Finite Group Databasebeta

← #27295015

Subgroups — #27295015

Order 13568.

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

normal subgroups only

This group has 438 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, complemented
2C2×#16810349C211yesmaximal, characteristic, complemented
3#20272742C212yesmaximal, complemented
4#20272718C212yesmaximal, complemented
5C53×#2591C211yesmaximal, characteristic
6#16810587—21no—
7C2×C22⋊#88322—21no—
8#16810392—21no—
9#16810392C2212yes—
10#16810349—21no—
11#16810349C2212yes—
12C2×#121×C53C411yescharacteristic
13C22×C4⋊C4×C53C2211yescharacteristic
14C53×#400C411yescharacteristic
15C53×#400C2211yescharacteristic
16C53×#394C411yescharacteristic
17C53×#394C2211yescharacteristic
18C53×#388—21no—
19C53×#388—21no—
20C53×#386—21no—
21C53×#386—21no—
22C4×C22⋊C4×C53C411yescharacteristic
23C2×#11719917—22no—
24C22×#88322—41no—
25C22⋊#88322—22no—
26C22⋊#88322—41no—
27C22⋊#88322—41no—
28C22⋊#88322—41no—
29C22×D4×C53—21no—
30C22×D4×C53C2×C411yescharacteristic
31C23×C4×C53C2×C411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C23×C4×C53D411yescharacteristic
34#121×C53—41no—
35#121×C53—41no—
36C2×C4⋊C4×C53—41no—
37C2×C4⋊C4×C53—41no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C4⋊C4×C53C2311yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
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×C53—21no—
57C24×C53C22⋊C411yescharacteristic
58C2×D4×C53—41no—
59C2×D4×C53—42no—
60C2×D4×C53—42no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53—81no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—41no—
75C22×C4×C53—81no—
76C22×C4×C53—81no—
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22×C411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—41no—
88C22⋊C4×C53—42no—
89C22⋊C4×C53—42no—
90C22⋊C4×C53—42no—
91C22⋊C4×C53—42no—
92C22⋊C4×C53—81no—
93C22⋊C4×C53—81no—
94C42×C53—81no—
95#11719917—42no—
96#11719917—42no—
97C2×#88322—42no—
98C2×#88322—42no—
99C2×#88322—82no—
100C23×C53—22no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53#9911yescharacteristic
112C23×C53#9911yescharacteristic
113C23×C53C2×C22⋊C411yescharacteristic
114D4×C53—82no—
115D4×C53—82no—
116D4×C53—82no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—81no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53#13512yes—
146#88322—82no—
147#88322—82no—
148#11852—531nomaximal, Sylow
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—42no—
157C22×C53—42no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53—81no—
163C22×C53—81no—
164C22×C53—81no—
165C22×C53#35512yes—
166C22×C53C2×#9911yescharacteristic
167C22×C53#41611yescharacteristic
168C22×C53#42111yescharacteristic
169C4×C53—22no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53—81no—
176C4×C53—81no—
177C4×C53—81no—
178#2591D5311yescharacteristic
179#1821—531no—
180C2×#324—531no—
181#1353—532no—
182#1345—532no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—22no—
186C2×C53—41no—
187C2×C53—161no—
188C2×C53#151611yescharacteristic
189C2×C53#212111yescharacteristic
190C2×C53C2×#35511yescharacteristic
191C2×#121Dic5311yescharacteristic
192C22×C4⋊C4C2×D5311yescharacteristic
193#407—1061no—
194C2×C22⋊C8—1061no—
195#400Dic5311yescharacteristic
196#400C2×D5311yescharacteristic
197#394Dic5311yescharacteristic
198#394C2×D5311yescharacteristic
199#388—21no—
200#388—21no—
201#386—21no—
202#386—21no—
203C4×C22⋊C4Dic5311yescharacteristic
204#343—532no—
205#343—1061no—
206#324—532no—
207#324—1061no—
208C53#1185211yescharacteristic, complemented, Sylow
209C22×D4—21no—
210C22×D4C2×Dic5311yescharacteristic
211C23×C4C2×Dic5311yescharacteristic
212C23×C4C22⋊D5311yescharacteristic
213C23×C4C22⋊D5311yescharacteristic
214C2×M16—1062no—
215C22×C8—2121no—
216#121—41no—
217#121—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4C2×Dic5311yescharacteristic
221C2×C4⋊C4C2×Dic5311yescharacteristic
222C2×C4⋊C4C22⋊D5311yescharacteristic
223C2×C4⋊C4C22⋊D5311yescharacteristic
224C2×C4⋊C4C22×D5311yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C42—21no—
232C22⋊C8—1062no—
233C22⋊C8—2121no—
234C22⋊C8—2121no—
235C22⋊C8—2121no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95C2×Dic5311yescharacteristic
241#95C2×Dic5311yescharacteristic
242C24—21no—
243C24C22⋊Dic5311yescharacteristic
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—81no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4C22⋊Dic5311yescharacteristic
264C22×C4C22⋊Dic5311yescharacteristic
265C22×C4C22⋊Dic5311yescharacteristic
266C22×C4C22×Dic5311yescharacteristic
267C22×C4C2×C22⋊D5311yescharacteristic
268C22×C4C2×C22⋊D5311yescharacteristic
269M16—2122no—
270M16—2122no—
271C2×C8—4242no—
272C2×C8—2122no—
273C2×C8—2122no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—81no—
284C22⋊C4—81no—
285C42—81no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23#1313313211yescharacteristic
298C23#1313313211yescharacteristic
299C23C2×C22⋊Dic5311yescharacteristic
300D4—82no—
301D4—82no—
302D4—82no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4#1313323712yes—
332C8—4242no—
333C8—4242no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—42no—
342C22—42no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22#1681042012yes—
351C22C2×#1313313211yescharacteristic
352C22#1681061911yescharacteristic
353C22#1681064011yescharacteristic
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—81no—
361C4—81no—
362C4—81no—
363C2—21no—
364C2—21no—
365C2—22no—
366C2—41no—
367C2—161no—
368C2#2027327111yescharacteristic
369C2#2027636111yescharacteristic
370C2C2×#1681042011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2591C2
socleC22×C53C2×#99
prime radicalC2×#121×C53C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2591D53
hypocenterC53#11852
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
1. #20274795C2semidirect
2. C2×#16810349C2semidirect
3. #20272742C2semidirect
4. #20272718C2semidirect
208. C53#11852semidirect

Maximal subgroups

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

subgroupindex
#202747952
C2×#168103492
#202727422
#202727182
C53×#25912
#1185253