Finite Group Databasebeta

← #27279057

Subgroups — #27279057

Order 13568.

398 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 103 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#20283821C211yesmaximal, characteristic, complemented
2#20273380C211yesmaximal, characteristic, complemented
3#20273184C212yesmaximal
4#20272614C212yesmaximal, complemented
5C53×#3135C211yesmaximal, characteristic
6#16810754—24no—
7#16810684—22no—
8#16810680C2212yes—
9#16810676C2212yes—
10#16810369—22no—
11D4⋊#88322—24no—
12C53×#533C2211yescharacteristic
13C53×#532C2211yescharacteristic
14C53×#491C412yes—
15C53×#490C412yes—
16C53×#489—22no—
17C53×#416—22no—
18C2×#97×C53C2211yescharacteristic
19C4⋊#88322—24no—
20C4⋊#88322—22no—
21C4⋊#88322D414yes—
22C22⋊#88322—22no—
23C22⋊#88322—22no—
24#13133128—22no—
25#13133128—22no—
26C4×#88322—22no—
27C4×#88322—22no—
28C2×#41×C53—21no—
29C22×C8×C53—21no—
30C4⋊Q8×C53C2×C411yescharacteristic
31C4⋊Q8×C53D412yes—
32#127×C53C2×C411yescharacteristic
33#125×C53C2×C412yes—
34#124×C53—21no—
35C22⋊Q8×C53—22no—
36#121×C53—22no—
37C4×Q8×C53—22no—
38C4×D4×C53—22no—
39#117×C53D412yes—
40C2×C42×C53C2311yescharacteristic
41#103×C53—24no—
42#102×C53—24no—
43#97×C53—21no—
44#97×C53C2×C412yes—
45#41×C53—44no—
46C2×Q8×C53—22no—
47C2×Q8×C53—21no—
48C2×D4×C53—21no—
49C2×D4×C53—22no—
50C2×D4×C53—41no—
51C22×C4×C53—41no—
52C22×C4×C53C2×D411yescharacteristic
53C22×C4×C53#4112yes—
54C2×C8×C53—22no—
55C2×C8×C53—42no—
56C4⋊C4×C53—24no—
57C4⋊C4×C53—22no—
58C4⋊C4×C53—22no—
59C4⋊C4×C53C22⋊C414yes—
60C22⋊C4×C53—22no—
61C22⋊C4×C53—42no—
62C42×C53—22no—
63C42×C53C22×C411yescharacteristic
64C42×C53C2×D411yescharacteristic
65C42×C53C2×D412yes—
66C2×#88322—24no—
67C2×#88322—42no—
68C23×C53—41no—
69C23×C53#12311yescharacteristic
70Q8×C53—42no—
71Q8×C53—82no—
72D4×C53—44no—
73D4×C53—42no—
74D4×C53—82no—
75C2×C4×C53—22no—
76C2×C4×C53—24no—
77C2×C4×C53—42no—
78C2×C4×C53—41no—
79C2×C4×C53—82no—
80C2×C4×C53C2×C22⋊C411yescharacteristic
81C2×C4×C53C4×D412yes—
82C2×C4×C53#12011yescharacteristic
83C2×C4×C53#12112yes—
84C8×C53—42no—
85#88322—44no—
86#88322—82no—
87#9008—531nomaximal, Sylow
88C22×C53—22no—
89C22×C53—21no—
90C22×C53—82no—
91C22×C53—81no—
92C22×C53#38611yescharacteristic
93C4×C53—22no—
94C4×C53—22no—
95C4×C53—21no—
96C4×C53—44no—
97C4×C53—81no—
98C4×C53#45412yes—
99#3135D5311yescharacteristic
100#2987—531no—
101#1544—531no—
102#1496—532no—
103#1302—532no—
104C2×C53—41no—
105C2×C53—161no—
106C2×C53#188112yes—
107C2×C53#188811yescharacteristic
108#533C2×D5311yescharacteristic
109#532C2×D5311yescharacteristic
110#491Dic5312yes—
111#490Dic5312yes—
112#489—22no—
113#446—1064no—
114#433—1062no—
115#432—532no—
116#431—532no—
117#416—22no—
118C2×#97C2×D5311yescharacteristic
119#332—1062no—
120D4⋊C8—1064no—
121C53#900811yescharacteristic, complemented, Sylow
122C2×#41—21no—
123C22×C8—21no—
124C4⋊Q8C2×Dic5311yescharacteristic
125C4⋊Q8C22⋊D5312yes—
126#127C2×Dic5311yescharacteristic
127#125C2×Dic5312yes—
128#124—21no—
129C22⋊Q8—22no—
130#121—22no—
131C4×Q8—22no—
132C4×D4—22no—
133#117C22⋊D5312yes—
134C2×C42C22×D5311yescharacteristic
135C4⋊C8—534no—
136C4⋊C8—1064no—
137C4⋊C8—1062no—
138#103—24no—
139#102—24no—
140C22⋊C8—1062no—
141C22⋊C8—1062no—
142#97—21no—
143#97—1062no—
144#97—1062no—
145#97C2×Dic5312yes—
146C4×C8—1062no—
147C4×C8—1062no—
148#41—44no—
149C2×Q8—22no—
150C2×Q8—21no—
151C2×D4—21no—
152C2×D4—22no—
153C2×D4—41no—
154C22×C4—41no—
155C22×C4#1171994712yes—
156C22×C4C2×C22⋊D5311yescharacteristic
157C2×C8—22no—
158C2×C8—42no—
159C2×C8—1064no—
160C2×C8—2122no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4C22⋊Dic5314yes—
165C22⋊C4—22no—
166C22⋊C4—42no—
167C42—22no—
168C42D4×D5312yes—
169C42C22×Dic5311yescharacteristic
170C42C2×C22⋊D5311yescharacteristic
171C23—41no—
172C23#1313319811yescharacteristic
173Q8—42no—
174Q8—82no—
175D4—44no—
176D4—42no—
177D4—82no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—82no—
183C2×C4C2×C22⋊Dic5311yescharacteristic
184C2×C4D4×Dic5312yes—
185C2×C4#1313318511yescharacteristic
186C2×C4#1313318912yes—
187C8—42no—
188C8—4242no—
189C8—2124no—
190C22—22no—
191C22—21no—
192C22—82no—
193C22—81no—
194C22#1681050311yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—44no—
199C4—81no—
200C4#1681079512yes—
201C2—41no—
202C2—161no—
203C2#2027507912yes—
204C2#2027511911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3135C2
socleC22×C53#386
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#3135D53
hypocenterC53#9008
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3135

Complements

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

NH ≅ G/Ntype
1. #20283821C2semidirect
2. #20273380C2semidirect
4. #20272614C2semidirect
121. C53#9008semidirect

Maximal subgroups

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

subgroupindex
#202838212
#202733802
#202731842
#202726142
C53×#31352
#900853