Finite Group Databasebeta

← #27271850

Subgroups — #27271850

Order 13568.

276 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 63 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
3C2×C53×#333C211yesmaximal, characteristic
4#16811101C411yescharacteristic
5#16810632—21no—
6C2×#13133136—21no—
7C2×#13133136C411yescharacteristic
8#16810422—21no—
9#16810422C412yescomplemented
10#16810383—21no—
11C2×C4⋊Q8×C53C2211yescharacteristic
12C2×C4⋊C8×C53—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
26C22×C8×C53—41no—
27C4⋊Q8×C53—41no—
28C4⋊Q8×C53C2×C411yescharacteristic
29C4⋊Q8×C53C2×C411yescharacteristic
30C4⋊Q8×C53D411yescharacteristic
31C4⋊Q8×C53Q811yescharacteristic
32C2×C4⋊C4×C53—41no—
33C2×C42×C53D411yescharacteristic
34C4⋊C8×C53—22no—
35C4⋊C8×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—
51C2×C8×C53—82no—
52C4⋊C4×C53—41no—
53C4⋊C4×C53—41no—
54C4⋊C4×C53—41no—
55C4⋊C4×C53—41no—
56C42×C53—21no—
57C42×C53C22⋊C411yescharacteristic
58C42×C53C4⋊C411yescharacteristic
59C22⋊Dic53—41no—
60C4⋊Dic53—22no—
61C4⋊Dic53—42no—
62C4×Dic53—41no—
63C4×Dic53—42no—
64#11719917—42no—
65#11719917—41no—
66#11719917—41no—
67#11719917—81no—
68C2×#88322—41no—
69C2×#88322—81no—
70C23×C53#9911yescharacteristic
71Q8×C53—42no—
72Q8×C53—41no—
73Q8×C53—41no—
74Q8×C53—41no—
75Q8×C53—41no—
76Q8×C53—81no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53#9511yescharacteristic
92C2×C4×C53#9911yescharacteristic
93C2×Dic53—42no—
94C8×C53—82no—
95#88322—81no—
96#88322—82no—
97#7491—531nomaximal, Sylow
98C22×C53—22no—
99C22×C53#34211yescharacteristic
100C22×C53AΣL(1,𝔽16)11yescharacteristic
101C22×C53#35211yescharacteristic
102C4×C53—21no—
103C4×C53—21no—
104C4×C53—22no—
105C4×C53—21no—
106C4×C53—21no—
107C4×C53—41no—
108C4×C53—41no—
109C4×C53—41no—
110C4×C53—41no—
111Dic53—82no—
112C2×#356—531no—
113#1886—531no—
114C2×#333D5311yescharacteristic
115C2×C53—21no—
116C2×C53—21no—
117C2×C53#139011yescharacteristic
118C2×C53#140512yes—
119C2×C4⋊Q8C2×D5311yescharacteristic
120#519—531no—
121C2×C4⋊C8—21no—
122#419—1061no—
123C2×#101—531no—
124C2×#101—1061no—
125#356—532no—
126#356—1061no—
127#339—1061no—
128#333—21no—
129#333Dic5312yescomplemented
130C53#749111yescharacteristic, complemented, Sylow
131C22×Q8C4×D5311yescharacteristic
132C22×Q8D21211yescharacteristic
133C2×M16—1061no—
134C2×M16—2121no—
135C22×C8—41no—
136C4⋊Q8—41no—
137C4⋊Q8Dic10611yescharacteristic
138C4⋊Q8C4×D5311yescharacteristic
139C4⋊Q8C2×Dic5311yescharacteristic
140C4⋊Q8C22⋊D5311yescharacteristic
141C4×Q8—2122no—
142#117—1061no—
143#117—2121no—
144C2×C4⋊C4—41no—
145C2×C42C22⋊D5311yescharacteristic
146C4⋊C8—22no—
147C4⋊C8—41no—
148#103—2122no—
149#101—1062no—
150#101—2121no—
151#101—2121no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8C4×Dic5311yescharacteristic
159C2×Q8#1171991911yescharacteristic
160C2×Q8C4⋊Dic5311yescharacteristic
161C2×Q8#1171992111yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4#1171992111yescharacteristic
166M16—4241no—
167M16—2122no—
168M16—2121no—
169M16—2121no—
170C2×C8—42no—
171C2×C8—82no—
172C2×C8—4241no—
173C2×C8—2121no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—1062no—
179C4⋊C4—2122no—
180C22⋊C4—2121no—
181C42—21no—
182C42—2121no—
183C42—2122no—
184C42#1171991911yescharacteristic
185C42C22⋊Dic5311yescharacteristic
186C23#1313313111yescharacteristic
187Q8—41no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191Q8—42no—
192Q8—81no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—2122no—
208C2×C4#1313312411yescharacteristic
209C2×C4#1313313111yescharacteristic
210C8—82no—
211C8—4241no—
212C8—4242no—
213C22—22no—
214C22#1681038911yescharacteristic
215C22#1681041111yescharacteristic
216C22#1681041411yescharacteristic
217C4—21no—
218C4—22no—
219C4—21no—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—4242no—
227C2—21no—
228C2—21no—
229C2#2027283411yescharacteristic
230C2#2027287012yes—
subgroupG/H
centerC22#16810411
commutatorC2×Q8×C53C42
FrattiniC2×C4⋊Q8C2×D53
FittingC2×C53×#333C2
socleC22×C53AΣL(1,𝔽16)
prime radicalC23×C53#99
cosocleC2×C4⋊Q8×C53C22
hypercenterC2×#333D53
hypocenterC53#7491
LCSG ▷ C2×Q8×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
9. #16810422C4semidirect
14. C53×#333C4semidirect
129. #333Dic53semidirect
130. C53#7491semidirect

Maximal subgroups

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

subgroupindex
C2×#168104222
#202751092
C2×C53×#3332
#749153