Finite Group Databasebeta

← #27271826

Subgroups — #27271826

Order 13568.

262 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C2×#16810422C211yesmaximal, characteristic
2#20275109C211yesmaximal, characteristic
3C53×#1558C211yesmaximal, characteristic
4#16811101C411yescharacteristic
5#16810632—21no—
6C2×#13133136—21no—
7C2×#13133136C411yescharacteristic
8#16810422—21no—
9#16810422C412yescomplemented
10#16810383—21no—
11C2×C4⋊Q8×C53C2211yescharacteristic
12C53×#432—21no—
13C53×#333—21no—
14C53×#333C412yescomplemented
15C2×#11719917—21no—
16C2×#11719917—41no—
17Q8×Dic53—42no—
18#13133176—21no—
19#13133176—41no—
20#13133141—42no—
21#13133136—22no—
22#13133136—41no—
23#13133136—41no—
24C22×Q8×C53C2×C411yescharacteristic
25C22×Q8×C53D411yescharacteristic
26C4⋊Q8×C53—41no—
27C4⋊Q8×C53C2×C411yescharacteristic
28C4⋊Q8×C53C2×C411yescharacteristic
29C4⋊Q8×C53D411yescharacteristic
30C4⋊Q8×C53Q811yescharacteristic
31C2×C4⋊C4×C53—41no—
32C2×C42×C53D411yescharacteristic
33C4⋊C8×C53—22no—
34C22⋊C8×C53—41no—
35#97×C53—41no—
36C2×Q8×C53—21no—
37C2×Q8×C53—21no—
38C2×Q8×C53—41no—
39C2×Q8×C53—41no—
40C2×Q8×C53—41no—
41C2×Q8×C53—41no—
42C2×Q8×C53C4211yescharacteristic
43C2×Q8×C53C22⋊C411yescharacteristic
44C2×Q8×C53C4⋊C411yescharacteristic
45C2×Q8×C53C4⋊C411yescharacteristic
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53C22⋊C411yescharacteristic
50C2×C8×C53—42no—
51C4⋊C4×C53—41no—
52C4⋊C4×C53—41no—
53C4⋊C4×C53—41no—
54C4⋊C4×C53—41no—
55C42×C53—21no—
56C42×C53C22⋊C411yescharacteristic
57C42×C53C4⋊C411yescharacteristic
58C22⋊Dic53—41no—
59C4⋊Dic53—22no—
60C4⋊Dic53—42no—
61C4×Dic53—41no—
62C4×Dic53—42no—
63#11719917—42no—
64#11719917—41no—
65#11719917—41no—
66#11719917—81no—
67C2×#88322—41no—
68C2×#88322—81no—
69C23×C53#9911yescharacteristic
70Q8×C53—42no—
71Q8×C53—41no—
72Q8×C53—41no—
73Q8×C53—41no—
74Q8×C53—41no—
75Q8×C53—81no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—42no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53#9511yescharacteristic
90C2×C4×C53#9911yescharacteristic
91C2×Dic53—42no—
92C8×C53—82no—
93#88322—81no—
94#88322—82no—
95#7483—531nomaximal, Sylow
96C22×C53—41no—
97C22×C53#34211yescharacteristic
98C22×C53AΣL(1,𝔽16)11yescharacteristic
99C22×C53#35211yescharacteristic
100C4×C53—22no—
101C4×C53—21no—
102C4×C53—21no—
103C4×C53—41no—
104C4×C53—41no—
105C4×C53—41no—
106C4×C53—41no—
107C4×C53—41no—
108Dic53—82no—
109C2×#356—531no—
110#1886—531no—
111#1558D5311yescharacteristic
112C2×C53—21no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53#139011yescharacteristic
116C2×C4⋊Q8C2×D5311yescharacteristic
117#519—531no—
118#432—21no—
119#419—1061no—
120C2×#101—531no—
121C2×#101—1061no—
122#356—532no—
123#356—1061no—
124#339—1061no—
125#333—21no—
126#333Dic5312yescomplemented
127C53#748311yescharacteristic, complemented, Sylow
128C22×Q8C4×D5311yescharacteristic
129C22×Q8D21211yescharacteristic
130C2×M16—1061no—
131C2×M16—2121no—
132C4⋊Q8—41no—
133C4⋊Q8Dic10611yescharacteristic
134C4⋊Q8C4×D5311yescharacteristic
135C4⋊Q8C2×Dic5311yescharacteristic
136C4⋊Q8C22⋊D5311yescharacteristic
137C4×Q8—2122no—
138#117—1061no—
139#117—2121no—
140C2×C4⋊C4—41no—
141C2×C42C22⋊D5311yescharacteristic
142C4⋊C8—22no—
143#103—2122no—
144#101—1062no—
145#101—2121no—
146#101—2121no—
147C22⋊C8—41no—
148#97—41no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8C4×Dic5311yescharacteristic
156C2×Q8#1171991911yescharacteristic
157C2×Q8C4⋊Dic5311yescharacteristic
158C2×Q8#1171992111yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4#1171992111yescharacteristic
163M16—4241no—
164M16—2122no—
165M16—2121no—
166M16—2121no—
167C2×C8—42no—
168C2×C8—4241no—
169C2×C8—2121no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—1062no—
175C4⋊C4—2122no—
176C22⋊C4—2121no—
177C42—21no—
178C42—2121no—
179C42—2122no—
180C42#1171991911yescharacteristic
181C42C22⋊Dic5311yescharacteristic
182C23#1313313111yescharacteristic
183Q8—41no—
184Q8—41no—
185Q8—41no—
186Q8—41no—
187Q8—42no—
188Q8—81no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—2122no—
203C2×C4#1313312411yescharacteristic
204C2×C4#1313313111yescharacteristic
205C8—82no—
206C8—4241no—
207C8—4242no—
208C22—41no—
209C22#1681038911yescharacteristic
210C22#1681041111yescharacteristic
211C22#1681041411yescharacteristic
212C4—22no—
213C4—21no—
214C4—21no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—4242no—
221C2—21no—
222C2—21no—
223C2—21no—
224C2#2027283411yescharacteristic
subgroupG/H
centerC2#20272834
commutatorC2×Q8×C53C42
FrattiniC2×C4⋊Q8C2×D53
FittingC53×#1558C2
socleC2×C53#1390
prime radicalC23×C53#99
cosocleC2×C4⋊Q8×C53C22
hypercenter#1558D53
hypocenterC53#7483
LCSG ▷ C2×Q8×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ #1558

Complements

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

NH ≅ G/Ntype
9. #16810422C4semidirect
14. C53×#333C4semidirect
126. #333Dic53semidirect
127. C53#7483semidirect

Maximal subgroups

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

subgroupindex
C2×#168104222
#202751092
C53×#15582
#748353