Finite Group Databasebeta

← #27304443

Subgroups — #27304443

Order 13568.

388 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 79 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×#16810416C212yesmaximal
2#20272886C214yesmaximal, complemented
3C53×#3138C211yesmaximal, characteristic
4C2×#13133132—22no—
5#16810416—22no—
6#16810416C2214yes—
7C53×#533C411yescharacteristic, complemented
8C2×#127×C53C2211yescharacteristic
9C53×#490C412yescomplemented
10C53×#490C2212yes—
11C53×#486—22no—
12C53×#418—22no—
13C4×M16×C53C411yescharacteristic, complemented
14C2×C22⋊Dic53—42no—
15#13133132—24no—
16#13133132—42no—
17C22×D4×C53—41no—
18C22×D4×C53D412yes—
19C2×M16×C53—21no—
20C4⋊Q8×C53—21no—
21#127×C53—41no—
22#127×C53C2×C411yescharacteristic
23#127×C53D412yes—
24#127×C53C2311yescharacteristic
25#125×C53C2×C412yescomplemented
26C22⋊Q8×C53—41no—
27C4×Q8×C53—41no—
28#117×C53—21no—
29C2×C42×C53C2×C411yescharacteristic
30#102×C53—24no—
31#102×C53—24no—
32#97×C53C2×C412yescomplemented
33C4×C8×C53—21no—
34C24×C53—22no—
35C2×Q8×C53—41no—
36C2×D4×C53—22no—
37C2×D4×C53—42no—
38C2×D4×C53—42no—
39C2×D4×C53—41no—
40C2×D4×C53—42no—
41C2×D4×C53—41no—
42C2×D4×C53—84no—
43C2×D4×C53C22⋊C412yes—
44C2×D4×C53C2×D412yes—
45C22×C4×C53—21no—
46C22×C4×C53C22⋊C411yescharacteristic
47C22×Dic53—82no—
48M16×C53—42no—
49C2×C8×C53—22no—
50C4⋊C4×C53—22no—
51C4⋊C4×C53—41no—
52C4⋊C4×C53—42no—
53C22⋊C4×C53—41no—
54C42×C53—21no—
55C42×C53—41no—
56C42×C53C22⋊C411yescharacteristic
57C42×C53C22×C411yescharacteristic
58C22⋊Dic53—44no—
59C22⋊Dic53—82no—
60C23×C53—22no—
61C23×C53—22no—
62C23×C53—42no—
63C23×C53—42no—
64C23×C53—84no—
65C23×C53#9911yescharacteristic
66Q8×C53—82no—
67D4×C53—42no—
68D4×C53—42no—
69D4×C53—82no—
70D4×C53—84no—
71D4×C53—82no—
72D4×C53—82no—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—42no—
77C2×C4×C53—42no—
78C2×C4×C53#9911yescharacteristic
79C2×C4×C53C2×C22⋊C411yescharacteristic
80C2×Dic53—84no—
81C2×Dic53—164no—
82C8×C53—42no—
83#13595—531nomaximal, Sylow
84C22×C53—41no—
85C22×C53—42no—
86C22×C53—42no—
87C22×C53—82no—
88C22×C53—82no—
89C22×C53—84no—
90C22×C53—162no—
91C22×C53#35412yes—
92C22×C53C2×#9911yescharacteristic
93C4×C53—22no—
94C4×C53—21no—
95C4×C53—21no—
96C4×C53—41no—
97C4×C53—82no—
98Dic53—164no—
99#3138D5311yescharacteristic
100C2×#353—532no—
101#1412—534no—
102C2×C53—21no—
103C2×C53—22no—
104C2×C53—82no—
105C2×C53—82no—
106C2×C53C2×#35411yescharacteristic
107#533Dic5311yescharacteristic, complemented
108C2×#127C2×D5311yescharacteristic
109#490Dic5312yescomplemented
110#490C2×D5312yes—
111#486—22no—
112#418—22no—
113C2×#99—1062no—
114C4×M16Dic5311yescharacteristic, complemented
115#353—534no—
116#353—1062no—
117C53#1359511yescharacteristic, complemented, Sylow
118C22×D4—41no—
119C22×D4C22⋊D5312yes—
120C2×M16—21no—
121C4⋊Q8—21no—
122#127—41no—
123#127C2×Dic5311yescharacteristic
124#127C22⋊D5312yes—
125#127C22×D5311yescharacteristic
126#125C2×Dic5312yescomplemented
127C22⋊Q8—41no—
128C4×Q8—41no—
129#117—21no—
130C2×C22⋊C4—2122no—
131C2×C42C2×Dic5311yescharacteristic
132#102—24no—
133#102—24no—
134#99—1064no—
135#99—2122no—
136#97C2×Dic5312yescomplemented
137C4×C8—21no—
138C24—22no—
139C2×Q8—41no—
140C2×D4—22no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—41no—
144C2×D4—42no—
145C2×D4—41no—
146C2×D4—84no—
147C2×D4C22⋊Dic5312yes—
148C2×D4C2×C22⋊D5312yes—
149C22×C4—21no—
150C22×C4—4242no—
151C22×C4C22⋊Dic5311yescharacteristic
152M16—42no—
153C2×C8—22no—
154C4⋊C4—22no—
155C4⋊C4—42no—
156C4⋊C4—41no—
157C22⋊C4—41no—
158C22⋊C4—4242no—
159C22⋊C4—2124no—
160C42—21no—
161C42—41no—
162C42C22⋊Dic5311yescharacteristic
163C42C22×Dic5311yescharacteristic
164C23—22no—
165C23—22no—
166C23—42no—
167C23—42no—
168C23—84no—
169C23#1313313211yescharacteristic
170Q8—82no—
171D4—42no—
172D4—42no—
173D4—82no—
174D4—84no—
175D4—82no—
176D4—82no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—8484no—
183C2×C4—4244no—
184C2×C4#1313313211yescharacteristic
185C2×C4C2×C22⋊Dic5311yescharacteristic
186C8—42no—
187C22—41no—
188C22—42no—
189C22—42no—
190C22—82no—
191C22—82no—
192C22—84no—
193C22—162no—
194C22#1681041812yes—
195C22C2×#1313313211yescharacteristic
196C4—22no—
197C4—21no—
198C4—21no—
199C4—41no—
200C4—82no—
201C4—8484no—
202C2—21no—
203C2—22no—
204C2—82no—
205C2—82no—
206C2C2×#1681041811yescharacteristic
subgroupG/H
centerC2C2×#16810418
commutatorC42×C53C22×C4
Frattini#127C22×D53
FittingC53×#3138C2
socleC2×C53C2×#354
prime radicalC2×#127×C53C22
cosocle#127×C53C23
hypercenter#3138D53
hypocenterC53#13595
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ #3138

Complements

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

NH ≅ G/Ntype
2. #20272886C2semidirect
7. C53×#533C4semidirect
9. C53×#490C4semidirect
13. C4×M16×C53C4semidirect
25. #125×C53C2×C4semidirect
32. #97×C53C2×C4semidirect
107. #533Dic53semidirect
109. #490Dic53semidirect
114. C4×M16Dic53semidirect
117. C53#13595semidirect
126. #125C2×Dic53semidirect
136. #97C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104162
#202728862
C53×#31382
#1359553