Finite Group Databasebeta

← #28299733

Subgroups — #28299733

Order 13568.

624 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 114 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#20290631C212yesmaximal, complemented
2#20289532C214yesmaximal, complemented
3#20277277C211yesmaximal, characteristic
4C2×#13133284—22no—
5#16811489—22no—
6#16811489C2214yes—
7C2×#13133214C411yescharacteristic
8C2×#13133179—22no—
9C2×#13133179C412yes—
10#16810524—22no—
11#16810524C2212yes—
12C42×Dic53C2211yescharacteristic
13C53×#384C411yescharacteristic
14C22×#88324—24no—
15#13133284—44no—
16C22×D212—22no—
17C22×D212C2×C412yes—
18C22×C4×D53—22no—
19#13133214—24no—
20#13133179—44no—
21#13133179—44no—
22#13133179—44no—
23C2×#11719921—22no—
24C2×#11719921—24no—
25C2×C4×Dic53C2×C411yescharacteristic
26C2×C4×Dic53D412yes—
27C2×C4×Dic53D412yes—
28C2×C4×Dic53C2311yescharacteristic
29C2×C4⋊C4×C53—22no—
30C2×C4⋊C4×C53C2×C412yes—
31C2×C42×C53C2×C411yescharacteristic
32C23×D53—22no—
33C22×C4×C53—22no—
34C22×C4×C53C22⋊C412yes—
35C22×C4×C53C22×C411yescharacteristic
36C22×Dic53C22⋊C411yescharacteristic
37C22×Dic53C22⋊C411yescharacteristic
38C22×Dic53C2×D412yes—
39C2×D212—28no—
40C2×D212—22no—
41C2×D212—44no—
42C2×D212—42no—
43C2×D212—42no—
44C2×D212—44no—
45C2×D212M1614yes—
46C2×C4×D53—44no—
47C2×C4×D53—48no—
48C2×#88324—44no—
49C2×#88324—44no—
50C2×#88324—44no—
51C4⋊C4×C53—42no—
52C4⋊C4×C53—44no—
53C42×C53—22no—
54#11719921—44no—
55#11719921—44no—
56#11719921—44no—
57C4×Dic53—22no—
58C4×Dic53—42no—
59C4×Dic53—41no—
60C4×Dic53D814yes—
61C4×Dic53SD1614yes—
62C23×C53C2×C22⋊C411yescharacteristic
63C22×D53—24no—
64C22×D53—42no—
65C22×D53—42no—
66C22×D53—44no—
67C22×D53—44no—
68C2×C4×C53—22no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—22no—
72C2×C4×C53—42no—
73C2×C4×C53—44no—
74C2×C4×C53#10214yes—
75C2×C4×C53C2×M1612yes—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—21no—
79C2×Dic53—22no—
80C2×Dic53#10012yes—
81C2×Dic53#10214yes—
82C2×Dic53C2×D812yes—
83C2×Dic53C2×SD1612yes—
84D212—44no—
85D212—44no—
86D212—416no—
87C4×D53—88no—
88#88324—84no—
89#7651—531nomaximal, Sylow
90C22×C53#40711yescharacteristic
91C22×C53#41011yescharacteristic
92C22×C53C2×#10011yescharacteristic
93C22×C53C2×#10212yes—
94C22×C53#42112yes—
95C2×D53—44no—
96C2×D53—48no—
97C2×D53—82no—
98C2×D53—82no—
99C2×D53—84no—
100C4×C53—22no—
101C4×C53—24no—
102C4×C53—82no—
103Dic53—22no—
104Dic53—22no—
105Dic53—41no—
106Dic53#33114yes—
107C4×#127—531no—
108#1831—532no—
109#1270—534no—
110C2×C53#146611yescharacteristic
111C2×C53#148612yes—
112C2×C53#151712yes—
113C2×C53#153611yescharacteristic
114C2×C53C2×#33111yescharacteristic
115D53—84no—
116C2×#127—531no—
117C2×C4×D4—532no—
118C2×C4×D4—1062no—
119C2×C4⋊C8—1062no—
120#390—532no—
121#390—1062no—
122#384#615711yescharacteristic
123C43—531no—
124#336—534no—
125#336—1062no—
126C53#765111yescharacteristic, complemented, Sylow
127C22×D4—532no—
128C22×D4—1062no—
129C23×C4—1062no—
130C22×C8—1064no—
131#127—1064no—
132C4×D4—2124no—
133C4×D4—2124no—
134C4×D4—2124no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4C2×#615712yes—
137C2×C22⋊C4—1062no—
138C2×C22⋊C4—1064no—
139C2×C42—532no—
140C2×C42—531no—
141C2×C42—532no—
142C2×C42—531no—
143C2×C42C2×#615711yescharacteristic
144C4⋊C8—2124no—
145C24—1062no—
146C2×D4—534no—
147C2×D4—1068no—
148C2×D4—1062no—
149C2×D4—2122no—
150C2×D4—2122no—
151C2×D4—2124no—
152C2×D4—2124no—
153C22×C4—22no—
154C22×C4—531no—
155C22×C4—532no—
156C22×C4—531no—
157C22×C4—2128no—
158C22×C4—2124no—
159C22×C4C22⋊#615712yes—
160C22×C4C22×#615711yescharacteristic
161C2×C8—2124no—
162C2×C8—2124no—
163C2×C8—2124no—
164C4⋊C4—42no—
165C4⋊C4—44no—
166C22⋊C4—2124no—
167C22⋊C4—2124no—
168C22⋊C4—2124no—
169C42—22no—
170C42—534no—
171C42—534no—
172C42—1062no—
173C42—2121no—
174C42—2122no—
175C23—1064no—
176C23—2122no—
177C23—2122no—
178C23—2124no—
179C23—2124no—
180C23C2×C22⋊#615711yescharacteristic
181D4—2124no—
182D4—2124no—
183D4—21216no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—532no—
191C2×C4—534no—
192C2×C4—532no—
193C2×C4—532no—
194C2×C4—1062no—
195C2×C4—1062no—
196C2×C4—1061no—
197C2×C4—1062no—
198C2×C4—4248no—
199C2×C4#1313327914yes—
200C2×C4D53⋊M1612yes—
201C8—4244no—
202C22—4244no—
203C22—4242no—
204C22—4242no—
205C22—2124no—
206C22—2128no—
207C22#1681159511yescharacteristic
208C22#1681160411yescharacteristic
209C22D53⋊#10011yescharacteristic
210C22C2×#1313327912yes—
211C22#1681162212yes—
212C4—22no—
213C4—24no—
214C4—82no—
215C4—534no—
216C4—1062no—
217C4—1062no—
218C4—2121no—
219C2—4244no—
220C2#2028997611yescharacteristic
221C2#2029004212yes—
222C2#2029014612yes—
223C2#2029020511yescharacteristic
224C2D53⋊#33111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133214C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#7651
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
1. #20290631C2semidirect
2. #20289532C2semidirect
126. C53#7651semidirect

Maximal subgroups

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

subgroupindex
#202906312
#202895322
#202772772
#765153