Finite Group Databasebeta

← #27285430

Subgroups — #27285430

Order 13568.

500 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#16811032C212yesmaximal
2#20282783C211yesmaximal, characteristic
3#20275534C211yesmaximal, characteristic
4C2×#16810364C212yesmaximal
5C53×#1295C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7C2×#13133209C2212yes—
8#16811032—24no—
9#16811032—24no—
10C2×#13133151—22no—
11C2×#13133149—22no—
12C2×#13133140—22no—
13C2×#13133128C2211yescharacteristic
14#16810563—22no—
15#16810555—22no—
16#16810553—22no—
17#16810364—24no—
18C2×#97×C53C2212yes—
19C53×#382C2211yescharacteristic
20C22×Dic106—21no—
21C22×#88322—21no—
22#13133217—21no—
23#13133217—22no—
24#13133217D412yes—
25#13133209—22no—
26#13133209—22no—
27#13133209—22no—
28#13133209—22no—
29C2×C4⋊Dic53—22no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53D412yes—
32C2×C4⋊Dic53D412yes—
33C2×#11719919—22no—
34C2×#11719919—22no—
35#13133151—28no—
36#13133149—28no—
37#13133140—44no—
38#13133128—22no—
39#13133124—24no—
40#13133124—22no—
41C22×C8×C53—22no—
42C2×C4⋊C4×C53D412yes—
43C2×C42×C53C2311yescharacteristic
44#97×C53—22no—
45#97×C53D414yes—
46#95×C53—22no—
47C22×C4×C53—22no—
48C22×C4×C53C2×D412yes—
49C22×C4×C53C2×D411yescharacteristic
50C22×Dic53—22no—
51C22×Dic53—22no—
52C22×Dic53—41no—
53C2×Dic106—22no—
54C2×Dic106—42no—
55C2×Dic106—42no—
56C2×Dic106—41no—
57C2×C8×C53—28no—
58C2×C8×C53—24no—
59C4⋊C4×C53—42no—
60C42×C53C2×D411yescharacteristic
61C42×C53C2×D412yes—
62C42×C53#4111yescharacteristic
63C4⋊Dic53—22no—
64C4⋊Dic53—24no—
65C4⋊Dic53—22no—
66C4⋊Dic53—22no—
67C4⋊Dic53—22no—
68C4⋊Dic53—42no—
69C4⋊Dic53—82no—
70#11719919—42no—
71#11719919—42no—
72#11719919—42no—
73#11719919—42no—
74C2×#88322—22no—
75C2×#88322—41no—
76C2×#88322—41no—
77C23×C53#12011yescharacteristic
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—24no—
81C2×C4×C53—42no—
82C2×C4×C53—44no—
83C2×C4×C53#12012yes—
84C2×C4×C53#12111yescharacteristic
85C2×C4×C53#12112yes—
86C2×C4×C53#12711yescharacteristic
87C2×Dic53—42no—
88C2×Dic53—42no—
89C2×Dic53—42no—
90C2×Dic53—42no—
91C2×Dic53—42no—
92C2×Dic53—44no—
93C2×Dic53—81no—
94C2×Dic53—81no—
95Dic106—44no—
96Dic106—82no—
97C8×C53—28no—
98#88322—42no—
99#10087—531nomaximal, Sylow
100C22×C53#39211yescharacteristic
101C22×C53#45311yescharacteristic
102C22×C53#45411yescharacteristic
103C22×C53#45614yes—
104C4×C53—22no—
105C4×C53—22no—
106C4×C53—21no—
107C4×C53—21no—
108C4×C53—82no—
109Dic53—82no—
110Dic53—82no—
111Dic53—82no—
112C2×#501—532no—
113#2822—531no—
114#1960—531no—
115C2×#330—532no—
116#1295D5311yescharacteristic
117C2×C53#166012yes—
118C2×C53#166312yes—
119C2×C53#200512yes—
120C2×C53#200611yescharacteristic
121C2×C4⋊Q8—531no—
122C2×#125—532no—
123#501—1064no—
124#501—1064no—
125C2×#107—1062no—
126C2×#106—1062no—
127C2×#103—1062no—
128C2×#97—531no—
129C2×#97C2×D5312yes—
130#400—1062no—
131#398—1062no—
132#398—1062no—
133#382C2×D5311yescharacteristic
134#330—1064no—
135C53#1008711yescharacteristic, complemented, Sylow
136C22×Q8—1061no—
137C22×C8—22no—
138C22×C8—1061no—
139C4⋊Q8—532no—
140C4⋊Q8—1062no—
141C4⋊Q8—1061no—
142#125—1062no—
143#125—1062no—
144#125—1062no—
145#125—1062no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4—1062no—
149C2×C4⋊C4—1062no—
150C2×C4⋊C4—1062no—
151C2×C4⋊C4—1062no—
152C2×C4⋊C4D21212yes—
153C2×C42C22×D5311yescharacteristic
154#107—1068no—
155#106—1068no—
156#103—2124no—
157#97—22no—
158#97—1062no—
159#97D21214yes—
160#95—22no—
161#95—1064no—
162#95—1062no—
163C2×Q8—1062no—
164C2×Q8—2122no—
165C2×Q8—2122no—
166C2×Q8—2121no—
167C22×C4—22no—
168C22×C4—1062no—
169C22×C4—1062no—
170C22×C4—2121no—
171C22×C4C2×D21211yescharacteristic
172C22×C4D4×D5312yes—
173C2×C8—28no—
174C2×C8—24no—
175C2×C8—1062no—
176C2×C8—2121no—
177C2×C8—2121no—
178C4⋊C4—42no—
179C4⋊C4—1062no—
180C4⋊C4—1064no—
181C4⋊C4—1062no—
182C4⋊C4—1062no—
183C4⋊C4—1062no—
184C4⋊C4—4242no—
185C4⋊C4—2122no—
186C4⋊C4—2122no—
187C4⋊C4—2122no—
188C4⋊C4—2122no—
189C4⋊C4—2122no—
190C42C2×D21212yes—
191C42D4×D5311yescharacteristic
192C42#1171994911yescharacteristic
193C23#1313318711yescharacteristic
194Q8—4242no—
195Q8—2124no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—4241no—
202C2×C4—4241no—
203C2×C4—2122no—
204C2×C4—2122no—
205C2×C4—2122no—
206C2×C4—2122no—
207C2×C4—2122no—
208C2×C4—2124no—
209C2×C4#1313318712yes—
210C2×C4#1313319212yes—
211C2×C4#1313319211yescharacteristic
212C2×C4#1313321411yescharacteristic
213C8—28no—
214C8—2122no—
215C22#1681053311yescharacteristic
216C22#1681079311yescharacteristic
217C22#1681079811yescharacteristic
218C22#1681080714yes—
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—82no—
224C4—4242no—
225C4—4242no—
226C4—4242no—
227C2#2027402712yes—
228C2#2027404412yes—
229C2#2027575412yes—
230C2#2027575911yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1295C2
socle, prime radicalC23×C53#120
hypercenter#1295D53
hypocenterC53#10087
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
135. C53#10087semidirect

Maximal subgroups

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

subgroupindex
C2×#168110322
#202827832
#202755342
C2×#168103642
C53×#12952
#1008753