Finite Group Databasebeta

← #27271474

Subgroups — #27271474

Order 13568.

268 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 43 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#20275520C211yesmaximal, characteristic
2#20272661C211yesmaximal, characteristic, complemented
3C53×#1325C211yesmaximal, characteristic, complemented
4C2×#13133214C411yescharacteristic
5C2×#13133137—21no—
6C2×#13133137—21no—
7C2×C4×#88322C411yescharacteristic
8#16810349—21no—
9C53×#384C2211yescharacteristic
10C53×#324—21no—
11C22×D212—21no—
12C22×D212—41no—
13C22×#88322—21no—
14#13133214—21no—
15#13133214—41no—
16#13133214C812yescomplemented
17#13133137—42no—
18#13133137—42no—
19C22⋊#88322—41no—
20C4×#88322—41no—
21C2×C4⋊C4×C53—41no—
22C2×C4⋊C4×C53D411yescharacteristic
23C2×C4⋊C4×C53D411yescharacteristic
24C2×C42×C53C2×C411yescharacteristic
25C22⋊C8×C53—41no—
26C23×D53—41no—
27C22×C4×C53—21no—
28C22×C4×C53—21no—
29C22×C4×C53—21no—
30C22×C4×C53C22⋊C411yescharacteristic
31C2×D212—22no—
32C2×D212—41no—
33C2×D212—41no—
34C2×D212—41no—
35C2×D212—41no—
36C2×D212—42no—
37C2×D212—42no—
38C2×D212—41no—
39C2×D212—82no—
40C2×D212—82no—
41C2×C8×C53—81no—
42C4⋊C4×C53—41no—
43C4⋊C4×C53—41no—
44C4⋊C4×C53—82no—
45C42×C53—21no—
46C42×C53C2×C811yescharacteristic
47C42×C53M1611yescharacteristic
48C2×#88322—22no—
49C2×#88322—41no—
50C2×#88322—41no—
51C2×#88322—81no—
52C23×C53#9911yescharacteristic
53C22×D53—42no—
54C22×D53—82no—
55C22×D53—82no—
56C22×D53—81no—
57C22×D53—81no—
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53—81no—
67C2×C4×C53—81no—
68C2×C4×C53C22⋊C811yescharacteristic
69C2×C4×C53#10011yescharacteristic
70D212—42no—
71D212—42no—
72D212—81no—
73D212—81no—
74D212—84no—
75D212—84no—
76C8×C53—161no—
77#88322—42no—
78#88322—161no—
79#7354—531nomaximal, Sylow
80C22×C53—21no—
81C22×C53—21no—
82C22×C53#32311yescharacteristic
83C22×C53#35311yescharacteristic
84C22×C53#35511yescharacteristic
85C2×D53—84no—
86C2×D53—82no—
87C2×D53—162no—
88C2×D53—161no—
89C2×D53—161no—
90C4×C53—22no—
91C4×C53—22no—
92C4×C53—21no—
93C4×C53—21no—
94C4×C53—81no—
95C4×C53—81no—
96#1957—531no—
97#1325—531no—
98#1325D5311yescharacteristic, complemented
99C2×C53—21no—
100C2×C53—21no—
101C2×C53#131811yescharacteristic
102C2×C53#140811yescharacteristic
103C2×C53#141011yescharacteristic
104D53—162no—
105C2×#127—531no—
106C2×#102—1061no—
107C2×#102—1061no—
108C2×C4×C8—531no—
109#384C2×D5311yescharacteristic
110#324—21no—
111#324—1061no—
112C53#735411yescharacteristic, complemented, Sylow
113C22×D4—1061no—
114C22×D4—2121no—
115C22×C8—1061no—
116#127—532no—
117#127—1061no—
118#127—2121no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4D21211yescharacteristic
121C2×C4⋊C4C22⋊D5311yescharacteristic
122C2×C42C4×D5311yescharacteristic
123#102—2122no—
124#102—2122no—
125C22⋊C8—41no—
126C22⋊C8—2121no—
127C4×C8—2121no—
128C24—2121no—
129C2×D4—1062no—
130C2×D4—4242no—
131C2×D4—4242no—
132C2×D4—2122no—
133C2×D4—2122no—
134C2×D4—2121no—
135C2×D4—2121no—
136C2×D4—2121no—
137C2×D4—2121no—
138C2×D4—2121no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4#1171992111yescharacteristic
143C2×C8—81no—
144C2×C8—1062no—
145C2×C8—4241no—
146C2×C8—2121no—
147C2×C8—2121no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—82no—
151C42—21no—
152C42C8×D5311yescharacteristic
153C42#1171991211yescharacteristic
154C23—4242no—
155C23—4242no—
156C23—4241no—
157C23—4241no—
158C23—2122no—
159C23#1313313111yescharacteristic
160D4—4244no—
161D4—4244no—
162D4—4241no—
163D4—4241no—
164D4—2122no—
165D4—2122no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4#1313312911yescharacteristic
177C2×C4#1313313311yescharacteristic
178C8—161no—
179C8—8481no—
180C8—2122no—
181C22—21no—
182C22—21no—
183C22—8482no—
184C22—8481no—
185C22—8481no—
186C22—4244no—
187C22—4242no—
188C22#1681034611yescharacteristic
189C22#1681041511yescharacteristic
190C22#1681041911yescharacteristic
191C4—22no—
192C4—22no—
193C4—21no—
194C4—21no—
195C4—81no—
196C4—81no—
197C2—21no—
198C2—21no—
199C2—8482no—
200C2#2027264611yescharacteristic
201C2#2027287711yescharacteristic
202C2#2027288111yescharacteristic
subgroupG/H
centerC22#16810415
commutatorC42×C53C2×C8
Frattini#384C2×D53
FittingC53×#1325C2
socleC22×C53#353
prime radicalC2×#13133214C4
cosocleC53×#384C22
hypercenter#1325D53
hypocenterC53#7354
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ #1325

Complements

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

NH ≅ G/Ntype
2. #20272661C2semidirect
3. C53×#1325C2semidirect
16. #13133214C8semidirect
98. #1325D53semidirect
112. C53#7354semidirect

Maximal subgroups

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

subgroupindex
#202755202
#202726612
C53×#13252
#735453