Finite Group Databasebeta

← #27287324

Subgroups — #27287324

Order 13568.

602 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 75 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#20280380C211yesmaximal, characteristic, complemented
2#20274659C212yesmaximal
3#20272623C211yesmaximal, characteristic, complemented
4#20272567C212yesmaximal, complemented
5C53×#1782C211yesmaximal, characteristic
6C2×#13133144—22no—
7C2×#13133144—22no—
8#16810587C2212yes—
9#16810559—21no—
10#16810535—21no—
11#16810527—22no—
12#16810510—22no—
13C4×C22⋊Dic53C2212yes—
14C22×#41×C53C2211yescharacteristic
15C2×#99×C53—21no—
16C2×#99×C53—21no—
17C53×#381C2211yescharacteristic
18C53×#381C2211yescharacteristic
19C22×Dic106—21no—
20C2×#11719917—22no—
21C2×C4⋊Dic53—21no—
22C2×#11719919—22no—
23C2×C22⋊Dic53—22no—
24C2×C4×Dic53D412yes—
25C2×C4×Dic53D412yes—
26#13133144—44no—
27#13133144—44no—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30#13133124—21no—
31#13133124—22no—
32#13133124—21no—
33#13133124—22no—
34C2×#41×C53—22no—
35C2×#41×C53—22no—
36C2×#41×C53—42no—
37C2×#41×C53—42no—
38C22×Q8×C53Q811yescharacteristic, complemented
39C22×D4×C53—21no—
40C22×D4×C53Q811yescharacteristic, complemented
41C23×C4×C53—21no—
42C23×C4×C53C2311yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53D411yescharacteristic
45C2×C22⋊C4×C53D411yescharacteristic
46#99×C53—42no—
47#99×C53—42no—
48#95×C53—22no—
49#95×C53—22no—
50C24×C53—21no—
51C24×C53C2×D411yescharacteristic
52#41×C53—24no—
53#41×C53—24no—
54#41×C53—42no—
55#41×C53—42no—
56#41×C53—44no—
57#41×C53—44no—
58C2×Q8×C53—22no—
59C2×Q8×C53—42no—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—42no—
64C2×D4×C53—41no—
65C2×D4×C53—41no—
66C2×D4×C53—41no—
67C2×D4×C53—41no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53—42no—
74C22×C4×C53—42no—
75C22×C4×C53—42no—
76C22×C4×C53—41no—
77C22×C4×C53C2×D412yes—
78C22×C4×C53C2×Q811yescharacteristic
79C22×C4×C53#4112yes—
80C22×C4×C53#4111yescharacteristic
81C22×Dic53—22no—
82C22×Dic53—22no—
83C2×Dic106—44no—
84C2×Dic106—81no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊Dic53—44no—
90C4⋊Dic53—81no—
91#11719919—44no—
92C4×Dic53—42no—
93C4×Dic53—42no—
94#11719917—42no—
95#11719917—42no—
96C2×#88322—42no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—22no—
100C23×C53—42no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—82no—
104C23×C53#12011yescharacteristic
105C23×C53C22⋊Q811yescharacteristic
106C23×C53#12411yescharacteristic
107Q8×C53—24no—
108Q8×C53—42no—
109D4×C53—24no—
110D4×C53—42no—
111D4×C53—42no—
112D4×C53—42no—
113D4×C53—42no—
114D4×C53—42no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—42no—
121C2×C4×C53—44no—
122C2×C4×C53—44no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—82no—
130C2×C4×C53—81no—
131C2×C4×C53C22⋊Q812yes—
132C2×C4×C53#12412yes—
133C2×Dic53—24no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139Dic106—84no—
140#88322—82no—
141#10386—531nomaximal, Sylow
142C22×C53—22no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—42no—
148C22×C53—81no—
149C22×C53—81no—
150C22×C53—82no—
151C22×C53—82no—
152C22×C53—82no—
153C22×C53#39311yescharacteristic
154C22×C53B(4,𝔽2)11yescharacteristic
155C22×C53B(4,𝔽2)11yescharacteristic
156C4×C53—22no—
157C4×C53—22no—
158C4×C53—42no—
159C4×C53—81no—
160C4×C53—81no—
161C4×C53—161no—
162Dic53—44no—
163Dic53—82no—
