Finite Group Databasebeta

← #27288222

Subgroups — #27288222

Order 13568.

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

normal subgroups only

This group has 382 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#20274831C211yesmaximal, characteristic
3#20273405C212yesmaximal
4#20272590C211yesmaximal, characteristic
5C53×#3249C211yesmaximal, characteristic
6#16810647C2212yes—
7C2×C4⋊#88322C2211yescharacteristic
8C42×Dic53C2211yescharacteristic
9#16810382—22no—
10#16810382—22no—
11#16810377—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—
28C2×Q16×C53—22no—
29C22×C8×C53D412yes—
30C4⋊Q8×C53D412yes—
31C4⋊Q8×C53D412yes—
32#125×C53—22no—
33C22⋊Q8×C53—22no—
34C22⋊Q8×C53—22no—
35C4×Q8×C53—22no—
36C4×Q8×C53—22no—
37#117×C53—22no—
38C2×C42×C53C2311yescharacteristic
39#107×C53—22no—
40C4⋊C8×C53—22no—
41#103×C53—24no—
42C2×Q8×C53—22no—
43C2×Q8×C53—22no—
44C22×C4×C53C2×D411yescharacteristic
45C22×C4×C53C2×D411yescharacteristic
46C22×C4×C53#4111yescharacteristic
47C22×Dic53—21no—
48C22×Dic53—21no—
49Q16×C53—82no—
50C2×C8×C53—22no—
51C2×C8×C53—42no—
52C4⋊C4×C53—22no—
53C4⋊C4×C53—22no—
54C4⋊C4×C53—22no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—42no—
57C4⋊C4×C53—42no—
58C22⋊C4×C53—42no—
59C42×C53—42no—
60C42×C53C2×D412yes—
61C42×C53#4112yes—
62C4×Dic53—21no—
63C4×Dic53—22no—
64C4×Dic53—42no—
65C4×Dic53—41no—
66C4×Dic53—41no—
67C4×Dic53Q1612yescomplemented
68#11719917—42no—
69#11719917—42no—
70C2×#88322—42no—
71C2×#88322—41no—
72C2×#88322—42no—
73C23×C53#12111yescharacteristic
74Q8×C53—42no—
75Q8×C53—42no—
76Q8×C53—82no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—41no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53#12011yescharacteristic
83C2×C4×C53#12112yes—
84C2×C4×C53#12312yes—
85C2×C4×C53#12411yescharacteristic
86C2×C4×C53C2×Q1611yescharacteristic
87C2×C4×C53#13511yescharacteristic
88C2×C4×C53#13712yes—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—41no—
92C2×Dic53—41no—
93C2×Dic53—41no—
94C2×Dic53—41no—
95C8×C53—42no—
96#88322—81no—
97#88322—82no—
98#10523—531nomaximal, Sylow
99C22×C53—21no—
100C22×C53—21no—
101C22×C53#39411yescharacteristic
102C22×C53#46711yescharacteristic
103C22×C53#47011yescharacteristic
104C4×C53—22no—
105C4×C53—21no—
106C4×C53—82no—
107C4×C53—82no—
108C4×C53#45411yescharacteristic
109C4×C53#45511yescharacteristic
110C4×C53#46212yes—
111C4×C53#46412yes—
112Dic53—42no—
113Dic53—42no—
114#3249D5311yescharacteristic
115#1994—532no—
116#1831—531no—
117#1550—532no—
118#1292—531no—
119C2×C53—21no—
120C2×C53—21no—
121C2×C53#204011yescharacteristic
122C2×C53#205011yescharacteristic
123C2×C53#205611yescharacteristic
124#533C2×D5312yes—
125#477—22no—
126#467—22no—
127#462—22no—
128#423—532no—
129C2×C4⋊C8—531no—
130C2×C4⋊C8C2×D5311yescharacteristic
131C43—531no—
132#339—1062no—
133#339—1062no—
134#336—1062no—
135#333—1062no—
136#332—1062no—
137C53#1052311yescharacteristic, complemented, Sylow
138C2×Q16—22no—
139C2×M16—1062no—
140C22×C8—1061no—
141C22×C8C22⋊D5312yes—
142C4⋊Q8C22⋊D5312yes—
143C4⋊Q8C22⋊D5312yes—
144#125—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149#117—22no—
150C2×C42—531no—
151C2×C42—531no—
152C2×C42—1061no—
153C2×C42—1061no—
154C2×C42C22×D5311yescharacteristic
155#107—22no—
156C4⋊C8—22no—
157C4⋊C8—1062no—
158C4⋊C8—1062no—
159C4⋊C8—1062no—
160#103—24no—
161C2×Q8—22no—
162C2×Q8—22no—
163C22×C4—1061no—
164C22×C4—1061no—
165C22×C4D4×D5311yescharacteristic
166C22×C4#1171994711yescharacteristic
167C22×C4C2×C22⋊D5311yescharacteristic
168Q16—82no—
169M16—2122no—
170M16—2122no—
171C2×C8—22no—
172C2×C8—42no—
173C2×C8—2122no—
174C2×C8—2121no—
175C2×C8—2122no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C22⋊C4—42no—
183C42—42no—
184C42—532no—
185C42—1061no—
186C42—1062no—
187C42—2121no—
188C42—2121no—
189C42—2122no—
190C42#1171994712yes—
191C42C2×C22⋊D5312yes—
192C23#1313318811yescharacteristic
193Q8—42no—
194Q8—42no—
195Q8—82no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—2121no—
204C2×C4—2121no—
205C2×C4—2121no—
206C2×C4—2121no—
207C2×C4#1313318611yescharacteristic
208C2×C4#1313318912yes—
209C2×C4#1313319812yes—
210C2×C4#1313320411yescharacteristic
211C2×C4Q16×D5311yescharacteristic
212C2×C4#1313324011yescharacteristic
213C2×C4#1313324912yes—
214C8—42no—
215C8—4241no—
216C8—4242no—
217C22—21no—
218C22—21no—
219C22#1681053711yescharacteristic
220C22#1681086111yescharacteristic
221C22#1681087811yescharacteristic
222C4—22no—
223C4—21no—
224C4—82no—
225C4—82no—
226C4—2122no—
227C4—2122no—
228C4#1681079611yescharacteristic
229C4#1681080111yescharacteristic
230C4#1681083112yes—
231C4#1681084712yes—
232C2—21no—
233C2—21no—
234C2#2027594111yescharacteristic
235C2#2027599911yescharacteristic
236C2#2027603511yescharacteristic
subgroupG/H
centerC22#16810537
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#3249C2
socleC22×C53#394
prime radicalC23×C53#121
hypercenter#3249D53
hypocenterC53#10523
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
67. C4×Dic53Q16semidirect
137. C53#10523semidirect

Maximal subgroups

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

subgroupindex
#202756932
#202748312
#202734052
#202725902
C53×#32492
#1052353