Finite Group Databasebeta

← #28310117

Subgroups — #28310117

Order 13568.

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

normal subgroups only

This group has 220 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#20290362C211yesmaximal, characteristic, complemented
2#20290313C211yesmaximal, characteristic
3#20289740C211yesmaximal, characteristic, complemented
4#20289675C211yesmaximal, characteristic, complemented
5#20289674C211yesmaximal, characteristic
6#20289634C211yesmaximal, characteristic, complemented
7#20276582C211yesmaximal, characteristic
8#16811674—21no—
9#16811651—21no—
10#16811512C2211yescharacteristic
11#16811484C2211yescharacteristic
12#16811484C2211yescharacteristic
13Q8⋊#88324—21no—
14#16811471—21no—
15#16811464C2211yescharacteristic
16C2×#13133160C411yescharacteristic
17Q16×Dic53C2211yescharacteristic
18#16810705C2211yescharacteristic
19#16810448C411yescharacteristic
20#16810425C411yescharacteristic
21#16810398C2211yescharacteristic
22C53×#358C411yescharacteristic
23C2×D53⋊C8—41no—
24C2×#11719910—21no—
25#13133284—41no—
26C4⋊#88324—21no—
27C4⋊#88324—21no—
28#13133274—41no—
29#13133272—41no—
30C4×#88324D411yescharacteristic
31C4×#88324D411yescharacteristic
32C2×D424C2×C411yescharacteristic
33Q8×Dic53—21no—
34#13133179—21no—
35#13133160—22no—
36C2×#11719908C2×C411yescharacteristic
37#13133151C2×C411yescharacteristic
38#13133141—21no—
39#13133137—21no—
40C8×Dic53C2311yescharacteristic
41C2×Q16×C53C2×C411yescharacteristic
42C2×C16×C53C2×C411yescharacteristic
43#107×C53C2×C411yescharacteristic
44C2×Q8×C53—21no—
45C2×D212—21no—
46C2×C4×D53—41no—
47C2×#88324—21no—
48C2×#88324—21no—
49C2×#88324—41no—
50C2×#88324—41no—
51D53⋊C8—82no—
52Q16×C53—21no—
53Q16×C53—21no—
54C2×C8×C53C22×C411yescharacteristic
55C4⋊C4×C53—21no—
56#11719921—41no—
57C4⋊Dic53—21no—
58C4⋊Dic53—41no—
59C4×Dic53—41no—
60C4×Dic53C2×D411yescharacteristic
61C2×#88322#4111yescharacteristic
62D424—21no—
63D424M1612yes—
64#11719910—22no—
65C16×C53—21no—
66#11719908—21no—
67C22×D53—41no—
68Q8×C53—41no—
69Q8×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53—41no—
72C2×C4×C53C4×D411yescharacteristic
73C2×Dic53—41no—
74C2×Dic53AGL(1,ℤ/8)11yescharacteristic
75C2×Dic53AGL(1,ℤ/8)11yescharacteristic
76D212—22no—
77D212—41no—
78C4×D53—81no—
79C4×D53—81no—
80#88324—41no—
81#88324—41no—
82#88324—81no—
83#88324—81no—
84C8×C53C2×M1611yescharacteristic
85C8×C53#13111yescharacteristic
86#88322—21no—
87#12077—531nomaximal, Sylow
88C22×C53#44211yescharacteristic
89C2×D53—42no—
90C2×D53—81no—
91C2×D53—81no—
92C4×C53—81no—
93C4×C53—81no—
94C4×C53#43511yescharacteristic
95C4×C53#44311yescharacteristic
96Dic53—21no—
97Dic53—21no—
98Dic53—81no—
99#2175—531no—
100#1593—531no—
101#1580—531no—
102#1368—531no—
103#1334—531no—
104#1333—531no—
105#1313—531no—
106C2×C53#158611yescharacteristic
107C2×C53#218011yescharacteristic
108C2×C53AGL(1,ℤ/16)11yescharacteristic
109D53—82no—
110C2×SD32—531no—
111#446—1061no—
112C4×Q16—531no—
113C4×D8—531no—
114C8×D4—1061no—
115#366—531no—
116#358#615711yescharacteristic
117#357—531no—
118#346—531no—
119#346—531no—
120#334—531no—
121#334—531no—
122Q8⋊C8—1061no—
123D4⋊C8—1061no—
124#322—531no—
125C53#1207711yescharacteristic, complemented, Sylow
126C2×Q16C2×#615711yescharacteristic
127C2×D8—531no—
128C22×C8—2121no—
129C4×Q8—1061no—
130C4×D4—1061no—
131SD32—1062no—
132C2×C16—531no—
133C2×C16—1061no—
134C2×C16C2×#615711yescharacteristic
135#107—531no—
136#107C2×#615711yescharacteristic
137C4⋊C8—1061no—
138C4⋊C8—1061no—
139C4⋊C8—2121no—
140#103—1061no—
141#102—1061no—
142C22⋊C8—2121no—
143#97—2121no—
144C4×C8—531no—
145C4×C8—531no—
146C4×C8—531no—
147C2×Q8—21no—
148C2×D4—1061no—
149C22×C4—2121no—
150Q16—21no—
151Q16—21no—
152D8—532no—
153D8—1061no—
154C2×C8—531no—
155C2×C8—1061no—
156C2×C8—1061no—
157C2×C8—4242no—
158C2×C8—2121no—
159C2×C8—2121no—
160C2×C8C22×#615711yescharacteristic
161C4⋊C4—21no—
162C4⋊C4—1061no—
163C4⋊C4—2121no—
164C22⋊C4—2121no—
165C42—531no—
166C42—2121no—
167C16—21no—
168C16—1062no—
169C16—1061no—
170C23—2121no—
171Q8—41no—
172Q8—41no—
173D4—1062no—
174D4—2121no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—531no—
178C2×C4—531no—
179C2×C4—4241no—
180C2×C4—4241no—
181C2×C4—2121no—
182C2×C4D4×#615711yescharacteristic
183C8—1061no—
184C8—4241no—
185C8—4241no—
186C8—2121no—
187C8—2121no—
188C8D53⋊M1611yescharacteristic
189C8#1313330411yescharacteristic
190C22—4241no—
191C22—4241no—
192C22—2122no—
193C22#1681166411yescharacteristic
194C4—81no—
195C4—81no—
196C4—1061no—
197C4—1061no—
198C4—4241no—
199C4#1681165511yescharacteristic
200C4#1681166611yescharacteristic
201C2—4242no—
202C2#2029033711yescharacteristic
203C2#2029112311yescharacteristic
204C2#2029112611yescharacteristic
subgroupG/H
centerC22#16811664
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC53×#358C4
socleC22×C53#442
prime radicalC2×D424C2×C4
cosocleC8×Dic53C23
hypercenter#358#6157
hypocenterC53#12077
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ #358

Complements

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

NH ≅ G/Ntype
1. #20290362C2semidirect
3. #20289740C2semidirect
4. #20289675C2semidirect
6. #20289634C2semidirect
125. C53#12077semidirect

Maximal subgroups

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

subgroupindex
#202903622
#202903132
#202897402
#202896752
#202896742
#202896342
#202765822
#1207753