Finite Group Databasebeta

← #28299057

Subgroups — #28299057

Order 13568.

190 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 32 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#20289662C211yesmaximal, characteristic, complemented
2#20289621C211yesmaximal, characteristic, complemented
3#20275999C211yesmaximal, characteristic
4#16811644—21no—
5#16811519—21no—
6#16810646C2211yescharacteristic
7C2×#13133144—21no—
8#16810607—21no—
9#16810381C411yescharacteristic
10C53×#575C411yescharacteristic
11#13133291—41no—
12C4⋊#88324—41no—
13C22⋊#88324—22no—
14#13133272—41no—
15C2×#11719917—21no—
16C2×#11719917D411yescharacteristic
17C2×C4×Dic53D411yescharacteristic
18#13133146—41no—
19#13133144—41no—
20#13133144—41no—
21#13133136—41no—
22#13133134—41no—
23C2×#41×C53—21no—
24#137×C53—41no—
25AGL(1,ℤ/8)×C53—41no—
26#135×C53—41no—
27C2×M16×C53C2×C411yescharacteristic
28#41×C53—41no—
29#41×C53—41no—
30#41×C53—41no—
31#41×C53—41no—
32C2×Q8×C53—21no—
33C2×D4×C53—21no—
34C2×D4×C53—41no—
35C22×C4×C53—41no—
36C22×C4×C53C22⋊C411yescharacteristic
37C22×Dic53—21no—
38C2×#88324—42no—
39Q16×C53—41no—
40SD16×C53—42no—
41D8×C53—41no—
42M16×C53—41no—
43C2×C8×C53—21no—
44C4×Dic53—21no—
45C4×Dic53D811yescharacteristic
46C4×Dic53SD1611yescharacteristic
47#11719917—41no—
48#11719917—41no—
49#11719917—41no—
50C2×#88322—21no—
51C2×#88322—41no—
52#11719910—81no—
53C23×C53—41no—
54C23×C53#10411yescharacteristic
55Q8×C53—41no—
56Q8×C53—41no—
57D4×C53—41no—
58D4×C53—41no—
59D4×C53—81no—
60D4×C53—81no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—41no—
64C2×C4×C53—81no—
65C2×C4×C53—81no—
66C2×C4×C53#9911yescharacteristic
67C2×C4×C53#10211yescharacteristic
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53—81no—
71#88324—82no—
72C8×C53—41no—
73#88322—41no—
74#88322—81no—
75#7415—531nomaximal, Sylow
76C22×C53—21no—
77C22×C53—41no—
78C22×C53—81no—
79C22×C53—161no—
80C22×C53#32711yescharacteristic
81C4×C53—21no—
82C4×C53—81no—
83C4×C53AΣL(1,𝔽16)11yescharacteristic
84C4×C53#35511yescharacteristic
85Dic53—41no—
86Dic53—41no—
87#2050—531no—
88#1327—531no—
89#1307—531no—
90C2×C53—21no—
91C2×C53—41no—
92C2×C53—161no—
93C2×C53#134511yescharacteristic
94#575#615711yescharacteristic
95#432—1061no—
96#423—531no—
97C2×#104—1061no—
98#413—1061no—
99#349—1061no—
100#338—531no—
101C53#741511yescharacteristic, complemented, Sylow
102C2×#41—21no—
103#137—41no—
104AGL(1,ℤ/8)—41no—
105#135—41no—
106C2×M16—531no—
107C2×M16—1061no—
108C2×M16C2×#615711yescharacteristic
109C2×C42—531no—
110M32—2121no—
111C4⋊C8—2121no—
112C4⋊C8—2121no—
113#104—2121no—
114#104—2121no—
115#101—2121no—
116#100—2121no—
117C22⋊C8—1062no—
118#97—2121no—
119#41—41no—
120#41—41no—
121#41—41no—
122#41—41no—
123C2×Q8—21no—
124C2×D4—21no—
125C2×D4—41no—
126C22×C4—41no—
127C22×C4—1061no—
128C22×C4C22⋊#615711yescharacteristic
129Q16—41no—
130SD16—42no—
131D8—41no—
132M16—41no—
133M16—2121no—
134M16—2121no—
135M16—2121no—
136C2×C8—21no—
137C2×C8—1061no—
138C2×C8—2121no—
139C2×C8—2122no—
140C42—531no—
141C42—531no—
142C42—1061no—
143C16—4241no—
144C23—41no—
145C23#1313328111yescharacteristic
146Q8—41no—
147Q8—41no—
148D4—41no—
149D4—41no—
150D4—81no—
151D4—81no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4—1061no—
158C2×C4—1061no—
159C2×C4—4241no—
160C2×C4#1313327511yescharacteristic
161C2×C4#1313327911yescharacteristic
162C8—41no—
163C8—4241no—
164C8—4242no—
165C8—2121no—
166C22—21no—
167C22—41no—
168C22—81no—
169C22—161no—
170C22#1681147411yescharacteristic
171C4—21no—
172C4—81no—
173C4—2121no—
174C4—2121no—
175C4#1681152311yescharacteristic
176C4#1681153111yescharacteristic
177C2—21no—
178C2—41no—
179C2—161no—
180C2#2028969711yescharacteristic
subgroupG/H
centerC2#20289697
commutatorC2×M16×C53C2×C4
FrattiniC2×M16C2×#6157
Fitting, prime radicalC53×#575C4
socleC2×C53#1345
cosocle#16810646C22
hypercenter#575#6157
hypocenterC53#7415
LCSG ▷ C2×M16×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #20289662C2semidirect
2. #20289621C2semidirect
101. C53#7415semidirect

Maximal subgroups

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

subgroupindex
#202896622
#202896212
#202759992
#741553