Finite Group Databasebeta

← #27541733

Subgroups — #27541733

Order 13568.

562 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 165 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
1Q8×D424C211yesmaximal, characteristic, complemented
2#20276938C213yesmaximal, complemented
3#20276852C213yesmaximal, complemented
4#20276747C213yesmaximal, complemented
5Q8×#11719908C211yesmaximal, characteristic, complemented
6C4×#13133158C213yesmaximal, complemented
7C53×#3391C211yesmaximal, characteristic, complemented
8Q8×D212—21no—
9C2×#13133158C2213yes—
10#16811028C2213yes—
11#16810904—23no—
12C4×D424C2213yes—
13#16810448C2213yes—
14C4⋊#11719908C2213yes—
15#16810424C2216yes—
16#16810424C2213yes—
17#16810423C2211yescharacteristic
18C4×#11719908C2213yes—
19C53×#550—21no—
20C53×#493C2213yes—
21C53×#459—23no—
22C8×Q8×C53C2211yescharacteristic
23C4×D8×C53C2213yes—
24C2×Q8×D53—41no—
25C2×D424C2311yescharacteristic
26#13133217—23no—
27#13133197—43no—
28C4×D212—23no—
29#13133158—26no—
30C2×#11719908C2313yes—
31C2×#11719908C2311yescharacteristic
32#13133151—23no—
33#13133151C2313yes—
34#13133139—23no—
35C2×#41×C53—41no—
36C2×D8×C53—23no—
37C2×D8×C53C2313yes—
38#127×C53—23no—
39#121×C53—43no—
40C4×Q8×C53D411yescharacteristic
41C4×D4×C53—23no—
42#107×C53C2311yescharacteristic
43C4⋊C8×C53D413yes—
44#102×C53—23no—
45C4×C8×C53C2313yes—
46#41×C53—84no—
47C2×Q8×C53D811yescharacteristic
48C2×D4×C53—23no—
49C2×D4×C53—43no—
50C2×D4×C53—43no—
51C22×C4×C53—43no—
52Q8×D53—84no—
53C2×D212—21no—
54C2×C4×D53—43no—
55C2×Dic106—43no—
56D8×C53—23no—
57D8×C53—23no—
58D8×C53—83no—
59C2×C8×C53C2×D413yes—
60C2×C8×C53C2411yescharacteristic
61C4⋊C4×C53—21no—
62C4⋊C4×C53D813yes—
63C22⋊C4×C53—43no—
64C42×C53C2×D413yes—
65#11719921—43no—
66C4⋊Dic53—23no—
67C4⋊Dic53—43no—
68D424—21no—
69D424#4112yescomplemented
70#11719908—23no—
71#11719908—21no—
72C23×C53—43no—
73C22×D53—41no—
74Q8×C53D1614yescomplemented
75D4×C53—43no—
76D4×C53—43no—
77D4×C53—86no—
78D4×C53—163no—
79C2×C4×C53—41no—
80C2×C4×C53—86no—
81C2×C4×C53C2×D813yes—
82C2×C4×C53C2×D813yes—
83C2×C4×C53C22×D411yescharacteristic
84C2×Dic53—43no—
85D212—22no—
86D212—41no—
87C4×D53—86no—
88Dic106—86no—
89C8×C53—23no—
90C8×C53C2×#4111yescharacteristic
91C8×C532−1+411yescharacteristic
92#33925—531nomaximal, Sylow
93C22×C53—83no—
94C22×C53—83no—
95C22×C53—83no—
96C22×C53C22×D811yescharacteristic
97C2×D53—42no—
98C2×D53—81no—
99C2×D53—81no—
100C4×C53—23no—
101C4×C53—81no—
102C4×C53C2×D1616yes—
103C4×C53#54811yescharacteristic
104C4×C53#57711yescharacteristic
105Dic53—83no—
106#3391D5311yescharacteristic, complemented
107Q8×D8—531no—
108#2240—533no—
109#2224—533no—
110#2206—533no—
111Q8×C16—531no—
112C4×D16—533no—
113C2×C53—163no—
114C2×C53#335811yescharacteristic
115C2×C53C22×D1611yescharacteristic
