Finite Group Databasebeta

← #27297866

Subgroups — #27297866

Order 13568.

336 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 67 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#20276886C212yesmaximal, complemented
2#20274346C211yesmaximal, characteristic, complemented
3#20272754C211yesmaximal, characteristic, complemented
4#20272691C212yesmaximal, complemented
5C53×#3293C211yesmaximal, characteristic
6#16811021C2211yescharacteristic
7#16811019—21no—
8#16810975—21no—
9#16810447—22no—
10#16810444C2212yes—
11#16810433—22no—
12#16810344C2211yescharacteristic
13C53×#578—22no—
14C53×#494C2211yescharacteristic
15C53×#473—22no—
16C53×#456—21no—
17C53×#456—21no—
18C53×#443C2212yes—
19#13133217—21no—
20#13133157—22no—
21#13133149—22no—
22#13133148—42no—
23#13133140—42no—
24C4×#88322—21no—
252−1+4×C53—21no—
262−1+4×C53—21no—
27#137×C53—22no—
28#137×C53—22no—
29#137×C53—42no—
30#137×C53—42no—
31#135×C53—42no—
32#135×C53D412yes—
33C2×Q16×C53—22no—
34C2×Q16×C53—42no—
35#131×C53—22no—
36C4⋊Q8×C53D411yescharacteristic
37C4⋊Q8×C53D411yescharacteristic
38#108×C53D412yes—
39#104×C53—22no—
40#101×C53—21no—
41#101×C53—21no—
42C4×C8×C53C2311yescharacteristic
43#41×C53—22no—
44#41×C53—42no—
45#41×C53—42no—
46#41×C53—81no—
47#41×C53—81no—
48C2×Q8×C53—21no—
49C2×Q8×C53—21no—
50C2×Q8×C53—42no—
51C2×Q8×C53—42no—
52C2×Dic106—41no—
53Q16×C53—42no—
54Q16×C53—42no—
55Q16×C53—42no—
56Q16×C53—42no—
57Q16×C53SD1612yes—
58SD16×C53—22no—
59SD16×C53—42no—
60SD16×C53—42no—
61D8×C53SD1612yes—
62M16×C53—22no—
63M16×C53—42no—
64C2×C8×C53—42no—
65C2×C8×C53C2×D412yes—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C42×C53C2×D411yescharacteristic
69C4⋊Dic53—41no—
70C4⋊Dic53—41no—
71C2×#88322—22no—
72#11719908—42no—
73Q8×C53—22no—
74Q8×C53—42no—
75Q8×C53—42no—
76Q8×C53—81no—
77Q8×C53—81no—
78D4×C53—22no—
79D4×C53—42no—
80D4×C53—81no—
81D4×C53—81no—
82C2×C4×C53—21no—
83C2×C4×C53—42no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—82no—
87C2×C4×C53—82no—
88C2×C4×C53#12011yescharacteristic
89C2×Dic53—81no—
90Dic106—82no—
91C8×C53—42no—
92C8×C53C2×SD1612yes—
93C8×C53AGL(1,ℤ/8)12yes—
94#88322—42no—
95#12367—531nomaximal, Sylow
96C22×C53—42no—
97C22×C53#45311yescharacteristic
98C4×C53—22no—
99C4×C53—42no—
100C4×C53—81no—
101C4×C53—81no—
102C4×C53#45012yes—
103Dic53—161no—
104#3293D5311yescharacteristic
105#2230—532no—
106#1716—531no—
107#1358—531no—
108#1336—532no—
109C2×C53—21no—
110C2×C53—82no—
111C2×C53#162911yescharacteristic
112#578—22no—
113#498—531no—
114#498—1061no—
115#494C2×D5311yescharacteristic
116#487—1061no—
117#473—22no—
118#456—21no—
119#456—21no—
120#443C2×D5312yes—
121#365—1062no—
122#364—532no—
123#360—1062no—
124#322—531no—
125C53#1236711yescharacteristic, complemented, Sylow
1262−1+4—21no—
1272−1+4—21no—
128#137—22no—
129#137—22no—
130#137—42no—
131#137—42no—
132#135—42no—
133#135C22⋊D5312yes—
134C2×Q16—22no—
135C2×Q16—42no—
136#131—22no—
137C4⋊Q8—1061no—
138C4⋊Q8C22⋊D5311yescharacteristic
139C4⋊Q8C22⋊D5311yescharacteristic
140M32—1062no—
141#108C22⋊D5312yes—
142#106—1062no—
143#106—2122no—
144#104—22no—
145#103—2122no—
146#101—21no—
147#101—21no—
148C4×C8—1061no—
149C4×C8C22×D5311yescharacteristic
150#41—22no—
151#41—42no—
152#41—42no—
153#41—81no—
154#41—81no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—2121no—
160Q16—42no—
161Q16—42no—
162Q16—42no—
163Q16—42no—
164Q16#1171992312yes—
165SD16—22no—
166SD16—42no—
167SD16—42no—
168D8#1171992312yes—
169M16—22no—
170M16—42no—
171C2×C8—42no—
172C2×C8—1062no—
173C2×C8C2×C22⋊D5312yes—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—2121no—
177C4⋊C4—2121no—
178C42C2×C22⋊D5311yescharacteristic
179C16—2122no—
180Q8—22no—
181Q8—42no—
182Q8—42no—
183Q8—81no—
184Q8—81no—
185Q8—4242no—
186D4—22no—
187D4—42no—
188D4—81no—
189D4—81no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—4241no—
197C2×C4#1313318611yescharacteristic
198C8—42no—
199C8—2122no—
200C8C2×#1171992312yes—
201C8#1313324312yes—
202C22—42no—
203C22#1681079111yescharacteristic
204C4—22no—
205C4—42no—
206C4—81no—
207C4—81no—
208C4—8481no—
209C4#1681077412yes—
210C2—21no—
211C2—82no—
212C2#2027383111yescharacteristic
subgroupG/H
centerC2#20273831
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
Fitting, prime radicalC53×#3293C2
socleC2×C53#1629
hypercenter#3293D53
hypocenterC53#12367
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3293

Complements

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

NH ≅ G/Ntype
1. #20276886C2semidirect
2. #20274346C2semidirect
3. #20272754C2semidirect
4. #20272691C2semidirect
125. C53#12367semidirect

Maximal subgroups

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

subgroupindex
#202768862
#202743462
#202727542
#202726912
C53×#32932
#1236753