Finite Group Databasebeta

← #27281505

Subgroups — #27281505

Order 13568.

570 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 115 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#20283756C211yesmaximal, characteristic
2#20279668C211yesmaximal, characteristic
3#20273401C212yesmaximal, complemented
4#20272616C212yesmaximal, complemented
5C53×#1800C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7C2×#13133196—22no—
8#16810701—24no—
9#16810676C2212yes—
10#16810587—22no—
11#16810559C414yes—
12#16810526—21no—
13#16810492—21no—
14C4×C4⋊Dic53C2211yescharacteristic
15#16810368—24no—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#423C2212yes—
18C2×#100×C53—21no—
19C53×#343—21no—
20C22×Dic106—21no—
21C2×#11719917—24no—
22#13133217—22no—
23#13133217—22no—
24#13133196—48no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53C2×C412yes—
28C2×#11719919—24no—
29C2×C22⋊Dic53—24no—
30C2×C4×Dic53—21no—
31C4⋊#88322D414yes—
32C22⋊#88322—24no—
33C22⋊#88322—22no—
34C4×#88322—22no—
35#13133124C2×C412yes—
36C22×D4×C53D411yescharacteristic
37C23×C4×C53D412yes—
38C2×M16×C53—22no—
39C4×D4×C53—22no—
40C4×D4×C53—22no—
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53C2×C412yes—
43C2×C42×C53C2311yescharacteristic
44C4⋊C8×C53—22no—
45C4⋊C8×C53—22no—
46#100×C53—42no—
47C24×C53C22⋊C412yes—
48C2×D4×C53—22no—
49C2×D4×C53—21no—
50C2×D4×C53—22no—
51C2×D4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—44no—
55C22×C4×C53C22⋊C412yes—
56C22×C4×C53C22×C411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C22×Dic53—22no—
60C22×Dic53—22no—
61C2×Dic106—44no—
62C2×Dic106—41no—
63C2×Dic106—41no—
64M16×C53—42no—
65M16×C53—42no—
66C2×C8×C53—42no—
67C4⋊C4×C53—22no—
68C22⋊C4×C53—24no—
69C42×C53C2×D412yes—
70C42×C53#4112yes—
71C22⋊Dic53—48no—
72C4⋊Dic53—24no—
73C4⋊Dic53—24no—
74C4⋊Dic53—42no—
75#11719919—48no—
76C4×Dic53—41no—
77C4×Dic53—41no—
78#11719917—84no—
79C2×#88322—24no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53—42no—
84C23×C53—42no—
85C23×C53C2×C22⋊C411yescharacteristic
86D4×C53—42no—
87D4×C53—42no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—41no—
91C2×C4×C53—44no—
92C2×C4×C53—42no—
93C2×C4×C53—82no—
94C2×C4×C53C4×D412yes—
95C2×C4×C53#12011yescharacteristic
96C2×C4×C53#12112yes—
97C2×C4×C53#12311yescharacteristic
98C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
99C2×C4×C53#13712yes—
100C2×C4×C53#13711yescharacteristic
101C2×Dic53—24no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—44no—
105C2×Dic53—42no—
106Dic106—82no—
107Dic106—82no—
108C8×C53—82no—
109#88322—44no—
110#9403—531nomaximal, Sylow
111C22×C53—21no—
112C22×C53—21no—
113C22×C53—44no—
114C22×C53—42no—
115C22×C53—42no—
116C22×C53—42no—
117C22×C53—82no—
118C22×C53#38611yescharacteristic
119C22×C53#41711yescharacteristic
120C22×C53#41911yescharacteristic
121C4×C53—21no—
122C4×C53—22no—
123C4×C53—42no—
124C4×C53#45312yes—
125C4×C53#46812yes—
126C4×C53#47012yes—
127Dic53—44no—
128Dic53—82no—
129#2975—531no—
130#2542—531no—
131#1800D5311yescharacteristic
132#1549—532no—
133#1303—532no—
134C2×C53—21no—
135C2×C53—21no—
136C2×C53—82no—
137C2×C53#188711yescharacteristic
138C2×C53#189411yescharacteristic
139C2×C53#191511yescharacteristic
140C2×C4⋊Q8—531no—
141C2×C22⋊Q8—1062no—
142C2×C4×D4C2×D5311yescharacteristic
143#436—1064no—
144#431—532no—
145#423C2×D5312yes—
146C2×#100—21no—
147#407—1062no—
148#399—534no—
149#390—1061no—
150#384—1061no—
151C4×C4⋊C4—531no—
152#343—21no—
153#332—1064no—
154C53#940311yescharacteristic, complemented, Sylow
155C22×Q8—1061no—
156C22×D4C22⋊D5311yescharacteristic
157C23×C4D21212yes—
158C2×M16—22no—
159C2×M16—1064no—
160C4⋊Q8—1062no—
161C4⋊Q8—1062no—
162C22⋊Q8—2128no—
163C4×D4—22no—
164C4×D4—22no—
165C2×C4⋊C4—532no—
166C2×C4⋊C4—1061no—
167C2×C4⋊C4—1061no—
168C2×C4⋊C4—1064no—
169C2×C4⋊C4C22⋊D5311yescharacteristic
170C2×C22⋊C4—1064no—
171C2×C22⋊C4C4×D5312yes—
172C2×C42—1061no—
173C2×C42C22×D5311yescharacteristic
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—534no—
177#100—42no—
178C22⋊C8—1064no—
179C22⋊C8—1062no—
180C4×C8—1062no—
181#95—532no—
182C24#1171992112yes—
183C2×Q8—2124no—
184C2×Q8—2121no—
185C2×Q8—2121no—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—44no—
193C22×C4—1062no—
194C22×C4—1062no—
195C22×C4#1171992112yes—
196C22×C4C2×C4×D5311yescharacteristic
197C22×C4C2×D21211yescharacteristic
198C22×C4C2×C22⋊D5311yescharacteristic
199M16—42no—
200M16—42no—
201M16—4244no—
202C2×C8—42no—
203C2×C8—1064no—
204C4⋊C4—22no—
205C4⋊C4—1064no—
206C4⋊C4—1064no—
207C4⋊C4—2128no—
208C4⋊C4—2122no—
209C22⋊C4—24no—
210C22⋊C4—2128no—
211C42—2121no—
212C42—2121no—
213C42D4×D5312yes—
214C42#1171994712yes—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—42no—
219C23—42no—
220C23C2×#1171992111yescharacteristic
221Q8—4242no—
222Q8—4242no—
223D4—42no—
224D4—42no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—82no—
231C2×C4—1064no—
232C2×C4—2124no—
233C2×C4—2122no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4#1313317712yes—
237C2×C4#1313318711yescharacteristic
238C2×C4#1313318912yes—
239C2×C4#1313320111yescharacteristic
240C2×C4#1313324311yescharacteristic
241C2×C4#1313324712yes—
242C2×C4#1313324811yescharacteristic
243C8—82no—
244C8—2124no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—82no—
252C22#1681050011yescharacteristic
253C22#1681062211yescharacteristic
254C22#1681063111yescharacteristic
255C4—22no—
256C4—21no—
257C4—42no—
258C4—4242no—
259C4—2124no—
260C4#1681079312yes—
261C4#1681086712yes—
262C4#1681087912yes—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2027511111yescharacteristic
267C2#2027514611yescharacteristic
268C2#2027526511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1800C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#1800D53
hypocenterC53#9403
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #20273401C2semidirect
4. #20272616C2semidirect
154. C53#9403semidirect

Maximal subgroups

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

subgroupindex
#202837562
#202796682
#202734012
#202726162
C53×#18002
#940353