Finite Group Databasebeta

← #27296012

Subgroups — #27296012

Order 13568.

240 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C4×#13133158C211yesmaximal, characteristic
2#20273599C211yesmaximal, characteristic, complemented
3D8×#88322C211yesmaximal, characteristic, complemented
4#20272788C211yesmaximal, characteristic, complemented
5#20272677C211yesmaximal, characteristic, complemented
6#20272634C211yesmaximal, characteristic, complemented
7C53×#1331C211yesmaximal, characteristic, complemented
8C2×#13133158C411yescharacteristic
9C4×D424C2211yescharacteristic
10#16810700—21no—
11D4×#88322—21no—
12#16810448C411yescharacteristic
13#16810424C411yescharacteristic
14#16810423C411yescharacteristic
15C4×#11719908C2211yescharacteristic
16#16810376C2211yescharacteristic
17D4⋊#88322—21no—
18#16810350—21no—
19C8×#88322C2211yescharacteristic
20C4×D8×C53C2211yescharacteristic
21C4×C16×C53C2211yescharacteristic
22C53×#335C2211yescharacteristic
23C2×D424C2×C411yescharacteristic
24C2×#11719912—41no—
25C22×#88322—41no—
26C4×D212—21no—
27#13133158—22no—
28C2×#11719908C2×C411yescharacteristic
29C2×#11719908C2×C411yescharacteristic
30#13133151C2×C411yescharacteristic
31C4⋊#88322—21no—
32#13133139—21no—
33C22⋊#88322—41no—
34#13133129—41no—
35C4×#88322D411yescharacteristic
36C4×#88322D411yescharacteristic
37C2×D8×C53C2×C411yescharacteristic
38C4×D4×C53—21no—
39C2×C16×C53—21no—
40#107×C53C2×C411yescharacteristic
41C4⋊C8×C53—21no—
42#102×C53—21no—
43C4×C8×C53C2311yescharacteristic
44C2×D4×C53—21no—
45C22×C4×C53—41no—
46C2×D212—21no—
47C2×C4×D53—41no—
48D8×C53—21no—
49D8×C53M1612yescomplemented
50C2×C8×C53—41no—
51C2×C8×C53C22×C411yescharacteristic
52C2×C8×C53#4111yescharacteristic
53C4⋊C4×C53—21no—
54C22⋊C4×C53—41no—
55C42×C53C2×D411yescharacteristic
56#11719921—41no—
57C4⋊Dic53—21no—
58C2×#88322—21no—
59C2×#88322—41no—
60C2×#88322—82no—
61C2×#88322D811yescharacteristic
62C2×#88322D811yescharacteristic
63D424—21no—
64D424—21no—
65#11719912—82no—
66C16×C53—41no—
67#11719908—21no—
68#11719908—21no—
69C23×C53—41no—
70C22×D53—41no—
71D4×C53—22no—
72D4×C53—41no—
73C2×C4×C53—41no—
74C2×C4×C53—81no—
75C2×C4×C53—81no—
76C2×C4×C53C4×D411yescharacteristic
77C2×C4×C53C2×D811yescharacteristic
78C2×C4×C53#13511yescharacteristic
79C2×Dic53—41no—
80D212—41no—
81D212—41no—
82C4×D53—81no—
83C4×D53—81no—
84C8×C53—21no—
85C8×C53—81no—
86C8×C53C2×M1611yescharacteristic
87C8×C53#13111yescharacteristic
88#88322—22no—
89#88322—21no—
90#88322—21no—
91#88322—81no—
92#12073—531nomaximal, Sylow
93C22×C53—42no—
94C22×C53—81no—
95C22×C53—81no—
96C22×C53C4×D811yescharacteristic
97C2×D53—81no—
98C2×D53—81no—
99C2×D53—81no—
100C4×C53—21no—
101C4×C53—81no—
102C4×C53#43511yescharacteristic
103C4×C53#44411yescharacteristic
104C4×C53#50911yescharacteristic
105C4×C53#50911yescharacteristic
106Dic53—81no—
107C4×D16—531no—
108#1589—531no—
109C8×D8—531no—
110#1372—531no—
111#1331—531no—
112#1331D5311yescharacteristic, complemented
113#1312—531no—
114C2×C53—82no—
