Finite Group Databasebeta

← #27294398

Subgroups — #27294398

Order 13568.

452 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 115 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#20274454C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272745C212yesmaximal, complemented
4#20272742C212yesmaximal, complemented
5C53×#2396C211yesmaximal, characteristic
6C22×#11719917—21no—
7#16810676—21no—
8#16810392—21no—
9#16810392C2212yes—
10#16810349C2212yes—
11#16810347—21no—
12C2×C4×Q8×C53C411yescharacteristic
13C2×#117×C53C2211yescharacteristic
14C53×#400C2211yescharacteristic
15C53×#399C411yescharacteristic
16C53×#398—21no—
17C53×#396—21no—
18C53×#395C2211yescharacteristic
19C53×#393C411yescharacteristic
20C4×C4⋊C4×C53—21no—
21C4×C4⋊C4×C53C411yescharacteristic
22C4×C22⋊C4×C53—21no—
23C2×#11719917—24no—
24C2×#11719917—22no—
25C2×#11719917—41no—
26C2×#11719917—42no—
27C22×#88322—41no—
28C4⋊#88322—41no—
29C22⋊#88322—22no—
30C4×#88322—41no—
31C22×Q8×C53C2×C411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C4×Q8×C53—41no—
34C4×Q8×C53—41no—
35#117×C53—41no—
36#117×C53—41no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C4⋊C4×C53C2311yescharacteristic
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53C2×C411yescharacteristic
47C2×C22⋊C4×C53D411yescharacteristic
48C2×C42×C53—21no—
49C2×C42×C53—21no—
50C2×C42×C53C2×C411yescharacteristic
51C2×C42×C53D411yescharacteristic
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53C2×C411yescharacteristic
57#95×C53C2×C411yescharacteristic
58C24×C53C22⋊C411yescharacteristic
59C2×Q8×C53—42no—
60C2×Q8×C53—41no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53SD1612yes—
72C22×C4×C53Q1612yes—
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C42×C53—21no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—41no—
92C42×C53—41no—
93C42×C53—42no—
94#11719917—24no—
95#11719917—44no—
96#11719917—42no—
97C2×#88322—42no—
98C2×#88322—44no—
99C2×#88322—42no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53#10312yes—
105C23×C53C2×C22⋊C411yescharacteristic
106Q8×C53—42no—
107Q8×C53—42no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
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#9911yescharacteristic
135C2×C4×C53#9911yescharacteristic
136C2×C4×C53#10312yes—
137C2×C4×C53#10412yes—
138C2×C4×C53C2×SD1611yescharacteristic
139C2×C4×C53C2×Q1611yescharacteristic
140#88322—44no—
141#88322—82no—
142#11721—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—22no—
146C22×C53—42no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—81no—
150C22×C53C2×#9911yescharacteristic
151C22×C53C2×#10311yescharacteristic
152C22×C53C2×#10411yescharacteristic
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—42no—
156C4×C53—41no—
157C4×C53—42no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—81no—
162C4×C53—81no—
163C4×C53#32812yes—
164C4×C53#32812yes—
165#2396D5311yescharacteristic
166#1742—531no—
167#1458—531no—
168#1354—532no—
169#1353—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—42no—
173C2×C53C2×#32811yescharacteristic
174C2×C53#212011yescharacteristic
175C2×C53#213211yescharacteristic
176C22×M16—1061no—
177C2×C4×Q8Dic5311yescharacteristic
178C2×#117C2×D5311yescharacteristic
179#431—1061no—
180#400C2×D5311yescharacteristic
181#399Dic5311yescharacteristic
182#398—21no—
183#396—21no—
184#395C2×D5311yescharacteristic
185#393Dic5311yescharacteristic
186C4×C4⋊C4—21no—
187C4×C4⋊C4Dic5311yescharacteristic
188C4×C22⋊C4—21no—
189#343—532no—
190#343—1061no—
191#324—532no—
192#323—1061no—
193C53#1172111yescharacteristic, complemented, Sylow
194C22×Q8C2×Dic5311yescharacteristic
195C23×C4C22⋊D5311yescharacteristic
196C2×M16—1064no—
197C2×M16—1062no—
198C2×M16—2121no—
199C2×M16—2122no—
200C22×C8—2121no—
201C4×Q8—41no—
202C4×Q8—41no—
203#117—41no—
204#117—41no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4C2×Dic5311yescharacteristic
210C2×C4⋊C4C22⋊D5311yescharacteristic
211C2×C4⋊C4C22×D5311yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4C2×Dic5311yescharacteristic
215C2×C22⋊C4C22⋊D5311yescharacteristic
216C2×C42—21no—
217C2×C42—21no—
218C2×C42C2×Dic5311yescharacteristic
219C2×C42C22⋊D5311yescharacteristic
220C4⋊C8—2121no—
221C22⋊C8—1062no—
222C4×C8—2121no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95C2×Dic5311yescharacteristic
228#95C2×Dic5311yescharacteristic
229C24C22⋊Dic5311yescharacteristic
230C2×Q8—42no—
231C2×Q8—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4#1171992412yes—
240C22×C4#1171992512yes—
241C22×C4C22⋊Dic5311yescharacteristic
242C22×C4C22⋊Dic5311yescharacteristic
243C22×C4C22⋊Dic5311yescharacteristic
244C22×C4C22×Dic5311yescharacteristic
245C22×C4C2×C22⋊D5311yescharacteristic
246C22×C4C2×C22⋊D5311yescharacteristic
247M16—1064no—
248M16—2124no—
249M16—2122no—
250C2×C8—2122no—
251C2×C8—2122no—
252C2×C8—2124no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—41no—
269C42—41no—
270C42—42no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23#1313314112yes—
276C23C2×C22⋊Dic5311yescharacteristic
277Q8—42no—
278Q8—42no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
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#1313313211yescharacteristic
306C2×C4#1313313211yescharacteristic
307C2×C4#1313314112yes—
308C2×C4#1313314412yes—
309C2×C4C2×#1171992411yescharacteristic
310C2×C4C2×#1171992511yescharacteristic
311C8—4242no—
312C8—2124no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—81no—
320C22C2×#1313313211yescharacteristic
321C22C2×#1313314111yescharacteristic
322C22C2×#1313314411yescharacteristic
323C4—21no—
324C4—21no—
325C4—42no—
326C4—41no—
327C4—42no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4#1681036012yes—
334C4#1681036012yes—
335C2—21no—
336C2—21no—
337C2—42no—
338C2C2×#1681036011yescharacteristic
339C2#2027635611yescharacteristic
340C2#2027640011yescharacteristic
subgroupG/H
centerC2×C4#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2396C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2396D53
hypocenterC53#11721
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #2396

Complements

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

NH ≅ G/Ntype
3. #20272745C2semidirect
4. #20272742C2semidirect
193. C53#11721semidirect

Maximal subgroups

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

subgroupindex
#202744542
#202730292
#202727452
#202727422
C53×#23962
#1172153