Finite Group Databasebeta

← #28308234

Subgroups — #28308234

Order 13568.

280 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 66 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#20291085C212yesmaximal, complemented
2#20289663C212yesmaximal, complemented
3#20289650C212yesmaximal
4#20283365C211yesmaximal, characteristic
5#16811545—22no—
6#16811519C2214yes—
7#16811516—22no—
8#16811091C411yescharacteristic
9#16810683—22no—
10#16810679—22no—
11#16810650C2211yescharacteristic
12#16810591C411yescharacteristic
13#16810589C2211yescharacteristic
14M16×Dic53C2211yescharacteristic
15C2×#13133127C411yescharacteristic
16C53×#424C411yescharacteristic
17#13133291—24no—
18C2×#11719945C2×C411yescharacteristic
19C2×#11719917C2311yescharacteristic
20C22×#88322C2×C411yescharacteristic
21#13133182—21no—
22#13133179—21no—
23#13133177—22no—
24#13133174—21no—
25C2×C4×Dic53C2×C411yescharacteristic
26C42×D53—21no—
27C4⋊#88322—22no—
28#13133129—22no—
29#13133129—22no—
30#13133127—21no—
31#13133127—21no—
32#13133127D412yes—
33C8×Dic53D412yes—
34C2×M16×C53C2×C411yescharacteristic
35C22×C8×C53C2×C411yescharacteristic
36#117×C53C2×C411yescharacteristic
37C4⋊C8×C53—22no—
38C22×C4×C53C22×C411yescharacteristic
39C2×C22⋊D53—21no—
40C22×Dic53—21no—
41#11719945—41no—
42#11719945—41no—
43C2×D212M1611yescharacteristic
44C2×C4×D53—21no—
45C2×C4×D53—22no—
46C2×Dic106M1611yescharacteristic
47M16×C53—41no—
48C2×C8×C53—21no—
49C2×C8×C53—41no—
50C2×C8×C53C22⋊C412yes—
51C4⋊C4×C53—21no—
52C22⋊C4×C53M1612yes—
53C42×C53—21no—
54#11719921—22no—
55#11719919—22no—
56C4×Dic53—22no—
57C4×Dic53—21no—
58C4×Dic53C22⋊C411yescharacteristic
59C4×Dic53C22⋊C411yescharacteristic
60#11719917—41no—
61C2×#88322—21no—
62C2×#88322—41no—
63C2×#88322C2×D412yes—
64#11719910—44no—
65C23×C53C2×M1611yescharacteristic
66C22×D53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53C2×C22⋊C411yescharacteristic
71C2×C4×C53C2×M1611yescharacteristic
72C22⋊D53—42no—
73C2×Dic53—21no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—81no—
77D212—41no—
78C4×D53—41no—
79C4×D53—41no—
80C4×D53—82no—
81Dic106—41no—
82C8×C53—22no—
83C8×C53—41no—
84#88322—22no—
85#88322—41no—
86#11523—531nomaximal, Sylow
87C22×C53—21no—
88C22×C53—41no—
89C22×C53#40711yescharacteristic
90C2×D53—42no—
91C2×D53—41no—
92C4×C53—21no—
93C4×C53—42no—
94C4×C53#41011yescharacteristic
95C4×C53#41311yescharacteristic
96Dic53—41no—
97Dic53—41no—
98Dic53—41no—
99#2923—531no—
100#2165—532no—
101#1328—532no—
102#1321—532no—
103C2×C53—21no—
104C2×C53—41no—
105C2×C53#146411yescharacteristic
106D53—81no—
107C4×#41—531no—
108#433—1062no—
109#432—1062no—
110#424—531no—
111#424#615711yescharacteristic
112#408—531no—
113#408—531no—
114C4×M16—531no—
115C2×#97—531no—
116#364—1062no—
117#349—534no—
118#347—1062no—
119C53#1152311yescharacteristic, complemented, Sylow
120C2×#41—531no—
121C2×M16—531no—
122C2×M16C2×#615711yescharacteristic
123C22×C8—531no—
124C22×C8C2×#615711yescharacteristic
125C4×Q8—1061no—
126C4×D4—1061no—
127C4×D4—1062no—
128#117—1061no—
129#117C2×#615711yescharacteristic
130C2×C42—531no—
131C2×C42—1061no—
132M32—1064no—
133C4⋊C8—22no—
134C4⋊C8—1062no—
135C22⋊C8—1062no—
136C22⋊C8—1062no—
137#97—532no—
138#97—1061no—
139#97—1061no—
140C4×C8—532no—
141#41—2121no—
142#41—2121no—
143C2×Q8—531no—
144C2×D4—531no—
145C2×D4—1061no—
146C22×C4—1062no—
147C22×C4—1061no—
148C22×C4—1061no—
149C22×C4C22×#615711yescharacteristic
150M16—41no—
151M16—2121no—
152C2×C8—21no—
153C2×C8—41no—
154C2×C8—532no—
155C2×C8—1061no—
156C2×C8—2121no—
157C2×C8C22⋊#615712yes—
158C4⋊C4—21no—
159C4⋊C4—1062no—
160C22⋊C4—1062no—
161C22⋊C4#1171993612yes—
162C42—21no—
163C42—531no—
164C42—531no—
165C42—1062no—
166C42—1061no—
167C16—2124no—
168C23—1061no—
169C23C2×#1171993611yescharacteristic
170Q8—2121no—
171D4—2121no—
172D4—2122no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—1061no—
177C2×C4—1061no—
178C2×C4—1061no—
179C2×C4—4242no—
180C2×C4—4241no—
181C2×C4—2121no—
182C2×C4—2121no—
183C2×C4C2×C22⋊#615711yescharacteristic
184C2×C4D53⋊M1611yescharacteristic
185C8—22no—
186C8—41no—
187C8—1062no—
188C8—2121no—
189C22—21no—
190C22—41no—
191C22—2122no—
192C22—2121no—
193C22#1681159411yescharacteristic
194C4—21no—
195C4—42no—
196C4—2121no—
197C4—2121no—
198C4—2121no—
199C4#1681160411yescharacteristic
200C4#1681161011yescharacteristic
201C2—21no—
202C2—41no—
203C2—4241no—
204C2#2028996911yescharacteristic
subgroupG/H
centerC4#16811604
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#424C4
socleC2×C53#1464
prime radicalC2×#11719945C2×C4
cosocleC2×#11719917C23
hypercenter#424#6157
hypocenterC53#11523
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #20291085C2semidirect
2. #20289663C2semidirect
119. C53#11523semidirect

Maximal subgroups

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

subgroupindex
#202910852
#202896632
#202896502
#202833652
#1152353