Finite Group Databasebeta

← #27271611

Subgroups — #27271611

Order 13568.

244 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 39 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#20272836C211yesmaximal, characteristic, complemented
2#20272641C211yesmaximal, characteristic, complemented
3C53×#2017C211yesmaximal, characteristic
4#16810597—21no—
5C22⋊#11719908—21no—
6C53×#575—21no—
7C53×#575C411yescharacteristic, complemented
8C53×#408—21no—
9C53×#408C2211yescharacteristic
10C53×#338C411yescharacteristic, complemented
11#13133176—41no—
12C2×#11719908—41no—
13#13133132—22no—
14#13133132—41no—
15C2×#41×C53—21no—
16C2×#41×C53D411yescharacteristic
17#137×C53—41no—
18#137×C53—41no—
19#137×C53—41no—
20AGL(1,ℤ/8)×C53—41no—
21AGL(1,ℤ/8)×C53—41no—
22AGL(1,ℤ/8)×C53—41no—
23#135×C53—42no—
24#135×C53—41no—
25C2×M16×C53—21no—
26C2×M16×C53C2×C411yescharacteristic
27C22×C8×C53D411yescharacteristic
28C22⋊C8×C53—41no—
29C22⋊C8×C53—42no—
30#41×C53—41no—
31#41×C53—41no—
32#41×C53—41no—
33#41×C53—41no—
34#41×C53—41no—
35#41×C53—41no—
36C2×Q8×C53—21no—
37C2×Q8×C53SD1611yescharacteristic
38C2×D4×C53—21no—
39C2×D4×C53—21no—
40C2×D4×C53—41no—
41C2×D4×C53D811yescharacteristic
42C22×C4×C53—21no—
43C22×C4×C53—41no—
44C22×C4×C53C22⋊C411yescharacteristic
45Q16×C53—42no—
46Q16×C53—41no—
47SD16×C53—42no—
48SD16×C53—42no—
49SD16×C53—42no—
50D8×C53—42no—
51D8×C53—41no—
52M16×C53—41no—
53M16×C53—41no—
54M16×C53—41no—
55C2×C8×C53—22no—
56C2×C8×C53—21no—
57C2×C8×C53—21no—
58C2×C8×C53—41no—
59C22⋊Dic53—42no—
60C4⋊Dic53—81no—
61C4×Dic53—81no—
62#11719908—81no—
63C23×C53—21no—
64C23×C53—41no—
65C23×C53#10411yescharacteristic
66Q8×C53—41no—
67Q8×C53—41no—
68Q8×C53—41no—
69D4×C53—41no—
70D4×C53—41no—
71D4×C53—41no—
72D4×C53—81no—
73D4×C53—81no—
74D4×C53—81no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53—81no—
82C2×C4×C53—81no—
83C2×C4×C53#9911yescharacteristic
84C2×C4×C53#10211yescharacteristic
85C2×Dic53—82no—
86C8×C53—42no—
87C8×C53—41no—
88C8×C53—41no—
89#7406—531nomaximal, Sylow
90C22×C53—21no—
91C22×C53—41no—
92C22×C53—41no—
93C22×C53—81no—
94C22×C53—82no—
95C22×C53—81no—
96C22×C53#32711yescharacteristic
97C4×C53—21no—
98C4×C53—41no—
99C4×C53—81no—
100C4×C53AΣL(1,𝔽16)11yescharacteristic
101C4×C53#35311yescharacteristic
102Dic53—162no—
103#2017D5311yescharacteristic
104#1391—531no—
105#1315—531no—
106C2×C53—21no—
107C2×C53—41no—
108C2×C53—81no—
109C2×C53—161no—
110C2×C53#134311yescharacteristic
111#575—21no—
112#575Dic5311yescharacteristic, complemented
113#410—1061no—
114#408—21no—
115#408C2×D5311yescharacteristic
116C22⋊C16—1061no—
117#338Dic5311yescharacteristic, complemented
118C53#740611yescharacteristic, complemented, Sylow
119C2×#41—21no—
120C2×#41C22⋊D5311yescharacteristic
121#137—41no—
122#137—41no—
123#137—41no—
124AGL(1,ℤ/8)—41no—
125AGL(1,ℤ/8)—41no—
126AGL(1,ℤ/8)—41no—
127#135—41no—
128#135—42no—
129C2×M16—21no—
130C2×M16C2×Dic5311yescharacteristic
131C22×C8C22⋊D5311yescharacteristic
132#117—2121no—
133C2×C16—2121no—
134#99—1062no—
135#99—2121no—
136C22⋊C8—41no—
137C22⋊C8—42no—
138#41—41no—
139#41—41no—
140#41—41no—
141#41—41no—
142#41—41no—
143#41—41no—
144C2×Q8—21no—
145C2×Q8#1171992311yescharacteristic
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—41no—
149C2×D4#1171992211yescharacteristic
150C22×C4—21no—
151C22×C4—41no—
152C22×C4C22⋊Dic5311yescharacteristic
153Q16—41no—
154Q16—42no—
155SD16—42no—
156SD16—42no—
157SD16—42no—
158D8—41no—
159D8—42no—
160M16—41no—
161M16—41no—
162M16—41no—
163C2×C8—22no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—41no—
167C4⋊C4—4241no—
168C22⋊C4—2122no—
169C42—4241no—
170C16—4241no—
171C23—21no—
172C23—41no—
173C23#1313314411yescharacteristic
174Q8—41no—
175Q8—41no—
176Q8—41no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—81no—
181D4—81no—
182D4—81no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—4242no—
192C2×C4#1313313211yescharacteristic
193C2×C4#1313313811yescharacteristic
194C8—42no—
195C8—41no—
196C8—41no—
197C22—21no—
198C22—41no—
199C22—41no—
200C22—81no—
201C22—82no—
202C22—81no—
203C22#1681035711yescharacteristic
204C4—21no—
205C4—41no—
206C4—81no—
207C4—8482no—
208C4#1681041011yescharacteristic
209C4#1681041611yescharacteristic
210C2—21no—
211C2—41no—
212C2—81no—
213C2—161no—
214C2#2027271211yescharacteristic
subgroupG/H
centerC4#16810416
commutatorC2×M16×C53C2×C4
Frattini#408C2×D53
Fitting, prime radicalC53×#2017C2
socleC2×C53#1343
cosocleC53×#408C22
hypercenter#2017D53
hypocenterC53#7406
LCSG ▷ C2×M16×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #408 ◁ #2017

Complements

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

NH ≅ G/Ntype
1. #20272836C2semidirect
2. #20272641C2semidirect
7. C53×#575C4semidirect
10. C53×#338C4semidirect
112. #575Dic53semidirect
117. #338Dic53semidirect
118. C53#7406semidirect

Maximal subgroups

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

subgroupindex
#202728362
#202726412
C53×#20172
#740653