Finite Group Databasebeta

← #27285085

Subgroups — #27285085

Order 13568.

476 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 105 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#20284750C212yesmaximal
2#20275655C211yesmaximal, characteristic
3#20273118C212yesmaximal
4#20272546C211yesmaximal, characteristic
5C53×#3162C211yesmaximal, characteristic
6C2×C4×Dic106C2211yescharacteristic
7#16810956—22no—
8#16810915—22no—
9#16810830—22no—
10#16810778—22no—
11C4×#11719925—24no—
12#16810676C2212yes—
13C2×C4⋊#88322C2211yescharacteristic
14#16810377—22no—
15Q8⋊#88322—24no—
16C53×#533C2212yes—
17C53×#501—22no—
18C53×#501—22no—
19C53×#427—22no—
20C2×#97×C53C2211yescharacteristic
21C22×Dic106D411yescharacteristic
22C2×#11719925—24no—
23C22×#88322—21no—
24C4×Dic106—22no—
25C4×Dic106—22no—
26C2×C4⋊Dic53D411yescharacteristic
27C2×#11719919—21no—
28C2×C4×Dic53—21no—
29#13133150—24no—
30C4⋊#88322—22no—
31C4⋊#88322—22no—
32#13133141—24no—
33#13133140—24no—
34C22⋊#88322—22no—
35C4×#88322D414yes—
36C22×C8×C53D412yes—
37C4⋊Q8×C53D412yes—
38C4⋊Q8×C53D412yes—
39#125×C53—22no—
40C22⋊Q8×C53—22no—
41C22⋊Q8×C53—22no—
42C4×Q8×C53—22no—
43C4×Q8×C53—22no—
44#117×C53—22no—
45C2×C42×C53C2311yescharacteristic
46#107×C53—22no—
47#107×C53—22no—
48#106×C53—24no—
49#97×C53—22no—
50C2×Q8×C53—22no—
51C2×Q8×C53—22no—
52C22×C4×C53C2×D411yescharacteristic
53C22×C4×C53C2×D411yescharacteristic
54C22×C4×C53#4111yescharacteristic
55C22×Dic53—21no—
56C22×Dic53—21no—
57C2×Dic106—22no—
58C2×Dic106—42no—
59C2×C8×C53—22no—
60C2×C8×C53—22no—
61C2×C8×C53Q1614yescomplemented
62C4⋊C4×C53—22no—
63C4⋊C4×C53—22no—
64C4⋊C4×C53—22no—
65C4⋊C4×C53—22no—
66C4⋊C4×C53—42no—
67C4⋊C4×C53—42no—
68C22⋊C4×C53—42no—
69C42×C53—42no—
70C42×C53C2×D412yes—
71C42×C53C2×D412yes—
72#11719925—48no—
73C4⋊Dic53—22no—
74#11719919—42no—
75C4×Dic53—41no—
76C4×Dic53—41no—
77C2×#88322—24no—
78C2×#88322—41no—
79C2×#88322—42no—
80C23×C53#12111yescharacteristic
81Q8×C53—44no—
82Q8×C53—44no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53#12012yes—
88C2×C4×C53#12011yescharacteristic
89C2×C4×C53#12112yes—
90C2×C4×C53#12711yescharacteristic
91C2×C4×C53C2×Q1612yes—
92C2×C4×C53#13712yes—
93C2×Dic53—22no—
94C2×Dic53—41no—
95C2×Dic53—41no—
96C2×Dic53—42no—
97C2×Dic53—41no—
98Dic106—24no—
99Dic106—42no—
100C8×C53—24no—
101#88322—44no—
102#88322—81no—
103#10037—531nomaximal, Sylow
104C22×C53—21no—
105C22×C53—21no—
106C22×C53#39211yescharacteristic
107C22×C53#46212yes—
108C4×C53—22no—
109C4×C53—22no—
110C4×C53—82no—
111C4×C53—82no—
112C4×C53#45112yes—
113C4×C53#49411yescharacteristic
114C4×C53#49711yescharacteristic
115Dic53—42no—
116Dic53—81no—
117#3162D5311yescharacteristic
118#3104—532no—
119#1985—531no—
120#1481—532no—
121#1275—531no—
