Finite Group Databasebeta

← #27279161

Subgroups — #27279161

Order 13568.

534 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277999C211yesmaximal, characteristic
2C4×#13133134C211yesmaximal, characteristic
3#20273161C212yesmaximal, complemented
4#20272607C212yesmaximal, complemented
5C53×#2973C211yesmaximal, characteristic
6C2×C4×Dic106C2211yescharacteristic
7C2×#13133134—21no—
8C4×#11719917C2212yes—
9#16810561—21no—
10#16810548C411yescharacteristic
11#16810548C411yescharacteristic
12#16810537—21no—
13#16810535C411yescharacteristic
14#16810535C411yescharacteristic
15C4×C22⋊Dic53—21no—
16C4×C22⋊Dic53—21no—
17#16810469C2211yescharacteristic
18#16810392—21no—
19#16810353—24no—
20C2×C4×D4×C53C2211yescharacteristic
21C53×#435—24no—
22C53×#432C2212yes—
23C53×#407—21no—
24C2×C22⋊C8×C53—21no—
25C22×Dic106C2×C411yescharacteristic
26C2×#11719917—22no—
27C4×Dic106—22no—
28C4×Dic106—22no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37#13133134—24no—
38#13133128—22no—
39C4×#88322—22no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124C2×C411yescharacteristic
46#13133124C2×C411yescharacteristic
47C22×D4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C23×C4×C53D411yescharacteristic
50C2×M16×C53—22no—
51C22×C8×C53—22no—
52C4×D4×C53—22no—
53C4×D4×C53—22no—
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C22⋊C4×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53C2×C411yescharacteristic
57C2×C42×C53C2311yescharacteristic
58C4⋊C8×C53D414yes—
59C22⋊C8×C53—22no—
60C22⋊C8×C53—22no—
61C22⋊C8×C53—22no—
62#97×C53—22no—
63C24×C53C22⋊C411yescharacteristic
64C24×C53C22⋊C411yescharacteristic
65C2×D4×C53—22no—
66C2×D4×C53—22no—
67C2×D4×C53—41no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—42no—
73C22×C4×C53—42no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C2×Dic106—22no—
84C2×Dic106—42no—
85M16×C53—82no—
86C2×C8×C53—24no—
87C2×C8×C53—82no—
88C4⋊C4×C53—22no—
89C22⋊C4×C53—22no—
90C22⋊C4×C53—22no—
91C42×C53C2×D412yes—
92C42×C53#4112yes—
93C22⋊Dic53—42no—
94C22⋊Dic53—42no—
95C22⋊Dic53—42no—
96C22⋊Dic53—42no—
97C4⋊Dic53—22no—
98#11719919—42no—
99C4×Dic53—42no—
100C4×Dic53—41no—
101C4×Dic53—41no—
102#11719917—24no—
103#11719917—42no—
104C2×#88322—42no—
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
116D4×C53—44no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—22no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—41no—
123C2×C4×C53—42no—
124C2×C4×C53—42no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53C4×D412yes—
130C2×C4×C53#12011yescharacteristic
131C2×C4×C53#12112yes—
132C2×C4×C53#12311yescharacteristic
133C2×Dic53—22no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145Dic106—24no—
146Dic106—42no—
147C8×C53—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#45512yes—
171Dic53—42no—
172Dic53—81no—
173Dic53—81no—
174Dic53—81no—
175#2973D5311yescharacteristic
176#2398—531no—
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×Q8—531no—
188C2×C4×D4C2×D5311yescharacteristic
189#435—24no—
190#432C2×D5312yes—
191C2×#100—1061no—
192#407—21no—
193C2×C22⋊C8—21no—
194C4×M16—532no—
195#400—1061no—
196#397—531no—
197#397—531no—
198#394—1061no—
199#393—531no—
200#393—531no—
201C4×C22⋊C4—1061no—
202C4×C22⋊C4—1061no—
203#376—531no—
204#343—1061no—
205Q8⋊C8—1064no—
206C53#902911yescharacteristic, complemented, Sylow
207C22×Q8—531no—
208C22×D4D21211yescharacteristic
209C23×C4C22⋊D5311yescharacteristic
210C23×C4C22⋊D5311yescharacteristic
211C2×M16—22no—
212C2×M16—1062no—
213C22×C8—22no—
214C4×Q8—1062no—
215C4×Q8—1062no—
216C4×D4—22no—
217C4×D4—22no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4D21211yescharacteristic
221C2×C22⋊C4—1061no—
222C2×C22⋊C4—1061no—
223C2×C22⋊C4—1061no—
224C2×C22⋊C4—1061no—
225C2×C22⋊C4C4×D5311yescharacteristic
226C2×C22⋊C4C4×D5311yescharacteristic
227C2×C42—1061no—
228C2×C42—1061no—
229C2×C42C22×D5311yescharacteristic
230C4⋊C8C22⋊D5314yes—
231#100—1064no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235#97—22no—
236#97—1062no—
237C4×C8—1062no—
238#95—531no—
239#95—531no—
240#95—1061no—
241#95—1061no—
242#95—1061no—
243#95—1061no—
244#95—1061no—
245C24#1171992111yescharacteristic
246C24#1171992111yescharacteristic
247C2×Q8—1062no—
248C2×Q8—2122no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4#1171992111yescharacteristic
263C22×C4#1171992111yescharacteristic
264C22×C4C2×C4×D5311yescharacteristic
265C22×C4C2×D21211yescharacteristic
266C22×C4C2×C22⋊D5311yescharacteristic
267M16—82no—
268M16—1064no—
269M16—2122no—
270C2×C8—24no—
271C2×C8—82no—
272C2×C8—2122no—
273C4⋊C4—22no—
274C4⋊C4—1062no—
275C4⋊C4—2122no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—2122no—
279C22⋊C4—2122no—
280C22⋊C4—2122no—
281C22⋊C4—2122no—
282C42—2122no—
283C42—2121no—
284C42—2121no—
285C42#1171994512yes—
286C42C2×C22⋊D5312yes—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23#1313314312yes—
297C23C2×#1171992111yescharacteristic
298Q8—1064no—
299Q8—2122no—
300D4—44no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—1062no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4C4×C22⋊D5312yes—
326C2×C4#1313318611yescharacteristic
327C2×C4#1313319112yes—
328C2×C4#1313320011yescharacteristic
329C8—44no—
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#1681050211yescharacteristic
344C22C2×#1313314311yescharacteristic
345C22#1681063911yescharacteristic
346C4—22no—
347C4—21no—
348C4—22no—
349C4—41no—
350C4—41no—
351C4—4241no—
352C4—4241no—
353C4—4241no—
354C4—2122no—
355C4#1681080112yes—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—81no—
360C2#2027501711yescharacteristic
361C2#2027510311yescharacteristic
362C2#2027513311yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2973C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#2973D53
hypocenterC53#9029
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
3. #20273161C2semidirect
4. #20272607C2semidirect
206. C53#9029semidirect

Maximal subgroups

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

subgroupindex
#202779992
C4×#131331342
#202731612
#202726072
C53×#29732
#902953