164#2607—531no—
165#1789—532no—
166#1782D5311yescharacteristic
167#1306—531no—
168#1284—532no—
169C2×C53—21no—
170C2×C53—21no—
171C2×C53—41no—
172C2×C53—41no—
173C2×C53—82no—
174C2×C53#202911yescharacteristic
175C2×C53#203112yes—
176C22×#41C2×D5311yescharacteristic
177C2×#104—1062no—
178C2×#104—1062no—
179C2×#99—21no—
180C2×#99—21no—
181#407—532no—
182#399—1061no—
183#393—1061no—
184#391—1062no—
185#387—1062no—
186#381C2×D5311yescharacteristic
187#381C2×D5311yescharacteristic
188C4×C22⋊C4—532no—
189C53#1038611yescharacteristic, complemented, Sylow
190C2×#41—22no—
191C2×#41—22no—
192C2×#41—42no—
193C2×#41—42no—
194C22×Q8—1061no—
195C22×Q8Dic10611yescharacteristic, complemented
196C22×D4—21no—
197C22×D4Dic10611yescharacteristic, complemented
198C23×C4—21no—
199C23×C4C22×D5311yescharacteristic
200C2×M16—1062no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4—1062no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—1062no—
205C2×C22⋊C4C22⋊D5311yescharacteristic
206C2×C22⋊C4C22⋊D5311yescharacteristic
207C2×C42—532no—
208C2×C42—532no—
209#104—2124no—
210#104—2124no—
211#99—42no—
212#99—42no—
213C22⋊C8—1062no—
214C22⋊C8—1062no—
215#95—22no—
216#95—22no—
217#95—1061no—
218#95—1062no—
219#95—1061no—
220#95—1062no—
221C24—21no—
222C24C2×C22⋊D5311yescharacteristic
223#41—24no—
224#41—24no—
225#41—42no—
226#41—42no—
227#41—44no—
228#41—44no—
229C2×Q8—22no—
230C2×Q8—42no—
231C2×Q8—4241no—
232C2×Q8—2124no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—1062no—
251C22×C4—1062no—
252C22×C4C2×Dic10611yescharacteristic
253C22×C4#1171994511yescharacteristic
254C22×C4D4×D5312yes—
255C22×C4#1171994712yes—
256M16—2122no—
257M16—2122no—
258C2×C8—2122no—
259C4⋊C4—4241no—
260C4⋊C4—2124no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—2124no—
266C42—2122no—
267C42—2122no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—42no—
272C23—41no—
273C23—41no—
274C23—82no—
275C23#1313318511yescharacteristic
276C23#1313319611yescharacteristic
277C23#1313320411yescharacteristic
278Q8—24no—
279Q8—42no—
280Q8—4244no—
281D4—24no—
282D4—42no—
283D4—42no—
284D4—42no—
285D4—42no—
286D4—42no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—1064no—
304C2×C4—2122no—
305C2×C4—2122no—
306C2×C4—2122no—
307C2×C4—2122no—
308C2×C4—2122no—
309C2×C4#1313319412yes—
310C2×C4#1313320312yes—
311C8—4242no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—81no—
319C22—81no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22#1681053511yescharacteristic
324C22#1681080911yescharacteristic
325C22#1681080911yescharacteristic
326C4—22no—
327C4—22no—
328C4—42no—
329C4—81no—
330C4—81no—
331C4—161no—
332C4—4242no—
333C4—2124no—
334C2—21no—
335C2—21no—
336C2—41no—
337C2—41no—
338C2—82no—
339C2#2027587911yescharacteristic
340C2#2027588912yes—
subgroupG/H
centerC22#16810535
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1782C2
socleC22×C53#393
prime radicalC22×#41×C53C22
hypercenter#1782D53
hypocenterC53#10386
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #20280380C2semidirect
3. #20272623C2semidirect
4. #20272567C2semidirect
38. C22×Q8×C53Q8semidirect
40. C22×D4×C53Q8semidirect
189. C53#10386semidirect
195. C22×Q8Dic106semidirect
197. C22×D4Dic106semidirect

Maximal subgroups

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

subgroupindex
#202803802
#202746592
#202726232
#202725672
C53×#17822
#1038653