Finite Group Databasebeta

← #27271640

Subgroups — #27271640

Order 13568.

224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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#20275944C211yesmaximal, characteristic
2#20272659C211yesmaximal, characteristic, complemented
3C53×#1295C211yesmaximal, characteristic, complemented
4#16811204C411yescharacteristic
5C2×#13133145—21no—
6#16810622—21no—
7#16810392C411yescharacteristic
8#16810349—21no—
9C2×#97×C53—21no—
10C53×#382C2211yescharacteristic
11C2×#11719945—21no—
12C2×#11719917—21no—
13#13133197—41no—
14#13133192—41no—
15#13133191—41no—
16#13133145—41no—
17#13133145—41no—
18#13133140—41no—
19#13133137—41no—
20C22⋊#88322—41no—
21C22×C8×C53—41no—
22C2×C4⋊C4×C53C2×C411yescharacteristic
23C2×C4⋊C4×C53D411yescharacteristic
24C2×C42×C53D411yescharacteristic
25#97×C53—22no—
26#97×C53—41no—
27#95×C53—41no—
28C22×C4×C53—21no—
29C22×C4×C53—21no—
30C22×C4×C53—21no—
31C22×C4×C53C22⋊C411yescharacteristic
32C2×C22⋊D53—41no—
33#11719945—41no—
34#11719945—41no—
35#11719945—41no—
36#11719945—41no—
37C2×D212—21no—
38C2×D212—41no—
39C2×C4×D53—41no—
40C2×Dic106—21no—
41C2×C8×C53—42no—
42C2×C8×C53—42no—
43C2×C8×C53—42no—
44C4⋊C4×C53—41no—
45C4⋊C4×C53—41no—
46C42×C53—21no—
47C42×C53D811yescharacteristic
48C42×C53SD1611yescharacteristic
49#11719921—42no—
50C4⋊Dic53—41no—
51#11719917—41no—
52#11719917—41no—
53C2×#88322—41no—
54C2×#88322—81no—
55C23×C53#9911yescharacteristic
56C22×D53—41no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—81no—
66C2×C4×C53—81no—
67C2×C4×C53#10211yescharacteristic
68C2×C4×C53#10411yescharacteristic
69C22⋊D53—81no—
70C22⋊D53—81no—
71C2×Dic53—41no—
72D212—41no—
73D212—41no—
74D212—161no—
75C4×D53—81no—
76C4×D53—81no—
77Dic106—41no—
78Dic106—41no—
79C8×C53—44no—
80#88322—81no—
81#88322—161no—
82#7416—531nomaximal, Sylow
83C22×C53—21no—
84C22×C53—21no—
85C22×C53#32711yescharacteristic
86C22×C53AΣL(1,𝔽16)11yescharacteristic
87C22×C53#35511yescharacteristic
88C2×D53—81no—
89C2×D53—81no—
90C2×D53—81no—
91C4×C53—21no—
92C4×C53—21no—
93C4×C53—41no—
94C4×C53—41no—
95C4×C53—81no—
96C4×C53—81no—
97Dic53—81no—
98#2040—531no—
99#1324—531no—
100#1295D5311yescharacteristic, complemented
101C2×C53—21no—
102C2×C53—21no—
103C2×C53#134511yescharacteristic
104C2×C53#140311yescharacteristic
105C2×C53#140411yescharacteristic
106D53—161no—
107#537—531no—
108C2×#104—1061no—
109#417—1061no—
110C2×#97—21no—
111#382C2×D5311yescharacteristic
112#343—531no—
113#324—1061no—
114C53#741611yescharacteristic, complemented, Sylow
115C2×#41—1061no—
116C2×M16—1061no—
117C22×C8—41no—
118C22⋊Q8—2121no—
119#121—2121no—
120#121—2121no—
121C2×C4⋊C4C4×D5311yescharacteristic
122C2×C4⋊C4D21211yescharacteristic
123C2×C42C22⋊D5311yescharacteristic
124#104—2121no—
125#104—2121no—
126#103—2121no—
127#102—2121no—
128C22⋊C8—2121no—
129#97—22no—
130#97—41no—
131#95—41no—
132#41—2121no—
133#41—2121no—
134#41—2121no—
135#41—2121no—
136C2×Q8—1061no—
137C2×D4—1061no—
138C2×D4—2121no—
139C2×D4—2121no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—2121no—
144C22×C4#1171992111yescharacteristic
145M16—2121no—
146M16—2121no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—4241no—
151C2×C8—2121no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—2121no—
155C22⋊C4—2122no—
156C42—21no—
157C42#1171992211yescharacteristic
158C42#1171992411yescharacteristic
159C23—2121no—
160C23#1313313111yescharacteristic
161Q8—2121no—
162Q8—2121no—
163D4—8481no—
164D4—4241no—
165D4—4241no—
166D4—2121no—
167D4—2121no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—4241no—
179C2×C4—4241no—
180C2×C4—2121no—
181C2×C4#1313313711yescharacteristic
182C2×C4#1313314511yescharacteristic
183C8—44no—
184C8—8481no—
185C8—4241no—
186C22—21no—
187C22—21no—
188C22—4241no—
189C22—4241no—
190C22—4241no—
191C22#1681035611yescharacteristic
192C22#1681041111yescharacteristic
193C22#1681041911yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—41no—
198C4—81no—
199C4—81no—
200C4—4241no—
201C2—21no—
202C2—21no—
203C2—8481no—
204C2#2027271711yescharacteristic
205C2#2027286411yescharacteristic
206C2#2027286711yescharacteristic
subgroupG/H
centerC22#16810411
commutatorC2×C4⋊C4×C53C2×C4
Frattini#382C2×D53
FittingC53×#1295C2
socleC22×C53AΣL(1,𝔽16)
prime radical#16811204C4
cosocleC53×#382C22
hypercenter#1295D53
hypocenterC53#7416
LCSG ▷ C2×C4⋊C4×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #382 ◁ #1295

Complements

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

NH ≅ G/Ntype
2. #20272659C2semidirect
3. C53×#1295C2semidirect
100. #1295D53semidirect
114. C53#7416semidirect

Maximal subgroups

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

subgroupindex
#202759442
#202726592
C53×#12952
#741653