Finite Group Databasebeta

← #27288487

Subgroups — #27288487

Order 13568.

490 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 99 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#20275701C212yesmaximal
2#20275139C212yesmaximal
3#20274892C211yesmaximal, characteristic
4#20272583C211yesmaximal, characteristic
5C53×#3189C211yesmaximal, characteristic
6C23×#88322C2211yescharacteristic
7C2×#13133146—21no—
8C2×#13133141—22no—
9#16810483C2212yes—
10C4×C22⋊Dic53C2211yescharacteristic
11#16810390—22no—
12#16810390—22no—
13#16810385—22no—
14#16810377—21no—
15C2×C22⋊Q8×C53C2212yes—
16C53×#484—22no—
17C53×#467—21no—
18C53×#467—21no—
19C53×#451—22no—
20C2×C22⋊C8×C53C2211yescharacteristic
21C22×#88322—22no—
22C22×#88322—21no—
23C22×#88322—42no—
24C2×C4⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53D411yescharacteristic
27C2×C22⋊Dic53D411yescharacteristic
28C2×C4×Dic53—21no—
29#13133146—42no—
30#13133141—44no—
31#13133124—21no—
32#13133124—22no—
33C22×Q8×C53D412yes—
34C23×C4×C53C2311yescharacteristic
35C2×Q16×C53—21no—
36C2×Q16×C53—21no—
37C22×C8×C53D411yescharacteristic
38C22×C8×C53D411yescharacteristic
39C22⋊Q8×C53—22no—
40C22⋊Q8×C53—22no—
41C22⋊Q8×C53—22no—
42C22⋊Q8×C53—22no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53D412yes—
45C2×C22⋊C4×C53—22no—
46#107×C53—21no—
47#107×C53—21no—
48#103×C53—22no—
49#103×C53—22no—
50C22⋊C8×C53—22no—
51C24×C53C2×D411yescharacteristic
52C2×Q8×C53—22no—
53C2×Q8×C53—22no—
54C2×Q8×C53—42no—
55C2×Q8×C53—42no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53C2×D412yes—
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53#4111yescharacteristic
63C22×C4×C53#4112yes—
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—42no—
67Q16×C53—81no—
68Q16×C53—81no—
69C2×C8×C53—21no—
70C2×C8×C53—21no—
71C2×C8×C53—41no—
72C2×C8×C53—41no—
73C4⋊C4×C53—22no—
74C4⋊C4×C53—22no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—42no—
77C22⋊C4×C53—42no—
78C22⋊C4×C53—42no—
79C22⋊Dic53—21no—
80C22⋊Dic53—22no—
81C22⋊Dic53—42no—
82C22⋊Dic53—42no—
83C22⋊Dic53Q1612yescomplemented
84C4⋊Dic53—42no—
85C4⋊Dic53—42no—
86C4×Dic53—41no—
87C4×Dic53—41no—
88C2×#88322—24no—
89C2×#88322—22no—
90C2×#88322—42no—
91C2×#88322—41no—
92C2×#88322—41no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53#12111yescharacteristic
96C23×C53#12112yes—
97C23×C53C2×Q1611yescharacteristic
98C23×C53#13511yescharacteristic
99C23×C53#13712yes—
100Q8×C53—44no—
101Q8×C53—42no—
102Q8×C53—42no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—82no—
112C2×C4×C53—82no—
113C2×C4×C53#12011yescharacteristic
114C2×C4×C53#12312yes—
115C2×C4×C53#12411yescharacteristic
116C2×Dic53—22no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—82no—
123C2×Dic53—82no—
124C2×Dic53—82no—
125C8×C53—41no—
126C8×C53—41no—
127#88322—42no—
128#10566—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—22no—
131C22×C53—21no—
132C22×C53—22no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—42no—
136C22×C53#39411yescharacteristic
137C22×C53#46212yes—
138C22×C53#46412yes—
139C22×C53#46711yescharacteristic
140C22×C53#47011yescharacteristic
141C4×C53—22no—
142C4×C53—21no—
143C4×C53—82no—
144C4×C53—82no—
145C4×C53B(4,𝔽2)11yescharacteristic
146C4×C53#45811yescharacteristic
147Dic53—42no—
148Dic53—81no—
149Dic53—162no—
150#3189D5311yescharacteristic
151#1996—532no—
152#1892—532no—
153#1850—531no—
