplemented
241C24C2×C22⋊D5311yescharacteristic
242C2×Q8—4241no—
243C2×Q8—4241no—
244C2×Q8—2122no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—1062no—
270C22×C4—1062no—
271C22×C4#1171992212yes—
272C22×C4#1171992512yes—
273C22×C4C2×Dic10611yescharacteristic
274C22×C4#1171994511yescharacteristic
275C22×C4D4×D5312yes—
276C22×C4Q8×D5312yes—
277C2×C8—1064no—
278C2×C8—2122no—
279C2×C8—2122no—
280C4⋊C4—24no—
281C4⋊C4—42no—
282C4⋊C4—48no—
283C4⋊C4—4241no—
284C4⋊C4—4241no—
285C4⋊C4—2122no—
286C4⋊C4—2122no—
287C4⋊C4—2124no—
288C4⋊C4—2122no—
289C22⋊C4—2122no—
290C22⋊C4—2122no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#1313318511yescharacteristic
296C23#1313319611yescharacteristic
297C23#1313321511yescharacteristic
298C23C2×#1171992211yescharacteristic
299C23C2×#1171992511yescharacteristic
300C23#1313324211yescharacteristic
301C23#1313324711yescharacteristic
302Q8—4244no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4#1313319512yes—
339C2×C4#1313321612yes—
340C8—2124no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22#1681054311yescharacteristic
354C22#1681075611yescharacteristic
355C22#1681077811yescharacteristic
356C22#1681079011yescharacteristic
357C22#1681080511yescharacteristic
358C22#1681080911yescharacteristic
359C22#1681081211yescharacteristic
360C22#1681090311yescharacteristic
361C22#1681091811yescharacteristic
362C22#1681102611yescharacteristic
363C22#1681103311yescharacteristic
364C4—22no—
365C4—22no—
366C4—44no—
367C4—42no—
368C4—42no—
369C4—82no—
370C4—4242no—
371C4—4242no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2#2027363211yescharacteristic
377C2#2027368811yescharacteristic
378C2#2027371711yescharacteristic
379C2#2027373911yescharacteristic
380C2#2027604311yescharacteristic
381C2#2027606011yescharacteristic
382C2#2027606811yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1444C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenter#1444D53
hypocenterC53#10567
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1444

Complements

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

NH ≅ G/Ntype
32. C22⋊#88322Q8semidirect
48. #95×C53Q8semidirect
210. C53#10567semidirect
240. #95Dic106semidirect

Maximal subgroups

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

subgroupindex
#202808292
#202749032
#202725942
#202725582
C53×#14442
#1056753