Finite Group Databasebeta

← #27297293

Subgroups — #27297293

Order 13568.

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

normal subgroups only

This group has 396 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#20276893C212yesmaximal, complemented
2#20276138C211yesmaximal, characteristic, complemented
3#20272724C212yesmaximal, complemented
4#20272639C211yesmaximal, characteristic, complemented
5C53×#3186C211yesmaximal, characteristic
6C2×#13133151C2211yescharacteristic, complemented
7#16810483—21no—
8#16810448—22no—
9#16810424—22no—
10C22⋊#11719908C2212yescomplemented
11#16810386—21no—
12#16810386C2211yescharacteristic, complemented
13C2×#121×C53—21no—
14C53×#480—22no—
15C53×#466—21no—
16C53×#466C2212yes—
17C53×#447—22no—
18C2×C22⋊C8×C53C2211yescharacteristic
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—21no—
21C2×C22⋊Dic53—41no—
22C2×#11719908—22no—
23#13133151—22no—
24#13133124—41no—
25C22×D4×C53—21no—
26C22×D4×C53—41no—
27C23×C4×C53D411yescharacteristic
28C2×D8×C53—21no—
29C2×D8×C53D412yes—
30C22×C8×C53D411yescharacteristic
31C22×C8×C53C2311yescharacteristic
32#121×C53—22no—
33#121×C53—22no—
34#121×C53—41no—
35#121×C53—41no—
36C2×C4⋊C4×C53—21no—
37C2×C22⋊C4×C53—41no—
38#107×C53—21no—
39#107×C53D412yes—
40#102×C53—22no—
41#102×C53—22no—
42C22⋊C8×C53—22no—
43C24×C53—41no—
44C24×C53D811yescharacteristic
45C2×D4×C53—22no—
46C2×D4×C53—41no—
47C2×D4×C53—42no—
48C2×D4×C53—42no—
49C2×D4×C53—81no—
50C2×D4×C53—81no—
51C2×D4×C53—82no—
52C2×D4×C53—82no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—41no—
56C22×C4×C53—41no—
57C22×C4×C53—41no—
58C22×C4×C53D811yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×Dic53—41no—
61C22×Dic53—41no—
62D8×C53—42no—
63D8×C53—81no—
64C2×C8×C53—21no—
65C2×C8×C53—41no—
66C2×C8×C53—41no—
67C2×C8×C53C2×D412yes—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—41no—
70C22⋊C4×C53—42no—
71C22⋊C4×C53—42no—
72C22⋊Dic53—82no—
73C4⋊Dic53—41no—
74C4⋊Dic53—41no—
75C4⋊Dic53—41no—
76C4⋊Dic53—41no—
77#11719908—42no—
78C23×C53—21no—
79C23×C53—21no—
80C23×C53—41no—
81C23×C53—41no—
82C23×C53—42no—
83C23×C53—82no—
84C23×C53—82no—
85C23×C53—82no—
86C23×C53D1612yes—
87C23×C53C2×D811yescharacteristic
88D4×C53—42no—
89D4×C53—42no—
90D4×C53—82no—
91D4×C53—84no—
92D4×C53—82no—
93D4×C53—82no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53—81no—
102C2×C4×C53—82no—
103C2×C4×C53#12011yescharacteristic
104C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
105C2×Dic53—81no—
106C2×Dic53—81no—
107C2×Dic53—81no—
108C2×Dic53—81no—
109C2×Dic53—81no—
110C2×Dic53—81no—
111C8×C53—22no—
112C8×C53—41no—
113#12278—531nomaximal, Sylow
114C22×C53—21no—
115C22×C53—21no—
116C22×C53—22no—
117C22×C53—42no—
118C22×C53—41no—
119C22×C53—41no—
120C22×C53—81no—
121C22×C53—82no—
122C22×C53—82no—
123C22×C53—82no—
124C22×C53—164no—
125C22×C53#44711yescharacteristic
126C22×C53C2×D1611yescharacteristic
127C22×C53#50911yescharacteristic
128C4×C53—21no—
129C4×C53—41no—
130C4×C53—82no—
131C4×C53#45511yescharacteristic
132C4×C53#45811yescharacteristic
133Dic53—161no—
134Dic53—161no—
135#3186D5311yescharacteristic
136#2232—532no—
137#2075—531no—
138#1347—532no—
