Finite Group Databasebeta

← #28299059

Subgroups — #28299059

Order 13568.

224 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 36 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#20289656C211yesmaximal, characteristic, complemented
2#20289591C211yesmaximal, characteristic, complemented
3#20275941C211yesmaximal, characteristic
4C2×#13133272—21no—
5#16811468—21no—
6C2×#13133144—21no—
7#16810624—21no—
8#16810488C2211yescharacteristic
9#16810392C411yescharacteristic
10C53×#537C411yescharacteristic
11C22×#88324—41no—
12C22⋊#88324—41no—
13#13133272—22no—
14#13133272—41no—
15C2×#11719917—21no—
16C2×C4⋊Dic53D411yescharacteristic
17C2×C4×Dic53D411yescharacteristic
18#13133144—41no—
19#13133144—41no—
20#13133141—41no—
21#13133138—41no—
22#13133124—41no—
23C2×#41×C53—21no—
24C22⋊Q8×C53—41no—
25#121×C53—41no—
26#121×C53—41no—
27C2×C4⋊C4×C53C2×C411yescharacteristic
28#41×C53—41no—
29#41×C53—41no—
30#41×C53—41no—
31#41×C53—41no—
32C2×Q8×C53—21no—
33C2×D4×C53—21no—
34C2×D4×C53—41no—
35C2×D4×C53—41no—
36C22×C4×C53—21no—
37C22×C4×C53—41no—
38C22×C4×C53C22⋊C411yescharacteristic
39C22×Dic53—21no—
40C22×Dic53—21no—
41C2×#88324—42no—
42C2×#88324—42no—
43C2×#88324—42no—
44C2×#88324—81no—
45C4⋊C4×C53—41no—
46C4⋊C4×C53—41no—
47C22⋊C4×C53—42no—
48C4⋊Dic53—41no—
49C4×Dic53—21no—
50C4×Dic53D811yescharacteristic
51C4×Dic53SD1611yescharacteristic
52#11719917—41no—
53#11719917—41no—
54C2×#88322—41no—
55C23×C53—41no—
56C23×C53#9911yescharacteristic
57Q8×C53—41no—
58Q8×C53—41no—
59D4×C53—41no—
60D4×C53—41no—
61D4×C53—81no—
62D4×C53—81no—
63D4×C53—161no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53—81no—
69C2×C4×C53—81no—
70C2×C4×C53—81no—
71C2×C4×C53#10211yescharacteristic
72C2×C4×C53#10411yescharacteristic
73C2×Dic53—21no—
74C2×Dic53—21no—
75C2×Dic53—41no—
76C2×Dic53—41no—
77C2×Dic53—41no—
78C2×Dic53—81no—
79#88324—44no—
80#88324—161no—
81#88322—81no—
82#7416—531nomaximal, Sylow
83C22×C53—21no—
84C22×C53—21no—
85C22×C53—81no—
86C22×C53—81no—
87C22×C53—81no—
88C22×C53#32711yescharacteristic
89C22×C53AΣL(1,𝔽16)11yescharacteristic
90C22×C53#35511yescharacteristic
91C4×C53—21no—
92C4×C53—21no—
93C4×C53—81no—
94C4×C53—81no—
95Dic53—41no—
96Dic53—41no—
97Dic53—81no—
98#2040—531no—
99#1324—531no—
100#1295—531no—
101C2×C53—21no—
102C2×C53—21no—
103C2×C53—161no—
104C2×C53#134511yescharacteristic
105C2×C53#140311yescharacteristic
106C2×C53#140411yescharacteristic
107#537#615711yescharacteristic
108C2×#104—1061no—
109#417—1061no—
110C2×#97—1061no—
111#382—531no—
112#343—531no—
113#324—1061no—
114C53#741611yescharacteristic, complemented, Sylow
115C2×#41—21no—
116C2×M16—1061no—
117C22×C8—2121no—
118C22⋊Q8—41no—
119#121—41no—
120#121—41no—
121C2×C4⋊C4—531no—
122C2×C4⋊C4C2×#615711yescharacteristic
123C2×C42—531no—
124#104—2121no—
125#104—2121no—
126#103—2121no—
127#102—2121no—
128C22⋊C8—2121no—
129#97—1062no—
130#97—2121no—
131#95—2121no—
132#41—41no—
133#41—41no—
134#41—41no—
135#41—41no—
136C2×Q8—21no—
137C2×D4—21no—
138C2×D4—41no—
139C2×D4—41no—
140C22×C4—21no—
141C22×C4—41no—
142C22×C4—1061no—
143C22×C4—1061no—
144C22×C4C22⋊#615711yescharacteristic
145M16—2121no—
146M16—2121no—
147C2×C8—4241no—
148C2×C8—2121no—
149C2×C8—2122no—
150C2×C8—2122no—
151C2×C8—2122no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—2121no—
155C22⋊C4—42no—
156C42—531no—
157C42—531no—
158C42—1061no—
159C23—41no—
160C23#1313327511yescharacteristic
161Q8—41no—
162Q8—41no—
163D4—41no—
164D4—41no—
165D4—81no—
166D4—81no—
167D4—161no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4—1061no—
176C2×C4—1061no—
177C2×C4—4241no—
178C2×C4—2121no—
179C2×C4—2121no—
180C2×C4—2121no—
181C2×C4#1313327911yescharacteristic
182C2×C4#1313328111yescharacteristic
183C8—8481no—
184C8—4241no—
185C8—2124no—
186C22—21no—
187C22—21no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22#1681147411yescharacteristic
192C22#1681152311yescharacteristic
193C22#1681153111yescharacteristic
194C4—21no—
195C4—21no—
196C4—81no—
197C4—81no—
198C4—4241no—
199C4—2121no—
200C4—2121no—
201C2—21no—
202C2—21no—
203C2—161no—
204C2#2028969711yescharacteristic
205C2#2028983611yescharacteristic
206C2#2028983811yescharacteristic
subgroupG/H
centerC22#16811523
commutatorC2×C4⋊C4×C53C2×C4
FrattiniC2×C4⋊C4C2×#6157
Fitting, prime radicalC53×#537C4
socleC22×C53AΣL(1,𝔽16)
cosocle#16810488C22
hypercenter#537#6157
hypocenterC53#7416
LCSG ▷ C2×C4⋊C4×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #20289656C2semidirect
2. #20289591C2semidirect
114. C53#7416semidirect

Maximal subgroups

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

subgroupindex
#202896562
#202895912
#202759412
#741653