Finite Group Databasebeta

← #27288231

Subgroups — #27288231

Order 13568.

386 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 73 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#20286096C211yesmaximal, characteristic
2#20274889C211yesmaximal, characteristic, complemented
3#20273438C212yesmaximal, complemented
4#20273327C212yesmaximal, complemented
5C53×#1307C211yesmaximal, characteristic, complemented
6#16811173C2212yes—
7#16810914—22no—
8#16810876—22no—
9#16810819—22no—
10C2×#13133152—22no—
11#16810647C2211yescharacteristic
12C2×#13133128C2211yescharacteristic
13#16810370—22no—
14#16810366—22no—
15#16810362—24no—
16C53×#432C2212yes—
17C53×#423C2211yescharacteristic
18C2×#11719945—22no—
19C2×D424—22no—
20C2×#11719917—21no—
21C22×#88322—21no—
22#13133217D412yes—
23#13133214D412yes—
24#13133205—22no—
25#13133191—22no—
26#13133191—22no—
27C4×D212—22no—
28C4×D212—22no—
29#13133152—44no—
30#13133149—22no—
31C4⋊#88322—22no—
32#13133139—24no—
33#13133128—21no—
34#13133128D412yes—
35C2×M16×C53D412yes—
36C2×C42×C53C2311yescharacteristic
37C4⋊C8×C53—22no—
38C4⋊C8×C53—22no—
39C22⋊C8×C53—22no—
40#97×C53—22no—
41C22×C4×C53C2×D411yescharacteristic
42C22×C4×C53#4112yes—
43C2×C22⋊D53—42no—
44#11719945—84no—
45C2×D212—22no—
46C2×D212—22no—
47C2×D212—22no—
48C2×C4×D53—42no—
49C2×Dic106—22no—
50M16×C53—42no—
51C2×C8×C53—22no—
52C2×C8×C53—42no—
53C42×C53C2×D412yes—
54C42×C53C2×D411yescharacteristic
55C42×C53#4111yescharacteristic
56#11719921—42no—
57#11719921—42no—
58C4⋊Dic53—22no—
59C4⋊Dic53—22no—
60#11719917—42no—
61C2×#88322—22no—
62C2×#88322—22no—
63C2×#88322—22no—
64C2×#88322—41no—
65D424—82no—
66C23×C53#12411yescharacteristic
67C22×D53—42no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—42no—
71C2×C4×C53#12011yescharacteristic
72C2×C4×C53#12112yes—
73C2×C4×C53#12111yescharacteristic
74C2×C4×C53#12312yes—
75C22⋊D53—84no—
76C2×Dic53—42no—
77D212—42no—
78D212—42no—
79D212—82no—
80D212—82no—
81C4×D53—84no—
82Dic106—82no—
83C8×C53—42no—
84C8×C53—82no—
85#88322—24no—
86#88322—81no—
87#10524—531nomaximal, Sylow
88C22×C53—22no—
89C22×C53—21no—
90C22×C53#39411yescharacteristic
91C2×D53—82no—
92C2×D53—82no—
93C2×D53—82no—
94C4×C53—22no—
95C4×C53—22no—
96C4×C53—21no—
97C4×C53#45311yescharacteristic
98C4×C53#45611yescharacteristic
99Dic53—82no—
100#3253—531no—
101#1849—531no—
102#1557—532no—
103#1530—532no—
104#1307D5311yescharacteristic, complemented
105C2×C53—41no—
106C2×C53#203911yescharacteristic
107C2×C53#204812yes—
108D53—162no—
109#532—532no—
110#476—1062no—
111#469—1062no—
112#459—1062no—
113#432C2×D5312yes—
114C2×#108—1062no—
115#423—531no—
116#423C2×D5311yescharacteristic
117C2×#97—531no—
118#332—1062no—
119#331—1062no—
120#329—1064no—
121C53#1052411yescharacteristic, complemented, Sylow
122C2×#41—1062no—
123C2×D8—1062no—
124C2×M16—1061no—
125C2×M16D21212yes—
126C22×C8—1061no—
127C4⋊Q8—532no—
128#127—532no—
129#124—1062no—
130#121—1062no—
131#121—1062no—
132C4×D4—1062no—
133C4×D4—1062no—
134C2×C42C22×D5311yescharacteristic
135#108—2124no—
136#106—1062no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—1062no—
140#102—1064no—
141C22⋊C8—22no—
142#97—22no—
143#97—532no—
144#97—1061no—
145#41—4244no—
146C2×Q8—1062no—
147C2×D4—1062no—
148C2×D4—1062no—
149C2×D4—1062no—
150C2×D4—2122no—
151C22×C4—2122no—
152C22×C4C2×D21211yescharacteristic
153C22×C4#1171994512yes—
154D8—4242no—
155M16—42no—
156M16—2122no—
157C2×C8—22no—
158C2×C8—42no—
159C2×C8—1062no—
160C2×C8—1062no—
161C2×C8—1062no—
162C2×C8—2121no—
163C4⋊C4—1062no—
164C4⋊C4—1062no—
165C22⋊C4—2122no—
166C22⋊C4—2122no—
167C42D4×D5312yes—
168C42D4×D5311yescharacteristic
169C42#1171994911yescharacteristic
170C23—2122no—
171C23#1313320511yescharacteristic
172Q8—4242no—
173D4—4242no—
174D4—4242no—
175D4—4244no—
176D4—2122no—
177D4—2122no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—4244no—
182C2×C4—2122no—
183C2×C4#1313318711yescharacteristic
184C2×C4#1313319012yes—
185C2×C4#1313319211yescharacteristic
186C2×C4#1313320212yes—
187C8—42no—
188C8—82no—
189C8—1064no—
190C8—4241no—
191C22—22no—
192C22—21no—
193C22—4242no—
194C22—4242no—
195C22—4242no—
196C22#1681054111yescharacteristic
197C4—22no—
198C4—22no—
199C4—21no—
200C4—4242no—
201C4#1681079311yescharacteristic
202C4#1681080811yescharacteristic
203C2—41no—
204C2—8482no—
205C2#2027594011yescharacteristic
206C2#2027599112yes—
subgroupG/H
centerC22#16810541
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1307C2
socleC22×C53#394
prime radical#20286096C2
hypercenter#1307D53
hypocenterC53#10524
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1307

Complements

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

NH ≅ G/Ntype
2. #20274889C2semidirect
3. #20273438C2semidirect
4. #20273327C2semidirect
5. C53×#1307C2semidirect
104. #1307D53semidirect
121. C53#10524semidirect

Maximal subgroups

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

subgroupindex
#202860962
#202748892
#202734382
#202733272
C53×#13072
#1052453