139#1314—531no—
140C2×C53—21no—
141C2×C53—21no—
142C2×C53—42no—
143C2×C53—162no—
144C2×C53#160711yescharacteristic
145C2×C53#218411yescharacteristic
146C2×C53#219111yescharacteristic
147C2×#121—21no—
148#480—22no—
149#466—21no—
150#466C2×D5312yes—
151#447—22no—
152C2×#107—531no—
153C2×C22⋊C8C2×D5311yescharacteristic
154#380—1061no—
155#366—1062no—
156#357—1062no—
157C22⋊C16—532no—
158#340—531no—
159#340—1061no—
160C53#1227811yescharacteristic, complemented, Sylow
161C22×D4—21no—
162C22×D4—41no—
163C23×C4C22⋊D5311yescharacteristic
164C2×D8—21no—
165C2×D8C22⋊D5312yes—
166C22×C8C22⋊D5311yescharacteristic
167C22×C8C22×D5311yescharacteristic
168#121—22no—
169#121—22no—
170#121—41no—
171#121—41no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—1061no—
174C2×C4⋊C4—1061no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—2121no—
177C2×C16—1062no—
178#107—21no—
179#107—1062no—
180#107C22⋊D5312yes—
181#102—22no—
182#102—22no—
183C22⋊C8—22no—
184#95—2121no—
185C24—41no—
186C24#1171992211yescharacteristic
187C2×D4—22no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4—82no—
194C2×D4—82no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—2121no—
201C22×C4—2121no—
202C22×C4#1171992211yescharacteristic
203C22×C4C2×C22⋊D5311yescharacteristic
204D8—42no—
205D8—81no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8C2×C22⋊D5312yes—
210C4⋊C4—22no—
211C4⋊C4—41no—
212C4⋊C4—2121no—
213C4⋊C4—2121no—
214C4⋊C4—2121no—
215C4⋊C4—2121no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—4242no—
219C16—2122no—
220C23—21no—
221C23—21no—
222C23—41no—
223C23—41no—
224C23—42no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23#1313315812yes—
229C23C2×#1171992211yescharacteristic
230D4—42no—
231D4—42no—
232D4—82no—
233D4—84no—
234D4—82no—
235D4—82no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—81no—
244C2×C4—82no—
245C2×C4—4241no—
246C2×C4—4241no—
247C2×C4—4241no—
248C2×C4—4241no—
249C2×C4—4241no—
250C2×C4—4241no—
251C2×C4#1313318611yescharacteristic
252C2×C4#1313324311yescharacteristic
253C8—22no—
254C8—41no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—164no—
266C22#1681075711yescharacteristic
267C22C2×#1313315811yescharacteristic
268C22#1681105911yescharacteristic
269C4—21no—
270C4—41no—
271C4—82no—
272C4—8481no—
273C4—8481no—
274C4#1681080111yescharacteristic
275C4#1681081311yescharacteristic
276C2—21no—
277C2—21no—
278C2—42no—
279C2—162no—
280C2#2027370111yescharacteristic
281C2#2027662711yescharacteristic
282C2#2027666611yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
Fitting, prime radicalC53×#3186C2
socleC22×C53#447
hypercenter#3186D53
hypocenterC53#12278
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #3186

Complements

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

NH ≅ G/Ntype
1. #20276893C2semidirect
2. #20276138C2semidirect
3. #20272724C2semidirect
4. #20272639C2semidirect
6. C2×#13133151C22semidirect
10. C22⋊#11719908C22semidirect
12. #16810386C22semidirect
160. C53#12278semidirect

Maximal subgroups

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

subgroupindex
#202768932
#202761382
#202727242
#202726392
C53×#31862
#1227853