Finite Group Databasebeta

← #27279162

Subgroups — #27279162

Order 13568.

534 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283744C211yesmaximal, characteristic
2C4×#13133134C211yesmaximal, characteristic
3#20273162C212yesmaximal, complemented
4#20272608C212yesmaximal, complemented
5C53×#2398C211yesmaximal, characteristic
6#16810695—24no—
7#16810680C2212yes—
8C2×#13133134—21no—
9#16810587—21no—
10C2×C22⋊#88322—21no—
11C4×#11719917C2212yes—
12#16810392—21no—
13Q8⋊#88322—24no—
14C2×C4×Q8×C53C2211yescharacteristic
15C2×C4×D4×C53C2211yescharacteristic
16C53×#400—21no—
17C53×#397C411yescharacteristic
18C53×#397C411yescharacteristic
19C53×#394—21no—
20C53×#393C411yescharacteristic
21C53×#393C411yescharacteristic
22C4×C22⋊C4×C53—21no—
23C4×C22⋊C4×C53—21no—
24C53×#376C2211yescharacteristic
25C2×#11719917—22no—
26C2×#11719917—22no—
27C22×#88322—22no—
28C4⋊#88322D414yes—
29#13133134—24no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33#13133128—22no—
34#13133128—22no—
35C4×#88322—22no—
36C22×Q8×C53C2×C411yescharacteristic
37C22×D4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C4×Q8×C53—22no—
41C4×Q8×C53—22no—
42C4×D4×C53—22no—
43C4×D4×C53—22no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C42×C53—21no—
54C2×C42×C53—21no—
55C2×C42×C53C2311yescharacteristic
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53C2×C411yescharacteristic
62#95×C53C2×C411yescharacteristic
63C24×C53C22⋊C411yescharacteristic
64C24×C53C22⋊C411yescharacteristic
65C2×Q8×C53—22no—
66C2×Q8×C53—42no—
67C2×D4×C53—22no—
68C2×D4×C53—22no—
69C2×D4×C53—41no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—42no—
79C22×C4×C53—42no—
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—42no—
88C22⋊C4×C53—22no—
89C22⋊C4×C53—22no—
90C22⋊C4×C53—42no—
91C22⋊C4×C53—42no—
92C22⋊C4×C53—42no—
93C22⋊C4×C53—42no—
94C42×C53—42no—
95C42×C53—41no—
96C42×C53—41no—
97C42×C53C2×D412yes—
98C42×C53#4112yes—
99#11719917—24no—
100#11719917—42no—
101#11719917—82no—
102C2×#88322—24no—
103C2×#88322—42no—
104C2×#88322—82no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53#10412yes—
115C23×C53C2×C22⋊C411yescharacteristic
116Q8×C53—24no—
117Q8×C53—42no—
118D4×C53—44no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
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—81no—
142C2×C4×C53—81no—
143C2×C4×C53C4×D412yes—
144C2×C4×C53#12011yescharacteristic
145C2×C4×C53#12112yes—
146C2×C4×C53#12311yescharacteristic
147#88322—44no—
148#88322—44no—
149#9029—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—22no—
153C22×C53—42no—
154C22×C53—42no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53#38611yescharacteristic
163C22×C53C2×#10411yescharacteristic
164C22×C53#42111yescharacteristic
165C4×C53—22no—
166C4×C53—21no—
167C4×C53—22no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—42no—
171C4×C53—81no—
172C4×C53—81no—
173C4×C53—81no—
174C4×C53#45512yes—
175#2973—531no—
176#2398D5311yescharacteristic
177C4×#100—531no—
178#1491—532no—
179#1300—532no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—42no—
183C2×C53—81no—
184C2×C53#187111yescharacteristic
185C2×C53#188511yescharacteristic
186C2×C53#189111yescharacteristic
187C2×C4×Q8C2×D5311yescharacteristic
188C2×C4×D4C2×D5311yescharacteristic
189#435—1064no—
190#432—532no—
191C2×#100—1061no—
192#407—1061no—
193C2×C22⋊C8—1061no—
194C4×M16—532no—
195#400—21no—
196#397Dic5311yescharacteristic
197#397Dic5311yescharacteristic
198#394—21no—
199#393Dic5311yescharacteristic
200#393Dic5311yescharacteristic
201C4×C22⋊C4—21no—
202C4×C22⋊C4—21no—
203#376C2×D5311yescharacteristic
204#343—1061no—
205Q8⋊C8—1064no—
206C53#902911yescharacteristic, complemented, Sylow
207C22×Q8C2×Dic5311yescharacteristic
208C22×D4C22⋊D5311yescharacteristic
209C23×C4C22⋊D5311yescharacteristic
210C23×C4C22⋊D5311yescharacteristic
211C2×M16—1062no—
212C2×M16—1062no—
213C22×C8—1062no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×D4—22no—
217C4×D4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C2×Dic5311yescharacteristic
220C2×C4⋊C4C22⋊D5311yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C2×Dic5311yescharacteristic
226C2×C22⋊C4C2×Dic5311yescharacteristic
227C2×C42—21no—
228C2×C42—21no—
229C2×C42C22×D5311yescharacteristic
230C4⋊C8—534no—
231#100—1064no—
232C22⋊C8—1062no—
233C22⋊C8—1062no—
234C22⋊C8—1062no—
235#97—1062no—
236#97—1062no—
237C4×C8—1062no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95C2×Dic5311yescharacteristic
244#95C2×Dic5311yescharacteristic
245C24C22⋊Dic5311yescharacteristic
246C24C22⋊Dic5311yescharacteristic
247C2×Q8—22no—
248C2×Q8—42no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4C22⋊Dic5311yescharacteristic
263C22×C4C22⋊Dic5311yescharacteristic
264C22×C4C22×Dic5311yescharacteristic
265C22×C4C2×C22⋊D5311yescharacteristic
266C22×C4C2×C22⋊D5311yescharacteristic
267M16—1064no—
268M16—4242no—
269M16—2122no—
270C2×C8—1064no—
271C2×C8—4242no—
272C2×C8—2122no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—42no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C42—42no—
283C42—41no—
284C42—41no—
285C42D4×D5312yes—
286C42#1171994712yes—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23#1313314412yes—
297C23C2×C22⋊Dic5311yescharacteristic
298Q8—24no—
299Q8—42no—
300D4—44no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
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—81no—
324C2×C4—81no—
325C2×C4D4×Dic5312yes—
326C2×C4#1313318511yescharacteristic
327C2×C4#1313318912yes—
328C2×C4#1313319811yescharacteristic
329C8—2124no—
330C8—2124no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—81no—
342C22—81no—
343C22#1681050311yescharacteristic
344C22C2×#1313314411yescharacteristic
345C22#1681064011yescharacteristic
346C4—22no—
347C4—21no—
348C4—22no—
349C4—41no—
350C4—41no—
351C4—42no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4#1681080012yes—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—81no—
360C2#2027501811yescharacteristic
361C2#2027510411yescharacteristic
362C2#2027513411yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2398C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#2398D53
hypocenterC53#9029
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2398

Complements

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

NH ≅ G/Ntype
3. #20273162C2semidirect
4. #20272608C2semidirect
206. C53#9029semidirect

Maximal subgroups

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

subgroupindex
#202837442
C4×#131331342
#202731622
#202726082
C53×#23982
#902953