Finite Group Databasebeta

← #27271838

Subgroups — #27271838

Order 13568.

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

normal subgroups only

This group has 298 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×#16810420C211yesmaximal, characteristic
2#20275104C211yesmaximal, characteristic
3C2×C53×#329C211yesmaximal, characteristic
4#16811098C411yescharacteristic
5#16810632—21no—
6C2×#13133136—21no—
7C2×#13133134C411yescharacteristic
8#16810420—21no—
9#16810420C412yescomplemented
10#16810383—21no—
11C2×#124×C53C2211yescharacteristic
12C2×#97×C53—21no—
13C53×#329—21no—
14C53×#329C412yescomplemented
15C2×#11719917—21no—
16C2×#11719917—41no—
17D4×Dic53—42no—
18#13133176—21no—
19C2×C22⋊Dic53—41no—
20#13133141—42no—
21#13133136—22no—
22#13133136—41no—
23#13133134—41no—
24C22×Q8×C53D411yescharacteristic
25C22×D4×C53C2×C411yescharacteristic
26C22×C8×C53—41no—
27#124×C53—41no—
28#124×C53C2×C411yescharacteristic
29#124×C53C2×C411yescharacteristic
30#124×C53D411yescharacteristic
31#124×C53Q811yescharacteristic
32C2×C22⋊C4×C53—41no—
33C2×C42×C53D411yescharacteristic
34#97×C53—22no—
35#97×C53—41no—
36C24×C53—21no—
37C2×Q8×C53—21no—
38C2×Q8×C53—41no—
39C2×Q8×C53—41no—
40C2×Q8×C53C22⋊C411yescharacteristic
41C2×Q8×C53C4⋊C411yescharacteristic
42C2×D4×C53—21no—
43C2×D4×C53—41no—
44C2×D4×C53—41no—
45C2×D4×C53C4211yescharacteristic
46C2×D4×C53C4⋊C411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53C22⋊C411yescharacteristic
50C22×Dic53—42no—
51C2×C8×C53—42no—
52C2×C8×C53—44no—
53C22⋊C4×C53—41no—
54C22⋊C4×C53—41no—
55C22⋊C4×C53—41no—
56C22⋊C4×C53—41no—
57C42×C53—21no—
58C42×C53C22⋊C411yescharacteristic
59C42×C53C4⋊C411yescharacteristic
60C22⋊Dic53—42no—
61C22⋊Dic53—41no—
62C4⋊Dic53—22no—
63C4×Dic53—41no—
64#11719917—42no—
65#11719917—41no—
66#11719917—41no—
67#11719917—81no—
68C2×#88322—41no—
69C2×#88322—81no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53—41no—
73C23×C53—41no—
74C23×C53—82no—
75C23×C53#9911yescharacteristic
76Q8×C53—41no—
77Q8×C53—41no—
78Q8×C53—81no—
79D4×C53—41no—
80D4×C53—41no—
81D4×C53—42no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—42no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53#9511yescharacteristic
92C2×C4×C53#9911yescharacteristic
93C2×Dic53—42no—
94C2×Dic53—162no—
95C8×C53—44no—
96#88322—81no—
97#88322—82no—
98#7487—531nomaximal, Sylow
99C22×C53—22no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—81no—
103C22×C53—81no—
104C22×C53—82no—
105C22×C53—161no—
106C22×C53#34211yescharacteristic
107C22×C53AΣL(1,𝔽16)11yescharacteristic
108C22×C53#35211yescharacteristic
109C4×C53—21no—
110C4×C53—21no—
111C4×C53—41no—
112C4×C53—41no—
113C4×C53—41no—
114C4×C53—41no—
115Dic53—82no—
116C2×#355—531no—
117#1885—531no—
118C2×#329D5311yescharacteristic
119C2×C53—21no—
120C2×C53—21no—
121C2×C53—81no—
122C2×C53—81no—
123C2×C53#139011yescharacteristic
124C2×C53#140312yes—
125C2×#124C2×D5311yescharacteristic
126#518—531no—
127#419—1061no—
128C2×#101—1061no—
129C2×#100—531no—
130C2×#97—21no—
131#355—532no—
132#355—1061no—
133#339—1061no—
134#329—21no—
135#329Dic5312yescomplemented
136C53#748711yescharacteristic, complemented, Sylow
137C22×Q8D21211yescharacteristic
138C22×D4C4×D5311yescharacteristic
139C2×M16—1061no—
140C2×M16—2121no—
141C22×C8—41no—
142#124—41no—
143#124Dic10611yescharacteristic
144#124C4×D5311yescharacteristic
145#124C2×Dic5311yescharacteristic
146#124C22⋊D5311yescharacteristic
147C4×D4—2122no—
148#117—1061no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—2121no—
151C2×C42C22⋊D5311yescharacteristic
152#103—2122no—
153#101—1062no—
154#101—2121no—
155#100—2121no—
156#97—22no—
157#97—41no—
158C24—21no—
159C2×Q8—21no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8C4⋊Dic5311yescharacteristic
163C2×Q8#1171992111yescharacteristic
164C2×D4—21no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4C4×Dic5311yescharacteristic
168C2×D4#1171991911yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—2122no—
172C22×C4#1171992111yescharacteristic
173M16—4241no—
174M16—2122no—
175M16—2121no—
176M16—2121no—
177C2×C8—42no—
178C2×C8—44no—
179C2×C8—4241no—
180C2×C8—2121no—
181C4⋊C4—1062no—
182C22⋊C4—41no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—2122no—
187C22⋊C4—2121no—
188C42—21no—
189C42—2121no—
190C42#1171991911yescharacteristic
191C42C22⋊Dic5311yescharacteristic
192C23—21no—
193C23—21no—
194C23—41no—
195C23—41no—
196C23—82no—
197C23#1313313111yescharacteristic
198Q8—41no—
199Q8—41no—
200Q8—81no—
201D4—41no—
202D4—41no—
203D4—42no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—8482no—
214C2×C4—2122no—
215C2×C4#1313312411yescharacteristic
216C2×C4#1313313111yescharacteristic
217C8—44no—
218C8—4241no—
219C8—4242no—
220C22—22no—
221C22—41no—
222C22—41no—
223C22—81no—
224C22—81no—
225C22—82no—
226C22—161no—
227C22#1681038911yescharacteristic
228C22#1681041111yescharacteristic
229C22#1681041411yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—41no—
234C4—41no—
235C4—41no—
236C4—4242no—
237C2—21no—
238C2—21no—
239C2—81no—
240C2—81no—
241C2#2027283411yescharacteristic
242C2#2027286412yes—
subgroupG/H
centerC22#16810411
commutatorC2×D4×C53C42
FrattiniC2×#124C2×D53
FittingC2×C53×#329C2
socleC22×C53AΣL(1,𝔽16)
prime radicalC22×D4×C53C2×C4
cosocleC2×#124×C53C22
hypercenterC2×#329D53
hypocenterC53#7487
LCSG ▷ C2×D4×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
9. #16810420C4semidirect
14. C53×#329C4semidirect
135. #329Dic53semidirect
136. C53#7487semidirect

Maximal subgroups

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

subgroupindex
C2×#168104202
#202751042
C2×C53×#3292
#748753