Finite Group Databasebeta

← #27271518

Subgroups — #27271518

Order 13568.

228 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 47 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#20272647C211yesmaximal, characteristic
2#20272536C211yesmaximal, characteristic
3C53×#1856C211yesmaximal, characteristic
4#16810378—21no—
5#16810349—21no—
6C53×#518C411yescharacteristic
7C53×#428—21no—
8C2×#100×C53C411yescharacteristic
9C53×#390C2211yescharacteristic
10C53×#339—21no—
11C22×#88322—41no—
12C22⋊#88322—41no—
13C22×D4×C53C2×C411yescharacteristic
14C2×M16×C53—21no—
15C4×D4×C53—41no—
16C4×D4×C53—41no—
17#117×C53—21no—
18C2×C4⋊C4×C53D411yescharacteristic
19C2×C22⋊C4×C53—41no—
20C2×C22⋊C4×C53—41no—
21C2×C42×C53D411yescharacteristic
22#107×C53—41no—
23#106×C53—41no—
24#100×C53—21no—
25#100×C53C812yescomplemented
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×C53—41no—
35C22×C4×C53—41no—
36C22×C4×C53C22⋊C411yescharacteristic
37M16×C53—22no—
38M16×C53—41no—
39C2×C8×C53—41no—
40C4⋊C4×C53—21no—
41C4⋊C4×C53—21no—
42C4⋊C4×C53—41no—
43C22⋊C4×C53—42no—
44C22⋊C4×C53—41no—
45C22⋊C4×C53—81no—
46C22⋊C4×C53—81no—
47C42×C53—41no—
48C42×C53—41no—
49C2×#88322—42no—
50C2×#88322—81no—
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—41no—
71C2×C4×C53—41no—
72C2×C4×C53—81no—
73C2×C4×C53—161no—
74C2×C4×C53—161no—
75C2×C4×C53C22⋊C811yescharacteristic
76C2×C4×C53#10111yescharacteristic
77C8×C53—42no—
78#88322—82no—
79#88322—161no—
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—
98C4×C53—81no—
99C4×C53—81no—
100#1856D5311yescharacteristic
101#1318—531no—
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#518Dic5311yescharacteristic
111#428—21no—
112C2×#100Dic5311yescharacteristic
113#390C2×D5311yescharacteristic
114#339—21no—
115#336—1061no—
116#324—1061no—
117C53#737511yescharacteristic, complemented, Sylow
118C22×D4C2×Dic5311yescharacteristic
119C2×M16—21no—
120C22×C8—2121no—
121C4×D4—41no—
122C4×D4—41no—
123#117—21no—
124C2×C4⋊C4C22⋊D5311yescharacteristic
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C42C22⋊D5311yescharacteristic
128#107—41no—
129#106—41no—
130#100—21no—
131#100#8832212yescomplemented
132C22⋊C8—2121no—
133C24—21no—
134C2×D4—21no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4C2×#8832211yescharacteristic
138C2×D4#1171991711yescharacteristic
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4C22⋊Dic5311yescharacteristic
144M16—22no—
145M16—41no—
146C2×C8—41no—
147C2×C8—4241no—
148C2×C8—4241no—
149C2×C8—4241no—
150C2×C8—2122no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—41no—
154C22⋊C4—42no—
155C22⋊C4—41no—
156C22⋊C4—81no—
157C22⋊C4—81no—
158C42—41no—
159C42—41no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—81no—
165C23—81no—
166C23#1313313211yescharacteristic
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—41no—
178C2×C4—41no—
179C2×C4—81no—
180C2×C4—161no—
181C2×C4—161no—
182C2×C4C22⋊#8832211yescharacteristic
183C2×C4#1313313611yescharacteristic
184C8—42no—
185C8—8481no—
186C8—4242no—
187C22—21no—
188C22—21no—
189C22—41no—
190C22—41no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—161no—
196C22#1681034911yescharacteristic
197C22#1681041011yescharacteristic
198C22#1681041311yescharacteristic
199C4—21no—
200C4—21no—
201C4—41no—
202C4—41no—
203C4—81no—
204C4—81no—
205C4—81no—
206C2—21no—
207C2—21no—
208C2—81no—
209C2—81no—
210C2#2027266311yescharacteristic
211C2#2027286011yescharacteristic
212C2#2027286311yescharacteristic
subgroupG/H
centerC22#16810410
commutatorC2×D4×C53C2×C8
Frattini#390C2×D53
FittingC53×#1856C2
socleC22×C53AΣL(1,𝔽16)
prime radicalC22×D4×C53C2×C4
cosocleC53×#390C22
hypercenter#1856D53
hypocenterC53#7375
LCSG ▷ C2×D4×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #1856

Complements

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

NH ≅ G/Ntype
25. #100×C53C8semidirect
117. C53#7375semidirect
131. #100#88322semidirect

Maximal subgroups

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

subgroupindex
#202726472
#202725362
C53×#18562
#737553