Finite Group Databasebeta

← #27295025

Subgroups — #27295025

Order 13568.

412 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 91 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#20274795C211yesmaximal, characteristic
2C2×#16810349C211yesmaximal, characteristic
3#20272745C212yesmaximal, complemented
4#20272721C212yesmaximal, complemented
5C53×#2583C211yesmaximal, characteristic
6#16810587—21no—
7C2×C22⋊#88322—21no—
8#16810392—21no—
9#16810392C2212yes—
10#16810349—21no—
11#16810349C2212yes—
12C2×C22⋊Q8×C53C411yescharacteristic
13C22×C4⋊C4×C53C2211yescharacteristic
14C53×#398C411yescharacteristic
15C53×#398C2211yescharacteristic
16C53×#395C411yescharacteristic
17C53×#395C2211yescharacteristic
18C53×#389—21no—
19C53×#389—21no—
20C53×#385—21no—
21C53×#385—21no—
22C4×C22⋊C4×C53C411yescharacteristic
23C2×#11719917—22no—
24C22×#88322—41no—
25C22⋊#88322—22no—
26C22⋊#88322—41no—
27C22⋊#88322—41no—
28C22⋊#88322—41no—
29C22×Q8×C53C2×C411yescharacteristic
30C23×C4×C53C2×C411yescharacteristic
31C23×C4×C53D411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C22⋊Q8×C53—41no—
34C22⋊Q8×C53—41no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—41no—
41C2×C4⋊C4×C53—41no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53C2311yescharacteristic
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C42×C53—21no—
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53C2×C411yescharacteristic
55#95×C53C2×C411yescharacteristic
56C24×C53C22⋊C411yescharacteristic
57C2×Q8×C53—41no—
58C2×Q8×C53—81no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—22no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—41no—
71C22×C4×C53—81no—
72C22×C4×C53—81no—
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—42no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—81no—
88C4⋊C4×C53—81no—
89C22⋊C4×C53—24no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C42×C53—81no—
93#11719917—42no—
94#11719917—42no—
95C2×#88322—42no—
96C2×#88322—42no—
97C2×#88322—82no—
98C23×C53—22no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—41no—
102C23×C53#9911yescharacteristic
103C23×C53#9911yescharacteristic
104C23×C53C2×C22⋊C411yescharacteristic
105Q8×C53—82no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53—81no—
137C2×C4×C53#13512yes—
138#88322—82no—
139#88322—82no—
140#11854—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53#35612yes—
151C22×C53C2×#9911yescharacteristic
152C22×C53#41611yescharacteristic
153C22×C53#42111yescharacteristic
154C4×C53—22no—
155C4×C53—42no—
156C4×C53—42no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—81no—
163C4×C53—81no—
164#2583D5311yescharacteristic
165#1821—531no—
166C2×#324—531no—
167#1354—532no—
168#1346—532no—
169C2×C53—21no—
170C2×C53—21no—
171C2×C53—22no—
172C2×C53—41no—
173C2×C53#151611yescharacteristic
174C2×C53#212111yescharacteristic
175C2×C53C2×#35611yescharacteristic
176C2×C22⋊Q8Dic5311yescharacteristic
177C22×C4⋊C4C2×D5311yescharacteristic
178#407—1061no—
179C2×C22⋊C8—1061no—
180#398Dic5311yescharacteristic
181#398C2×D5311yescharacteristic
182#395Dic5311yescharacteristic
183#395C2×D5311yescharacteristic
184#389—21no—
185#389—21no—
186#385—21no—
187#385—21no—
188C4×C22⋊C4Dic5311yescharacteristic
189#343—532no—
190#343—1061no—
191#324—532no—
192#324—1061no—
193C53#1185411yescharacteristic, complemented, Sylow
194C22×Q8C2×Dic5311yescharacteristic
195C23×C4C2×Dic5311yescharacteristic
196C23×C4C22⋊D5311yescharacteristic
197C23×C4C22⋊D5311yescharacteristic
198C2×M16—1062no—
199C22×C8—2121no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4C2×Dic5311yescharacteristic
210C2×C4⋊C4C2×Dic5311yescharacteristic
211C2×C4⋊C4C22⋊D5311yescharacteristic
212C2×C4⋊C4C22⋊D5311yescharacteristic
213C2×C4⋊C4C22×D5311yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C42—21no—
217C22⋊C8—1062no—
218C22⋊C8—2121no—
219C22⋊C8—2121no—
220C22⋊C8—2121no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95C2×Dic5311yescharacteristic
226#95C2×Dic5311yescharacteristic
227C24C22⋊Dic5311yescharacteristic
228C2×Q8—41no—
229C2×Q8—81no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—41no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4C22⋊Dic5311yescharacteristic
245C22×C4C22⋊Dic5311yescharacteristic
246C22×C4C22⋊Dic5311yescharacteristic
247C22×C4C22×Dic5311yescharacteristic
248C22×C4C2×C22⋊D5311yescharacteristic
249C22×C4C2×C22⋊D5311yescharacteristic
250M16—2122no—
251M16—2122no—
252C2×C8—4242no—
253C2×C8—2122no—
254C2×C8—2122no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C22⋊C4—24no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C42—81no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—41no—
273C23#1313313211yescharacteristic
274C23#1313313211yescharacteristic
275C23C2×C22⋊Dic5311yescharacteristic
276Q8—82no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4#1313323712yes—
309C8—4242no—
310C8—4242no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22#1681042212yes—
321C22C2×#1313313211yescharacteristic
322C22#1681061911yescharacteristic
323C22#1681064011yescharacteristic
324C4—22no—
325C4—42no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—81no—
333C4—81no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—41no—
338C2#2027327111yescharacteristic
339C2#2027636111yescharacteristic
340C2C2×#1681042211yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2583C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2583D53
hypocenterC53#11854
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2583

Complements

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

NH ≅ G/Ntype
3. #20272745C2semidirect
4. #20272721C2semidirect
193. C53#11854semidirect

Maximal subgroups

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

subgroupindex
#202747952
C2×#168103492
#202727452
#202727212
C53×#25832
#1185453