Finite Group Databasebeta

← #27298251

Subgroups — #27298251

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#20276853C211yesmaximal, characteristic
2#20276761C211yesmaximal, characteristic
3#20274312C211yesmaximal, characteristic, complemented
4C8×D424C211yesmaximal, characteristic
5#20272769C211yesmaximal, characteristic, complemented
6#20272765C211yesmaximal, characteristic, complemented
7C53×#1367C211yesmaximal, characteristic, complemented
8C2×#13133162—21no—
9#16811028C2211yescharacteristic
10#16811021—21no—
11#16811003C2211yescharacteristic
12#16810974—21no—
13C4×D424C2211yescharacteristic
14C8×D212—21no—
15#16810449—21no—
16#16810428—21no—
17#16810425—21no—
18#16810376C2211yescharacteristic
19#16810352—21no—
20C53×C4⋊C16C2211yescharacteristic
21C53×C4⋊C16C2211yescharacteristic
22C82×C53C2211yescharacteristic
23C2×Dic212—21no—
24C2×Dic212—21no—
25C2×D424D411yescharacteristic
26C2×C8×D53—41no—
27#13133217—21no—
28#13133217—21no—
29C4×D212—21no—
30#13133162—42no—
31#13133151—21no—
32#13133151—21no—
33#13133151D411yescharacteristic
34#13133149—42no—
35C4⋊#88322—21no—
36#13133142—41no—
37#13133142—41no—
38#13133139—21no—
39#13133129—41no—
40C2×C16×C53—21no—
41C2×C16×C53—21no—
42C4×C8×C53D411yescharacteristic
43C4×C8×C53D411yescharacteristic
44C4×C8×C53C2311yescharacteristic
45C2×D212—21no—
46C2×C4×D53—41no—
47C2×Dic106—41no—
48C2×Dic106—41no—
49C2×C8×C53—21no—
50C2×C8×C53D811yescharacteristic
51C2×C8×C53D811yescharacteristic
52C2×C8×C53C2×D411yescharacteristic
53C2×C8×C53#4111yescharacteristic
54C42×C53C2×D411yescharacteristic
55#11719921—41no—
56C4⋊Dic53—21no—
57C4⋊Dic53—41no—
58C4⋊Dic53—41no—
59C4⋊Dic53—41no—
60C4⋊Dic53—41no—
61C2×#88322—41no—
62Dic212—41no—
63Dic212—41no—
64Dic212—81no—
65D424—21no—
66D424SD1612yes—
67C8×D53—82no—
68C16×C53—41no—
69C16×C53—41no—
70C22×D53—41no—
71C2×C4×C53#12111yescharacteristic
72C2×C4×C53C2×D811yescharacteristic
73C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
74C2×Dic53—41no—
75C2×Dic53—81no—
76C2×Dic53—81no—
77D212—22no—
78D212—41no—
79C4×D53—81no—
80C4×D53—81no—
81Dic106—81no—
82Dic106—81no—
83Dic106—81no—
84Dic106—81no—
85C8×C53—22no—
86C8×C53—21no—
87C8×C53—21no—
88C8×C53—21no—
89C8×C53C2×SD1611yescharacteristic
90C8×C53#13511yescharacteristic
91#88322—81no—
92#12422—531nomaximal, Sylow
93C22×C53#45911yescharacteristic
94C2×D53—42no—
95C2×D53—81no—
96C2×D53—81no—
97C4×C53—21no—
98C4×C53#46511yescharacteristic
99C4×C53#47111yescharacteristic
100C4×C53#50811yescharacteristic
101C4×C53#50911yescharacteristic
102Dic53—81no—
103Dic53—161no—
104Dic53—161no—
105#2224—531no—
106#2208—531no—
107#1707—531no—
108C8×D8—531no—
109#1367D5311yescharacteristic, complemented
110#1364—531no—
111#1362—531no—
112C2×C53#166511yescharacteristic
113C2×C53#221011yescharacteristic
