Finite Group Databasebeta

← #27296049

Subgroups — #27296049

Order 13568.

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

normal subgroups only

This group has 234 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#20276592C211yesmaximal, characteristic
2#20273628C211yesmaximal, characteristic, complemented
3#20273538C211yesmaximal, characteristic, complemented
4#20272681C211yesmaximal, characteristic, complemented
5#20272681C211yesmaximal, characteristic, complemented
6#20272637C211yesmaximal, characteristic, complemented
7C53×#1368C211yesmaximal, characteristic, complemented
8C2×D848C411yescharacteristic
9C4×D424C2211yescharacteristic
10C4×D424C2211yescharacteristic
11#16810700—21no—
12#16810688—21no—
13#16810451C411yescharacteristic
14#16810425C411yescharacteristic
15#16810425C411yescharacteristic
16#16810399C2211yescharacteristic
17#16810350—21no—
18#16810350—21no—
19#16810344C2211yescharacteristic
20C53×#346C2211yescharacteristic
21C53×#334C2211yescharacteristic
22C53×#334C2211yescharacteristic
23C2×D424C2×C411yescharacteristic
24C2×D424C2×C411yescharacteristic
25C2×#11719912—41no—
26C2×C8×D53—41no—
27C4×D212—21no—
28C4×D212—21no—
29D848—22no—
30C2×#11719908—21no—
31#13133151C2×C411yescharacteristic
32#13133151C2×C411yescharacteristic
33#13133139—21no—
34#13133139—21no—
35#13133129—41no—
36#13133129—41no—
37C4×#88322D411yescharacteristic
38C4×#88322D411yescharacteristic
39C2×C16×C53C2×C411yescharacteristic
40C2×C16×C53C2×C411yescharacteristic
41C4⋊C8×C53—21no—
42C4⋊C8×C53—21no—
43C4×C8×C53C2311yescharacteristic
44C2×D212—21no—
45C2×D212—21no—
46C2×C4×D53—41no—
47C2×C4×D53—41no—
48C2×C8×C53—41no—
49C2×C8×C53—41no—
50C2×C8×C53C22×C411yescharacteristic
51C2×C8×C53#4111yescharacteristic
52C42×C53C2×D411yescharacteristic
53#11719921—41no—
54#11719921—41no—
55C4⋊Dic53—21no—
56C4⋊Dic53—21no—
57C2×#88322—21no—
58C2×#88322—21no—
59D424—21no—
60D424—21no—
61D424—21no—
62D424M1612yescomplemented
63#11719912—82no—
64C8×D53—82no—
65C16×C53—21no—
66C16×C53—21no—
67#11719908—22no—
68C22×D53—41no—
69C22×D53—41no—
70C2×C4×C53C4×D411yescharacteristic
71C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
72C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
73C2×Dic53—41no—
74C2×Dic53—41no—
75D212—22no—
76D212—41no—
77D212—41no—
78D212—41no—
79C4×D53—81no—
80C4×D53—81no—
81C4×D53—81no—
82C4×D53—81no—
83C8×C53—21no—
84C8×C53—81no—
85C8×C53—81no—
86C8×C53C2×M1611yescharacteristic
87C8×C53#13111yescharacteristic
88#88322—41no—
89#88322—41no—
90#12078—531nomaximal, Sylow
91C22×C53#44211yescharacteristic
92C2×D53—42no—
93C2×D53—81no—
94C2×D53—81no—
95C2×D53—81no—
96C2×D53—81no—
97C2×D53—81no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53#43511yescharacteristic
101C4×C53#44311yescharacteristic
102Dic53—81no—
103Dic53—81no—
104#2177—531no—
105#1594—531no—
106#1580—531no—
107#1368D5311yescharacteristic, complemented
108#1333—531no—
109#1333—531no—
110#1313—531no—
111C2×C53#158611yescharacteristic
112C2×C53AGL(1,ℤ/16)11yescharacteristic
113C2×C53AGL(1,ℤ/16)11yescharacteristic
114D53—82no—
115D53—161no—
116C2×D16—531no—
117C4×D8—531no—
118C4×D8—531no—
119#436—1061no—
120C8×D4—1061no—
121#367—531no—
122#357—531no—
123#357—531no—
124#346—531no—
125#346C2×D5311yescharacteristic
126#334C2×D5311yescharacteristic
127#334C2×D5311yescharacteristic
128D4⋊C8—1061no—
129D4⋊C8—1061no—
130#322—531no—
131C53#1207811yescharacteristic, complemented, Sylow
132C2×D8—531no—
133C2×D8—531no—
134C2×M16—2121no—
135C22×C8—2121no—
136C4×D4—1061no—
137C4×D4—1061no—
138D16—1062no—
139C2×C16—1061no—
140C2×C16C4×D5311yescharacteristic
141C2×C16C4×D5311yescharacteristic
142#107—531no—
143#107—531no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146#102—1061no—
147#102—1061no—
148C22⋊C8—2121no—
149C22⋊C8—2121no—
150C4×C8—531no—
151C4×C8—531no—
152C4×C8C22×D5311yescharacteristic
153C2×D4—1061no—
154C2×D4—1061no—
155C22×C4—2121no—
156C22×C4—2121no—
157D8—532no—
158D8—1061no—
159D8—1061no—
160D8—1061no—
161M16—4242no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—1061no—
165C2×C8—1061no—
166C2×C8—4242no—
167C2×C8C2×C4×D5311yescharacteristic
168C2×C8#1171994911yescharacteristic
169C4⋊C4—1061no—
170C4⋊C4—1061no—
171C22⋊C4—2121no—
172C22⋊C4—2121no—
173C42D4×D5311yescharacteristic
174C16—21no—
175C16—21no—
176C16—1062no—
177C23—2121no—
178C23—2121no—
179D4—1062no—
180D4—2121no—
181D4—2121no—
182D4—2121no—
183C2×C4—4241no—
184C2×C4—4241no—
185C2×C4—4241no—
186C2×C4—4241no—
187C2×C4—2121no—
188C2×C4—2121no—
189C2×C4#1313317911yescharacteristic
190C2×C4#1313324411yescharacteristic
191C2×C4#1313324411yescharacteristic
192C8—21no—
193C8—81no—
194C8—81no—
195C8—2121no—
196C8—2121no—
197C8M16×D5311yescharacteristic
198C8#1313322311yescharacteristic
199C22—4241no—
200C22—4241no—
201C22—4241no—
202C22—4241no—
203C22—4241no—
204C22—2122no—
205C22#1681073411yescharacteristic
206C4—21no—
207C4—21no—
208C4—4241no—
209C4—4241no—
210C4#1681069311yescharacteristic
211C4#1681073911yescharacteristic
212C2—8481no—
213C2—4242no—
214C2#2027357811yescharacteristic
215C2#2027661211yescharacteristic
216C2#2027661211yescharacteristic
subgroupG/H
centerC22#16810734
commutatorC2×C8×C53C22×C4
FrattiniC4×C8C22×D53
FittingC53×#1368C2
socleC22×C53#442
prime radicalC2×D848C4
cosocleC4×C8×C53C23
hypercenter#1368D53
hypocenterC53#12078
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1368

Complements

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

NH ≅ G/Ntype
2. #20273628C2semidirect
3. #20273538C2semidirect
4. #20272681C2semidirect
5. #20272681C2semidirect
6. #20272637C2semidirect
7. C53×#1368C2semidirect
62. D424M16semidirect
107. #1368D53semidirect
131. C53#12078semidirect

Maximal subgroups

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

subgroupindex
#202765922
#202736282
#202735382
#202726812
#202726372
C53×#13682
#1207853