Finite Group Databasebeta

← #27298253

Subgroups — #27298253

Order 13568.

234 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 63 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#20276849C211yesmaximal, characteristic
2#20276761C211yesmaximal, characteristic
3#20274313C211yesmaximal, characteristic, complemented
4#20273509C211yesmaximal, characteristic
5#20272778C211yesmaximal, characteristic, complemented
6#20272763C211yesmaximal, characteristic, complemented
7C53×#1364C211yesmaximal, characteristic, complemented
8C2×#13133162—21no—
9#16811019—21no—
10#16811003C2211yescharacteristic
11#16810973—21no—
12C4×D424C2211yescharacteristic
13#16810688—21no—
14#16810448—21no—
15C4⋊#11719908C2211yescharacteristic
16#16810428—21no—
17#16810423—21no—
18#16810350—21no—
19C8×#88322C2211yescharacteristic
20C53×#500C2211yescharacteristic
21C53×C4⋊C16C2211yescharacteristic
22C53×#335C2211yescharacteristic
23C2×Dic212—21no—
24C2×Dic212—21no—
25C2×D424D411yescharacteristic
26C2×C8×D53—41no—
27#13133217—21no—
28C4×D212—21no—
29#13133162—42no—
30C2×#11719908—21no—
31#13133151D411yescharacteristic
32#13133148—42no—
33#13133141—41no—
34#13133141—41no—
35#13133139—21no—
36#13133129—41no—
37C4×#88322D411yescharacteristic
38C4×#88322D411yescharacteristic
39C4⋊Q8×C53—21no—
40C2×C16×C53—21no—
41#107×C53—21no—
42#107×C53—21no—
43C4⋊C8×C53—21no—
44C4×C8×C53C2311yescharacteristic
45C2×Q8×C53—41no—
46C2×D212—21no—
47C2×C4×D53—41no—
48C2×Dic106—41no—
49C2×C8×C53—41no—
50C2×C8×C53C2×D411yescharacteristic
51C2×C8×C53#4111yescharacteristic
52C4⋊C4×C53—41no—
53C4⋊C4×C53—41no—
54C42×C53C2×D411yescharacteristic
55#11719921—41no—
56C4⋊Dic53—21no—
57C4⋊Dic53—41no—
58C4⋊Dic53—41no—
59C2×#88322—21no—
60C2×#88322D811yescharacteristic
61C2×#88322D811yescharacteristic
62Dic212—41no—
63Dic212—41no—
64Dic212—81no—
65D424—21no—
66D424SD1612yes—
67C8×D53—82no—
68C16×C53—41no—
69#11719908—41no—
70C22×D53—41no—
71Q8×C53—81no—
72Q8×C53—81no—
73C2×C4×C53—81no—
74C2×C4×C53#12111yescharacteristic
75C2×C4×C53C2×D811yescharacteristic
76C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
77C2×Dic53—41no—
78C2×Dic53—81no—
79D212—22no—
80D212—41no—
81C4×D53—81no—
82C4×D53—81no—
83Dic106—81no—
84Dic106—81no—
85C8×C53—21no—
86C8×C53—81no—
87C8×C53C2×SD1611yescharacteristic
88C8×C53#13511yescharacteristic
89#88322—22no—
90#88322—21no—
91#88322—21no—
92#12422—531nomaximal, Sylow
93C22×C53#45911yescharacteristic
94C2×D53—42no—
95C2×D53—81no—
96C2×D53—81no—
97C4×C53—21no—
98C4×C53—161no—
99C4×C53#46511yescharacteristic
100C4×C53#47111yescharacteristic
101C4×C53#50811yescharacteristic
102C4×C53#50911yescharacteristic
103Dic53—81no—
104Dic53—161no—
105#2224—531no—
106#2208—531no—
107#1707—531no—
108C8×D8—531no—
109#1367—531no—
110#1364D5311yescharacteristic, complemented
111#1362—531no—
112C2×C53#166511yescharacteristic
113C2×C53#221011yescharacteristic
114C2×C53#221411yescharacteristic
115D53—82no—
116C2×SD32—1061no—
117#500C2×D5311yescharacteristic
118#498—1061no—
119#494—531no—
120#487—1061no—
121C4×D8—531no—
122C8×D4—1061no—
123#366—1061no—
124C4⋊C16—531no—
125C4⋊C16C2×D5311yescharacteristic
126#358—1061no—
127#357—1061no—
128#335C2×D5311yescharacteristic
129D4⋊C8—1061no—
130C82—531no—
131C53#1242211yescharacteristic, complemented, Sylow
132C2×Q16—1061no—
133C2×Q16—1061no—
134C2×D8—531no—
135C22×C8—2121no—
136C4⋊Q8—21no—
137C4⋊Q8—1061no—
138C4×D4—1061no—
139SD32—2122no—
140C2×C16—21no—
141C2×C16—1061no—
142#107—21no—
143#107—21no—
144#107—531no—
145#106—2122no—
146C4⋊C8—21no—
147#103—2121no—
148#103—2121no—
149#102—1061no—
150C22⋊C8—2121no—
151C4×C8—531no—
152C4×C8—531no—
153C4×C8C22×D5311yescharacteristic
154C2×Q8—41no—
155C2×Q8—2121no—
156C2×D4—1061no—
157C22×C4—2121no—
158Q16—4241no—
159Q16—2121no—
160Q16—2121no—
161D8—532no—
162D8—1061no—
163C2×C8—41no—
164C2×C8—531no—
165C2×C8—531no—
166C2×C8—1061no—
167C2×C8—4242no—
168C2×C8#1171994511yescharacteristic
169C2×C8D4×D5311yescharacteristic
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—1061no—
173C4⋊C4—2121no—
174C4⋊C4—2121no—
175C22⋊C4—2121no—
176C42D4×D5311yescharacteristic
177C16—41no—
178C16—2121no—
179C23—2121no—
180Q8—81no—
181Q8—81no—
182Q8—4241no—
183Q8—4241no—
184D4—1062no—
185D4—2121no—
186C2×C4—81no—
187C2×C4—4241no—
188C2×C4—4241no—
189C2×C4—4241no—
190C2×C4—2121no—
191C2×C4#1313319011yescharacteristic
192C2×C4D8×D5311yescharacteristic
193C2×C4#1313324411yescharacteristic
194C8—21no—
195C8—81no—
196C8—1062no—
197C8—1061no—
198C8—1061no—
199C8SD16×D5311yescharacteristic
200C8#1313324011yescharacteristic
201C22—4241no—
202C22—4241no—
203C22—2122no—
204C22#1681081711yescharacteristic
205C4—21no—
206C4—161no—
207C4—8481no—
208C4—4241no—
209C4#1681085511yescharacteristic
210C4#1681088911yescharacteristic
211C4#1681105411yescharacteristic
212C4#1681105711yescharacteristic
213C2—4242no—
214C2#2027405911yescharacteristic
215C2#2027676911yescharacteristic
216C2#2027679511yescharacteristic
subgroupG/H
centerC22#16810817
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#1364C2
socleC22×C53#459
prime radicalC2×D424D4
hypercenter#1364D53
hypocenterC53#12422
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
3. #20274313C2semidirect
5. #20272778C2semidirect
6. #20272763C2semidirect
7. C53×#1364C2semidirect
110. #1364D53semidirect
131. C53#12422semidirect

Maximal subgroups

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

subgroupindex
#202768492
#202767612
#202743132
#202735092
#202727782
#202727632
C53×#13642
#1242253