Finite Group Databasebeta

← #27271722

Subgroups — #27271722

Order 13568.

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

normal subgroups only

This group has 198 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#20272661C211yesmaximal, characteristic
2#20272598C211yesmaximal, characteristic
3C53×#2060C211yesmaximal, characteristic
4C2×C4⋊#88322—21no—
5#16810349—21no—
6C53×#544C411yescharacteristic
7C53×#428—21no—
8C53×#384C2211yescharacteristic
9C53×#343C411yescharacteristic
10C53×#339—21no—
11C22×#88322—41no—
12C4⋊#88322—22no—
13C4⋊#88322—41no—
14C22⋊#88322—41no—
15C2×M16×C53—21no—
16C4⋊Q8×C53—41no—
17#125×C53—41no—
18C22⋊Q8×C53—41no—
19#117×C53—21no—
20C2×C4⋊C4×C53—41no—
21C2×C4⋊C4×C53C2×C411yescharacteristic
22C2×C4⋊C4×C53D411yescharacteristic
23C2×C42×C53D411yescharacteristic
24#107×C53—41no—
25#106×C53—41no—
26C2×Q8×C53—41no—
27C22×C4×C53—21no—
28C22×C4×C53—21no—
29C22×C4×C53—21no—
30C22×C4×C53C22⋊C411yescharacteristic
31M16×C53—22no—
32M16×C53—41no—
33C2×C8×C53—41no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53—41no—
37C4⋊C4×C53—41no—
38C4⋊C4×C53—42no—
39C4⋊C4×C53—41no—
40C4⋊C4×C53—81no—
41C4⋊C4×C53—81no—
42C22⋊C4×C53—41no—
43C42×C53—21no—
44C42×C53—41no—
45C42×C53SD1611yescharacteristic
46C42×C53Q1611yescharacteristic
47C2×#88322—42no—
48C2×#88322—81no—
49C2×#88322—81no—
50C2×#88322—81no—
51C23×C53#9911yescharacteristic
52Q8×C53—161no—
53C2×C4×C53—21no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
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#10311yescharacteristic
68C2×C4×C53#10411yescharacteristic
69C8×C53—42no—
70#88322—82no—
71#88322—161no—
72#7443—531nomaximal, Sylow
73C22×C53—21no—
74C22×C53—21no—
75C22×C53#32811yescharacteristic
76C22×C53AΣL(1,𝔽16)11yescharacteristic
77C22×C53#35611yescharacteristic
78C4×C53—21no—
79C4×C53—21no—
80C4×C53—21no—
81C4×C53—21no—
82C4×C53—21no—
83C4×C53—21no—
84C4×C53—81no—
85C4×C53—81no—
86C4×C53—81no—
87C4×C53—81no—
88#2060D5311yescharacteristic
89#1325—531no—
90#1296—531no—
91C2×C53—21no—
92C2×C53—21no—
93C2×C53#135411yescharacteristic
94C2×C53#140211yescharacteristic
95C2×C53#140511yescharacteristic
96#544Dic5311yescharacteristic
97#428—21no—
98C2×C4⋊C8—1061no—
99#384C2×D5311yescharacteristic
100#343Dic5311yescharacteristic
101#339—21no—
102#324—1061no—
103C53#744311yescharacteristic, complemented, Sylow
104C2×M16—21no—
105C22×C8—2121no—
106C4⋊Q8—41no—
107#125—41no—
108C22⋊Q8—41no—
109#117—21no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4C2×Dic5311yescharacteristic
112C2×C4⋊C4C22⋊D5311yescharacteristic
113C2×C42C22⋊D5311yescharacteristic
114#107—41no—
115#106—41no—
116C4⋊C8—1062no—
117C4⋊C8—2121no—
118C22⋊C8—2121no—
119C2×Q8—41no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4C22⋊Dic5311yescharacteristic
124M16—22no—
125M16—41no—
126C2×C8—41no—
127C2×C8—4241no—
128C2×C8—4241no—
129C2×C8—4241no—
130C2×C8—2122no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4—42no—
137C4⋊C4—81no—
138C4⋊C4—81no—
139C22⋊C4—41no—
140C42—21no—
141C42—41no—
142C42#1171992411yescharacteristic
143C42#1171992511yescharacteristic
144C23#1313313211yescharacteristic
145Q8—161no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4#1313314111yescharacteristic
161C2×C4#1313314411yescharacteristic
162C8—42no—
163C8—8481no—
164C8—4242no—
165C22—21no—
166C22—21no—
167C22#1681036011yescharacteristic
168C22#1681041011yescharacteristic
169C22#1681042211yescharacteristic
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—81no—
177C4—81no—
178C4—81no—
179C4—81no—
180C2—21no—
181C2—21no—
182C2#2027274511yescharacteristic
183C2#2027286011yescharacteristic
184C2#2027286911yescharacteristic
subgroupG/H
centerC22#16810410
commutatorC2×C4⋊C4×C53C2×C4
Frattini#384C2×D53
FittingC53×#2060C2
socleC22×C53AΣL(1,𝔽16)
prime radicalC23×C53#99
cosocleC53×#384C22
hypercenter#2060D53
hypocenterC53#7443
LCSG ▷ C2×C4⋊C4×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ #2060

Complements

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

NH ≅ G/Ntype
103. C53#7443semidirect

Maximal subgroups

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

subgroupindex
#202726612
#202725982
C53×#20602
#744353