Finite Group Databasebeta

← #27271675

Subgroups — #27271675

Order 13568.

214 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 37 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#20275882C211yesmaximal, characteristic
2#20272623C211yesmaximal, characteristic
3C53×#1317C211yesmaximal, characteristic
4#16811249C411yescharacteristic
5#16810587—21no—
6#16810486—21no—
7#16810390C411yescharacteristic
8#16810389—21no—
9C53×#381C2211yescharacteristic
10C53×#323—21no—
11C2×#11719917—41no—
12#13133196—41no—
13#13133194—42no—
14C2×C22⋊Dic53—21no—
15C2×C22⋊Dic53—21no—
16C22⋊#88322—22no—
17C22⋊#88322—41no—
18#13133124—42no—
19C23×C4×C53D411yescharacteristic
20C2×C22⋊C4×C53C2×C411yescharacteristic
21C2×C22⋊C4×C53D411yescharacteristic
22C22⋊C8×C53—41no—
23#95×C53—41no—
24C24×C53C22⋊C411yescharacteristic
25C22×C4×C53—21no—
26C22×C4×C53—21no—
27C22×C4×C53—21no—
28C22×C4×C53—21no—
29C22×C4×C53—41no—
30C22×C4×C53—41no—
31C22×C4×C53SD1611yescharacteristic
32C22×C4×C53Q1611yescharacteristic
33C22×Dic53—22no—
34C22×Dic53—41no—
35C2×Dic106—41no—
36C2×C8×C53—81no—
37C22⋊C4×C53—41no—
38C22⋊C4×C53—41no—
39C22⋊Dic53—41no—
40C22⋊Dic53—41no—
41C22⋊Dic53—41no—
42C22⋊Dic53—41no—
43C4⋊Dic53—41no—
44#11719919—42no—
45#11719917—81no—
46#11719917—81no—
47C2×#88322—42no—
48C23×C53—21no—
49C23×C53—21no—
50C23×C53—81no—
51C23×C53#9911yescharacteristic
52C23×C53#10311yescharacteristic
53C23×C53#10411yescharacteristic
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53—41no—
59C2×C4×C53—41no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53—81no—
63C2×C4×C53—81no—
64C2×C4×C53—81no—
65C2×Dic53—42no—
66C2×Dic53—82no—
67C2×Dic53—81no—
68C2×Dic53—82no—
69Dic106—161no—
70C8×C53—161no—
71#88322—82no—
72#7427—531nomaximal, Sylow
73C22×C53—21no—
74C22×C53—21no—
75C22×C53—41no—
76C22×C53—41no—
77C22×C53—41no—
78C22×C53—42no—
79C22×C53—81no—
80C22×C53#32811yescharacteristic
81C22×C53AΣL(1,𝔽16)11yescharacteristic
82C22×C53#35411yescharacteristic
83C4×C53—41no—
84C4×C53—41no—
85C4×C53—81no—
86C4×C53—81no—
87Dic53—82no—
88Dic53—161no—
89#2030—531no—
90#1317D5311yescharacteristic
91#1306—531no—
92C2×C53—21no—
93C2×C53—21no—
94C2×C53—21no—
95C2×C53—41no—
96C2×C53—41no—
97C2×C53#135111yescharacteristic
98#543—531no—
99#407—1061no—
100#381—1061no—
101#381C2×D5311yescharacteristic
102#342—531no—
103#342—1061no—
104#323—21no—
105C53#742711yescharacteristic, complemented, Sylow
106C23×C4D21211yescharacteristic
107C2×M16—2121no—
108C22⋊Q8—2122no—
109C22⋊Q8—2121no—
110C2×C22⋊C4—1061no—
111C2×C22⋊C4—1061no—
112C2×C22⋊C4C4×D5311yescharacteristic
113C2×C22⋊C4C22⋊D5311yescharacteristic
114C22⋊C8—41no—
115C22⋊C8—1062no—
116C22⋊C8—2121no—
117#95—41no—
118#95—2122no—
119C24#1171992111yescharacteristic
120C2×Q8—2121no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—1062no—
128C22×C4—2121no—
129C22×C4#1171991311yescharacteristic
130C22×C4Dic21211yescharacteristic
131M16—4241no—
132M16—4241no—
133C2×C8—81no—
134C2×C8—2122no—
135C4⋊C4—2122no—
136C4⋊C4—2121no—
137C22⋊C4—41no—
138C22⋊C4—41no—
139C22⋊C4—2121no—
140C22⋊C4—2121no—
141C22⋊C4—2121no—
142C22⋊C4—2121no—
143C23—21no—
144C23—21no—
145C23—81no—
146C23#1313313111yescharacteristic
147C23#1313314211yescharacteristic
148C23#1313314311yescharacteristic
149Q8—8481no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4—81no—
161C2×C4—4241no—
162C2×C4—4242no—
163C2×C4—4242no—
164C2×C4—2122no—
165C8—161no—
166C8—4242no—
167C22—21no—
168C22—21no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22—42no—
173C22—81no—
174C22#1681036111yescharacteristic
175C22#1681041111yescharacteristic
176C22#1681041711yescharacteristic
177C4—41no—
178C4—41no—
179C4—81no—
180C4—81no—
181C4—8481no—
182C4—4242no—
183C2—21no—
184C2—21no—
185C2—21no—
186C2—41no—
187C2—41no—
188C2#2027273711yescharacteristic
subgroupG/H
centerC2#20272737
commutatorC2×C22⋊C4×C53C2×C4
Frattini#381C2×D53
FittingC53×#1317C2
socleC2×C53#1351
prime radicalC24×C53C22⋊C4
cosocleC53×#381C22
hypercenter#1317D53
hypocenterC53#7427
LCSG ▷ C2×C22⋊C4×C53 ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ #1317

Complements

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

NH ≅ G/Ntype
105. C53#7427semidirect

Maximal subgroups

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

subgroupindex
#202758822
#202726232
C53×#13172
#742753