Finite Group Databasebeta

← #27304787

Subgroups — #27304787

Order 13568.

284 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 55 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#20287853C211yesmaximal, characteristic
2C2×#16810460C211yesmaximal, characteristic, complemented
3#20276837C211yesmaximal, characteristic, complemented
4#20272906C211yesmaximal, characteristic, complemented
5#20272899C211yesmaximal, characteristic, complemented
6#20272858C211yesmaximal, characteristic, complemented
7C53×#2264C211yesmaximal, characteristic, complemented
8#16811425—21no—
9#16811060—22no—
10#16811058—21no—
11#16811056—21no—
12C2×D848C2211yescharacteristic
13D16×D53—21no—
14#16811039C2211yescharacteristic
15#16810895—21no—
16#16810460—22no—
17C2×#13133123C2211yescharacteristic, complemented
18#16810453C2211yescharacteristic, complemented
19#16810439—21no—
20C53×#508C2211yescharacteristic
21C2×Q32×C53C2211yescharacteristic, complemented
22C53×Q64—21no—
23C53×SD64—21no—
24C53×M64C2211yescharacteristic, complemented
25#13133263—41no—
26#13133248—41no—
27#13133244—42no—
28#13133242—41no—
29#13133240—22no—
30C2×D424—21no—
31D8×D53—22no—
32#13133223D411yescharacteristic
33#13133160—22no—
34D848—21no—
35D848D412yes—
36#13133158—22no—
37#13133157D411yescharacteristic
38#13133156—21no—
39C16×D53—21no—
40#13133153—21no—
41#13133136—41no—
42#13133123—21no—
43#135×C53—21no—
44C2×Q16×C53—21no—
45Q32×C53—21no—
46Q32×C53—21no—
47Q32×C53D411yescharacteristic
48SD32×C53—21no—
49D16×C53D411yescharacteristic
50C2×C16×C53C2311yescharacteristic
51C32×C53—21no—
52#41×C53—41no—
53C2×Q8×C53—41no—
54#11719949—42no—
55D4×D53—42no—
56D4×D53—82no—
57#11719945—81no—
58#11719945D811yescharacteristic
59C2×D212—41no—
60C2×D212—81no—
61Q16×C53—21no—
62Q16×C53—41no—
63Q16×C53—41no—
64SD16×C53—41no—
65D8×C53—21no—
66C2×C8×C53C2×D411yescharacteristic
67#11719925—81no—
68#11719924—42no—
69#11719924—81no—
70#11719922—42no—
71#11719917—41no—
72#11719917D811yescharacteristic
73D424—22no—
74D424—41no—
75#11719912—21no—
76C8×D53—21no—
77C16×C53C2×D411yescharacteristic
78C16×C53C2×D411yescharacteristic
79#11719908—21no—
80C22×D53—82no—
81C22×D53—81no—
82Q8×C53—41no—
83Q8×C53—81no—
84Q8×C53—81no—
85D4×C53—41no—
86D4×C53—81no—
87C2×C4×C53—81no—
88C2×C4×C53—81no—
89C2×C4×C53C2×D811yescharacteristic
90C22⋊D53—21no—
91C22⋊D53—82no—
92D212—42no—
93D212—82no—
94D212—81no—
95D212D1611yescharacteristic
96C4×D53—21no—
97C4×D53—82no—
98Dic106D1611yescharacteristic
99C8×C53#12011yescharacteristic
100C8×C53AGL(1,ℤ/8)11yescharacteristic
101#88322—21no—
102#88322—81no—
103#13671—531nomaximal, Sylow
104C22×C53—81no—
105C22×C53#50911yescharacteristic
106C2×D53—21no—
107C2×D53—82no—
108C2×D53—322no—
109C2×D53—321no—
110C4×C53—81no—
111C4×C53—161no—
112C4×C53#44711yescharacteristic
113C4×C53C2×D1611yescharacteristic
114Dic53—21no—
115#3415—531no—
116#2264D5311yescharacteristic, complemented
117C2×SD64—531no—
118#2222—531no—
119#1419—531no—
120#1417—531no—
121#1401—531no—
122C2×C53—21no—
123C2×C53—161no—
