Finite Group Databasebeta

← #28299564

Subgroups — #28299564

Order 13568.

608 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 118 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#20290642C212yesmaximal, complemented
2#20289531C214yesmaximal, complemented
3#127×Dic53C211yesmaximal, characteristic
4#16811489—22no—
5#16811489—22no—
6#16811489C2214yes—
7C2×D4×Dic53—22no—
8C2×D4×Dic53C412yes—
9#16810526—22no—
10#16810526C2212yes—
11#16810493C411yescharacteristic
12C42×Dic53C2211yescharacteristic
13C2×#127×C53C411yescharacteristic
14C22×#88324—24no—
15C23×Dic53—22no—
16D4×Dic53—48no—
17D4×Dic53—44no—
18C2×C4⋊Dic53—22no—
19C2×C4⋊Dic53C2×C412yes—
20C2×C22⋊Dic53—22no—
21C2×C22⋊Dic53—24no—
22C2×C4×Dic53C2×C411yescharacteristic
23C2×C4×Dic53D412yes—
24C2×C4×Dic53D412yes—
25C2×C4×Dic53C2311yescharacteristic
26C22×D4×C53—22no—
27C22×D4×C53C2×C412yes—
28#127×C53—22no—
29#127×C53C814yescomplemented
30C2×C42×C53C2×C411yescharacteristic
31C24×C53—22no—
32C2×D4×C53—28no—
33C2×D4×C53—22no—
34C2×D4×C53—44no—
35C2×D4×C53—44no—
36C2×D4×C53—42no—
37C2×D4×C53—42no—
38C2×D4×C53C2×C814yes—
39C22×C4×C53C22⋊C412yes—
40C22×C4×C53C22×C411yescharacteristic
41C22×Dic53—22no—
42C22×Dic53—48no—
43C22×Dic53—44no—
44C22×Dic53C22⋊C411yescharacteristic
45C22×Dic53C22⋊C411yescharacteristic
46C22×Dic53C2×D412yes—
47C2×#88324—44no—
48C2×#88324—44no—
49C2×#88324—44no—
50C42×C53C2×C812yes—
51C42×C53M1612yes—
52C22⋊Dic53—44no—
53C22⋊Dic53—48no—
54C4⋊Dic53—44no—
55C4⋊Dic53—42no—
56C4×Dic53—22no—
57C4×Dic53—24no—
58C4×Dic53—42no—
59C4×Dic53—41no—
60C23×C53—24no—
61C23×C53—42no—
62C23×C53—42no—
63C23×C53—44no—
64C23×C53—44no—
65C23×C53C2×C22⋊C411yescharacteristic
66D4×C53—44no—
67D4×C53—44no—
68D4×C53—416no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53C22⋊C814yes—
74C2×C4×C53C22×C811yescharacteristic
75C2×C4×C53C2×M1611yescharacteristic
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—21no—
80C2×Dic53—22no—
81C2×Dic53—42no—
82C2×Dic53—44no—
83C2×Dic53—88no—
84C2×Dic53#10012yes—
85C2×Dic53AGL(1,ℤ/8)14yes—
86#88324—84no—
87#7603—531nomaximal, Sylow
88C22×C53—44no—
89C22×C53—48no—
90C22×C53—82no—
91C22×C53—82no—
92C22×C53—84no—
93C22×C53C2×C22⋊C811yescharacteristic
94C22×C53#41011yescharacteristic
95C22×C53C2×#10011yescharacteristic
96C22×C53#41812yes—
97C22×C53#42112yes—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—24no—
101Dic53—22no—
102Dic53—24no—
103Dic53—41no—
104Dic53—82no—
105C4×#127—531no—
106#1839—532no—
107#1270—534no—
108C2×C53—84no—
109C2×C53#146011yescharacteristic
110C2×C53#147812yes—
111C2×C53#151412yes—
112C2×C53#153611yescharacteristic
113C2×C53#155511yescharacteristic
114C2×#127#615711yescharacteristic
115C2×C4×D4—532no—
116C2×C4×D4—1062no—
117#390—532no—
118#390—1062no—
119#384—531no—
120C43—531no—
121#336—534no—
122#336—1062no—
123#336—1062no—
124C53#760311yescharacteristic, complemented, Sylow
125C22×D4—22no—
126C22×D4C2×#615712yes—
127C23×C4—1062no—
128C22×C8—1064no—
129#127—22no—
130#127#8832414yescomplemented
131C4×D4—2128no—
132C4×D4—2124no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—1062no—
135C2×C22⋊C4—1062no—
136C2×C22⋊C4—1064no—
137C2×C42—532no—
138C2×C42—531no—
139C2×C42—532no—
140C2×C42—531no—
141C2×C42C2×#615711yescharacteristic
142C24—22no—
143C2×D4—28no—
144C2×D4—22no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—44no—
148C2×D4—44no—
149C2×D4C2×#8832414yes—
150C22×C4—531no—
151C22×C4—532no—
152C22×C4—531no—
153C22×C4—1062no—
154C22×C4—2124no—
155C22×C4—2128no—
156C22×C4C22⋊#615712yes—
157C22×C4C22×#615711yescharacteristic
158C2×C8—2124no—
159C2×C8—2124no—
160C2×C8—2124no—
161C4⋊C4—2122no—
162C4⋊C4—2124no—
163C22⋊C4—2128no—
164C22⋊C4—2124no—
165C42—1062no—
166C42—1064no—
167C42—2121no—
168C42—2122no—
169C42C2×#8832412yes—
170C42#1171994012yes—
171C23—24no—
172C23—42no—
173C23—42no—
174C23—44no—
175C23—44no—
176C23C2×C22⋊#615711yescharacteristic
177D4—44no—
178D4—44no—
179D4—416no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—532no—
185C2×C4—534no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—1061no—
190C2×C4—1062no—
191C2×C4—4248no—
192C2×C4—2122no—
193C2×C4—2124no—
194C2×C4C22⋊#8832414yes—
195C2×C4C22×#8832411yescharacteristic
196C2×C4C2×#1171994011yescharacteristic
197C8—4244no—
198C22—44no—
199C22—48no—
200C22—84no—
201C22—82no—
202C22—82no—
203C22C2×C22⋊#8832411yescharacteristic
204C22#1681160311yescharacteristic
205C22D53⋊#10011yescharacteristic
206C22#1681161512yes—
207C22#1681162112yes—
208C4—21no—
209C4—21no—
210C4—24no—
211C4—1064no—
212C4—1062no—
213C4—4242no—
214C4—2121no—
215C2—84no—
216C2#2028995511yescharacteristic
217C2#2029001412yes—
218C2#2029013512yes—
219C2#2029020411yescharacteristic
220C2#2029024311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#127×C53C4
socleC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#127#6157
hypocenterC53#7603
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #20290642C2semidirect
2. #20289531C2semidirect
29. #127×C53C8semidirect
124. C53#7603semidirect
130. #127#88324semidirect

Maximal subgroups

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

subgroupindex
#202906422
#202895312
#127×Dic532
#760353