Finite Group Databasebeta

← #27288430

Subgroups — #27288430

Order 13568.

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

normal subgroups only

This group has 500 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#20275179C212yesmaximal, complemented
2#20274897C211yesmaximal, characteristic, complemented
3#20273231C212yesmaximal, complemented
4#20272592C211yesmaximal, characteristic, complemented
5C53×#3186C211yesmaximal, characteristic
6C22×#11719917C2211yescharacteristic
7C2×#13133146—21no—
8C2×#13133134—22no—
9#16810587C2212yes—
10C4×C22⋊Dic53C2211yescharacteristic
11#16810392—22no—
12#16810377—21no—
13#16810357—22no—
14#16810357—22no—
15C2×#121×C53C2212yes—
16C53×#480—22no—
17C53×#466—21no—
18C53×#466—21no—
19C53×#447—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#13133134—44no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133124D411yescharacteristic
32#13133124D411yescharacteristic
33C22×D4×C53—22no—
34C22×D4×C53D412yes—
35C23×C4×C53C2311yescharacteristic
36C2×D8×C53—21no—
37C2×D8×C53—21no—
38C22×C8×C53D411yescharacteristic
39C22×C8×C53D411yescharacteristic
40#121×C53—22no—
41#121×C53—22no—
42#121×C53—22no—
43#121×C53—22no—
44C2×C4⋊C4×C53D412yes—
45C2×C22⋊C4×C53—22no—
46#107×C53—21no—
47#107×C53—21no—
48#102×C53—22no—
49#102×C53—22no—
50C22⋊C8×C53—22no—
51C24×C53—22no—
52C24×C53#4111yescharacteristic
53C2×D4×C53—22no—
54C2×D4×C53—22no—
55C2×D4×C53—42no—
56C2×D4×C53—42no—
57C2×D4×C53—42no—
58C2×D4×C53—42no—
59C2×D4×C53—82no—
60C2×D4×C53—82no—
61C22×C4×C53—22no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53C2×D412yes—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53#4112yes—
68C22×Dic53—21no—
69C22×Dic53—21no—
70D8×C53—81no—
71D8×C53—81no—
72C2×C8×C53—21no—
73C2×C8×C53—21no—
74C2×C8×C53—41no—
75C2×C8×C53—41no—
76C4⋊C4×C53—22no—
77C4⋊C4×C53—22no—
78C22⋊C4×C53—42no—
79C22⋊C4×C53—42no—
80C22⋊Dic53—42no—
81C4×Dic53—41no—
82C4×Dic53—41no—
83#11719917—24no—
84#11719917—42no—
85#11719917—42no—
86#11719917—42no—
87C2×#88322—22no—
88C2×#88322—41no—
89C2×#88322—41no—
90C2×#88322—42no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53—44no—
96C23×C53—42no—
97C23×C53—82no—
98C23×C53—82no—
99C23×C53#12111yescharacteristic
100C23×C53#12312yes—
101D4×C53—42no—
102D4×C53—42no—
103D4×C53—82no—
104D4×C53—84no—
105D4×C53—82no—
106D4×C53—82no—
107C2×C4×C53—22no—
108C2×C4×C53—21no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—82no—
115C2×C4×C53#12011yescharacteristic
116C2×C4×C53#12112yes—
117C2×C4×C53#12411yescharacteristic
118C2×Dic53—22no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C8×C53—41no—
125C8×C53—41no—
126#88322—42no—
127#88322—82no—
128#10557—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—22no—
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—81no—
135C22×C53—81no—
136C22×C53—82no—
137C22×C53—82no—
138C22×C53—82no—
139C22×C53—164no—
140C22×C53#39411yescharacteristic
141C22×C53#47211yescharacteristic
142C22×C53#47211yescharacteristic
143C4×C53—22no—
144C4×C53—21no—
145C4×C53—82no—
146C4×C53B(4,𝔽2)11yescharacteristic
147C4×C53#45811yescharacteristic
148Dic53—42no—
149Dic53—81no—
150#3186D5311yescharacteristic
151#1900—532no—
152#1851—531no—
153#1507—532no—
154#1293—531no—
155C2×C53—21no—
156C2×C53—21no—
157C2×C53—42no—
