Finite Group Databasebeta

← #27279187

Subgroups — #27279187

Order 13568.

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

normal subgroups only

This group has 482 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#20283353C211yesmaximal, characteristic
2#20273144C212yesmaximal, complemented
3#20273029C211yesmaximal, characteristic
4#20272618C212yesmaximal, complemented
5C53×#2396C211yesmaximal, characteristic
6#16810684—24no—
7#16810676C2212yes—
8#16810647C2212yes—
9#16810587—21no—
10C4×#11719917—21no—
11Q8⋊#88322—24no—
12#16810349—21no—
13#16810347—21no—
14C2×C4×Q8×C53C2211yescharacteristic
15C2×#117×C53C2211yescharacteristic
16C53×#400C411yescharacteristic
17C53×#399C411yescharacteristic
18C53×#398—21no—
19C53×#396—21no—
20C53×#395C411yescharacteristic
21C53×#393C411yescharacteristic
22C4×C4⋊C4×C53—21no—
23C4×C4⋊C4×C53C2211yescharacteristic
24C4×C22⋊C4×C53—21no—
25C2×#11719917—22no—
26C4⋊#88322—22no—
27C4⋊#88322—22no—
28C4⋊#88322D414yes—
29C22⋊#88322—24no—
30C22⋊#88322—22no—
31#13133128—22no—
32C4×#88322—22no—
33C4×#88322—22no—
34C22×Q8×C53C2×C411yescharacteristic
35C23×C4×C53D411yescharacteristic
36C4×Q8×C53—22no—
37C4×Q8×C53—22no—
38#117×C53—22no—
39#117×C53—22no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C42×C53—21no—
52C2×C42×C53—21no—
53C2×C42×C53D411yescharacteristic
54C2×C42×C53C2311yescharacteristic
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53C2×C411yescharacteristic
60#95×C53C2×C411yescharacteristic
61C24×C53C22⋊C411yescharacteristic
62C2×Q8×C53—22no—
63C2×Q8×C53—42no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—22no—
69C22×C4×C53—22no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C4⋊C4×C53—22no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—42no—
83C4⋊C4×C53—42no—
84C4⋊C4×C53—42no—
85C22⋊C4×C53—22no—
86C22⋊C4×C53—22no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—42no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—41no—
92C42×C53—41no—
93C42×C53—42no—
94C42×C53C2×D412yes—
95C42×C53#4112yes—
96#11719917—42no—
97#11719917—42no—
98C2×#88322—24no—
99C2×#88322—42no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53C2×C22⋊C411yescharacteristic
106Q8×C53—24no—
107Q8×C53—42no—
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—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
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—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53C4×D412yes—
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53#12112yes—
134C2×C4×C53#12311yescharacteristic
135C2×C4×C53#13512yes—
136C2×C4×C53#13712yes—
137#88322—44no—
138#88322—82no—
139#9034—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—42no—
143C22×C53—41no—
144C22×C53—41no—
145C22×C53—41no—
146C22×C53—81no—
147C22×C53#38611yescharacteristic
148C22×C53#41611yescharacteristic
149C22×C53#41911yescharacteristic
150C4×C53—22no—
151C4×C53—22no—
152C4×C53—42no—
153C4×C53—41no—
154C4×C53—41no—
155C4×C53—41no—
156C4×C53—42no—
157C4×C53—81no—
158C4×C53—81no—
159C4×C53#45212yes—
160C4×C53#48512yes—
161#2921—531no—
162#2396D5311yescharacteristic
163#1487—532no—
164#1458—531no—
165#1303—532no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53—81no—
169C2×C53#187311yescharacteristic
170C2×C53#188411yescharacteristic
171C2×C53#189111yescharacteristic
172C2×C4×Q8C2×D5311yescharacteristic
173C2×#117C2×D5311yescharacteristic
174#433—1064no—
175#431—532no—
176#423—532no—
177#407—1061no—
178C4×M16—1061no—
179#400Dic5311yescharacteristic
180#399Dic5311yescharacteristic
181#398—21no—
182#396—21no—
183#395Dic5311yescharacteristic
184#393Dic5311yescharacteristic
185C4×C4⋊C4—21no—
186C4×C4⋊C4C2×D5311yescharacteristic
187C4×C22⋊C4—21no—
188Q8⋊C8—1064no—
189#324—1061no—
190#323—1061no—
191C53#903411yescharacteristic, complemented, Sylow
192C22×Q8C2×Dic5311yescharacteristic
193C23×C4C22⋊D5311yescharacteristic
194C2×M16—1062no—
195C4×Q8—22no—
196C4×Q8—22no—
197#117—22no—
198#117—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4C2×Dic5311yescharacteristic
204C2×C4⋊C4C2×Dic5311yescharacteristic
205C2×C4⋊C4C22⋊D5311yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4C2×Dic5311yescharacteristic
209C2×C22⋊C4C22⋊D5311yescharacteristic
210C2×C42—21no—
211C2×C42—21no—
212C2×C42C22⋊D5311yescharacteristic
213C2×C42C22×D5311yescharacteristic
214C4⋊C8—534no—
215C4⋊C8—1062no—
216C4⋊C8—1062no—
217C22⋊C8—1064no—
218C22⋊C8—1062no—
219#97—1062no—
220C4×C8—1062no—
221C4×C8—1062no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95C2×Dic5311yescharacteristic
227#95C2×Dic5311yescharacteristic
228C24C22⋊Dic5311yescharacteristic
229C2×Q8—22no—
230C2×Q8—42no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4C22⋊Dic5311yescharacteristic
240C22×C4C22⋊Dic5311yescharacteristic
241C22×C4C22⋊Dic5311yescharacteristic
242C22×C4C22×Dic5311yescharacteristic
243C22×C4C2×C22⋊D5311yescharacteristic
244C22×C4C2×C22⋊D5311yescharacteristic
245M16—2122no—
246M16—2122no—
247C2×C8—1064no—
248C2×C8—2122no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C42—21no—
261C42—21no—
262C42—41no—
263C42—41no—
264C42—42no—
265C42D4×D5312yes—
266C42#1171994712yes—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23C2×C22⋊Dic5311yescharacteristic
273Q8—24no—
274Q8—42no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4D4×Dic5312yes—
299C2×C4#1313318511yescharacteristic
300C2×C4#1313318912yes—
301C2×C4#1313319811yescharacteristic
302C2×C4#1313323712yes—
303C2×C4#1313324812yes—
304C8—4242no—
305C8—2124no—
306C22—21no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—81no—
313C22#1681050311yescharacteristic
314C22#1681061911yescharacteristic
315C22#1681063211yescharacteristic
316C4—22no—
317C4—22no—
318C4—42no—
319C4—41no—
320C4—41no—
321C4—42no—
322C4—41no—
323C4—81no—
324C4—81no—
325C4#1681078312yes—
326C4#1681096112yes—
327C2—21no—
328C2—21no—
329C2—81no—
330C2#2027502811yescharacteristic
331C2#2027509911yescharacteristic
332C2#2027513411yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2396C2
socleC22×C53#386
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2396D53
hypocenterC53#9034
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2396

Complements

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

NH ≅ G/Ntype
2. #20273144C2semidirect
4. #20272618C2semidirect
191. C53#9034semidirect

Maximal subgroups

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

subgroupindex
#202833532
#202731442
#202730292
#202726182
C53×#23962
#903453