Finite Group Databasebeta

← #28312460

Subgroups — #28312460

Order 13568.

398 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 60 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#20291008C212yesmaximal
2#20289852C214yesmaximal, complemented
3#20285172C211yesmaximal, characteristic
4#16811609—22no—
5#16811534C2214yes—
6#16811532—22no—
7C2×#13133248—22no—
8#16811173C2211yescharacteristic
9#16811015—22no—
10#16811002C2212yes—
11#16810997C412yes—
12C4×#11719917C411yescharacteristic
13C2×C4⋊Q8×C53C411yescharacteristic
14D53⋊M16—42no—
15#13133278—24no—
16#13133277—42no—
17C2×#11719945D412yes—
18#13133248—44no—
19#13133248—44no—
20C2×#11719925—24no—
21C2×#11719924—24no—
22C2×#11719917—21no—
23#13133217C2311yescharacteristic
24#13133214C2×C411yescharacteristic
25#13133205D412yes—
26#13133191—41no—
27C4×D212—41no—
28#13133128—21no—
29C4×#88322C2×C412yes—
30C22×Q8×C53—21no—
31C4⋊Q8×C53—21no—
32C4⋊Q8×C53—41no—
33C4⋊Q8×C53C2×C412yes—
34C2×C4⋊C4×C53—41no—
35C2×C42×C53C2×C411yescharacteristic
36C2×Q8×C53—22no—
37C2×Q8×C53—42no—
38C2×Q8×C53—42no—
39C2×Q8×C53—41no—
40C22×C4×C53—21no—
41C22×C4×C53—41no—
42C22×C4×C53C22⋊C411yescharacteristic
43C2×C22⋊D53—22no—
44#11719945—42no—
45#11719945—42no—
46C2×D212—41no—
47C2×D212C22⋊C412yes—
48C2×C4×D53—22no—
49C2×Dic106C2×D412yes—
50#11719940—44no—
51#11719936—82no—
52D53⋊C8—82no—
53C4⋊C4×C53—42no—
54C4⋊C4×C53—42no—
55C42×C53—21no—
56C42×C53C22⋊C411yescharacteristic
57C42×C53C22×C411yescharacteristic
58#11719925—48no—
59#11719924—48no—
60#11719921—42no—
61C4⋊Dic53—41no—
62#11719917—24no—
63C2×#88322—22no—
64C23×C53#9911yescharacteristic
65C22×D53—22no—
66Q8×C53—44no—
67Q8×C53—82no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—21no—
71C2×C4×C53—42no—
72C2×C4×C53—42no—
73C2×C4×C53—81no—
74C2×C4×C53—81no—
75C2×C4×C53#9911yescharacteristic
76C2×C4×C53C2×C22⋊C411yescharacteristic
77C22⋊D53—82no—
78C2×Dic53—22no—
79D212—42no—
80D212—82no—
81C4×D53—42no—
82C4×D53—42no—
83Dic106—42no—
84#88324—84no—
85#88322—24no—
86#13592—531nomaximal, Sylow
87C22×C53—21no—
88C22×C53—41no—
89C22×C53C2×#9911yescharacteristic
90C2×D53—42no—
91C2×D53—82no—
92C4×C53—22no—
93C4×C53—21no—
94C4×C53—41no—
95C4×C53—82no—
96C4×C53#35312yes—
97Dic53—42no—
98#3152—531no—
99#2132—532no—
100#1411—534no—
101C2×C53—21no—
102C2×C53—41no—
103C2×C53C2×#35311yescharacteristic
104D53—82no—
105C2×#137—1062no—
106#532—531no—
107C2×C4⋊Q8#615711yescharacteristic
108#497—1062no—
109#494—532no—
110#492—532no—
111#413—1062no—
112C4×M16—531no—
113#356—534no—
114#355—1062no—
115C53#1359211yescharacteristic, complemented, Sylow
116C2×#41—532no—
117C22×Q8—21no—
118#137—2124no—
119#137—2124no—
120C2×Q16—1064no—
121C2×SD16—1064no—
122C2×M16—1061no—
123C2×M16—2122no—
124C4⋊Q8—21no—
125C4⋊Q8—41no—
126C4⋊Q8—531no—
127C4⋊Q8C2×#615712yes—
128#127—531no—
129#124—532no—
130#121—2121no—
131C4×D4—2121no—
132C2×C4⋊C4—41no—
133C2×C42C2×#615711yescharacteristic
134#101—1064no—
135#100—2122no—
136#97—1061no—
137C4×C8—532no—
138#41—2122no—
139#41—2122no—
140C2×Q8—22no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—41no—
144C2×Q8—532no—
145C2×D4—532no—
146C2×D4—1062no—
147C2×D4—2121no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—1062no—
151C22×C4C22⋊#615711yescharacteristic
152Q16—2128no—
153SD16—2128no—
154M16—1064no—
155M16—4242no—
156M16—2124no—
157C2×C8—1062no—
158C2×C8—4242no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—2121no—
162C22⋊C4—2122no—
163C42—21no—
164C42C22⋊#615711yescharacteristic
165C42C22×#615711yescharacteristic
166C23—1062no—
167C23#1313327511yescharacteristic
168Q8—44no—
169Q8—82no—
170Q8—2122no—
171D4—4242no—
172D4—4242no—
173D4—2122no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4—1062no—
182C2×C4—2122no—
183C2×C4—2122no—
184C2×C4#1313327511yescharacteristic
185C2×C4C2×C22⋊#615711yescharacteristic
186C8—1064no—
187C8—4244no—
188C22—21no—
189C22—41no—
190C22—4242no—
191C22—2122no—
192C22C2×#1313327511yescharacteristic
193C4—22no—
194C4—21no—
195C4—41no—
196C4—82no—
197C4—2122no—
198C4#1681152712yes—
199C2—21no—
200C2—41no—
201C2—4242no—
202C2C2×#1681152711yescharacteristic
subgroupG/H
centerC2C2×#16811527
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC2×C53C2×#353
prime radical#16811173C22
cosocle#13133217C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#13592
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20289852C2semidirect
115. C53#13592semidirect

Maximal subgroups

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

subgroupindex
#202910082
#202898522
#202851722
#1359253