122C2×C53—22no—
123C2×C53#201011yescharacteristic
124C2×C53#201611yescharacteristic
125C2×C53#201711yescharacteristic
126#533C2×D5312yes—
127C2×C4×Q8—531no—
128#501—22no—
129#501—22no—
130#484—1062no—
131#477—1062no—
132#462—1062no—
133#451—1062no—
134C4×Q16—1064no—
135#431—532no—
136#427—22no—
137C2×C4⋊C8—531no—
138C2×#97C2×D5311yescharacteristic
139#336—1062no—
140Q8⋊C8—1064no—
141C53#1003711yescharacteristic, complemented, Sylow
142C22×Q8—531no—
143C2×Q16—1064no—
144C22×C8—1061no—
145C22×C8C22⋊D5312yes—
146C4⋊Q8C22⋊D5312yes—
147C4⋊Q8C22⋊D5312yes—
148#125—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—1062no—
154C4×Q8—1062no—
155#117—22no—
156C2×C4⋊C4—531no—
157C2×C4⋊C4—1061no—
158C2×C42—1061no—
159C2×C42C22×D5311yescharacteristic
160#107—22no—
161#107—22no—
162#107—1064no—
163#106—24no—
164C4⋊C8—1062no—
165C4⋊C8—1062no—
166#103—1064no—
167#103—1064no—
168C22⋊C8—1062no—
169#97—22no—
170C4×C8—534no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—1062no—
174C2×Q8—2122no—
175C22×C4—1061no—
176C22×C4—1061no—
177C22×C4D4×D5311yescharacteristic
178C22×C4#1171994711yescharacteristic
179C22×C4C2×C22⋊D5311yescharacteristic
180Q16—2128no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—1064no—
184C2×C8—2121no—
185C2×C8—2122no—
186C2×C8#1171992514yescomplemented
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—1062no—
194C4⋊C4—2122no—
195C22⋊C4—42no—
196C42—42no—
197C42—2121no—
198C42—2121no—
199C42D4×D5312yes—
200C42C2×C22⋊D5312yes—
201C23#1313318811yescharacteristic
202Q8—44no—
203Q8—44no—
204Q8—1064no—
205Q8—2122no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—1062no—
211C2×C4—2122no—
212C2×C4—2121no—
213C2×C4—2121no—
214C2×C4—2121no—
215C2×C4#1313318512yes—
216C2×C4#1313318611yescharacteristic
217C2×C4#1313318912yes—
218C2×C4#1313321311yescharacteristic
219C2×C4C2×#1171992512yes—
220C2×C4#1313324812yes—
221C8—24no—
222C8—4241no—
223C8—2124no—
224C22—21no—
225C22—21no—
226C22#1681053111yescharacteristic
227C22#1681083012yes—
228C4—22no—
229C4—22no—
230C4—82no—
231C4—82no—
232C4—4241no—
233C4—2122no—
234C4#1681077912yes—
235C4#1681100211yescharacteristic
236C4#1681101511yescharacteristic
237C2—22no—
238C2#2027577911yescharacteristic
239C2#2027581511yescharacteristic
240C2#2027581811yescharacteristic
subgroupG/H
centerC2×C4#13133213
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#3162C2
socleC22×C53#392
prime radicalC23×C53#121
hypercenter#3162D53
hypocenterC53#10037
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
61. C2×C8×C53Q16semidirect
141. C53#10037semidirect
186. C2×C8#11719925semidirect

Maximal subgroups

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

subgroupindex
#202847502
#202756552
#202731182
#202725462
C53×#31622
#1003753