Finite Group Databasebeta

← #27271517

Subgroups — #27271517

Order 13568.

228 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 43 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#20274924C211yesmaximal, characteristic
2#20272536C211yesmaximal, characteristic
3C53×#1318C211yesmaximal, characteristic
4#16811098C411yescharacteristic
5#16810663—21no—
6C2×#13133134C411yescharacteristic
7#16810383—21no—
8#16810378—21no—
9C53×#390C2211yescharacteristic
10C53×#324—21no—
11C2×#11719917—21no—
12C22×#88322—41no—
13D4×Dic53—41no—
14D4×Dic53—41no—
15#13133176—21no—
16C2×C22⋊Dic53—41no—
17#13133150—41no—
18#13133148—41no—
19#13133134—21no—
20#13133134C812yescomplemented
21C22×D4×C53C2×C411yescharacteristic
22C2×C4⋊C4×C53D411yescharacteristic
23C2×C22⋊C4×C53—41no—
24C2×C42×C53D411yescharacteristic
25C22⋊C8×C53—41no—
26C24×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—41no—
29C2×D4×C53—41no—
30C2×D4×C53C2×C811yescharacteristic
31C2×D4×C53M1611yescharacteristic
32C22×C4×C53—21no—
33C22×C4×C53—21no—
34C22×C4×C53C22⋊C411yescharacteristic
35C22×Dic53—41no—
36C22×Dic53—41no—
37C2×C8×C53—81no—
38C4⋊C4×C53—41no—
39C22⋊C4×C53—81no—
40C22⋊C4×C53—81no—
41C42×C53—41no—
42C22⋊Dic53—42no—
43C22⋊Dic53—41no—
44C4⋊Dic53—21no—
45C4⋊Dic53—21no—
46C4×Dic53—41no—
47#11719917—22no—
48#11719917—41no—
49C2×#88322—41no—
50C2×#88322—42no—
51C2×#88322—81no—
52C2×#88322—81no—
53C23×C53—21no—
54C23×C53—21no—
55C23×C53—41no—
56C23×C53—41no—
57C23×C53—81no—
58C23×C53—81no—
59C23×C53#9911yescharacteristic
60D4×C53—41no—
61D4×C53—41no—
62D4×C53—42no—
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—41no—
68C2×C4×C53—41no—
69C2×C4×C53—41no—
70C2×C4×C53—81no—
71C2×C4×C53C22⋊C811yescharacteristic
72C2×C4×C53#10111yescharacteristic
73C2×Dic53—41no—
74C2×Dic53—41no—
75C2×Dic53—161no—
76C2×Dic53—161no—
77C8×C53—161no—
78#88322—42no—
79#88322—82no—
80#7375—531nomaximal, Sylow
81C22×C53—21no—
82C22×C53—21no—
83C22×C53—41no—
84C22×C53—41no—
85C22×C53—81no—
86C22×C53—81no—
87C22×C53—81no—
88C22×C53—81no—
89C22×C53—161no—
90C22×C53#32411yescharacteristic
91C22×C53AΣL(1,𝔽16)11yescharacteristic
92C22×C53#35211yescharacteristic
93C4×C53—21no—
94C4×C53—21no—
95C4×C53—41no—
96C4×C53—41no—
97C4×C53—81no—
98Dic53—81no—
99Dic53—81no—
100#1856—531no—
101#1318D5311yescharacteristic
102#1270—531no—
103C2×C53—21no—
104C2×C53—21no—
105C2×C53—81no—
106C2×C53—81no—
107C2×C53#132611yescharacteristic
108C2×C53#140211yescharacteristic
109C2×C53#140311yescharacteristic
110#518—531no—
111#428—1061no—
112C2×#100—531no—
113#390C2×D5311yescharacteristic
114#339—1061no—
115#336—1061no—
116#324—21no—
117C53#737511yescharacteristic, complemented, Sylow
118C22×D4C4×D5311yescharacteristic
119C2×M16—1061no—
120C22×C8—2121no—
121C4×D4—2121no—
122C4×D4—2121no—
123#117—1061no—
124C2×C4⋊C4C22⋊D5311yescharacteristic
125C2×C22⋊C4—41no—
126C2×C22⋊C4—2121no—
127C2×C42D21211yescharacteristic
128#107—2121no—
129#106—2121no—
130#100—532no—
131#100—1061no—
132C22⋊C8—41no—
133C24—21no—
134C2×D4—21no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4C8×D5311yescharacteristic
138C2×D4#1171991211yescharacteristic
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—2121no—
142C22×C4—2121no—
143C22×C4#1171992111yescharacteristic
144M16—1062no—
145M16—2121no—
146C2×C8—81no—
147C2×C8—4241no—
148C2×C8—4241no—
149C2×C8—2122no—
150C2×C8—2121no—
151C4⋊C4—41no—
152C4⋊C4—1061no—
153C4⋊C4—1061no—
154C22⋊C4—81no—
155C22⋊C4—81no—
156C22⋊C4—2122no—
157C22⋊C4—2121no—
158C42—41no—
159C42—2121no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—81no—
165C23—81no—
166C23#1313313111yescharacteristic
167D4—41no—
168D4—41no—
169D4—42no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—81no—
178C2×C4—8481no—
179C2×C4—8481no—
180C2×C4—2121no—
181C2×C4—2121no—
182C2×C4#1313312911yescharacteristic
183C2×C4#1313313511yescharacteristic
184C8—161no—
185C8—4242no—
186C8—2122no—
187C22—21no—
188C22—21no—
189C22—41no—
190C22—41no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—161no—
196C22#1681034811yescharacteristic
197C22#1681040911yescharacteristic
198C22#1681041411yescharacteristic
199C4—21no—
200C4—21no—
201C4—41no—
202C4—41no—
203C4—81no—
204C4—4241no—
205C4—4241no—
206C2—21no—
207C2—21no—
208C2—81no—
209C2—81no—
210C2#2027266211yescharacteristic
211C2#2027285911yescharacteristic
212C2#2027286211yescharacteristic
subgroupG/H
centerC22#16810409
commutatorC2×D4×C53C2×C8
Frattini#390C2×D53
FittingC53×#1318C2
socleC22×C53AΣL(1,𝔽16)
prime radicalC22×D4×C53C2×C4
cosocleC53×#390C22
hypercenter#1318D53
hypocenterC53#7375
LCSG ▷ C2×D4×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #1318

Complements

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

NH ≅ G/Ntype
20. #13133134C8semidirect
117. C53#7375semidirect

Maximal subgroups

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

subgroupindex
#202749242
#202725362
C53×#13182
#737553