Finite Group Databasebeta

← #27297354

Subgroups — #27297354

Order 13568.

354 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 67 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#20286025C211yesmaximal, characteristic
2#20276895C212yesmaximal, complemented
3#20272723C212yesmaximal, complemented
4#20272655C211yesmaximal, characteristic, complemented
5C53×#2086C211yesmaximal, characteristic, complemented
6#16811173—21no—
7#16810904—22no—
8#16810859—21no—
9#16810859C2212yes—
10#16810819—22no—
11#16810448—22no—
12#16810423—22no—
13C22⋊#11719908C2212yes—
14C2×#107×C53C2211yescharacteristic
15C2×C4⋊C8×C53C2211yescharacteristic
16C53×#389—21no—
17C53×#340—21no—
18C53×#340C2211yescharacteristic
19C2×#11719945—41no—
20C2×D424—21no—
21C2×D424D412yes—
22#13133217—21no—
23#13133214—21no—
24#13133205—41no—
25#13133191—22no—
26#13133191—41no—
27C4×D212—22no—
28C4×D212—41no—
29C2×#11719908—22no—
30#13133151—21no—
31#13133151D412yes—
32#13133139—22no—
33#13133139—22no—
34C22×C8×C53D411yescharacteristic
35C22×C8×C53C2311yescharacteristic
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—41no—
39C2×C42×C53D411yescharacteristic
40#107×C53—22no—
41C4⋊C8×C53—22no—
42#95×C53—41no—
43C22×C4×C53—41no—
44C22×C4×C53—41no—
45C22×C4×C53D811yescharacteristic
46C22×C4×C53D811yescharacteristic
47C22×C4×C53C2×D411yescharacteristic
48C2×C22⋊D53—42no—
49#11719945—82no—
50#11719945—82no—
51C2×D212—22no—
52C2×D212—41no—
53C2×D212—41no—
54C2×C4×D53—42no—
55C2×Dic106—41no—
56C2×C8×C53—21no—
57C2×C8×C53—41no—
58C2×C8×C53—41no—
59C2×C8×C53C2×D412yes—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—82no—
65C42×C53—21no—
66C42×C53—21no—
67#11719921—42no—
68#11719921—42no—
69C4⋊Dic53—22no—
70C4⋊Dic53—41no—
71D424—42no—
72D424—81no—
73#11719908—42no—
74C23×C53C2×D811yescharacteristic
75C22×D53—42no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—41no—
82C2×C4×C53—42no—
83C2×C4×C53—81no—
84C2×C4×C53—81no—
85C2×C4×C53—81no—
86C2×C4×C53—81no—
87C2×C4×C53—81no—
88C2×C4×C53D1612yes—
89C2×C4×C53#12011yescharacteristic
90C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
91C22⋊D53—82no—
92C22⋊D53—82no—
93C2×Dic53—42no—
94D212—42no—
95D212—42no—
96D212—82no—
97D212—82no—
98C4×D53—84no—
99Dic106—82no—
100C8×C53—22no—
101C8×C53—41no—
102#12288—531nomaximal, Sylow
103C22×C53—21no—
104C22×C53—21no—
105C22×C53#44711yescharacteristic
106C22×C53C2×D1611yescharacteristic
107C22×C53#50911yescharacteristic
108C2×D53—82no—
109C2×D53—82no—
110C2×D53—82no—
111C4×C53—21no—
112C4×C53—22no—
113C4×C53—41no—
114C4×C53—82no—
115C4×C53—161no—
116C4×C53#45611yescharacteristic
117C4×C53#45811yescharacteristic
118Dic53—82no—
119#3245—531no—
120#2232—532no—
121#2086D5311yescharacteristic, complemented
122#1347—532no—
123#1322—531no—
124C2×C53—21no—
125C2×C53—21no—
126C2×C53#161011yescharacteristic
127C2×C53#218411yescharacteristic
128C2×C53#219111yescharacteristic
129D53—162no—
130#532—1061no—
131#474—1062no—
132#466—532no—
133#466—1061no—
134#459—1062no—
135C2×#107C2×D5311yescharacteristic
136C2×C4⋊C8C2×D5311yescharacteristic
137#389—21no—
138#366—1062no—
139#357—1062no—
140C22⋊C16—532no—
141#340—21no—
142#340C2×D5311yescharacteristic
143C53#1228811yescharacteristic, complemented, Sylow
144C2×#41—2121no—
145C2×D8—532no—
146C2×D8—1061no—
147C22×C8C22⋊D5311yescharacteristic
148C22×C8C22×D5311yescharacteristic
149C4⋊Q8—1061no—
150#127—1061no—
151#124—2121no—
152#121—1062no—
153#121—2121no—
154C4×D4—1062no—
155C4×D4—2121no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—41no—
159C2×C42C22⋊D5311yescharacteristic
160C2×C16—1062no—
161#107—22no—
162#107—532no—
163#107—1061no—
164C4⋊C8—22no—
165#102—1062no—
166#102—1062no—
167#95—41no—
168#41—4242no—
169#41—4242no—
170C2×Q8—2121no—
171C2×D4—1062no—
172C2×D4—2121no—
173C2×D4—2121no—
174C2×D4—2122no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—2122no—
178C22×C4#1171992211yescharacteristic
179C22×C4#1171992211yescharacteristic
180C22×C4C2×C22⋊D5311yescharacteristic
181D8—4241no—
182D8—2122no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8D4×D5312yes—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—82no—
192C4⋊C4—1062no—
193C4⋊C4—2121no—
194C22⋊C4—2122no—
195C22⋊C4—2122no—
196C42—21no—
197C42—21no—
198C16—2122no—
199C23—2122no—
200C23C2×#1171992211yescharacteristic
201Q8—4242no—
202D4—4242no—
203D4—4242no—
204D4—4242no—
205D4—4242no—
206D4—2122no—
207D4—2122no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—4244no—
221C2×C4—2122no—
222C2×C4#1313315812yes—
223C2×C4#1313318511yescharacteristic
224C2×C4#1313324211yescharacteristic
225C8—22no—
226C8—41no—
227C22—21no—
228C22—21no—
229C22—4242no—
230C22—4242no—
231C22—4242no—
232C22#1681075611yescharacteristic
233C22C2×#1313315811yescharacteristic
234C22#1681105811yescharacteristic
235C4—21no—
236C4—22no—
237C4—41no—
238C4—82no—
239C4—161no—
240C4—4242no—
241C4#1681080511yescharacteristic
242C4#1681081211yescharacteristic
243C2—21no—
244C2—21no—
245C2—8482no—
246C2#2027371711yescharacteristic
247C2#2027662611yescharacteristic
248C2#2027666511yescharacteristic
subgroupG/H
centerC22#16810756
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#2086C2
socleC22×C53#447
prime radical#20286025C2
hypercenter#2086D53
hypocenterC53#12288
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2086

Complements

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

NH ≅ G/Ntype
2. #20276895C2semidirect
3. #20272723C2semidirect
4. #20272655C2semidirect
5. C53×#2086C2semidirect
121. #2086D53semidirect
143. C53#12288semidirect

Maximal subgroups

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

subgroupindex
#202860252
#202768952
#202727232
#202726552
C53×#20862
#1228853