Finite Group Databasebeta

← #27271639

Subgroups — #27271639

Order 13568.

190 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 37 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#20276002C211yesmaximal, characteristic
2#20272626C211yesmaximal, characteristic, complemented
3C53×#1327C211yesmaximal, characteristic, complemented
4#16811423C411yescharacteristic
5#16810680—21no—
6C2×#13133145—21no—
7#16810605—21no—
8#16810381C411yescharacteristic
9C53×#423C2211yescharacteristic
10C53×#349—21no—
11C2×#11719945—21no—
12#13133250—41no—
13#13133245—41no—
14#13133239—41no—
15C2×#11719917—21no—
16C4⋊#88322—41no—
17#13133145—41no—
18#13133145—41no—
19#13133135—41no—
20#13133133—41no—
21C22⋊#88322—22no—
22#13133128—41no—
23C2×M16×C53C2×C411yescharacteristic
24C2×M16×C53D411yescharacteristic
25C2×C42×C53D411yescharacteristic
26M32×C53—41no—
27C4⋊C8×C53—41no—
28C22×C4×C53—21no—
29C22×C4×C53C22⋊C411yescharacteristic
30C2×C22⋊D53—41no—
31#11719945—41no—
32#11719945—41no—
33#11719945—41no—
34#11719945—41no—
35C2×D212—21no—
36C2×C4×D53—41no—
37C2×Dic106—21no—
38M16×C53—41no—
39M16×C53—41no—
40C2×C8×C53—21no—
41C2×C8×C53—21no—
42C42×C53—21no—
43C42×C53D811yescharacteristic
44C42×C53SD1611yescharacteristic
45#11719917—41no—
46#11719917—41no—
47C2×#88322—42no—
48C2×#88322—41no—
49Dic212—41no—
50D424—41no—
51#11719913—42no—
52C16×C53—81no—
53C23×C53#10411yescharacteristic
54C22×D53—41no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—81no—
60C2×C4×C53#9911yescharacteristic
61C2×C4×C53#10211yescharacteristic
62C22⋊D53—81no—
63C22⋊D53—81no—
64C2×Dic53—41no—
65D212—41no—
66D212—41no—
67C4×D53—81no—
68C4×D53—81no—
69Dic106—41no—
70Dic106—41no—
71C8×C53—41no—
72C8×C53—41no—
73#88322—81no—
74#88322—82no—
75#7415—531nomaximal, Sylow
76C22×C53—21no—
77C22×C53—41no—
78C22×C53#32711yescharacteristic
79C2×D53—81no—
80C2×D53—161no—
81C4×C53—21no—
82C4×C53—41no—
83C4×C53—41no—
84C4×C53AΣL(1,𝔽16)11yescharacteristic
85C4×C53#35511yescharacteristic
86Dic53—81no—
87#2050—531no—
88#1327D5311yescharacteristic, complemented
89#1307—531no—
90C2×C53—21no—
91C2×C53—41no—
92C2×C53#134511yescharacteristic
93D53—161no—
94#575—531no—
95#432—1061no—
96#423C2×D5311yescharacteristic
97C2×#104—1061no—
98#413—1061no—
99#349—21no—
100#338—531no—
101C53#741511yescharacteristic, complemented, Sylow
102C2×#41—1061no—
103#137—2121no—
104AGL(1,ℤ/8)—2121no—
105#135—2121no—
106C2×M16—1061no—
107C2×M16C4×D5311yescharacteristic
108C2×M16C22⋊D5311yescharacteristic
109C2×C42D21211yescharacteristic
110M32—41no—
111C4⋊C8—41no—
112C4⋊C8—2121no—
113#104—2121no—
114#104—2121no—
115#101—2121no—
116#100—2121no—
117C22⋊C8—1062no—
118#97—2121no—
119#41—2121no—
120#41—2121no—
121#41—2121no—
122#41—2121no—
123C2×Q8—1061no—
124C2×D4—1061no—
125C2×D4—2121no—
126C22×C4—21no—
127C22×C4—2121no—
128C22×C4#1171992111yescharacteristic
129Q16—2121no—
130SD16—2122no—
131D8—2121no—
132M16—41no—
133M16—41no—
134M16—2121no—
135M16—2121no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—2122no—
139C2×C8—2121no—
140C42—21no—
141C42#1171991311yescharacteristic
142C42D42411yescharacteristic
143C16—81no—
144C23—2121no—
145C23#1313314311yescharacteristic
146Q8—2121no—
147Q8—2121no—
148D4—4241no—
149D4—4241no—
150D4—2121no—
151D4—2121no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—81no—
157C2×C4—4241no—
158C2×C4—4241no—
159C2×C4—2121no—
160C2×C4#1313313111yescharacteristic
161C2×C4#1313313911yescharacteristic
162C8—41no—
163C8—41no—
164C8—4241no—
165C8—4242no—
166C22—21no—
167C22—41no—
168C22—8481no—
169C22—4241no—
170C22#1681035811yescharacteristic
171C4—21no—
172C4—41no—
173C4—41no—
174C4—4241no—
175C4#1681040911yescharacteristic
176C4#1681041911yescharacteristic
177C2—21no—
178C2—41no—
179C2—8481no—
180C2#2027271911yescharacteristic
subgroupG/H
centerC2#20272719
commutatorC2×M16×C53C2×C4
Frattini#423C2×D53
FittingC53×#1327C2
socleC2×C53#1345
prime radical#16811423C4
cosocleC53×#423C22
hypercenter#1327D53
hypocenterC53#7415
LCSG ▷ C2×M16×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #423 ◁ #1327

Complements

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

NH ≅ G/Ntype
2. #20272626C2semidirect
3. C53×#1327C2semidirect
88. #1327D53semidirect
101. C53#7415semidirect

Maximal subgroups

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

subgroupindex
#202760022
#202726262
C53×#13272
#741553