115C2×C53#158111yescharacteristic
116C2×C53#217711yescharacteristic
117C2×C53#217811yescharacteristic
118D53—161no—
119C2×D16—531no—
120C4×D8—531no—
121C4×D8C2×D5311yescharacteristic
122#436—1061no—
123C8×D4—1061no—
124#366—531no—
125#357—531no—
126#357—531no—
127C4×C16—531no—
128C4×C16C2×D5311yescharacteristic
129#335—531no—
130#335C2×D5311yescharacteristic
131D4⋊C8—1061no—
132D4⋊C8—1061no—
133C82—531no—
134C53#1207311yescharacteristic, complemented, Sylow
135C2×D8—531no—
136C2×D8C4×D5311yescharacteristic
137C2×M16—2121no—
138C22×C8—2121no—
139C4×D4—21no—
140C4×D4—1061no—
141D16—1062no—
142C2×C16—21no—
143C2×C16—531no—
144C2×C16—531no—
145#107—531no—
146#107C4×D5311yescharacteristic
147C4⋊C8—21no—
148C4⋊C8—1061no—
149#102—21no—
150#102—1061no—
151C22⋊C8—2121no—
152C22⋊C8—2121no—
153C4×C8—531no—
154C4×C8—531no—
155C4×C8C22×D5311yescharacteristic
156C2×D4—21no—
157C2×D4—1061no—
158C22×C4—41no—
159C22×C4—2121no—
160D8—21no—
161D8—1061no—
162D8—1061no—
163D8#1171991212yescomplemented
164M16—4242no—
165C2×C8—41no—
166C2×C8—531no—
167C2×C8—531no—
168C2×C8—1061no—
169C2×C8—4242no—
170C2×C8—2121no—
171C2×C8C2×C4×D5311yescharacteristic
172C2×C8#1171994711yescharacteristic
173C4⋊C4—21no—
174C4⋊C4—1061no—
175C22⋊C4—41no—
176C22⋊C4—2121no—
177C42D4×D5311yescharacteristic
178C16—41no—
179C16—1061no—
180C16—1061no—
181C23—41no—
182C23—2121no—
183D4—22no—
184D4—41no—
185D4—2121no—
186D4—2121no—
187C2×C4—41no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—4241no—
191C2×C4—4241no—
192C2×C4—2121no—
193C2×C4#1313317711yescharacteristic
194C2×C4D8×D5311yescharacteristic
195C2×C4#1313323811yescharacteristic
196C8—21no—
197C8—81no—
198C8—1062no—
199C8—1061no—
200C8—1061no—
201C8—4241no—
202C8C2×#1171991211yescharacteristic
203C8#1313322511yescharacteristic
204C22—42no—
205C22—81no—
206C22—81no—
207C22—4241no—
208C22—4241no—
209C22—4241no—
210C22#1681070311yescharacteristic
211C4—21no—
212C4—81no—
213C4—4241no—
214C4#1681069111yescharacteristic
215C4#1681074211yescharacteristic
216C4#1681105711yescharacteristic
217C4#1681106011yescharacteristic
218C2—82no—
219C2—8481no—
220C2#2027354111yescharacteristic
221C2#2027658811yescharacteristic
222C2#2027659311yescharacteristic
subgroupG/H
centerC2×C4D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC4×C8C22×D53
FittingC53×#1331C2
socleC22×C53C4×D8
prime radicalC2×#13133158C4
cosocleC4×C8×C53C23
hypercenter#1331D53
hypocenterC53#12073
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ #1331

Complements

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

NH ≅ G/Ntype
2. #20273599C2semidirect
3. D8×#88322C2semidirect
4. #20272788C2semidirect
5. #20272677C2semidirect
6. #20272634C2semidirect
7. C53×#1331C2semidirect
49. D8×C53M16semidirect
112. #1331D53semidirect
134. C53#12073semidirect
163. D8#11719912semidirect

Maximal subgroups

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

subgroupindex
C4×#131331582
#202735992
D8×#883222
#202727882
#202726772
#202726342
C53×#13312
#1207353