158C2×C53—162no—
159C2×C53#204411yescharacteristic
160C2×C53#204511yescharacteristic
161C2×C53#205011yescharacteristic
162C22×M16—531no—
163C2×#121C2×D5312yes—
164#480—22no—
165#466—21no—
166#466—21no—
167#447—22no—
168C2×C4⋊C8—1061no—
169C2×#100—1062no—
170#407—532no—
171C2×C22⋊C8C2×D5311yescharacteristic
172C4×C22⋊C4—531no—
173#343—1062no—
174#336—1061no—
175#327—1062no—
176#327—1062no—
177C53#1055711yescharacteristic, complemented, Sylow
178C22×D4—22no—
179C22×D4C22⋊D5312yes—
180C23×C4C22×D5311yescharacteristic
181C2×D8—21no—
182C2×D8—21no—
183C2×M16—1062no—
184C2×M16—1062no—
185C2×M16—2122no—
186C22×C8—1061no—
187C22×C8C22⋊D5311yescharacteristic
188C22×C8C22⋊D5311yescharacteristic
189#121—22no—
190#121—22no—
191#121—22no—
192#121—22no—
193C2×C4⋊C4C22⋊D5312yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—1061no—
196C2×C42—1061no—
197#107—21no—
198#107—21no—
199C4⋊C8—2122no—
200#102—22no—
201#102—22no—
202#100—2124no—
203C22⋊C8—22no—
204C22⋊C8—1062no—
205C22⋊C8—1062no—
206#95—531no—
207#95—531no—
208C24—22no—
209C24#1171994711yescharacteristic
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—82no—
217C2×D4—82no—
218C22×C4—22no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—1061no—
222C22×C4—1061no—
223C22×C4D4×D5311yescharacteristic
224C22×C4#1171994712yes—
225C22×C4C2×C22⋊D5312yes—
226C22×C4C2×C22⋊D5311yescharacteristic
227D8—81no—
228D8—81no—
229M16—1064no—
230M16—2122no—
231M16—2122no—
232M16—2122no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—1062no—
238C2×C8—2122no—
239C2×C8—2121no—
240C2×C8—2121no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—2122no—
246C42—2121no—
247C42—2121no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—44no—
253C23—42no—
254C23—82no—
255C23—82no—
256C23#1313318811yescharacteristic
257C23#1313319812yes—
258D4—42no—
259D4—42no—
260D4—82no—
261D4—84no—
262D4—82no—
263D4—82no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—82no—
272C2×C4—1062no—
273C2×C4—2121no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4#1313318611yescharacteristic
279C2×C4#1313318912yes—
280C2×C4#1313320411yescharacteristic
281C8—41no—
282C8—41no—
283C8—4242no—
284C8—2122no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—81no—
291C22—81no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—164no—
296C22#1681053711yescharacteristic
297C22#1681089011yescharacteristic
298C22#1681089011yescharacteristic
299C4—22no—
300C4—21no—
301C4—82no—
302C4—4241no—
303C4—2122no—
304C4#1681081011yescharacteristic
305C4#1681081311yescharacteristic
306C2—21no—
307C2—21no—
308C2—42no—
309C2—162no—
310C2#2027596311yescharacteristic
311C2#2027596811yescharacteristic
312C2#2027599911yescharacteristic
subgroupG/H
centerC22#16810537
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#3186C2
socleC22×C53#394
hypercenter#3186D53
hypocenterC53#10557
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3186

Complements

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

NH ≅ G/Ntype
1. #20275179C2semidirect
2. #20274897C2semidirect
3. #20273231C2semidirect
4. #20272592C2semidirect
177. C53#10557semidirect

Maximal subgroups

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

subgroupindex
#202751792
#202748972
#202732312
#202725922
C53×#31862
#1055753