154#1289—531no—
155C2×C53—21no—
156C2×C53—21no—
157C2×C53—22no—
158C2×C53—22no—
159C2×C53#204411yescharacteristic
160C2×C53#205311yescharacteristic
161C2×C53#205611yescharacteristic
162C23×C8—531no—
163C2×C22⋊Q8C2×D5312yes—
164#484—22no—
165#467—21no—
166#467—21no—
167#451—22no—
168C2×C4⋊C8—1061no—
169C2×#103—1062no—
170C2×C22⋊C8C2×D5311yescharacteristic
171#380—532no—
172C4×C22⋊C4—531no—
173#342—1062no—
174#342—1062no—
175#340—1062no—
176#336—1061no—
177C53#1056611yescharacteristic, complemented, Sylow
178C22×Q8C22⋊D5312yes—
179C23×C4C22×D5311yescharacteristic
180C2×Q16—21no—
181C2×Q16—21no—
182C22×C8—1061no—
183C22×C8—1062no—
184C22×C8—2122no—
185C22×C8C22⋊D5311yescharacteristic
186C22×C8C22⋊D5311yescharacteristic
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—1062no—
193C2×C4⋊C4C22⋊D5312yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—531no—
196C2×C22⋊C4—531no—
197C2×C22⋊C4—1062no—
198C2×C42—1061no—
199#107—21no—
200#107—21no—
201C4⋊C8—2122no—
202#103—22no—
203#103—22no—
204#103—2124no—
205C22⋊C8—22no—
206#95—1061no—
207#95—1062no—
208C24D4×D5311yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—42no—
212C2×Q8—42no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—1061no—
218C22×C4—1061no—
219C22×C4—2122no—
220C22×C4#1171994711yescharacteristic
221C22×C4#1171994712yes—
222C22×C4C2×C22⋊D5312yes—
223C22×C4C2×C22⋊D5311yescharacteristic
224Q16—81no—
225Q16—81no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—1064no—
231C2×C8—1062no—
232C2×C8—2121no—
233C2×C8—2122no—
234C2×C8—2121no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—2122no—
240C4⋊C4—2122no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—532no—
244C22⋊C4—1061no—
245C22⋊C4—1062no—
246C22⋊C4—2122no—
247C22⋊C4—2122no—
248C42—2121no—
249C42—2121no—
250C23—41no—
251C23—41no—
252C23#1313318811yescharacteristic
253C23#1313318912yes—
254C23Q16×D5311yescharacteristic
255C23#1313324011yescharacteristic
256C23#1313324912yes—
257Q8—44no—
258Q8—42no—
259Q8—42no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—1062no—
271C2×C4—4242no—
272C2×C4—4242no—
273C2×C4—4242no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4#1313318611yescharacteristic
280C2×C4#1313319812yes—
281C2×C4#1313320411yescharacteristic
282C8—41no—
283C8—41no—
284C8—2122no—
285C22—22no—
286C22—21no—
287C22—22no—
288C22—21no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22#1681053711yescharacteristic
293C22#1681083112yes—
294C22#1681084712yes—
295C22#1681086111yescharacteristic
296C22#1681087811yescharacteristic
297C4—22no—
298C4—21no—
299C4—82no—
300C4—82no—
301C4—8482no—
302C4—4241no—
303C4—2122no—
304C4#1681081011yescharacteristic
305C4#1681081311yescharacteristic
306C2—22no—
307C2—21no—
308C2—21no—
309C2—22no—
310C2#2027596311yescharacteristic
311C2#2027601811yescharacteristic
312C2#2027603511yescharacteristic
subgroupG/H
centerC22#16810537
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#3189C2
socleC22×C53#394
prime radicalC24×C53C2×D4
hypercenter#3189D53
hypocenterC53#10566
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

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

NH ≅ G/Ntype
83. C22⋊Dic53Q16semidirect
177. C53#10566semidirect

Maximal subgroups

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

subgroupindex
#202757012
#202751392
#202748922
#202725832
C53×#31892
#1056653