Finite Group Databasebeta

← #27288217

Subgroups — #27288217

Order 13568.

402 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 99 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#20275693C212yesmaximal
2#20274866C211yesmaximal, characteristic
3#20273405C212yesmaximal
4#20272590C211yesmaximal, characteristic
5C53×#3249C211yesmaximal, characteristic
6#16810647C2212yes—
7C2×C4⋊#88322C2211yescharacteristic
8C2×#13133146—22no—
9C42×Dic53C2211yescharacteristic
10#16810382—22no—
11#16810382—22no—
12#16810372—22no—
13#16810369—22no—
14C53×#533C2212yes—
15C53×#477—22no—
16C53×#467—22no—
17C53×#462—22no—
18C2×C4⋊C8×C53C2211yescharacteristic
19C2×#11719917—22no—
20C22×#88322—21no—
21C2×C4×Dic53—21no—
22C2×C4×Dic53—21no—
23C2×C4×Dic53D411yescharacteristic
24C2×C4×Dic53D411yescharacteristic
25C4⋊#88322—22no—
26C4⋊#88322—22no—
27C4⋊#88322—22no—
28#13133146—44no—
29C2×Q16×C53—22no—
30C22×C8×C53D412yes—
31C4⋊Q8×C53D412yes—
32C4⋊Q8×C53D412yes—
33#125×C53—22no—
34C22⋊Q8×C53—22no—
35C22⋊Q8×C53—22no—
36C4×Q8×C53—22no—
37C4×Q8×C53—22no—
38#117×C53—22no—
39C2×C42×C53C2311yescharacteristic
40#107×C53—22no—
41C4⋊C8×C53—22no—
42#103×C53—24no—
43C2×Q8×C53—22no—
44C2×Q8×C53—22no—
45C22×C4×C53C2×D411yescharacteristic
46C22×C4×C53C2×D411yescharacteristic
47C22×C4×C53#4111yescharacteristic
48C22×Dic53—21no—
49C22×Dic53—21no—
50Q16×C53—82no—
51C2×C8×C53—22no—
52C2×C8×C53—42no—
53C4⋊C4×C53—22no—
54C4⋊C4×C53—22no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—22no—
57C4⋊C4×C53—42no—
58C4⋊C4×C53—42no—
59C22⋊C4×C53—42no—
60C42×C53—42no—
61C42×C53C2×D412yes—
62C42×C53#4112yes—
63C4×Dic53—22no—
64C4×Dic53—22no—
65C4×Dic53—21no—
66C4×Dic53—22no—
67C4×Dic53—42no—
68C4×Dic53Q1612yescomplemented
69#11719917—42no—
70#11719917—42no—
71C2×#88322—42no—
72C2×#88322—41no—
73C2×#88322—42no—
74C23×C53#12111yescharacteristic
75Q8×C53—42no—
76Q8×C53—42no—
77Q8×C53—82no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—41no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53#12011yescharacteristic
84C2×C4×C53#12112yes—
85C2×C4×C53#12312yes—
86C2×C4×C53#12411yescharacteristic
87C2×C4×C53C2×Q1611yescharacteristic
88C2×C4×C53#13511yescharacteristic
89C2×C4×C53#13712yes—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—41no—
95C2×Dic53—41no—
96C8×C53—42no—
97#88322—81no—
98#88322—82no—
99#10522—531nomaximal, Sylow
100C22×C53—21no—
101C22×C53—21no—
102C22×C53#39411yescharacteristic
103C22×C53#46711yescharacteristic
104C22×C53#47011yescharacteristic
105C4×C53—22no—
106C4×C53—21no—
107C4×C53—82no—
108C4×C53—82no—
109C4×C53#45311yescharacteristic
110C4×C53#45611yescharacteristic
111C4×C53#46212yes—
112C4×C53#46412yes—
113Dic53—24no—
114Dic53—42no—
115#3249D5311yescharacteristic
116#1994—532no—
117#1842—531no—
118#1550—532no—
119#1292—531no—
120C2×C53—21no—
121C2×C53—21no—
122C2×C53#203911yescharacteristic
123C2×C53#205011yescharacteristic
124C2×C53#205611yescharacteristic
125#533C2×D5312yes—
126#477—22no—
127#467—22no—
128#462—22no—
129#423—532no—
130C2×C4⋊C8—531no—
131C2×C4⋊C8—1062no—
132C2×C4⋊C8C2×D5311yescharacteristic
133C43—531no—
134#339—1062no—
135#339—1062no—
136#333—1062no—
137#332—1062no—
138C53#1052211yescharacteristic, complemented, Sylow
139C2×Q16—22no—
140C2×M16—1062no—
141C22×C8—1061no—
142C22×C8C22⋊D5312yes—
143C4⋊Q8C22⋊D5312yes—
144C4⋊Q8C22⋊D5312yes—
145#125—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150#117—22no—
151C2×C42—531no—
152C2×C42—531no—
153C2×C42—1061no—
154C2×C42—1061no—
155C2×C42C22×D5311yescharacteristic
156#107—22no—
157C4⋊C8—22no—
158C4⋊C8—1062no—
159C4⋊C8—1062no—
160C4⋊C8—1062no—
161C4⋊C8—2124no—
162#103—24no—
163C2×Q8—22no—
164C2×Q8—22no—
165C22×C4—1061no—
166C22×C4—1061no—
167C22×C4D4×D5311yescharacteristic
168C22×C4#1171994711yescharacteristic
169C22×C4C2×C22⋊D5311yescharacteristic
170Q16—82no—
171M16—2122no—
172M16—2122no—
173C2×C8—22no—
174C2×C8—42no—
175C2×C8—2122no—
176C2×C8—2121no—
177C2×C8—2122no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C22⋊C4—42no—
185C42—42no—
186C42—532no—
187C42—1062no—
188C42—1062no—
189C42—1061no—
190C42—1062no—
191C42—2122no—
192C42#1171994712yes—
193C42C2×C22⋊D5312yes—
194C23#1313318811yescharacteristic
195Q8—42no—
196Q8—42no—
197Q8—82no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—2121no—
208C2×C4—2121no—
209C2×C4#1313318611yescharacteristic
210C2×C4#1313318912yes—
211C2×C4#1313319812yes—
212C2×C4#1313320411yescharacteristic
213C2×C4Q16×D5311yescharacteristic
214C2×C4#1313324011yescharacteristic
215C2×C4#1313324912yes—
216C8—42no—
217C8—4241no—
218C8—4242no—
219C22—21no—
220C22—21no—
221C22#1681053711yescharacteristic
222C22#1681086111yescharacteristic
223C22#1681087811yescharacteristic
224C4—22no—
225C4—21no—
226C4—82no—
227C4—82no—
228C4—1064no—
229C4—2122no—
230C4#1681079111yescharacteristic
231C4#1681080611yescharacteristic
232C4#1681083112yes—
233C4#1681084712yes—
234C2—21no—
235C2—21no—
236C2#2027593611yescharacteristic
237C2#2027599911yescharacteristic
238C2#2027603511yescharacteristic
subgroupG/H
centerC22#16810537
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#3249C2
socleC22×C53#394
prime radicalC23×C53#121
hypercenter#3249D53
hypocenterC53#10522
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
68. C4×Dic53Q16semidirect
138. C53#10522semidirect

Maximal subgroups

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

subgroupindex
#202756932
#202748662
#202734052
#202725902
C53×#32492
#1052253