Finite Group Databasebeta

← #27271849

Subgroups — #27271849

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
2C2×#16810372C211yesmaximal, characteristic
3C53×#1886C211yesmaximal, characteristic
4C2×C4⋊#88322—21no—
5C2×#13133136—21no—
6#16810422—21no—
7#16810422C412yescomplemented
8#16810372—21no—
9#16810372C412yescomplemented
10C2×C4⋊Q8×C53C2211yescharacteristic
11C53×#519C411yescharacteristic
12C53×#419—21no—
13C2×#101×C53C411yescharacteristic
14C53×#339—21no—
15C2×#11719917—41no—
16C22×#88322—41no—
17C4⋊#88322—22no—
18C4⋊#88322—41no—
19#13133136—22no—
20#13133136—41no—
21C22×Q8×C53C2×C411yescharacteristic
22C22×Q8×C53D411yescharacteristic
23C2×M16×C53—21no—
24C4⋊Q8×C53—41no—
25C4⋊Q8×C53C2×C411yescharacteristic
26C4⋊Q8×C53C2×C411yescharacteristic
27C4⋊Q8×C53D411yescharacteristic
28C4⋊Q8×C53Q811yescharacteristic
29C4×Q8×C53—42no—
30#117×C53—21no—
31#117×C53—41no—
32C2×C4⋊C4×C53—41no—
33C2×C42×C53D411yescharacteristic
34#103×C53—42no—
35#101×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
50M16×C53—41no—
51M16×C53—41no—
52C2×C8×C53—41no—
53C4⋊C4×C53—22no—
54C4⋊C4×C53—41no—
55C4⋊C4×C53—41no—
56C4⋊C4×C53—41no—
57C4⋊C4×C53—41no—
58C4⋊C4×C53—42no—
59C22⋊C4×C53—41no—
60C42×C53—21no—
61C42×C53—41no—
62C42×C53—42no—
63C42×C53C22⋊C411yescharacteristic
64C42×C53C4⋊C411yescharacteristic
65#11719917—42no—
66#11719917—81no—
67C2×#88322—42no—
68C2×#88322—82no—
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—42no—
92C2×C4×C53#9511yescharacteristic
93C2×C4×C53#9911yescharacteristic
94C8×C53—81no—
95#88322—82no—
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—
111C4×C53—82no—
112C2×#356—531no—
113#1886D5311yescharacteristic
114C2×#333—531no—
115C2×C53—21no—
116C2×C53—21no—
117C2×C53#139011yescharacteristic
118C2×C53#140512yes—
119C2×C4⋊Q8C2×D5311yescharacteristic
120#519Dic5311yescharacteristic
121C2×C4⋊C8—1061no—
122#419—21no—
123C2×#101—1061no—
124C2×#101Dic5311yescharacteristic
125#356—532no—
126#356—1061no—
127#339—21no—
128#333—532no—
129#333—1061no—
130C53#749111yescharacteristic, complemented, Sylow
131C22×Q8C2×Dic5311yescharacteristic
132C22×Q8C22⋊D5311yescharacteristic
133C2×M16—21no—
134C2×M16—2121no—
135C22×C8—2121no—
136C4⋊Q8—41no—
137C4⋊Q8Dic10611yescharacteristic
138C4⋊Q8C4×D5311yescharacteristic
139C4⋊Q8C4×D5311yescharacteristic
140C4⋊Q8D21211yescharacteristic
141C4×Q8—42no—
142#117—21no—
143#117—41no—
144C2×C4⋊C4—41no—
145C2×C42C22⋊D5311yescharacteristic
146C4⋊C8—1062no—
147C4⋊C8—2121no—
148#103—42no—
149#101—41no—
150#101—1062no—
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×C4C22⋊Dic5311yescharacteristic
166M16—41no—
167M16—41no—
168M16—4241no—
169M16—2122no—
170C2×C8—41no—
171C2×C8—4242no—
172C2×C8—4241no—
173C2×C8—2122no—
174C4⋊C4—22no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C22⋊C4—41no—
181C42—21no—
182C42—41no—
183C42—42no—
184C42#1171991911yescharacteristic
185C42#1171992111yescharacteristic
186C23#1313313211yescharacteristic
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—42no—
208C2×C4#1313312411yescharacteristic
209C2×C4#1313313211yescharacteristic
210C8—81no—
211C8—4242no—
212C8—4242no—
213C22—22no—
214C22#1681039011yescharacteristic
215C22#1681041011yescharacteristic
216C22#1681041311yescharacteristic
217C4—21no—
218C4—22no—
219C4—21no—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—82no—
227C2—21no—
228C2—21no—
229C2#2027283311yescharacteristic
230C2#2027286912yes—
subgroupG/H
centerC22#16810410
commutatorC2×Q8×C53C42
FrattiniC2×C4⋊Q8C2×D53
FittingC53×#1886C2
socleC22×C53AΣL(1,𝔽16)
prime radicalC23×C53#99
cosocleC2×C4⋊Q8×C53C22
hypercenter#1886D53
hypocenterC53#7491
LCSG ▷ C2×Q8×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ #1886

Complements

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

NH ≅ G/Ntype
7. #16810422C4semidirect
9. #16810372C4semidirect
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
C2×#168103722
C53×#18862
#749153