Finite Group Databasebeta

← #27297495

Subgroups — #27297495

Order 13568.

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

normal subgroups only

This group has 372 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#20274320C211yesmaximal, characteristic, complemented
3#20272752C211yesmaximal, characteristic, complemented
4#20272688C212yesmaximal, complemented
5C53×#3292C211yesmaximal, characteristic
6#16811063—22no—
7#16811003C2211yescharacteristic, complemented
8#16810969—22no—
9#16810444C2212yescomplemented
10#16810438—22no—
11C8×#88322C2211yescharacteristic, complemented
12C53×#576—22no—
13C53×#493C2211yescharacteristic
14C53×#472—22no—
15C53×#453—22no—
16C53×#443C2212yes—
17C2×Dic212—22no—
18#13133217—21no—
19#13133163—42no—
20#13133161—42no—
21#13133157—22no—
22#13133138—44no—
23C4×#88322—21no—
24H(5,𝔽2)×C53—22no—
25AGL(1,ℤ/8)×C53—24no—
26AGL(1,ℤ/8)×C53—44no—
27#135×C53—42no—
28#135×C53D412yes—
29C2×D8×C53—22no—
30C2×D8×C53—42no—
31#131×C53—22no—
32#127×C53D412yes—
33#108×C53D412yes—
34#104×C53—22no—
35#100×C53—22no—
36C4×C8×C53C2311yescharacteristic
37#41×C53—22no—
38#41×C53—44no—
39C2×D4×C53—22no—
40C2×D4×C53—42no—
41C2×D4×C53—44no—
42C2×D4×C53—82no—
43C2×Dic106—41no—
44Q16×C53D812yes—
45SD16×C53—22no—
46SD16×C53—44no—
47D8×C53—44no—
48D8×C53—42no—
49D8×C53—42no—
50D8×C53D812yes—
51M16×C53—22no—
52M16×C53—42no—
53C2×C8×C53—42no—
54C2×C8×C53C2×D412yes—
55C42×C53C2×D411yescharacteristic
56C4⋊Dic53—42no—
57C2×#88322—22no—
58Dic212—42no—
59Dic212—42no—
60#11719908—42no—
61C23×C53—42no—
62C23×C53—82no—
63Q8×C53—22no—
64D4×C53—22no—
65D4×C53—42no—
66D4×C53—44no—
67D4×C53—84no—
68D4×C53—84no—
69C2×C4×C53—21no—
70C2×C4×C53—42no—
71C2×C4×C53—82no—
72C2×C4×C53#12011yescharacteristic
73C2×Dic53—81no—
74Dic106—82no—
75C8×C53—42no—
76C8×C53C2×D812yes—
77C8×C53AGL(1,ℤ/8)12yes—
78#88322—42no—
79#12312—531nomaximal, Sylow
80C22×C53—42no—
81C22×C53—82no—
82C22×C53—162no—
83C22×C53—322no—
84C22×C53#45311yescharacteristic
85C4×C53—22no—
86C4×C53—42no—
87C4×C53#44712yes—
88Dic53—161no—
89#3292D5311yescharacteristic
90#2214—532no—
91#1709—531no—
92#1357—531no—
93#1335—532no—
94C2×C53—21no—
95C2×C53—82no—
96C2×C53—162no—
97C2×C53#161911yescharacteristic
98#576—22no—
99#510—1062no—
100#494—531no—
101#493C2×D5311yescharacteristic
102#486—1062no—
103#472—22no—
104#453—22no—
105#443C2×D5312yes—
106#364—532no—
107#362—1062no—
108C82—531no—
109C53#1231211yescharacteristic, complemented, Sylow
110H(5,𝔽2)—22no—
111AGL(1,ℤ/8)—24no—
112AGL(1,ℤ/8)—44no—
113#135—42no—
114#135C22⋊D5312yes—
115C2×Q16—1062no—
116C2×D8—22no—
117C2×D8—42no—
118#131—22no—
119C4⋊Q8—1061no—
120#127C22⋊D5312yes—
121Q32—2122no—
122SD32—2122no—
123M32—1062no—
124#108C22⋊D5312yes—
125#104—22no—
126#102—2124no—
127#100—22no—
128C4×C8—1061no—
129C4×C8C22×D5311yescharacteristic
130#41—22no—
131#41—44no—
132C2×Q8—2121no—
133C2×D4—22no—
134C2×D4—42no—
135C2×D4—44no—
136C2×D4—82no—
137Q16—2122no—
138Q16—2122no—
139Q16#1171992212yes—
140SD16—22no—
141SD16—44no—
142D8—44no—
143D8—42no—
144D8—42no—
145D8#1171992212yes—
146M16—22no—
147M16—42no—
148C2×C8—42no—
149C2×C8—1062no—
150C2×C8C2×C22⋊D5312yes—
151C4⋊C4—2122no—
152C42C2×C22⋊D5311yescharacteristic
153C16—2122no—
154C23—42no—
155C23—82no—
156Q8—22no—
157Q8—4242no—
158D4—22no—
159D4—42no—
160D4—44no—
161D4—84no—
162D4—84no—
163C2×C4—21no—
164C2×C4—42no—
165C2×C4—82no—
166C2×C4—4241no—
167C2×C4#1313318611yescharacteristic
168C8—42no—
169C8—2122no—
170C8C2×#1171992212yes—
171C8#1313324312yes—
172C22—42no—
173C22—82no—
174C22—162no—
175C22—322no—
176C22#1681079111yescharacteristic
177C4—22no—
178C4—42no—
179C4—8481no—
180C4#1681075712yes—
181C2—21no—
182C2—82no—
183C2—162no—
184C2#2027376911yescharacteristic
subgroupG/H
centerC2#20273769
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
Fitting, prime radicalC53×#3292C2
socleC2×C53#1619
hypercenter#3292D53
hypocenterC53#12312
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3292

Complements

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

NH ≅ G/Ntype
1. #20276796C2semidirect
2. #20274320C2semidirect
3. #20272752C2semidirect
4. #20272688C2semidirect
7. #16811003C22semidirect
9. #16810444C22semidirect
11. C8×#88322C22semidirect
109. C53#12312semidirect

Maximal subgroups

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

subgroupindex
#202767962
#202743202
#202727522
#202726882
C53×#32922
#1231253