Finite Group Databasebeta

← #27288442

Subgroups — #27288442

Order 13568.

448 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 81 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#20275184C212yesmaximal
2#20274897C211yesmaximal, characteristic
3#20273235C212yesmaximal
4#20272592C211yesmaximal, characteristic
5C53×#3189C211yesmaximal, characteristic
6C22×#11719917C2211yescharacteristic
7C2×#13133146—21no—
8C2×#13133136—22no—
9#16810587C2212yes—
10C4×C22⋊Dic53C2211yescharacteristic
11#16810392—22no—
12#16810377—21no—
13#16810360—22no—
14#16810360—22no—
15C2×C22⋊Q8×C53C2212yes—
16C53×#484—22no—
17C53×#467—21no—
18C53×#467—21no—
19C53×#451—22no—
20C2×C22⋊C8×C53C2211yescharacteristic
21C2×#11719917—22no—
22C2×#11719917—22no—
23C2×#11719917—42no—
24C22×#88322—21no—
25C2×C22⋊Dic53—21no—
26C2×C4×Dic53—21no—
27#13133146—42no—
28#13133136—44no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133124D411yescharacteristic
32#13133124D411yescharacteristic
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×C53#4111yescharacteristic
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×C53C2×D411yescharacteristic
63C22×C4×C53#4112yes—
64C22×Dic53—21no—
65C22×Dic53—21no—
66Q16×C53—81no—
67Q16×C53—81no—
68C2×C8×C53—21no—
69C2×C8×C53—21no—
70C2×C8×C53—41no—
71C2×C8×C53—41no—
72C4⋊C4×C53—22no—
73C4⋊C4×C53—22no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—42no—
76C22⋊C4×C53—42no—
77C22⋊C4×C53—42no—
78C22⋊Dic53—42no—
79C4×Dic53—41no—
80C4×Dic53—41no—
81#11719917—24no—
82#11719917—42no—
83#11719917—42no—
84#11719917—42no—
85C2×#88322—22no—
86C2×#88322—41no—
87C2×#88322—41no—
88C2×#88322—42no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53#12111yescharacteristic
94C23×C53#12312yes—
95Q8×C53—42no—
96Q8×C53—42no—
97Q8×C53—82no—
98C2×C4×C53—22no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—21no—
102C2×C4×C53—41no—
103C2×C4×C53—42no—
104C2×C4×C53—44no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—82no—
108C2×C4×C53#12011yescharacteristic
109C2×C4×C53#12112yes—
110C2×C4×C53#12411yescharacteristic
111C2×Dic53—22no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C8×C53—41no—
118C8×C53—41no—
119#88322—42no—
120#88322—82no—
121#10559—531nomaximal, Sylow
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—22no—
125C22×C53—22no—
126C22×C53—22no—
127C22×C53—81no—
128C22×C53—81no—
129C22×C53#39411yescharacteristic
130C22×C53#47311yescharacteristic
131C22×C53#47311yescharacteristic
132C4×C53—22no—
133C4×C53—21no—
134C4×C53—82no—
135C4×C53—82no—
136C4×C53B(4,𝔽2)11yescharacteristic
137C4×C53#45811yescharacteristic
138Dic53—42no—
139Dic53—81no—
140#3189D5311yescharacteristic
141#1901—532no—
142#1851—531no—
143#1508—532no—
144#1293—531no—
145C2×C53—21no—
146C2×C53—21no—
147C2×C53—42no—
148C2×C53#204411yescharacteristic
149C2×C53#204711yescharacteristic
150C2×C53#205011yescharacteristic
151C22×M16—531no—
152C2×C22⋊Q8C2×D5312yes—
153#484—22no—
154#467—21no—
155#467—21no—
156#451—22no—
157C2×C4⋊C8—1061no—
158C2×#101—1062no—
159#407—532no—
160C2×C22⋊C8C2×D5311yescharacteristic
161C4×C22⋊C4—531no—
162#343—1062no—
163#336—1061no—
164#328—1062no—
165#328—1062no—
166C53#1055911yescharacteristic, complemented, Sylow
167C22×Q8C22⋊D5312yes—
168C23×C4C22×D5311yescharacteristic
169C2×Q16—21no—
170C2×Q16—21no—
171C2×M16—1062no—
172C2×M16—1062no—
173C2×M16—2122no—
174C22×C8—1061no—
175C22×C8C22⋊D5311yescharacteristic
176C22×C8C22⋊D5311yescharacteristic
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4C22⋊D5312yes—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—1061no—
185C2×C42—1061no—
186#107—21no—
187#107—21no—
188C4⋊C8—2122no—
189#103—22no—
190#103—22no—
191#101—2124no—
192C22⋊C8—22no—
193C22⋊C8—1062no—
194C22⋊C8—1062no—
195#95—531no—
196#95—531no—
197C24#1171994711yescharacteristic
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—42no—
201C2×Q8—42no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—1061no—
207C22×C4—1061no—
208C22×C4D4×D5311yescharacteristic
209C22×C4#1171994712yes—
210C22×C4C2×C22⋊D5312yes—
211C22×C4C2×C22⋊D5311yescharacteristic
212Q16—81no—
213Q16—81no—
214M16—1064no—
215M16—2122no—
216M16—2122no—
217M16—2122no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—1062no—
223C2×C8—2122no—
224C2×C8—2121no—
225C2×C8—2121no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—2122no—
233C42—2121no—
234C42—2121no—
235C23—21no—
236C23—21no—
237C23—41no—
238C23—41no—
239C23#1313318811yescharacteristic
240C23#1313319812yes—
241Q8—42no—
242Q8—42no—
243Q8—82no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—82no—
254C2×C4—1062no—
255C2×C4—2121no—
256C2×C4—2121no—
257C2×C4—2121no—
258C2×C4—2121no—
259C2×C4—2121no—
260C2×C4#1313318611yescharacteristic
261C2×C4#1313318912yes—
262C2×C4#1313320411yescharacteristic
263C8—41no—
264C8—41no—
265C8—4242no—
266C8—2122no—
267C22—21no—
268C22—21no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—81no—
273C22—81no—
274C22#1681053711yescharacteristic
275C22#1681089511yescharacteristic
276C22#1681089511yescharacteristic
277C4—22no—
278C4—21no—
279C4—82no—
280C4—82no—
281C4—4241no—
282C4—2122no—
283C4#1681081011yescharacteristic
284C4#1681081311yescharacteristic
285C2—21no—
286C2—21no—
287C2—42no—
288C2#2027596311yescharacteristic
289C2#2027598011yescharacteristic
290C2#2027599911yescharacteristic
subgroupG/H
centerC22#16810537
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#3189C2
socleC22×C53#394
prime radicalC24×C53#41
hypercenter#3189D53
hypocenterC53#10559
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
166. C53#10559semidirect

Maximal subgroups

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

subgroupindex
#202751842
#202748972
#202732352
#202725922
C53×#31892
#1055953