Finite Group Databasebeta

← #27296572

Subgroups — #27296572

Order 13568.

350 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 63 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#20276643C212yesmaximal
2#20275971C211yesmaximal, characteristic
3#20272733C212yesmaximal, complemented
4#20272657C211yesmaximal, characteristic, complemented
5C53×#3261C211yesmaximal, characteristic, complemented
6C22×D424C2211yescharacteristic
7C2×#13133160—22no—
8#16810627—21no—
9C2×#13133133—21no—
10#16810423—22no—
11C22⋊#11719908C2212yes—
12#16810388C2211yescharacteristic
13C53×#541—21no—
14C53×#479—22no—
15C53×#470—21no—
16C53×#467C2212yes—
17C53×#462—22no—
18C53×#424C2211yescharacteristic
19C22×D212—21no—
20C2×D424—22no—
21C2×D424—42no—
22C2×#11719917—21no—
23#13133160—44no—
24C2×#11719908—22no—
25#13133137—42no—
26#13133133—42no—
27C2×Q16×C53—21no—
28C2×Q16×C53D412yes—
29C2×M16×C53D411yescharacteristic
30C22×C8×C53C2311yescharacteristic
31C4⋊Q8×C53—21no—
32#126×C53—42no—
33#125×C53—21no—
34#125×C53—42no—
35C22⋊Q8×C53—22no—
36C4×Q8×C53—22no—
37#117×C53D411yescharacteristic
38#107×C53D412yes—
39#106×C53—21no—
40C4⋊C8×C53—22no—
41#103×C53—22no—
42#103×C53—22no—
43C23×D53—41no—
44C2×Q8×C53—22no—
45C22×C4×C53C2×D411yescharacteristic
46C2×D212—22no—
47C2×D212—41no—
48C2×D212—41no—
49C2×D212—41no—
50C2×D212—41no—
51C2×D212—41no—
52Q16×C53—42no—
53Q16×C53—81no—
54M16×C53—41no—
55C2×C8×C53—21no—
56C2×C8×C53—41no—
57C2×C8×C53C2×D412yes—
58C4⋊C4×C53—22no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53D812yes—
64C22⋊C4×C53—21no—
65C22⋊C4×C53—42no—
66C42×C53—21no—
67C42×C53—42no—
68#11719917—41no—
69#11719917—41no—
70C2×#88322—41no—
71D424—24no—
72D424—42no—
73#11719908—42no—
74C23×C53AGL(1,ℤ/8)11yescharacteristic
75C22×D53—42no—
76C22×D53—81no—
77C22×D53—81no—
78C22×D53—162no—
79Q8×C53—42no—
80Q8×C53—42no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53#12011yescharacteristic
87C2×C4×C53C2×D811yescharacteristic
88D212—42no—
89D212—42no—
90D212—81no—
91D212—81no—
92C8×C53—22no—
93C8×C53—41no—
94#88322—81no—
95#12165—531nomaximal, Sylow
96C22×C53—21no—
97C22×C53—22no—
98C22×C53#44711yescharacteristic
99C2×D53—82no—
100C2×D53—162no—
101C2×D53—162no—
102C2×D53—162no—
103C4×C53—21no—
104C4×C53—42no—
105C4×C53—82no—
106C4×C53—82no—
107C4×C53#45411yescharacteristic
108C4×C53B(4,𝔽2)11yescharacteristic
109#3261D5311yescharacteristic, complemented
110#2187—532no—
111#2045—531no—
112#1350—532no—
113#1323—531no—
114C2×C53—41no—
115C2×C53#159811yescharacteristic
116C2×C53#219312yes—
117D53—162no—
118C22×D8—531no—
119#541—21no—
120C2×SD32—1062no—
121#479—22no—
122#470—21no—
123#467C2×D5312yes—
124#462—22no—
125#424C2×D5311yescharacteristic
126#418—1061no—
127C2×#100—1061no—
128#357—1062no—
129C22⋊C16—532no—
130#341—531no—
131C53#1216511yescharacteristic, complemented, Sylow
132C22×D4—1061no—
133C2×Q16—21no—
134C2×Q16C22⋊D5312yes—
135C2×D8—1062no—
136C2×D8—2122no—
137C2×M16—1061no—
138C2×M16C22⋊D5311yescharacteristic
139C22×C8C22×D5311yescharacteristic
140C4⋊Q8—21no—
141#126—42no—
142#125—21no—
143#125—42no—
144C22⋊Q8—22no—
145C4×Q8—22no—
146#117C22⋊D5311yescharacteristic
147SD32—2124no—
148C2×C16—1062no—
149#107C22⋊D5312yes—
150#106—21no—
151C4⋊C8—22no—
152#103—22no—
153#103—22no—
154#102—2122no—
155#100—2122no—
156C24—2121no—
157C2×Q8—22no—
158C2×D4—1062no—
159C2×D4—2121no—
160C2×D4—2121no—
161C2×D4—2121no—
162C2×D4—2121no—
163C2×D4—2121no—
164C22×C4C2×C22⋊D5311yescharacteristic
165Q16—42no—
166Q16—81no—
167D8—1064no—
168D8—2122no—
169M16—41no—
170M16—2121no—
171M16—2121no—
172C2×C8—21no—
173C2×C8—41no—
174C2×C8—2121no—
175C2×C8C2×C22⋊D5312yes—
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4#1171992212yes—
182C22⋊C4—21no—
183C22⋊C4—42no—
184C42—21no—
185C42—42no—
186C16—2122no—
187C23—8482no—
188C23—4241no—
189C23—4241no—
190C23—2122no—
191C23#1313324311yescharacteristic
192Q8—42no—
193Q8—42no—
194D4—4241no—
195D4—4241no—
196D4—2122no—
197D4—2122no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4#1313318611yescharacteristic
204C2×C4C2×#1171992211yescharacteristic
205C8—22no—
206C8—41no—
207C8—4241no—
208C22—22no—
209C22—21no—
210C22—8482no—
211C22—8482no—
212C22—8482no—
213C22—4242no—
214C22#1681075711yescharacteristic
215C4—21no—
216C4—42no—
217C4—82no—
218C4—82no—
219C4#1681079611yescharacteristic
220C4#1681081011yescharacteristic
221C2—41no—
222C2—8482no—
223C2#2027365011yescharacteristic
224C2#2027667912yes—
subgroupG/H
centerC22#16810757
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#3261C2
socleC22×C53#447
prime radicalC22×D424C22
hypercenter#3261D53
hypocenterC53#12165
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ #3261

Complements

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

NH ≅ G/Ntype
3. #20272733C2semidirect
4. #20272657C2semidirect
5. C53×#3261C2semidirect
109. #3261D53semidirect
131. C53#12165semidirect

Maximal subgroups

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

subgroupindex
#202766432
#202759712
#202727332
#202726572
C53×#32612
#1216553