Finite Group Databasebeta

← #28308105

Subgroups — #28308105

Order 13568.

368 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 82 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#20290952C212yesmaximal, complemented
2#20289663C212yesmaximal, complemented
3#20289648C212yesmaximal, complemented
4#20283420C211yesmaximal, characteristic
5D53⋊M32—22no—
6#16811520—22no—
7#16811519C2214yes—
8C2×#13133223C2211yescharacteristic
9C22×#11719912C411yescharacteristic
10#16811135C411yescharacteristic
11#16810702—22no—
12#16810696—22no—
13#16810650C2211yescharacteristic
14#16810591C411yescharacteristic
15#16810586C2211yescharacteristic
16C53×#405C411yescharacteristic
17#13133292—42no—
18#13133291—24no—
19D53⋊C16—42no—
20C2×#11719945C2×C411yescharacteristic
21C22×C4×D53C2×C411yescharacteristic
22#13133223—42no—
23C2×#11719917C2311yescharacteristic
24C2×#11719912—21no—
25C2×#11719912—21no—
26C2×#11719912—42no—
27C2×#11719912D412yes—
28C22×#88322C2×C411yescharacteristic
29C2×C8×D53D412yes—
30#13133201C812yes—
31#13133197—21no—
32#13133192—21no—
33#13133187C812yes—
34C4×D212—22no—
35C4⋊#88322—22no—
36#13133129—22no—
37#13133129—22no—
38C2×M16×C53C2×C411yescharacteristic
39C22×C8×C53C2×C411yescharacteristic
40#117×C53C2×C411yescharacteristic
41#97×C53—21no—
42C4×C8×C53—21no—
43C23×D53C2×C811yescharacteristic
44C22×C4×C53C22×C411yescharacteristic
45C2×C22⋊D53C2×C812yes—
46C22×Dic53C2×C811yescharacteristic
47#11719945—41no—
48#11719945—41no—
49C2×D212—22no—
50C2×D212M1611yescharacteristic
51C2×C4×D53—21no—
52C2×C4×D53—21no—
53C2×C4×D53—41no—
54C2×C4×D53—41no—
55C2×C4×D53C22⋊C412yes—
56C2×Dic106M1611yescharacteristic
57M16×C53—41no—
58C2×C8×C53—22no—
59C2×C8×C53—21no—
60C2×C8×C53C22⋊C411yescharacteristic
61C2×C8×C53C22⋊C411yescharacteristic
62C4⋊C4×C53—21no—
63C22⋊C4×C53C2×C812yes—
64C42×C53—21no—
65#11719921—22no—
66#11719921—22no—
67C4⋊Dic53—22no—
68#11719917—41no—
69C2×#88322—21no—
70C2×#88322—41no—
71C2×#88322C2×D412yes—
72#11719912—42no—
73#11719912—42no—
74#11719912—42no—
75C8×D53—22no—
76C8×D53—22no—
77#11719910—44no—
78C23×C53C22×C811yescharacteristic
79C22×D53—21no—
80C22×D53—41no—
81C22×D53—81no—
82C22×D53C22⋊C812yes—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53C2×C22⋊C411yescharacteristic
87C2×C4×C53C2×M1611yescharacteristic
88C22⋊D53—42no—
89C2×Dic53—21no—
90C2×Dic53—41no—
91C2×Dic53C22⋊C812yes—
92D212—41no—
93D212—82no—
94C4×D53—22no—
95C4×D53—22no—
96C4×D53—42no—
97C4×D53—41no—
98C4×D53—41no—
99Dic106—41no—
100C8×C53—21no—
101C8×C53—41no—
102C8×C53#9912yes—
103#88322—22no—
104#88322—41no—
105#11484—531nomaximal, Sylow
106C22×C53—21no—
107C22×C53—41no—
108C22×C53C2×C22⋊C811yescharacteristic
109C2×D53—22no—
110C2×D53—42no—
111C2×D53—41no—
112C2×D53—41no—
113C2×D53—41no—
114C2×D53—82no—
115C4×C53—21no—
116C4×C53—42no—
117C4×C53C2×#9911yescharacteristic
118C4×C53#41311yescharacteristic
119Dic53—22no—
120Dic53—41no—
121#2930—531no—
122#2115—532no—
123#1328—532no—
124#1320—532no—
