Finite Group Databasebeta

← #27288978

Subgroups — #27288978

Order 13568.

236 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 69 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#20273443C211yesmaximal, characteristic
2#20273413C211yesmaximal, characteristic
3#20273189C211yesmaximal, characteristic
4#20272626C211yesmaximal, characteristic
5#20272585C211yesmaximal, characteristic
6#20272577C211yesmaximal, characteristic
7C53×#1571C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810680C2211yescharacteristic
10#16810680C2211yescharacteristic
11#16810680C2211yescharacteristic
12#16810371—21no—
13#16810370—21no—
14#16810370—21no—
15#16810368—21no—
16#16810355—22no—
17C53×#431C2211yescharacteristic
18C53×#423C2211yescharacteristic
19C2×C4⋊C8×C53C2211yescharacteristic
20C53×#335—22no—
21#13133217D411yescharacteristic
22#13133217D411yescharacteristic
23#13133209—21no—
24#13133196—21no—
25#13133196—21no—
26C4×Dic106—21no—
27C4×Dic106—21no—
28#13133176—21no—
29C4⋊#88322—21no—
30C4⋊#88322—21no—
31C4⋊#88322—21no—
32C22⋊#88322—21no—
33C22⋊#88322—21no—
34C22⋊#88322—21no—
35#13133128—21no—
36#13133128—21no—
37#13133128—21no—
38C2×M16×C53—21no—
39C22×C8×C53—21no—
40C2×C42×C53C2311yescharacteristic
41C4⋊C8×C53—21no—
42C4⋊C8×C53—21no—
43C4⋊C8×C53—21no—
44C4⋊C8×C53—21no—
45C4⋊C8×C53—21no—
46C22⋊C8×C53—21no—
47C4×C8×C53D412yes—
48C22×C4×C53#4111yescharacteristic
49C22×C4×C53#4111yescharacteristic
50C22×C4×C53#4111yescharacteristic
51C2×Dic106—21no—
52C2×Dic106—21no—
53M16×C53—81no—
54C2×C8×C53—22no—
55C2×C8×C53—41no—
56C2×C8×C53—41no—
57C2×C8×C53—81no—
58C42×C53C2×D411yescharacteristic
59C42×C53C2×D411yescharacteristic
60C42×C53#4111yescharacteristic
61C42×C53#4111yescharacteristic
62C22⋊Dic53—41no—
63C4⋊Dic53—21no—
64C4⋊Dic53—21no—
65C4⋊Dic53—21no—
66C4⋊Dic53—21no—
67#11719919—41no—
68#11719919—41no—
69C4×Dic53—41no—
70C2×#88322—41no—
71C2×#88322—41no—
72C2×#88322—41no—
73C23×C53#12611yescharacteristic
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—41no—
77C2×C4×C53—41no—
78C2×C4×C53#12111yescharacteristic
79C2×C4×C53#12311yescharacteristic
80C2×C4×C53#12311yescharacteristic
81C2×C4×C53#12311yescharacteristic
82C2×C4×C53#12411yescharacteristic
83C2×C4×C53#12611yescharacteristic
84C2×Dic53—41no—
85C2×Dic53—41no—
86Dic106—41no—
87Dic106—41no—
88Dic106—81no—
89C8×C53—42no—
90C8×C53—81no—
91C8×C53—81no—
92#88322—81no—
93#88322—81no—
94#88322—81no—
95#10646—531nomaximal, Sylow
96C22×C53—21no—
97C22×C53—21no—
98C22×C53—21no—
99C22×C53#39711yescharacteristic
100C4×C53—21no—
101C4×C53—21no—
102C4×C53—21no—
103C4×C53—21no—
104C4×C53—21no—
105C4×C53#47211yescharacteristic
106C4×C53#47311yescharacteristic
107Dic53—81no—
108Dic53—81no—
109#1571D5311yescharacteristic
110#1559—531no—
111#1552—531no—
112#1497—531no—
113#1307—531no—
114#1290—531no—
115#1287—531no—
116C2×C53—41no—
117C2×C53#207811yescharacteristic
118C2×C53#208011yescharacteristic
119C2×C53#208211yescharacteristic
120#533—531no—
121#432—531no—
122#432—531no—
123#432—531no—
124#431C2×D5311yescharacteristic
125#423C2×D5311yescharacteristic
126C2×C4⋊C8C2×D5311yescharacteristic
127#335—22no—
128#333—1061no—
129#332—1061no—
130#332—1061no—
131#332—1061no—
132Q8⋊C8—1062no—
133C53#1064611yescharacteristic, complemented, Sylow
134C2×M16—21no—
135C22×C8—21no—
136C4⋊Q8—531no—
137C4⋊Q8—531no—
138#125—1061no—
139C22⋊Q8—1061no—
140C22⋊Q8—1061no—
141C4×Q8—1061no—
142C4×Q8—1061no—
143#117—1061no—
144C2×C42C22×D5311yescharacteristic
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—1061no—
151C4⋊C8—1061no—
152C4⋊C8—1061no—
153C22⋊C8—21no—
154C22⋊C8—1061no—
155C22⋊C8—1061no—
156C22⋊C8—1061no—
157#97—1061no—
158#97—1061no—
159#97—1061no—
160C4×C8C22⋊D5312yes—
161C2×Q8—1061no—
162C2×Q8—1061no—
163C22×C4#1171994511yescharacteristic
164C22×C4#1171994711yescharacteristic
165C22×C4#1171994911yescharacteristic
166M16—81no—
167C2×C8—22no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—81no—
171C2×C8—2121no—
172C2×C8—2121no—
173C2×C8—2121no—
174C4⋊C4—1061no—
175C4⋊C4—1061no—
176C4⋊C4—1061no—
177C4⋊C4—1061no—
178C4⋊C4—2121no—
179C4⋊C4—2121no—
180C22⋊C4—2121no—
181C42—2121no—
182C42#1171994511yescharacteristic
183C42D4×D5311yescharacteristic
184C42#1171994911yescharacteristic
185C42C2×C22⋊D5311yescharacteristic
186C23#1313321211yescharacteristic
187Q8—4241no—
188Q8—2121no—
189Q8—2121no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—2121no—
195C2×C4—2121no—
196C2×C4#1313318811yescharacteristic
197C2×C4#1313320011yescharacteristic
198C2×C4#1313320211yescharacteristic
199C2×C4#1313320211yescharacteristic
200C2×C4#1313320611yescharacteristic
201C2×C4#1313321211yescharacteristic
202C8—42no—
203C8—81no—
204C8—81no—
205C8—4241no—
206C8—4241no—
207C8—4241no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22#1681054911yescharacteristic
212C4—21no—
213C4—21no—
214C4—21no—
215C4—21no—
216C4—21no—
217C4—4241no—
218C4—4241no—
219C4#1681089011yescharacteristic
220C4#1681089511yescharacteristic
221C2—41no—
222C2#2027615611yescharacteristic
223C2#2027616611yescharacteristic
224C2#2027618011yescharacteristic
subgroupG/H
centerC22#16810549
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1571C2
socleC22×C53#397
prime radicalC23×C53#126
hypercenter#1571D53
hypocenterC53#10646
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
133. C53#10646semidirect

Maximal subgroups

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

subgroupindex
#202734432
#202734132
#202731892
#202726262
#202725852
#202725772
C53×#15712
#1064653