Finite Group Databasebeta

← #27304446

Subgroups — #27304446

Order 13568.

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

normal subgroups only

This group has 326 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
1#20276380C212yesmaximal
2#20272886C214yesmaximal, complemented
3C53×#3142C211yesmaximal, characteristic
4#16810597—22no—
5#16810418—22no—
6#16810416C2214yes—
7C53×#532C2211yescharacteristic
8C2×#125×C53C411yescharacteristic, complemented
9C53×#491C412yescomplemented
10C53×#490C2212yes—
11C53×#488—22no—
12C53×#417—22no—
13C4×M16×C53C411yescharacteristic, complemented
14#13133176—42no—
15#13133132—24no—
16#13133132—42no—
17C2×#41×C53D412yes—
18C2×M16×C53—21no—
19C4⋊Q8×C53C2×C411yescharacteristic
20#127×C53C2311yescharacteristic
21#125×C53—21no—
22#125×C53—41no—
23#125×C53C2×C412yes—
24#124×C53D412yes—
25#121×C53—41no—
26C4×D4×C53—41no—
27C2×C4⋊C4×C53—21no—
28C2×C4⋊C4×C53—41no—
29C2×C42×C53C2×C411yescharacteristic
30#103×C53—24no—
31#102×C53—24no—
32#97×C53C2×C412yes—
33C4×C8×C53—21no—
34#41×C53—42no—
35#41×C53—42no—
36C2×Q8×C53C22⋊C412yes—
37C2×D4×C53—22no—
38C2×D4×C53—41no—
39C2×D4×C53C2×D412yes—
40C22×C4×C53—21no—
41C22×C4×C53—22no—
42C22×C4×C53—41no—
43C22×C4×C53C22⋊C411yescharacteristic
44M16×C53—42no—
45C2×C8×C53—22no—
46C4⋊C4×C53—22no—
47C4⋊C4×C53—41no—
48C4⋊C4×C53—42no—
49C4⋊C4×C53—42no—
50C4⋊C4×C53—41no—
51C22⋊C4×C53—42no—
52C42×C53—21no—
53C42×C53C22⋊C411yescharacteristic
54C42×C53C22×C411yescharacteristic
55C22⋊Dic53—44no—
56C4⋊Dic53—82no—
57C4×Dic53—82no—
58C23×C53—22no—
59C23×C53#9911yescharacteristic
60Q8×C53—42no—
61D4×C53—42no—
62D4×C53—82no—
63D4×C53—82no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—42no—
69C2×C4×C53—42no—
70C2×C4×C53—42no—
71C2×C4×C53—81no—
72C2×C4×C53—82no—
73C2×C4×C53#9911yescharacteristic
74C2×C4×C53C2×C22⋊C411yescharacteristic
75C2×Dic53—84no—
76C8×C53—42no—
77#13596—531nomaximal, Sylow
78C22×C53—21no—
79C22×C53—41no—
80C22×C53—42no—
81C22×C53—82no—
82C22×C53C2×#9911yescharacteristic
83C4×C53—21no—
84C4×C53—41no—
85C4×C53—41no—
86C4×C53—42no—
87C4×C53—82no—
88C4×C53#35412yes—
89Dic53—164no—
90#3142D5311yescharacteristic
91#2126—532no—
92#1412—534no—
93C2×C53—21no—
94C2×C53—41no—
95C2×C53—82no—
96C2×C53C2×#35411yescharacteristic
97#532C2×D5311yescharacteristic
98C2×#125Dic5311yescharacteristic, complemented
99#491Dic5312yescomplemented
100#490C2×D5312yes—
101#488—22no—
102#417—22no—
103#410—1062no—
104C4×M16Dic5311yescharacteristic, complemented
105#354—1062no—
106#353—534no—
107C53#1359611yescharacteristic, complemented, Sylow
108C2×#41C22⋊D5312yes—
109C2×M16—21no—
110C4⋊Q8C2×Dic5311yescharacteristic
111#127C22×D5311yescharacteristic
112#125—21no—
113#125—41no—
114#125C2×Dic5312yes—
115#124C22⋊D5312yes—
116#121—41no—
117C4×D4—41no—
118#117—2122no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—41no—
121C2×C42C2×Dic5311yescharacteristic
122#103—24no—
123#102—24no—
124#99—1064no—
125#99—2122no—
126#97C2×Dic5312yes—
127C4×C8—21no—
128#41—42no—
129#41—42no—
130C2×Q8C22⋊Dic5312yes—
131C2×D4—22no—
132C2×D4—41no—
133C2×D4C2×C22⋊D5312yes—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—41no—
137C22×C4C22⋊Dic5311yescharacteristic
138M16—42no—
139C2×C8—22no—
140C4⋊C4—22no—
141C4⋊C4—41no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—41no—
145C4⋊C4—4242no—
146C22⋊C4—42no—
147C22⋊C4—2124no—
148C42—21no—
149C42—4242no—
150C42C22⋊Dic5311yescharacteristic
151C42C22×Dic5311yescharacteristic
152C23—22no—
153C23#1313313211yescharacteristic
154Q8—42no—
155D4—42no—
156D4—82no—
157D4—82no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—81no—
166C2×C4—82no—
167C2×C4—4244no—
168C2×C4#1313313211yescharacteristic
169C2×C4C2×C22⋊Dic5311yescharacteristic
170C8—42no—
171C22—21no—
172C22—41no—
173C22—42no—
174C22—82no—
175C22C2×#1313313211yescharacteristic
176C4—21no—
177C4—41no—
178C4—41no—
179C4—42no—
180C4—82no—
181C4—8484no—
182C4#1681041812yes—
183C2—21no—
184C2—41no—
185C2—82no—
186C2C2×#1681041811yescharacteristic
subgroupG/H
centerC2C2×#16810418
commutatorC42×C53C22×C4
Frattini#127C22×D53
FittingC53×#3142C2
socleC2×C53C2×#354
prime radicalC53×#532C22
cosocle#127×C53C23
hypercenter#3142D53
hypocenterC53#13596
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3142

Complements

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

NH ≅ G/Ntype
2. #20272886C2semidirect
8. C2×#125×C53C4semidirect
9. C53×#491C4semidirect
13. C4×M16×C53C4semidirect
98. C2×#125Dic53semidirect
99. #491Dic53semidirect
104. C4×M16Dic53semidirect
107. C53#13596semidirect

Maximal subgroups

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

subgroupindex
#202763802
#202728862
C53×#31422
#1359653