125C2×C53—21no—
126C2×C53—41no—
127C2×C53#145911yescharacteristic
128D53—42no—
129D53—81no—
130C2×#131—531no—
131C22×M16—531no—
132#525—531no—
133C2×M32—1062no—
134#436—1062no—
135#435—1062no—
136#424—531no—
137#408—531no—
138#407—531no—
139#405#615711yescharacteristic
140#349—534no—
141#349—1062no—
142C53#1148411yescharacteristic, complemented, Sylow
143C2×#41—531no—
144C23×C4—531no—
145#131—2122no—
146C2×M16—531no—
147C2×M16—532no—
148C2×M16—1061no—
149C2×M16—1061no—
150C2×M16—2122no—
151C2×M16C2×#615711yescharacteristic
152C22×C8—532no—
153C22×C8—531no—
154C22×C8C2×#615711yescharacteristic
155#123—532no—
156C22⋊Q8—1061no—
157#121—1061no—
158#120—532no—
159C4×D4—1062no—
160#117C2×#615711yescharacteristic
161M32—1064no—
162M32—2122no—
163C2×C16—2122no—
164C4⋊C8—1062no—
165C22⋊C8—1062no—
166C22⋊C8—1062no—
167#97—21no—
168C4×C8—21no—
169C24—531no—
170#41—2121no—
171#41—2121no—
172C2×Q8—531no—
173C2×D4—532no—
174C2×D4—531no—
175C2×D4—1062no—
176C22×C4—532no—
177C22×C4—531no—
178C22×C4—1061no—
179C22×C4—1061no—
180C22×C4—2121no—
181C22×C4—2121no—
182C22×C4C22×#615711yescharacteristic
183M16—41no—
184M16—2122no—
185M16—2122no—
186M16—2121no—
187M16—2122no—
188C2×C8—22no—
189C2×C8—21no—
190C2×C8—532no—
191C2×C8—1062no—
192C2×C8—1062no—
193C2×C8—1061no—
194C2×C8—2121no—
195C2×C8C22⋊#615711yescharacteristic
196C2×C8C22⋊#615711yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—1062no—
199C22⋊C4—1062no—
200C22⋊C4—1062no—
201C22⋊C4D53⋊C812yes—
202C42—21no—
203C16—2124no—
204C23—532no—
205C23—1061no—
206C23—4241no—
207C23—2121no—
208C23C2×D53⋊C811yescharacteristic
209Q8—2121no—
210D4—4242no—
211D4—2121no—
212D4—2122no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—532no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—1061no—
220C2×C4—2122no—
221C2×C4—2121no—
222C2×C4—2121no—
223C2×C4—2121no—
224C2×C4C2×C22⋊#615711yescharacteristic
225C2×C4D53⋊M1611yescharacteristic
226C8—21no—
227C8—41no—
228C8—1062no—
229C8—2121no—
230C8#1313327612yes—
231C22—21no—
232C22—41no—
233C22—1062no—
234C22—4242no—
235C22—2121no—
236C22—2121no—
237C22—2122no—
238C22—2121no—
239C22D53⋊(C22⋊C8)11yescharacteristic
240C4—21no—
241C4—42no—
242C4—1062no—
243C4—2121no—
244C4C2×#1313327611yescharacteristic
245C4#1681160911yescharacteristic
246C2—21no—
247C2—41no—
248C2—4241no—
249C2—2122no—
250C2#2028995211yescharacteristic
subgroupG/H
centerC4C2×#13133276
commutatorC23×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#405C4
socleC2×C53#1459
prime radical#16811135C4
cosocleC2×#11719917C23
hypercenter#405#6157
hypocenterC53#11484
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ #405

Complements

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

NH ≅ G/Ntype
1. #20290952C2semidirect
2. #20289663C2semidirect
3. #20289648C2semidirect
142. C53#11484semidirect

Maximal subgroups

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

subgroupindex
#202909522
#202896632
#202896482
#202834202
#1148453