Finite Group Databasebeta

← #27288227

Subgroups — #27288227

Order 13568.

386 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 77 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#20274889C211yesmaximal, characteristic, complemented
2#20273437C212yesmaximal, complemented
3#20273326C212yesmaximal, complemented
4#20272626C211yesmaximal, characteristic, complemented
5C53×#3253C211yesmaximal, characteristic
6#16810680C2212yes—
7C2×#13133152—22no—
8#16810647C2211yescharacteristic
9C2×#13133128C2211yescharacteristic
10#16810369—22no—
11#16810367—22no—
12#16810363—24no—
13C53×#532C2212yes—
14C53×#476—22no—
15C53×#469—22no—
16C53×#459—22no—
17C53×#423C2211yescharacteristic
18C2×#11719917—21no—
19C22×#88322—21no—
20#13133152—44no—
21C4⋊#88322—22no—
22C4⋊#88322—22no—
23C22⋊#88322—22no—
24#13133128—21no—
25#13133128—22no—
26#13133128D412yes—
27C2×#41×C53—22no—
28C2×D8×C53—22no—
29C2×M16×C53D412yes—
30C4⋊Q8×C53D412yes—
31#127×C53D412yes—
32#124×C53—22no—
33#121×C53—22no—
34#121×C53—22no—
35C4×D4×C53—22no—
36C4×D4×C53—22no—
37C2×C42×C53C2311yescharacteristic
38#106×C53—22no—
39C4⋊C8×C53—22no—
40#102×C53—24no—
41#41×C53—84no—
42C2×Q8×C53—22no—
43C2×D4×C53—22no—
44C2×D4×C53—22no—
45C2×D4×C53—22no—
46C2×D4×C53—42no—
47C22×C4×C53—42no—
48C22×C4×C53C2×D411yescharacteristic
49C22×C4×C53#4112yes—
50D8×C53—82no—
51M16×C53—42no—
52C2×C8×C53—22no—
53C4⋊C4×C53—22no—
54C4⋊C4×C53—22no—
55C22⋊C4×C53—42no—
56C22⋊C4×C53—42no—
57C42×C53C2×D412yes—
58C42×C53C2×D411yescharacteristic
59C42×C53#4111yescharacteristic
60#11719917—42no—
61C2×#88322—22no—
62C2×#88322—22no—
63C2×#88322—22no—
64C2×#88322—41no—
65C2×#88322—42no—
66C23×C53—42no—
67C23×C53#12411yescharacteristic
68Q8×C53—82no—
69D4×C53—42no—
70D4×C53—42no—
71D4×C53—82no—
72D4×C53—82no—
73D4×C53—84no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—42no—
77C2×C4×C53—42no—
78C2×C4×C53—84no—
79C2×C4×C53#12011yescharacteristic
80C2×C4×C53#12111yescharacteristic
81C2×C4×C53#12112yes—
82C2×C4×C53#12312yes—
83C8×C53—42no—
84#88322—24no—
85#88322—82no—
86#88322—81no—
87#10524—531nomaximal, Sylow
88C22×C53—22no—
89C22×C53—21no—
90C22×C53—82no—
91C22×C53—82no—
92C22×C53—82no—
93C22×C53#39411yescharacteristic
94C4×C53—22no—
95C4×C53—22no—
96C4×C53—21no—
97C4×C53—82no—
98C4×C53#45311yescharacteristic
99C4×C53#45611yescharacteristic
100#3253D5311yescharacteristic
101#1849—531no—
102#1557—532no—
103#1530—532no—
104#1307—531no—
105C2×C53—41no—
106C2×C53—162no—
107C2×C53#203911yescharacteristic
108C2×C53#204812yes—
109#532C2×D5312yes—
110#476—22no—
111#469—22no—
112#459—22no—
113#432—532no—
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—22no—
123C2×D8—22no—
124C2×M16—1061no—
125C2×M16C22⋊D5312yes—
126C22×C8—1061no—
127C4⋊Q8C22⋊D5312yes—
128#127C22⋊D5312yes—
129#124—22no—
130#121—22no—
131#121—22no—
132C4×D4—22no—
133C4×D4—22no—
134C2×C42C22×D5311yescharacteristic
135#108—2124no—
136#106—22no—
137C4⋊C8—22no—
138C4⋊C8—1062no—
139C4⋊C8—1062no—
140#102—24no—
141C22⋊C8—1062no—
142#97—532no—
143#97—1062no—
144#97—1061no—
145#41—84no—
146C2×Q8—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—42no—
151C22×C4—42no—
152C22×C4#1171994712yes—
153C22×C4C2×C22⋊D5311yescharacteristic
154D8—82no—
155M16—42no—
156M16—2122no—
157C2×C8—22no—
158C2×C8—1062no—
159C2×C8—1062no—
160C2×C8—1062no—
161C2×C8—2121no—
162C2×C8—2122no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C22⋊C4—42no—
166C22⋊C4—42no—
167C42D4×D5311yescharacteristic
168C42#1171994711yescharacteristic
169C42C2×C22⋊D5312yes—
170C23—42no—
171C23#1313320411yescharacteristic
172Q8—82no—
173D4—42no—
174D4—42no—
175D4—82no—
176D4—82no—
177D4—84no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—84no—
183C2×C4#1313318611yescharacteristic
184C2×C4#1313318811yescharacteristic
185C2×C4#1313318912yes—
186C2×C4#1313319812yes—
187C8—42no—
188C8—1064no—
189C8—4242no—
190C8—4241no—
191C22—22no—
192C22—21no—
193C22—82no—
194C22—82no—
195C22—82no—
196C22#1681053711yescharacteristic
197C4—22no—
198C4—22no—
199C4—21no—
200C4—82no—
201C4#1681079111yescharacteristic
202C4#1681080611yescharacteristic
203C2—41no—
204C2—162no—
205C2#2027593611yescharacteristic
206C2#2027598512yes—
subgroupG/H
centerC22#16810537
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3253C2
socleC22×C53#394
hypercenter#3253D53
hypocenterC53#10524
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3253

Complements

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

NH ≅ G/Ntype
1. #20274889C2semidirect
2. #20273437C2semidirect
3. #20273326C2semidirect
4. #20272626C2semidirect
121. C53#10524semidirect

Maximal subgroups

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

subgroupindex
#202748892
#202734372
#202733262
#202726262
C53×#32532
#1052453