124C2×C53#218411yescharacteristic
125D53—41no—
126D53—162no—
127#576—1061no—
128#509—1061no—
129#509—1062no—
130#508—1061no—
131#508C2×D5311yescharacteristic
132C2×Q32C2×D5311yescharacteristic, complemented
133C2×D16—531no—
134C2×D16—1061no—
135#504—531no—
136#473—1061no—
137Q64—21no—
138SD64—21no—
139SD64—1062no—
140M64C2×D5311yescharacteristic, complemented
141C2×C32—531no—
142#368—531no—
143#362—1061no—
144C53#1367111yescharacteristic, complemented, Sylow
145H(5,𝔽2)—2121no—
146#137—2121no—
147AGL(1,ℤ/8)—2121no—
148AGL(1,ℤ/8)—2122no—
149#135—21no—
150#135—1062no—
151C2×Q16—21no—
152C2×D8—1062no—
153C2×D8—1061no—
154#131—531no—
155Q32—21no—
156Q32—21no—
157Q32C22⋊D5311yescharacteristic
158SD32—21no—
159SD32—1062no—
160D16—532no—
161D16—1062no—
162D16—1061no—
163D16C22⋊D5311yescharacteristic
164M32—531no—
165M32—1061no—
166C2×C16—1061no—
167C2×C16C22×D5311yescharacteristic
168#108—1061no—
169#101—2121no—
170C32—21no—
171C32—1061no—
172#41—41no—
173#41—531no—
174#41—4241no—
175#41—2122no—
176C2×Q8—41no—
177C2×D4—4241no—
178C2×D4—4242no—
179C2×D4—2122no—
180C2×D4—2121no—
181Q16—21no—
182Q16—41no—
183Q16—41no—
184Q16—4241no—
185SD16—41no—
186SD16—4241no—
187SD16—2122no—
188D8—21no—
189D8—1062no—
190D8—2122no—
191D8—2121no—
192M16—531no—
193M16—1061no—
194M16—2121no—
195C2×C8—1061no—
196C2×C8D4×D5311yescharacteristic
197C16—1061no—
198C16D4×D5311yescharacteristic
199C16C2×C22⋊D5311yescharacteristic
200C23—4242no—
201C23—4241no—
202Q8—41no—
203Q8—81no—
204Q8—81no—
205Q8—531no—
206D4—41no—
207D4—81no—
208D4—531no—
209D4—1061no—
210D4—4242no—
211D4—4242no—
212D4—4241no—
213D4—2122no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—1061no—
217C2×C4—4242no—
218C2×C4D8×D5311yescharacteristic
219C8—1061no—
220C8—4241no—
221C8#1313318511yescharacteristic
222C8#1313324111yescharacteristic
223C22—81no—
224C22—1061no—
225C22—16961no—
226C22—16962no—
227C22—4242no—
228C22#1681105711yescharacteristic
229C4—81no—
230C4—161no—
231C4—1061no—
232C4#1681075511yescharacteristic
233C4D16×D5311yescharacteristic
234C2—21no—
235C2—161no—
236C2—8482no—
237C2—2121no—
238C2#2027662511yescharacteristic
subgroupG/H
centerC2#20276625
commutator, cosocleC2×C16×C53C23
FrattiniC2×C16C22×D53
FittingC53×#2264C2
socleC2×C53#2184
prime radical#20287853C2
hypercenter#2264D53
hypocenterC53#13671
LCSG ▷ C2×C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2264

Complements

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

NH ≅ G/Ntype
2. C2×#16810460C2semidirect
3. #20276837C2semidirect
4. #20272906C2semidirect
5. #20272899C2semidirect
6. #20272858C2semidirect
7. C53×#2264C2semidirect
17. C2×#13133123C22semidirect
18. #16810453C22semidirect
21. C2×Q32×C53C22semidirect
24. C53×M64C22semidirect
116. #2264D53semidirect
132. C2×Q32C2×D53semidirect
140. M64C2×D53semidirect
144. C53#13671semidirect

Maximal subgroups

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

subgroupindex
#202878532
C2×#168104602
#202768372
#202729062
#202728992
#202728582
C53×#22642
#1367153