116C2×C53#341611yescharacteristic
117D53—82no—
118#550—21no—
119D4×Q8—1061no—
120C2×D16—533no—
121#500—533no—
122#493C2×D5313yes—
123#474—1063no—
124#459—23no—
125C8×Q8C2×D5311yescharacteristic
126C4×D8—533no—
127C4×D8C2×D5313yes—
128#366—533no—
129C4⋊C16—533no—
130#357—533no—
131#357—536no—
132#357—531no—
133C4×C16—533no—
134C53#3392511yescharacteristic, complemented, Sylow
135C2×#41—41no—
136C22×Q8—2121no—
137C2×D8—23no—
138C2×D8—531no—
139C2×D8C22×D5313yes—
140C4⋊Q8—1063no—
141#127—23no—
142C22⋊Q8—2123no—
143#121—43no—
144C4×Q8C22⋊D5311yescharacteristic
145C4×D4—23no—
146C4×D4—1063no—
147D16—1066no—
148C2×C16—531no—
149C2×C16—533no—
150#107—533no—
151#107—1063no—
152#107C22×D5311yescharacteristic
153C4⋊C8C22⋊D5313yes—
154#102—23no—
155#102—1063no—
156C4×C8C22×D5313yes—
157#41—84no—
158C2×Q8—4244no—
159C2×Q8—2123no—
160C2×Q8#1171992211yescharacteristic
161C2×D4—23no—
162C2×D4—43no—
163C2×D4—43no—
164C2×D4—1061no—
165C22×C4—43no—
166C22×C4—2123no—
167D8—23no—
168D8—23no—
169D8—83no—
170D8—532no—
171D8—1061no—
172C2×C8C2×C22⋊D5313yes—
173C2×C8C23×D5311yescharacteristic
174C4⋊C4—21no—
175C4⋊C4—1063no—
176C4⋊C4—2123no—
177C4⋊C4#1171992213yes—
178C22⋊C4—43no—
179C22⋊C4—2123no—
180C42C2×C22⋊D5313yes—
181C16—1063no—
182C16—1061no—
183C23—43no—
184C23—2121no—
185Q8—4246no—
186Q8#1313315814yescomplemented
187D4—43no—
188D4—43no—
189D4—86no—
190D4—163no—
191D4—1062no—
192D4—2121no—
193C2×C4—41no—
194C2×C4—86no—
195C2×C4—4246no—
196C2×C4—2123no—
197C2×C4C2×#1171992213yes—
198C2×C4C2×#1171992213yes—
199C2×C4C22×C22⋊D5311yescharacteristic
200C8—23no—
201C8#41×D5311yescharacteristic
202C8#1313326411yescharacteristic
203C22—83no—
204C22—83no—
205C22—83no—
206C22—4241no—
207C22—4241no—
208C22—2122no—
209C22C22×#1171992211yescharacteristic
210C4—23no—
211C4—81no—
212C4—4243no—
213C4C2×#1313315816yes—
214C4#1681128011yescharacteristic
215C4#1681143411yescharacteristic
216C2—163no—
217C2—4242no—
218C2#2028734611yescharacteristic
219C2C22×#1313315811yescharacteristic
220C2#2028786211yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3391C2
socleC22×C53C22×D8
hypercenter#3391D53
hypocenterC53#33925
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3391

Complements

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

NH ≅ G/Ntype
1. Q8×D424C2semidirect
2. #20276938C2semidirect
3. #20276852C2semidirect
4. #20276747C2semidirect
5. Q8×#11719908C2semidirect
6. C4×#13133158C2semidirect
7. C53×#3391C2semidirect
69. D424#41semidirect
74. Q8×C53D16semidirect
106. #3391D53semidirect
134. C53#33925semidirect
186. Q8#13133158semidirect

Maximal subgroups

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

subgroupindex
Q8×D4242
#202769382
#202768522
#202767472
Q8×#117199082
C4×#131331582
C53×#33912
#3392553