Finite Group Databasebeta

← #27303562

Subgroups — #27303562

Order 13568.

444 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C2×#16810422C212yesmaximal
2#20275109C212yesmaximal
3#20274929C211yesmaximal, characteristic
4#20272842C211yesmaximal, characteristic
5C53×#2734C211yesmaximal, characteristic
6#16811101C2211yescharacteristic
7#16810663—21no—
8#16810632—22no—
9C2×#13133136C2212yes—
10C2×#13133136C2211yescharacteristic
11#16810422—24no—
12#16810383—22no—
13#16810383—21no—
14C2×C4⋊Q8×C53C2212yes—
15C53×#398—21no—
16C53×#395—22no—
17C53×#391C2211yescharacteristic
18C53×#389—21no—
19C53×#387—22no—
20C2×#11719917—21no—
21C2×#11719917—22no—
22Q8×Dic53—41no—
23Q8×Dic53—41no—
24#13133176—22no—
25#13133176—21no—
26#13133150—41no—
27#13133148—41no—
28#13133141—42no—
29#13133141—42no—
30#13133136—22no—
31#13133136—21no—
32#13133136—22no—
33#13133136D412yes—
34C22×Q8×C53D412yes—
35C22×Q8×C53C2311yescharacteristic
36C4⋊Q8×C53—22no—
37C4⋊Q8×C53—22no—
38C4⋊Q8×C53—22no—
39C4⋊Q8×C53—22no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C42×C53D412yes—
47C2×C42×C53D411yescharacteristic
48#95×C53—21no—
49#95×C53—22no—
50#95×C53—22no—
51#95×C53—22no—
52C2×Q8×C53—22no—
53C2×Q8×C53—22no—
54C2×Q8×C53—41no—
55C2×Q8×C53—41no—
56C2×Q8×C53—42no—
57C2×Q8×C53—42no—
58C2×Q8×C53C2×D411yescharacteristic
59C2×Q8×C53#4111yescharacteristic
60C2×Q8×C53#4112yes—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53C2×D412yes—
66C22×C4×C53C2×D411yescharacteristic
67C4⋊C4×C53—24no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—42no—
73C4⋊C4×C53—81no—
74C4⋊C4×C53—81no—
75C42×C53—22no—
76C42×C53—22no—
77C42×C53—41no—
78C22⋊Dic53—41no—
79C4⋊Dic53—22no—
80C4⋊Dic53—22no—
81C4⋊Dic53—21no—
82C4⋊Dic53—21no—
83C4×Dic53—42no—
84C4×Dic53—41no—
85#11719917—22no—
86#11719917—41no—
87#11719917—42no—
88#11719917—42no—
89C2×#88322—41no—
90C2×#88322—42no—
91C23×C53#12011yescharacteristic
92Q8×C53—41no—
93Q8×C53—41no—
94Q8×C53—42no—
95Q8×C53—44no—
96Q8×C53—82no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—24no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—41no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—41no—
111C2×C4×C53—42no—
112C2×C4×C53#12111yescharacteristic
113C2×C4×C53#12112yes—
114C2×C4×C53#12312yes—
115C2×C4×C53#12411yescharacteristic
116C2×Dic53—41no—
117C2×Dic53—41no—
118#88322—42no—
119#88322—82no—
120#13428—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53#39411yescharacteristic
124C22×C53B(4,𝔽2)11yescharacteristic
125C22×C53#45811yescharacteristic
126C4×C53—21no—
127C4×C53—22no—
128C4×C53—22no—
129C4×C53—21no—
130C4×C53—44no—
131C4×C53—42no—
132C4×C53—81no—
133C4×C53—82no—
134Dic53—81no—
135Dic53—81no—
136#2734D5311yescharacteristic
137C2×#356—532no—
138#1886—532no—
139#1857—531no—
140#1394—531no—
141C2×C53—21no—
142C2×C53—21no—
143C2×C53#204411yescharacteristic
144C2×C53#220311yescharacteristic
145C2×C53#220511yescharacteristic
146C2×C4⋊Q8C2×D5312yes—
147#519—531no—
148#428—1061no—
149#419—1062no—
150C2×#101—531no—
151C2×#101—532no—
152#398—21no—
153#395—22no—
154#391C2×D5311yescharacteristic
155#389—21no—
156#387—22no—
157#356—1064no—
158#339—1061no—
159#339—1062no—
160C53#1342811yescharacteristic, complemented, Sylow
161C22×Q8C22⋊D5312yes—
162C22×Q8C22×D5311yescharacteristic
163C2×M16—1061no—
164C2×M16—1062no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4×Q8—2121no—
170C4×Q8—2121no—
171#117—1062no—
172#117—1061no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4C22⋊D5311yescharacteristic
179C2×C42C22⋊D5312yes—
180C2×C42C22⋊D5311yescharacteristic
181#107—2121no—
182#106—2121no—
183#103—2122no—
184#103—2122no—
185#101—532no—
186#101—1061no—
187#101—1062no—
188#101—1062no—
189#95—21no—
190#95—22no—
191#95—22no—
192#95—22no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8D4×D5311yescharacteristic
200C2×Q8#1171994711yescharacteristic
201C2×Q8#1171994712yes—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4C2×C22⋊D5311yescharacteristic
207C22×C4C2×C22⋊D5312yes—
208M16—1062no—
209M16—2122no—
210M16—2122no—
211M16—2121no—
212C2×C8—2122no—
213C2×C8—2121no—
214C4⋊C4—24no—
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—1062no—
223C4⋊C4—1062no—
224C4⋊C4—1061no—
225C4⋊C4—1061no—
226C22⋊C4—2121no—
227C42—22no—
228C42—22no—
229C42—41no—
230C42—2121no—
231C42—2122no—
232C23#1313318611yescharacteristic
233Q8—41no—
234Q8—42no—
235Q8—41no—
236Q8—44no—
237Q8—82no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—2121no—
254C2×C4—2121no—
255C2×C4#1313318811yescharacteristic
256C2×C4#1313318912yes—
257C2×C4#1313319812yes—
258C2×C4#1313320411yescharacteristic
259C8—4242no—
260C8—2122no—
261C22—21no—
262C22—21no—
263C22#1681053711yescharacteristic
264C22#1681081011yescharacteristic
265C22#1681081311yescharacteristic
266C4—22no—
267C4—21no—
268C4—22no—
269C4—21no—
270C4—44no—
271C4—42no—
272C4—81no—
273C4—82no—
274C4—4241no—
275C4—4241no—
276C2—21no—
277C2—21no—
278C2#2027596311yescharacteristic
279C2#2027673611yescharacteristic
280C2#2027674511yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC22×Q8×C53C23
FrattiniC22×Q8C22×D53
FittingC53×#2734C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2734D53
hypocenterC53#13428
LCSG ▷ C22×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2734

Complements

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

NH ≅ G/Ntype
160. C53#13428semidirect

Maximal subgroups

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

subgroupindex
C2×#168104222
#202751092
#202749292
#202728422
C53×#27342
#1342853