Finite Group Databasebeta

← #28308335

Subgroups — #28308335

Order 13568.

496 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 72 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#20289963C211yesmaximal, characteristic, complemented
2C2×#16811468C211yesmaximal, characteristic, complemented
3#20289658C212yesmaximal, complemented
4#20289643C212yesmaximal, complemented
5#20279187C211yesmaximal, characteristic
6#16811593—21no—
7C2×C22⋊#88324—21no—
8#16811468—21no—
9#16811468—21no—
10#16811468C2212yes—
11#16811468C2212yes—
12C2×#13133198—21no—
13C2×#13133198—21no—
14C22×C4⋊Dic53C2211yescharacteristic
15#16810559—21no—
16#16810559—21no—
17#16810526C411yescharacteristic
18#16810526C2211yescharacteristic
19#16810492C411yescharacteristic
20#16810492C2211yescharacteristic
21C4×C22⋊Dic53C411yescharacteristic
22C2×#121×C53C411yescharacteristic
23C2×#11719940—41no—
24C22×#88324—41no—
25C22⋊#88324—22no—
26C22⋊#88324—22no—
27C22⋊#88324—41no—
28C22⋊#88324—41no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31#13133198—44no—
32#13133198—44no—
33C2×C4⋊Dic53—41no—
34C2×C4⋊Dic53—41no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×C4⋊Dic53D411yescharacteristic
38C2×C4⋊Dic53C2311yescharacteristic
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×C22⋊Dic53—21no—
44C2×C22⋊Dic53—21no—
45C2×C22⋊Dic53—21no—
46C2×C22⋊Dic53—21no—
47C2×C22⋊Dic53—21no—
48C2×C4×Dic53C2×C411yescharacteristic
49C2×C4×Dic53C2×C411yescharacteristic
50#13133124—21no—
51C22×D4×C53—21no—
52C22×D4×C53C2×C411yescharacteristic
53C23×C4×C53C2×C411yescharacteristic
54#121×C53—41no—
55#121×C53—41no—
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C22⋊C4×C53—21no—
58C24×C53—21no—
59C24×C53C22⋊C411yescharacteristic
60C2×D4×C53—22no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53—42no—
64C2×D4×C53—42no—
65C2×D4×C53—41no—
66C2×D4×C53—41no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—22no—
70C22×C4×C53—21no—
71C22×C4×C53—41no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—42no—
81C22×Dic53—42no—
82C22×Dic53C22⋊C411yescharacteristic
83C22×Dic53C22⋊C411yescharacteristic
84C22×Dic53C2×D411yescharacteristic
85C22×Dic53C2×D411yescharacteristic
86#11719940—82no—
87C2×#88324—42no—
88C2×#88324—42no—
89C2×#88324—82no—
90C4⋊C4×C53—41no—
91C22⋊C4×C53—42no—
92C22⋊Dic53—24no—
93C22⋊Dic53—42no—
94C22⋊Dic53—42no—
95C22⋊Dic53—42no—
96C22⋊Dic53—42no—
97C4⋊Dic53—41no—
98C4⋊Dic53—41no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101#11719919—42no—
102#11719919—42no—
103#11719919—42no—
104#11719919—42no—
105C4×Dic53—22no—
106C4×Dic53—41no—
107C23×C53—22no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53#9911yescharacteristic
119C23×C53#9911yescharacteristic
120C23×C53C2×C22⋊C411yescharacteristic
121D4×C53—82no—
122D4×C53—82no—
123D4×C53—82no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53#13112yes—
135C2×Dic53—22no—
136C2×Dic53—22no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—42no—
144C2×Dic53—42no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—81no—
152C2×Dic53—81no—
153#88324—82no—
154#88324—82no—
155#11549—531nomaximal, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—42no—
164C22×C53—42no—
165C22×C53—81no—
166C22×C53—81no—
167C22×C53—81no—
168C22×C53—81no—
169C22×C53—81no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53#35312yes—
173C22×C53#40811yescharacteristic
174C22×C53C2×#9911yescharacteristic
175C22×C53#41311yescharacteristic
176C4×C53—22no—
177C4×C53—41no—
178C4×C53—81no—