114C2×C53#221411yescharacteristic
115D53—82no—
116C2×SD32—1061no—
117#500—531no—
118#498—1061no—
119#494—531no—
120#487—1061no—
121C4×D8—531no—
122C8×D4—1061no—
123#366—1061no—
124C4⋊C16C2×D5311yescharacteristic
125C4⋊C16C2×D5311yescharacteristic
126#358—1061no—
127#357—1061no—
128#335—531no—
129D4⋊C8—1061no—
130C82C2×D5311yescharacteristic
131C53#1242211yescharacteristic, complemented, Sylow
132C2×Q16—1061no—
133C2×Q16—1061no—
134C2×D8—531no—
135C22×C8—2121no—
136C4⋊Q8—1061no—
137C4⋊Q8—1061no—
138C4×D4—1061no—
139SD32—2122no—
140C2×C16—21no—
141C2×C16—21no—
142#107—531no—
143#107—1061no—
144#107—1061no—
145#106—2122no—
146C4⋊C8—1061no—
147#103—2121no—
148#103—2121no—
149#102—1061no—
150C22⋊C8—2121no—
151C4×C8D21211yescharacteristic
152C4×C8D21211yescharacteristic
153C4×C8C22×D5311yescharacteristic
154C2×Q8—2121no—
155C2×Q8—2121no—
156C2×D4—1061no—
157C22×C4—2121no—
158Q16—4241no—
159Q16—2121no—
160Q16—2121no—
161D8—532no—
162D8—1061no—
163C2×C8—21no—
164C2×C8—4242no—
165C2×C8—2121no—
166C2×C8D42411yescharacteristic
167C2×C8D42411yescharacteristic
168C2×C8D4×D5311yescharacteristic
169C2×C8#1171994911yescharacteristic
170C4⋊C4—1061no—
171C4⋊C4—2121no—
172C4⋊C4—2121no—
173C4⋊C4—2121no—
174C4⋊C4—2121no—
175C22⋊C4—2121no—
176C42C2×D21211yescharacteristic
177C16—41no—
178C16—41no—
179C23—2121no—
180Q8—4241no—
181Q8—4241no—
182Q8—4241no—
183Q8—4241no—
184D4—1062no—
185D4—2121no—
186C2×C4—4241no—
187C2×C4—4241no—
188C2×C4—4241no—
189C2×C4—4241no—
190C2×C4—2121no—
191C2×C4#1313319211yescharacteristic
192C2×C4C2×D42411yescharacteristic
193C2×C4#1313324511yescharacteristic
194C8—22no—
195C8—21no—
196C8—21no—
197C8—21no—
198C8—4241no—
199C8SD16×D5311yescharacteristic
200C8#1313323811yescharacteristic
201C22—4241no—
202C22—4241no—
203C22—2122no—
204C22#1681081911yescharacteristic
205C4—21no—
206C4—8481no—
207C4—8481no—
208C4—4241no—
209C4#1681085311yescharacteristic
210C4#1681088711yescharacteristic
211C4#1681105511yescharacteristic
212C4#1681106111yescharacteristic
213C2—4242no—
214C2#2027405711yescharacteristic
215C2#2027677111yescharacteristic
216C2#2027679711yescharacteristic
subgroupG/H
centerC22#16810819
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#1367C2
socleC22×C53#459
prime radicalC2×D424D4
hypercenter#1367D53
hypocenterC53#12422
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C82 ◁ #1367

Complements

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

NH ≅ G/Ntype
3. #20274312C2semidirect
5. #20272769C2semidirect
6. #20272765C2semidirect
7. C53×#1367C2semidirect
109. #1367D53semidirect
131. C53#12422semidirect

Maximal subgroups

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

subgroupindex
#202768532
#202767612
#202743122
C8×D4242
#202727692
#202727652
C53×#13672
#1242253