Finite Group Databasebeta

← #27298370

Subgroups — #27298370

Order 13568.

212 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 63 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#20276967C211yesmaximal, characteristic
2#20276620C211yesmaximal, characteristic
3#20274086C211yesmaximal, characteristic
4#20274082C211yesmaximal, characteristic
5#20272788C211yesmaximal, characteristic
6#20272765C211yesmaximal, characteristic
7C53×#1362C211yesmaximal, characteristic
8C2×#13133163—21no—
9C2×#13133163—21no—
10#16811003C2211yescharacteristic
11#16810909—21no—
12#16810905—21no—
13C8×Dic106C2211yescharacteristic
14#16810441—21no—
15#16810398—21no—
16C4×#11719908C2211yescharacteristic
17#16810376C2211yescharacteristic
18#16810369—21no—
19#16810368—21no—
20C53×#494C2211yescharacteristic
21C53×C4⋊C16C2211yescharacteristic
22C53×#335C2211yescharacteristic
23C2×Dic212—21no—
24C2×Dic212—21no—
25#13133217—21no—
26C4×Dic106D411yescharacteristic
27#13133163—42no—
28#13133163—42no—
29C2×#11719908D411yescharacteristic
30C2×#11719908D411yescharacteristic
31#13133149—41no—
32#13133148—41no—
33C4⋊#88322—21no—
34C4⋊#88322D411yescharacteristic
35#13133146—21no—
36#13133142—41no—
37#13133140—41no—
38C8×Dic53—21no—
39C2×Q16×C53—21no—
40C2×Q16×C53—21no—
41C4⋊Q8×C53—21no—
42C2×C16×C53—21no—
43C4⋊C8×C53—21no—
44C4×C8×C53C2311yescharacteristic
45C2×Q8×C53—41no—
46C2×Dic106—41no—
47C2×Dic106D811yescharacteristic
48Q16×C53—41no—
49Q16×C53—41no—
50Q16×C53—81no—
51C2×C8×C53—41no—
52C2×C8×C53C2×D411yescharacteristic
53C2×C8×C53#4111yescharacteristic
54C4⋊C4×C53—41no—
55C4⋊C4×C53—41no—
56C42×C53C2×D411yescharacteristic
57C4⋊Dic53—41no—
58C4⋊Dic53—41no—
59C4⋊Dic53D811yescharacteristic
60#11719919—21no—
61C4×Dic53—21no—
62C2×#88322—21no—
63C2×#88322—41no—
64Dic212—41no—
65Dic212—41no—
66Dic212—81no—
67C16×C53—41no—
68#11719908—21no—
69#11719908SD1612yes—
70Q8×C53—81no—
71Q8×C53—81no—
72C2×C4×C53—81no—
73C2×C4×C53#12111yescharacteristic
74C2×C4×C53C2×D811yescharacteristic
75C2×C4×C53#13511yescharacteristic
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—81no—
79Dic106—21no—
80Dic106—21no—
81Dic106—81no—
82Dic106—81no—
83C8×C53—21no—
84C8×C53—81no—
85C8×C53C2×SD1611yescharacteristic
86C8×C53AGL(1,ℤ/8)11yescharacteristic
87#88322—41no—
88#88322—81no—
89#12443—531nomaximal, Sylow
90C22×C53#46611yescharacteristic
91C4×C53—21no—
92C4×C53—161no—
93C4×C53#46011yescharacteristic
94C4×C53#47211yescharacteristic
95C4×C53#51011yescharacteristic
96C4×C53#51011yescharacteristic
97Dic53—41no—
98Dic53—41no—
99Dic53—161no—
100#2247—531no—
101#2183—531no—
102#1670—531no—
103#1670—531no—
104#1372—531no—
105#1362—531no—
106#1362D5311yescharacteristic
107C2×C53#166811yescharacteristic
108C2×C53#221911yescharacteristic
109C2×C53#222211yescharacteristic
110C2×Q32—1061no—
111C2×Q32—1061no—
112#494—531no—
113#494C2×D5311yescharacteristic
114#475—1061no—
115#475—1061no—
116C8×Q8—531no—
117C4⋊C16—1061no—
118C4⋊C16C2×D5311yescharacteristic
119#346—1061no—
120C4×C16—531no—
121#335—531no—
122#335C2×D5311yescharacteristic
123#332—1061no—
124#332—1061no—
125C53#1244311yescharacteristic, complemented, Sylow
126C2×Q16—21no—
127C2×Q16—21no—
128C2×Q16—1061no—
129C2×Q16—1061no—
130C4⋊Q8—21no—
131C4⋊Q8—1061no—
132C4×Q8—531no—
133Q32—2122no—
134Q32—2122no—
135C2×C16—21no—
136C2×C16—531no—
137C2×C16—531no—
138#106—2121no—
139#106—2121no—
140C4⋊C8—21no—
141C4⋊C8—531no—
142C4⋊C8—1061no—
143C4⋊C8—1061no—
144#103—2121no—
145#103—2121no—
146C4×C8—1061no—
147C4×C8C22×D5311yescharacteristic
148C2×Q8—41no—
149C2×Q8—531no—
150C2×Q8—2121no—
151Q16—41no—
152Q16—41no—
153Q16—81no—
154Q16—4241no—
155Q16—2121no—
156Q16—2121no—
157C2×C8—41no—
158C2×C8—1061no—
159C2×C8—2121no—
160C2×C8#1171994511yescharacteristic
161C2×C8D4×D5311yescharacteristic
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—531no—
165C4⋊C4—1061no—
166C4⋊C4—2121no—
167C4⋊C4—2121no—
168C42—1061no—
169C42D4×D5311yescharacteristic
170C16—41no—
171C16—532no—
172C16—1061no—
173Q8—81no—
174Q8—81no—
175Q8—1061no—
176Q8—1061no—
177Q8—4241no—
178Q8—4241no—
179C2×C4—81no—
180C2×C4—1061no—
181C2×C4—1061no—
182C2×C4—4241no—
183C2×C4#1313319011yescharacteristic
184C2×C4D8×D5311yescharacteristic
185C2×C4#1313323811yescharacteristic
186C8—21no—
187C8—81no—
188C8—4241no—
189C8—2121no—
190C8SD16×D5311yescharacteristic
191C8#1313324111yescharacteristic
192C22#1681085811yescharacteristic
193C4—21no—
194C4—161no—
195C4—8481no—
196C4—2121no—
197C4—2121no—
198C4#1681082211yescharacteristic
199C4#1681089211yescharacteristic
200C4#1681106211yescharacteristic
201C4#1681106511yescharacteristic
202C2#2027407411yescharacteristic
203C2#2027682211yescharacteristic
204C2#2027683911yescharacteristic
subgroupG/H
centerC22#16810858
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#1362C2
socle, prime radicalC22×C53#466
hypercenter#1362D53
hypocenterC53#12443
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
125. C53#12443semidirect

Maximal subgroups

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

subgroupindex
#202769672
#202766202
#202740862
#202740822
#202727882
#202727652
C53×#13622
#1244353