179Dic53—42no—
180Dic53—42no—
181Dic53—41no—
182Dic53—41no—
183Dic53—41no—
184Dic53—41no—
185#2501—531no—
186#1463—531no—
187C2×#324—531no—
188#1325—532no—
189#1318—532no—
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—22no—
193C2×C53—41no—
194C2×C53—161no—
195C2×C53#146811yescharacteristic
196C2×C53C2×#35311yescharacteristic
197C2×C53#213311yescharacteristic
198C2×#123—1061no—
199C2×#123—1061no—
200C2×#121#615711yescharacteristic
201C22×C4⋊C4—531no—
202#407—1061no—
203C2×C22⋊C8—1061no—
204#399—1061no—
205#399—1061no—
206#390—531no—
207#390—531no—
208#384—531no—
209#384—531no—
210C4×C22⋊C4—531no—
211#324—532no—
212#324—532no—
213#324—1061no—
214#324—1061no—
215C53#1154911yescharacteristic, complemented, Sylow
216C22×D4—21no—
217C22×D4C2×#615711yescharacteristic
218C23×C4—531no—
219C23×C4—531no—
220C23×C4C2×#615711yescharacteristic
221C2×M16—2121no—
222C22×C8—2121no—
223#123—2124no—
224#123—2124no—
225#121—41no—
226#121—41no—
227C2×C4⋊C4—531no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4—531no—
230C2×C4⋊C4—531no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—2121no—
236C2×C4⋊C4—2121no—
237C2×C4⋊C4C2×#615711yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—1061no—
240C2×C22⋊C4—1061no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C42—531no—
245C2×C42—531no—
246C22⋊C8—1062no—
247C22⋊C8—1062no—
248C22⋊C8—2121no—
249C22⋊C8—2121no—
250#95—1061no—
251C24—21no—
252C24C22⋊#615711yescharacteristic
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—531no—
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—2122no—
276C22×C4—2122no—
277C22×C4C22⋊#615711yescharacteristic
278C22×C4C22×#615711yescharacteristic
279M16—4242no—
280C2×C8—4242no—
281C2×C8—2122no—
282C2×C8—2122no—
283C4⋊C4—41no—
284C4⋊C4—2121no—
285C4⋊C4—2121no—
286C4⋊C4—2121no—
287C4⋊C4—2121no—
288C4⋊C4—2122no—
289C4⋊C4—2122no—
290C4⋊C4—2122no—
291C4⋊C4—2122no—
292C22⋊C4—42no—
293C22⋊C4—1064no—
294C22⋊C4—2122no—
295C22⋊C4—2122no—
296C22⋊C4—2122no—
297C22⋊C4—2122no—
298C42—1062no—
299C42—2121no—
300C23—22no—
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23#1313327511yescharacteristic
312C23#1313327511yescharacteristic
313C23C2×C22⋊#615711yescharacteristic
314D4—82no—
315D4—82no—
316D4—82no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—1062no—
328C2×C4—1062no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—4241no—
334C2×C4—4241no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4#1313330412yes—
346C8—4242no—
347C8—4242no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—42no—
356C22—42no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22#1681152712yes—
365C22#1681159611yescharacteristic
366C22C2×#1313327511yescharacteristic
367C22#1681161011yescharacteristic
368C4—22no—
369C4—41no—
370C4—81no—
371C4—2122no—
372C4—2122no—
373C4—2121no—
374C4—2121no—
375C4—2121no—
376C4—2121no—
377C2—21no—
378C2—21no—
379C2—22no—
380C2—41no—
381C2—161no—
382C2#2028998011yescharacteristic
383C2C2×#1681152711yescharacteristic
384C2#2029101011yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#121×C53C4
socleC22×C53C2×#99
cosocleC2×C4⋊Dic53C23
hypercenterC2×#121#6157
hypocenterC53#11549
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #20289963C2semidirect
2. C2×#16811468C2semidirect
3. #20289658C2semidirect
4. #20289643C2semidirect
215. C53#11549semidirect

Maximal subgroups

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

subgroupindex
#202899632
C2×#168114682
#202896582
#202896432
#202791872
#1154953