Finite Group Databasebeta

← #27297500

Subgroups — #27297500

Order 13568.

356 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 67 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#20276796C212yesmaximal, complemented
2#20274331C211yesmaximal, characteristic, complemented
3#20272754C211yesmaximal, characteristic, complemented
4#20272688C212yesmaximal, complemented
5C53×#3293C211yesmaximal, characteristic
6#16811063—22no—
7#16811026—22no—
8#16811003C2211yescharacteristic
9#16810444C2212yes—
10#16810438—22no—
11C8×#88322C2211yescharacteristic
12C53×#578—22no—
13C53×#494C2211yescharacteristic
14C53×#473—22no—
15C53×#456—22no—
16C53×#443C2212yes—
17C2×Dic212—22no—
18#13133217—21no—
19#13133163—42no—
20#13133161—42no—
21#13133157—22no—
22#13133150—44no—
23C4×#88322—21no—
242−1+4×C53—22no—
25#137×C53—24no—
26#137×C53—44no—
27#135×C53—42no—
28#135×C53D412yes—
29C2×Q16×C53—22no—
30C2×Q16×C53—42no—
31#131×C53—22no—
32C4⋊Q8×C53D412yes—
33#108×C53D412yes—
34#104×C53—22no—
35#101×C53—22no—
36C4×C8×C53C2311yescharacteristic
37#41×C53—22no—
38#41×C53—44no—
39#41×C53—82no—
40C2×Q8×C53—22no—
41C2×Q8×C53—44no—
42C2×Dic106—41no—
43Q16×C53—44no—
44Q16×C53—42no—
45Q16×C53—42no—
46Q16×C53D812yes—
47SD16×C53—22no—
48SD16×C53—44no—
49D8×C53D812yes—
50M16×C53—22no—
51M16×C53—42no—
52C2×C8×C53—42no—
53C2×C8×C53C2×D412yes—
54C4⋊C4×C53—42no—
55C42×C53C2×D411yescharacteristic
56C4⋊Dic53—42no—
57C2×#88322—22no—
58Dic212—42no—
59Dic212—42no—
60#11719908—42no—
61Q8×C53—22no—
62Q8×C53—44no—
63Q8×C53—82no—
64D4×C53—22no—
65D4×C53—42no—
66D4×C53—82no—
67C2×C4×C53—21no—
68C2×C4×C53—42no—
69C2×C4×C53—42no—
70C2×C4×C53—84no—
71C2×C4×C53#12011yescharacteristic
72C2×Dic53—81no—
73Dic106—82no—
74C8×C53—42no—
75C8×C53C2×D812yes—
76C8×C53AGL(1,ℤ/8)12yes—
77#88322—24no—
78#12313—531nomaximal, Sylow
79C22×C53—42no—
80C22×C53#45311yescharacteristic
81C4×C53—22no—
82C4×C53—42no—
83C4×C53—82no—
84C4×C53#44712yes—
85Dic53—161no—
86#3293D5311yescharacteristic
87#2214—532no—
88#1713—531no—
89#1358—531no—
90#1335—532no—
91C2×C53—21no—
92C2×C53—82no—
93C2×C53#161911yescharacteristic
94#578—22no—
95#510—1062no—
96#500—1062no—
97#494—531no—
98#494C2×D5311yescharacteristic
99#473—22no—
100#456—22no—
101#443C2×D5312yes—
102#364—532no—
103#362—1062no—
104C82—531no—
105C53#1231311yescharacteristic, complemented, Sylow
1062−1+4—22no—
107#137—24no—
108#137—44no—
109#135—42no—
110#135C22⋊D5312yes—
111C2×Q16—22no—
112C2×Q16—42no—
113C2×Q16—1062no—
114#131—22no—
115C4⋊Q8—1061no—
116C4⋊Q8C22⋊D5312yes—
117Q32—2122no—
118SD32—2122no—
119M32—1062no—
120#108C22⋊D5312yes—
121#107—2124no—
122#104—22no—
123#101—22no—
124C4×C8—1061no—
125C4×C8C22×D5311yescharacteristic
126#41—22no—
127#41—44no—
128#41—82no—
129C2×Q8—22no—
130C2×Q8—44no—
131C2×Q8—2121no—
132Q16—44no—
133Q16—42no—
134Q16—42no—
135Q16—2122no—
136Q16—2122no—
137Q16#1171992212yes—
138SD16—22no—
139SD16—44no—
140D8#1171992212yes—
141M16—22no—
142M16—42no—
143C2×C8—42no—
144C2×C8—1062no—
145C2×C8C2×C22⋊D5312yes—
146C4⋊C4—42no—
147C4⋊C4—2122no—
148C42C2×C22⋊D5311yescharacteristic
149C16—2122no—
150Q8—22no—
151Q8—44no—
152Q8—82no—
153Q8—4242no—
154D4—22no—
155D4—42no—
156D4—82no—
157C2×C4—21no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—84no—
161C2×C4—4241no—
162C2×C4#1313318611yescharacteristic
163C8—42no—
164C8—1064no—
165C8C2×#1171992212yes—
166C8#1313324312yes—
167C22—42no—
168C22#1681079111yescharacteristic
169C4—22no—
170C4—42no—
171C4—82no—
172C4—8481no—
173C4#1681075712yes—
174C2—21no—
175C2—82no—
176C2#2027376911yescharacteristic
subgroupG/H
centerC2#20273769
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
Fitting, prime radicalC53×#3293C2
socleC2×C53#1619
hypercenter#3293D53
hypocenterC53#12313
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3293

Complements

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

NH ≅ G/Ntype
1. #20276796C2semidirect
2. #20274331C2semidirect
3. #20272754C2semidirect
4. #20272688C2semidirect
105. C53#12313semidirect

Maximal subgroups

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

subgroupindex
#202767962
#202743312
#202727542
#202726882
C53×#32932
#1231353