Finite Group Databasebeta

← #28298966

Subgroups — #28298966

Order 13568.

202 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 40 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#20289645C211yesmaximal, characteristic
2#20289535C211yesmaximal, characteristic
3#20274930C211yesmaximal, characteristic
4#16811489—21no—
5#16811468—21no—
6#16810666—21no—
7C2×#13133136C411yescharacteristic
8#16810530C2211yescharacteristic
9#16810382—21no—
10C53×#519C411yescharacteristic
11C22×#88324—41no—
12C22⋊#88324—41no—
13C2×#11719917—21no—
14C2×C4⋊Dic53D411yescharacteristic
15C2×C4×Dic53D411yescharacteristic
16#13133150—41no—
17#13133148—41no—
18#13133136—21no—
19#13133136C812yescomplemented
20#13133124—41no—
21C22×Q8×C53C2×C411yescharacteristic
22C4×Q8×C53—41no—
23C4×Q8×C53—41no—
24#117×C53—21no—
25#117×C53—41no—
26C2×Q8×C53—21no—
27C2×Q8×C53—41no—
28C2×Q8×C53—41no—
29C2×Q8×C53C2×C811yescharacteristic
30C2×Q8×C53M1611yescharacteristic
31C22×C4×C53—21no—
32C22×C4×C53C22⋊C411yescharacteristic
33C22×Dic53—21no—
34C22×Dic53—21no—
35C2×#88324—42no—
36C2×#88324—81no—
37C2×#88324—81no—
38C2×#88324—81no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53—21no—
41C4⋊C4×C53—41no—
42C4⋊C4×C53—41no—
43C22⋊C4×C53—41no—
44C42×C53—41no—
45C42×C53—42no—
46C4⋊Dic53—41no—
47C4×Dic53—41no—
48#11719917—22no—
49#11719917—41no—
50C2×#88322—41no—
51C23×C53#9911yescharacteristic
52Q8×C53—42no—
53Q8×C53—41no—
54Q8×C53—41no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—41no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53C22⋊C811yescharacteristic
64C2×C4×C53#10111yescharacteristic
65C2×Dic53—21no—
66C2×Dic53—21no—
67C2×Dic53—41no—
68C2×Dic53—41no—
69C2×Dic53—41no—
70C2×Dic53—81no—
71#88324—82no—
72#88324—161no—
73#88322—42no—
74#7376—531nomaximal, Sylow
75C22×C53—21no—
76C22×C53—21no—
77C22×C53#32411yescharacteristic
78C22×C53AΣL(1,𝔽16)11yescharacteristic
79C22×C53#35211yescharacteristic
80C4×C53—21no—
81C4×C53—21no—
82C4×C53—41no—
83C4×C53—41no—
84C4×C53—81no—
85C4×C53—81no—
86Dic53—41no—
87Dic53—41no—
88Dic53—81no—
89#1857—531no—
90#1319—531no—
91#1272—531no—
92C2×C53—21no—
93C2×C53—21no—
94C2×C53#132611yescharacteristic
95C2×C53#140411yescharacteristic
96C2×C53#140511yescharacteristic
97#519#615711yescharacteristic
98#428—1061no—
99C2×#101—531no—
100#391—531no—
101#339—1061no—
102#336—1061no—
103#324—1061no—
104C53#737611yescharacteristic, complemented, Sylow
105C22×Q8C2×#615711yescharacteristic
106C2×M16—1061no—
107C22×C8—2121no—
108C4×Q8—41no—
109C4×Q8—41no—
110#117—21no—
111#117—41no—
112C2×C4⋊C4—531no—
113C2×C42—531no—
114#107—2121no—
115#106—2121no—
116#101—532no—
117#101—1061no—
118C22⋊C8—2121no—
119#95—2121no—
120C2×Q8—21no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8D53⋊C811yescharacteristic
124C2×Q8#1171993611yescharacteristic
125C22×C4—21no—
126C22×C4—1061no—
127C22×C4—1061no—
128C22×C4C22⋊#615711yescharacteristic
129M16—1062no—
130M16—2121no—
131C2×C8—4241no—
132C2×C8—4241no—
133C2×C8—4241no—
134C2×C8—2122no—
135C2×C8—2121no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—2121no—
141C22⋊C4—41no—
142C42—41no—
143C42—42no—
144C42—2121no—
145C23#1313327511yescharacteristic
146Q8—41no—
147Q8—41no—
148Q8—42no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—1061no—
158C2×C4—1061no—
159C2×C4—4241no—
160C2×C4—2121no—
161C2×C4—2121no—
162C2×C4—2121no—
163C2×C4#1313327411yescharacteristic
164C2×C4#1313327811yescharacteristic
165C8—8481no—
166C8—4242no—
167C8—2122no—
168C22—21no—
169C22—21no—
170C22#1681146911yescharacteristic
171C22#1681152311yescharacteristic
172C22#1681152511yescharacteristic
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—81no—
178C4—81no—
179C4—4241no—
180C4—2121no—
181C4—2121no—
182C2—21no—
183C2—21no—
184C2#2028966011yescharacteristic
185C2#2028983811yescharacteristic
186C2#2028984011yescharacteristic
subgroupG/H
centerC22#16811523
commutatorC2×Q8×C53C2×C8
FrattiniC22×Q8C2×#6157
FittingC53×#519C4
socleC22×C53AΣL(1,𝔽16)
prime radicalC23×C53#99
cosocle#16810530C22
hypercenter#519#6157
hypocenterC53#7376
LCSG ▷ C2×Q8×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C22×Q8 ◁ #519

Complements

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

NH ≅ G/Ntype
19. #13133136C8semidirect
104. C53#7376semidirect

Maximal subgroups

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

subgroupindex
#202896452
#202895352
#202749302
#737653