Finite Group Databasebeta

← #27287164

Subgroups — #27287164

Order 13568.

400 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 93 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#20275687C212yesmaximal
2#20275534C211yesmaximal, characteristic
3#20272596C212yesmaximal
4#20272596C211yesmaximal, characteristic
5C53×#2844C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7C2×#13133140—21no—
8C2×#13133140—21no—
9C2×#13133128C2211yescharacteristic
10C2×#13133128C2212yes—
11#16810385—22no—
12#16810385—22no—
13C53×#400—21no—
14C53×#400—22no—
15C53×#395—21no—
16C53×#395—22no—
17C53×#382C2211yescharacteristic
18C53×#382C2212yes—
19C22×Dic106—21no—
20C22×#88322—21no—
21C22×#88322—22no—
22#13133217—21no—
23#13133217—22no—
24#13133217Q812yescomplemented
25C2×C4⋊Dic53—22no—
26#13133140—42no—
27#13133140—42no—
28#13133128—22no—
29#13133128—22no—
30#13133128—22no—
31C22×Q8×C53—21no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53D411yescharacteristic
35C2×C4⋊C4×C53D411yescharacteristic
36C2×C4⋊C4×C53D412yes—
37C2×C4⋊C4×C53D412yes—
38C2×C42×C53C2311yescharacteristic
39#95×C53—22no—
40#95×C53—21no—
41#95×C53—21no—
42#95×C53—22no—
43#95×C53—22no—
44#95×C53—22no—
45C2×Q8×C53—81no—
46C2×Q8×C53—82no—
47C22×C4×C53—21no—
48C22×C4×C53—22no—
49C22×C4×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53C2×D412yes—
52C22×C4×C53C2×D411yescharacteristic
53C22×Dic53—41no—
54C2×Dic106—22no—
55C2×Dic106—41no—
56C2×Dic106—41no—
57C2×Dic106—41no—
58C2×Dic106—41no—
59C2×Dic106—41no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—81no—
65C4⋊C4×C53—82no—
66C42×C53C2×Q811yescharacteristic
67C42×C53#4111yescharacteristic
68C42×C53#4112yes—
69C4⋊Dic53—24no—
70C4⋊Dic53—42no—
71C2×#88322—22no—
72C2×#88322—24no—
73C2×#88322—41no—
74C2×#88322—42no—
75C2×#88322—41no—
76C2×#88322—42no—
77C23×C53#12011yescharacteristic
78Q8×C53—84no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—41no—
84C2×C4×C53—42no—
85C2×C4×C53—41no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53C22⋊Q811yescharacteristic
94C2×C4×C53C22⋊Q812yes—
95C2×C4×C53#12411yescharacteristic
96C2×C4×C53#12412yes—
97C2×Dic53—42no—
98C2×Dic53—81no—
99C2×Dic53—81no—
100Dic106—42no—
101Dic106—42no—
102Dic106—81no—
103Dic106—81no—
104#88322—42no—
105#88322—44no—
106#10359—531nomaximal, Sylow
107C22×C53#39311yescharacteristic
108C22×C53#45311yescharacteristic
109C22×C53#45411yescharacteristic
110C22×C53#45512yes—
111C22×C53#45612yes—
112C4×C53—21no—
113C4×C53—21no—
114C4×C53—22no—
115C4×C53—22no—
116C4×C53—81no—
117C4×C53—82no—
118C4×C53—81no—
119C4×C53—82no—
120Dic53—82no—
121#2844D5311yescharacteristic
122#1993—532no—
123#1960—531no—
124#1295—531no—
125#1295—532no—
126C2×C53#165911yescharacteristic
127C2×C53#166011yescharacteristic
128C2×C53#166412yes—
129C2×C53#202711yescharacteristic
130C2×C53#202812yes—
131C2×C4⋊Q8—531no—
132C2×#103—1061no—
133C2×#103—1061no—
134C2×#97—531no—
135C2×#97—532no—
136#400—22no—
137#400—21no—
138#395—22no—
139#395—21no—
140#382C2×D5311yescharacteristic
141#382C2×D5312yes—
142#340—1062no—
143#340—1062no—
144C53#1035911yescharacteristic, complemented, Sylow
145C22×Q8—21no—
146C22×Q8—1061no—
147C22×C8—1062no—
148C22×C8—1061no—
149C4⋊Q8—532no—
150C4⋊Q8—1062no—
151C4⋊Q8—1061no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—1062no—
155C2×C4⋊C4C22⋊D5312yes—
156C2×C4⋊C4C22⋊D5312yes—
157C2×C4⋊C4C22⋊D5311yescharacteristic
158C2×C4⋊C4C22⋊D5311yescharacteristic
159C2×C42C22×D5311yescharacteristic
160#103—2122no—
161#103—2122no—
162#97—1062no—
163#97—1062no—
164#97—1062no—
165#95—22no—
166#95—21no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—22no—
171C2×Q8—81no—
172C2×Q8—82no—
173C2×Q8—1062no—
174C2×Q8—2121no—
175C2×Q8—2121no—
176C2×Q8—2121no—
177C2×Q8—2121no—
178C2×Q8—2121no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—2121no—
184C22×C4C2×C22⋊D5312yes—
185C22×C4C2×C22⋊D5311yescharacteristic
186C2×C8—1062no—
187C2×C8—1064no—
188C2×C8—2121no—
189C2×C8—2122no—
190C2×C8—2122no—
191C2×C8—2121no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—82no—
198C4⋊C4—1064no—
199C4⋊C4—2122no—
200C42Q8×D5311yescharacteristic
201C42#1171994912yes—
202C42#1171994911yescharacteristic
203C23#1313318611yescharacteristic
204Q8—84no—
205Q8—4241no—
206Q8—4241no—
207Q8—2122no—
208Q8—2122no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—4241no—
224C2×C4—4241no—
225C2×C4—2122no—
226C2×C4#1313319312yes—
227C2×C4#1313319311yescharacteristic
228C2×C4#1313320612yes—
229C2×C4#1313320611yescharacteristic
230C8—2122no—
231C8—2124no—
232C22#1681053411yescharacteristic
233C22#1681079111yescharacteristic
234C22#1681079611yescharacteristic
235C22#1681080112yes—
236C22#1681080612yes—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—81no—
242C4—82no—
243C4—81no—
244C4—82no—
245C4—4242no—
246C2#2027402111yescharacteristic
247C2#2027402611yescharacteristic
248C2#2027404912yes—
249C2#2027586811yescharacteristic
250C2#2027587312yes—
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2844C2
socle, prime radicalC23×C53#120
hypercenter#2844D53
hypocenterC53#10359
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2844

Complements

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

NH ≅ G/Ntype
24. #13133217Q8semidirect
144. C53#10359semidirect

Maximal subgroups

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

subgroupindex
#202756872
#202755342
#202725962
C53×#28442
#1035953