Finite Group Databasebeta

← #27271401

Subgroups — #27271401

Order 13568.

328 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 71 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#20275013C211yesmaximal, characteristic
2#20273053C211yesmaximal, characteristic
3C53×#1800C211yesmaximal, characteristic
4C22×#11719917C411yescharacteristic
5#16810587—21no—
6C2×C22⋊#88322—21no—
7C2×C22⋊#88322C411yescharacteristic
8#16810378—21no—
9#16810349—21no—
10#16810347C412yes—
11C2×C4×D4×C53C2211yescharacteristic
12C53×#423—21no—
13C2×#100×C53C412yes—
14C53×#343—21no—
15C2×#11719917—41no—
16C2×#11719917—41no—
17C2×#11719917—42no—
18C22×#88322—21no—
19C22×#88322—21no—
20C22⋊#88322—22no—
21C22⋊#88322—41no—
22C22⋊#88322—41no—
23C22⋊#88322—42no—
24C22×D4×C53C2×C411yescharacteristic
25C23×C4×C53C2×C411yescharacteristic
26C23×C4×C53D411yescharacteristic
27C2×M16×C53—22no—
28C4×D4×C53—41no—
29C4×D4×C53—41no—
30C2×C4⋊C4×C53D411yescharacteristic
31C2×C22⋊C4×C53C2×C411yescharacteristic
32C2×C22⋊C4×C53Q811yescharacteristic
33C2×C42×C53D411yescharacteristic
34C4⋊C8×C53—41no—
35C4⋊C8×C53—41no—
36#100×C53—42no—
37C24×C53C4211yescharacteristic
38C24×C53C4⋊C411yescharacteristic
39C2×D4×C53—21no—
40C2×D4×C53—21no—
41C2×D4×C53—41no—
42C2×D4×C53—41no—
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53—41no—
48C22×C4×C53—41no—
49C22×C4×C53—41no—
50C22×C4×C53—41no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C4⋊C411yescharacteristic
55C22×C4×C53C4⋊C411yescharacteristic
56M16×C53—42no—
57M16×C53—42no—
58C2×C8×C53—42no—
59C4⋊C4×C53—41no—
60C22⋊C4×C53—41no—
61C22⋊C4×C53—41no—
62C42×C53—21no—
63C42×C53—21no—
64#11719917—42no—
65#11719917—42no—
66#11719917—81no—
67#11719917—81no—
68C2×#88322—22no—
69C2×#88322—41no—
70C2×#88322—41no—
71C2×#88322—42no—
72C2×#88322—41no—
73C2×#88322—41no—
74C2×#88322—41no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—21no—
78C23×C53—21no—
79C23×C53—21no—
80C23×C53—21no—
81C23×C53—41no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53—41no—
85C23×C53#9511yescharacteristic
86D4×C53—42no—
87D4×C53—42no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—42no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—81no—
104C2×C4×C53#9911yescharacteristic
105C2×C4×C53#9911yescharacteristic
106C2×C4×C53#10011yescharacteristic
107C2×C4×C53#10111yescharacteristic
108C8×C53—82no—
109#88322—42no—
110#88322—81no—
111#88322—82no—
112#7321—531nomaximal, Sylow
113C22×C53—21no—
114C22×C53—21no—
115C22×C53—42no—
116C22×C53—41no—
117C22×C53—41no—
118C22×C53—41no—
119C22×C53—41no—
120C22×C53—41no—
121C22×C53—41no—
122C22×C53—81no—
123C22×C53—81no—
124C22×C53—81no—
125C22×C53#34211yescharacteristic
126C22×C53#34311yescharacteristic
127C22×C53#34411yescharacteristic
128C4×C53—21no—
129C4×C53—21no—
130C4×C53—21no—
131C4×C53—41no—
132C4×C53—41no—
133C4×C53—41no—
134C4×C53AΣL(1,𝔽16)11yescharacteristic
135C4×C53#35211yescharacteristic
136#1870—531no—
137#1800D5311yescharacteristic
138#1465—531no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53—81no—
142C2×C53—81no—
143C2×C53#130511yescharacteristic
144C2×C53#139011yescharacteristic
145C2×C53#139211yescharacteristic
146C22×M16—531no—
147C2×C4×D4C2×D5311yescharacteristic
148#423—21no—
149C2×#100Dic5312yes—
150#407—1061no—
151C2×C22⋊C8—531no—
152C2×C22⋊C8—1061no—
153#343—21no—
154#336—1061no—
155#324—1061no—
156#323—532no—
157C53#732111yescharacteristic, complemented, Sylow
158C22×D4C2×Dic5311yescharacteristic
159C23×C4C4×D5311yescharacteristic
160C23×C4D21211yescharacteristic
161C2×M16—22no—
162C2×M16—2122no—
163C2×M16—2121no—
164C2×M16—2121no—
165C22×C8—1061no—
166C22×C8—1061no—
167C4×D4—41no—
168C4×D4—41no—
169C2×C4⋊C4C22⋊D5311yescharacteristic
170C2×C22⋊C4Dic10611yescharacteristic
171C2×C22⋊C4C4×D5311yescharacteristic
172C2×C42C22⋊D5311yescharacteristic
173C4⋊C8—41no—
174C4⋊C8—41no—
175#100—42no—
176C22⋊C8—1062no—
177C22⋊C8—2122no—
178C22⋊C8—2121no—
179C22⋊C8—2121no—
180C24C4×Dic5311yescharacteristic
181C24C4⋊Dic5311yescharacteristic
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4#1171991911yescharacteristic
195C22×C4#1171991911yescharacteristic
196C22×C4#1171992111yescharacteristic
197C22×C4#1171992111yescharacteristic
198C22×C4C22⋊Dic5311yescharacteristic
199M16—42no—
200M16—42no—
201M16—4241no—
202M16—4241no—
203M16—2122no—
204M16—2122no—
205C2×C8—42no—
206C2×C8—1062no—
207C2×C8—2122no—
208C2×C8—2121no—
209C2×C8—2121no—
210C2×C8—2121no—
211C2×C8—2121no—
212C2×C8—2121no—
213C4⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C42—21no—
217C42—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23#1313312411yescharacteristic
229D4—42no—
230D4—42no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—81no—
247C2×C4#1313313111yescharacteristic
248C2×C4#1313313111yescharacteristic
249C2×C4#1313313411yescharacteristic
250C2×C4#1313313611yescharacteristic
251C8—82no—
252C8—4241no—
253C8—4242no—
254C8—2122no—
255C22—21no—
256C22—21no—
257C22—42no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22#1681038911yescharacteristic
268C22#1681039211yescharacteristic
269C22#1681039311yescharacteristic
270C4—21no—
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4#1681041111yescharacteristic
277C4#1681041411yescharacteristic
278C2—21no—
279C2—21no—
280C2—81no—
281C2—81no—
282C2#2027262111yescharacteristic
283C2#2027283411yescharacteristic
284C2#2027283711yescharacteristic
subgroupG/H
centerC22#16810389
commutatorC24×C53C42
FrattiniC2×C4×D4C2×D53
FittingC53×#1800C2
socleC22×C53#342
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×D4×C53C22
hypercenter#1800D53
hypocenterC53#7321
LCSG ▷ C24×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
157. C53#7321semidirect

Maximal subgroups

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

subgroupindex
#202750132
#202730532
C53×#18002
#732153