Finite Group Databasebeta

← #27297355

Subgroups — #27297355

Order 13568.

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

normal subgroups only

This group has 354 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#20276203C211yesmaximal, characteristic, complemented
3#20272724C212yesmaximal, complemented
4#20272655C211yesmaximal, characteristic, complemented
5C53×#3245C211yesmaximal, characteristic
6C2×#13133151C2211yescharacteristic
7#16810521—21no—
8#16810448—22no—
9#16810424—22no—
10C22⋊#11719908C2212yes—
11#16810386—21no—
12#16810386C2211yescharacteristic
13C53×#532—21no—
14C53×#474—22no—
15C53×#466—21no—
16C53×#466C2212yes—
17C53×#459—22no—
18C2×C4⋊C8×C53C2211yescharacteristic
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—21no—
21C2×#11719919—41no—
22C2×#11719908—22no—
23#13133151—22no—
24#13133124—41no—
25C2×#41×C53—41no—
26C2×D8×C53—21no—
27C2×D8×C53D412yes—
28C22×C8×C53D411yescharacteristic
29C22×C8×C53C2311yescharacteristic
30C4⋊Q8×C53—21no—
31#127×C53—21no—
32#124×C53—41no—
33#121×C53—22no—
34#121×C53—41no—
35C4×D4×C53—22no—
36C4×D4×C53—41no—
37C2×C42×C53D411yescharacteristic
38#107×C53—21no—
39#107×C53D412yes—
40C4⋊C8×C53—22no—
41#102×C53—22no—
42#102×C53—22no—
43#41×C53—82no—
44#41×C53—82no—
45C2×Q8×C53—41no—
46C2×D4×C53—22no—
47C2×D4×C53—41no—
48C2×D4×C53—41no—
49C2×D4×C53—42no—
50C22×C4×C53—42no—
51C22×C4×C53D811yescharacteristic
52C22×C4×C53D811yescharacteristic
53C22×C4×C53C2×D411yescharacteristic
54C22×Dic53—41no—
55C22×Dic53—41no—
56D8×C53—42no—
57D8×C53—81no—
58C2×C8×C53—21no—
59C2×C8×C53—41no—
60C2×C8×C53—41no—
61C2×C8×C53C2×D412yes—
62C4⋊C4×C53—22no—
63C4⋊C4×C53—41no—
64C22⋊C4×C53—42no—
65C22⋊C4×C53—42no—
66C42×C53—21no—
67C42×C53—21no—
68C4⋊Dic53—22no—
69C4⋊Dic53—41no—
70C4⋊Dic53—41no—
71C4⋊Dic53—41no—
72#11719919—82no—
73#11719908—42no—
74C23×C53—42no—
75C23×C53C2×D811yescharacteristic
76Q8×C53—82no—
77D4×C53—42no—
78D4×C53—42no—
79D4×C53—82no—
80D4×C53—82no—
81D4×C53—82no—
82D4×C53—82no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—41no—
89C2×C4×C53—42no—
90C2×C4×C53—84no—
91C2×C4×C53D1612yes—
92C2×C4×C53#12011yescharacteristic
93C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
94C2×Dic53—42no—
95C2×Dic53—81no—
96C2×Dic53—81no—
97C2×Dic53—81no—
98C2×Dic53—81no—
99C2×Dic53—81no—
100C8×C53—22no—
101C8×C53—41no—
102#12288—531nomaximal, Sylow
103C22×C53—21no—
104C22×C53—21no—
105C22×C53—82no—
106C22×C53—82no—
107C22×C53—82no—
108C22×C53#44711yescharacteristic
109C22×C53C2×D1611yescharacteristic
110C22×C53#50911yescharacteristic
111C4×C53—21no—
112C4×C53—22no—
113C4×C53—41no—
114C4×C53—82no—
115C4×C53#45611yescharacteristic
116C4×C53#45811yescharacteristic
117Dic53—82no—
118Dic53—161no—
119#3245D5311yescharacteristic
120#2232—532no—
121#2086—531no—
122#1347—532no—
123#1322—531no—
124C2×C53—21no—
125C2×C53—21no—
126C2×C53—162no—
127C2×C53#161011yescharacteristic
128C2×C53#218411yescharacteristic
129C2×C53#219111yescharacteristic
130#532—21no—
131#474—22no—
132#466—21no—
133#466C2×D5312yes—
134#459—22no—
135C2×#107—531no—
136C2×C4⋊C8C2×D5311yescharacteristic
137#389—1061no—
138#366—1062no—
139#357—1062no—
140C22⋊C16—532no—
141#340—531no—
142#340—1061no—
143C53#1228811yescharacteristic, complemented, Sylow
144C2×#41—41no—
145C2×D8—21no—
146C2×D8C22⋊D5312yes—
147C22×C8C22⋊D5311yescharacteristic
148C22×C8C22×D5311yescharacteristic
149C4⋊Q8—21no—
150#127—21no—
151#124—41no—
152#121—22no—
153#121—41no—
154C4×D4—22no—
155C4×D4—41no—
156C2×C4⋊C4—1061no—
157C2×C4⋊C4—1061no—
158C2×C4⋊C4—2121no—
159C2×C42C22⋊D5311yescharacteristic
160C2×C16—1062no—
161#107—21no—
162#107—1062no—
163#107C22⋊D5312yes—
164C4⋊C8—22no—
165#102—22no—
166#102—22no—
167#95—2121no—
168#41—82no—
169#41—82no—
170C2×Q8—41no—
171C2×D4—22no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—42no—
175C22×C4—42no—
176C22×C4—2121no—
177C22×C4—2121no—
178C22×C4#1171992211yescharacteristic
179C22×C4#1171992211yescharacteristic
180C22×C4C2×C22⋊D5311yescharacteristic
181D8—42no—
182D8—81no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8C2×C22⋊D5312yes—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—1062no—
190C4⋊C4—4242no—
191C4⋊C4—2121no—
192C4⋊C4—2121no—
193C4⋊C4—2121no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C42—21no—
197C42—21no—
198C16—2122no—
199C23—42no—
200C23C2×#1171992211yescharacteristic
201Q8—82no—
202D4—42no—
203D4—42no—
204D4—82no—
205D4—82no—
206D4—82no—
207D4—82no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—84no—
216C2×C4—4241no—
217C2×C4—4241no—
218C2×C4—4241no—
219C2×C4—4241no—
220C2×C4—4241no—
221C2×C4—2122no—
222C2×C4#1313315812yes—
223C2×C4#1313318611yescharacteristic
224C2×C4#1313324311yescharacteristic
225C8—22no—
226C8—41no—
227C22—21no—
228C22—21no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22#1681075711yescharacteristic
233C22C2×#1313315811yescharacteristic
234C22#1681105911yescharacteristic
235C4—21no—
236C4—22no—
237C4—41no—
238C4—82no—
239C4—8481no—
240C4—4242no—
241C4#1681080611yescharacteristic
242C4#1681081311yescharacteristic
243C2—21no—
244C2—21no—
245C2—162no—
246C2#2027371811yescharacteristic
247C2#2027662711yescharacteristic
248C2#2027666611yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
Fitting, prime radicalC53×#3245C2
socleC22×C53#447
hypercenter#3245D53
hypocenterC53#12288
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #20276893C2semidirect
2. #20276203C2semidirect
3. #20272724C2semidirect
4. #20272655C2semidirect
143. C53#12288semidirect

Maximal subgroups

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

subgroupindex
#202768932
#202762032
#202727242
#202726552
C53×#32452
#1228853