Finite Group Databasebeta

← #27293306

Subgroups — #27293306

Order 13568.

306 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 85 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#20283358C211yesmaximal, characteristic
2#20276515C212yesmaximal, complemented
3#20272657C211yesmaximal, characteristic, complemented
4#20272640C212yesmaximal
5C53×#1323C211yesmaximal, characteristic, complemented
6#16811091C411yescharacteristic
7#16810683—22no—
8#16810675—22no—
9#16810650C411yescharacteristic
10#16810591C411yescharacteristic
11#16810589C2211yescharacteristic
12M16×Dic53C411yescharacteristic
13C2×C8×Dic53C2211yescharacteristic
14#16810441—22no—
15C22⋊#11719908C2212yes—
16#16810398—22no—
17C53×#424C2211yescharacteristic
18C53×C22⋊C16C2212yes—
19C2×#11719945C2×C411yescharacteristic
20C2×#11719917C2×C411yescharacteristic
21C22×#88322C2×C411yescharacteristic
22#13133182C812yes—
23#13133179C812yes—
24#13133177—22no—
25#13133174—21no—
26C2×C4×Dic53C2×C411yescharacteristic
27C42×D53—21no—
28C2×#11719908—22no—
29C4⋊#88322—22no—
30#13133129—22no—
31#13133129—22no—
32#13133127—21no—
33C8×Dic53—21no—
34C8×Dic53D412yes—
35C8×Dic53D412yes—
36C2×M16×C53C2×C411yescharacteristic
37C22×C8×C53C2311yescharacteristic
38#117×C53C2×C411yescharacteristic
39C2×C16×C53—22no—
40C4⋊C8×C53—22no—
41C22×C4×C53C22×C411yescharacteristic
42C2×C22⋊D53—21no—
43C22×Dic53M1612yes—
44#11719945—41no—
45#11719945—41no—
46C2×D212C2×C811yescharacteristic
47C2×C4×D53—21no—
48C2×C4×D53—22no—
49C2×Dic106C2×C811yescharacteristic
50M16×C53—41no—
51C2×C8×C53—21no—
52C2×C8×C53—41no—
53C2×C8×C53C2×D412yes—
54C4⋊C4×C53C2×C812yes—
55C22⋊C4×C53—21no—
56C42×C53—21no—
57#11719921—22no—
58#11719919—22no—
59C4×Dic53—22no—
60C4×Dic53C22⋊C411yescharacteristic
61C4×Dic53C22⋊C411yescharacteristic
62C4×Dic53C2×C812yes—
63#11719917—41no—
64C2×#88322—21no—
65C2×#88322—41no—
66C2×#88322C22⋊C412yes—
67C16×C53—42no—
68#11719908—42no—
69C23×C53C2×M1611yescharacteristic
70C22×D53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53C2×C22⋊C411yescharacteristic
75C2×C4×C53C22×C811yescharacteristic
76C22⋊D53—42no—
77C2×Dic53—21no—
78C2×Dic53—42no—
79C2×Dic53C22⋊C812yes—
80C2×Dic53C22⋊C812yes—
81D212—41no—
82C4×D53—44no—
83C4×D53—41no—
84C4×D53—41no—
85Dic106—41no—
86C8×C53—22no—
87C8×C53—41no—
88#88322—22no—
89#88322—41no—
90#11463—531nomaximal, Sylow
91C22×C53—22no—
92C22×C53—21no—
93C22×C53C2×C22⋊C811yescharacteristic
94C2×D53—42no—
95C2×D53—41no—
96C4×C53—21no—
97C4×C53—42no—
98C4×C53#41011yescharacteristic
99C4×C53#41311yescharacteristic
100Dic53—22no—
101Dic53—22no—
102Dic53—41no—
103#2922—531no—
104#2163—532no—
105#1323—531no—
106#1323D5311yescharacteristic, complemented
107#1315—532no—
108C2×C53—41no—
109C2×C53#145811yescharacteristic
110C2×C53#211712yes—
111D53—81no—
112C4×#41—531no—
113#433—1062no—
114#431—1062no—
115#424—531no—
116#424C2×D5311yescharacteristic
117#408—531no—
118#408—531no—
119C4×M16—531no—
120C2×C4×C8—531no—
121C4⋊C16—1062no—
122C22⋊C16—532no—
123C22⋊C16C2×D5312yes—
124#346—1062no—
125C53#1146311yescharacteristic, complemented, Sylow
126C2×#41—531no—
127C2×M16—531no—
128C2×M16C4×D5311yescharacteristic
129C22×C8—531no—
130C22×C8C22×D5311yescharacteristic
131C4×Q8—532no—
132C4×D4—532no—
133C4×D4—1062no—
134#117—1061no—
135#117C4×D5311yescharacteristic
136C2×C42—531no—
137C2×C42—1061no—
138C2×C16—22no—
139C2×C16—1062no—
140C4⋊C8—22no—
141C4⋊C8—1062no—
142C22⋊C8—1062no—
143C22⋊C8—1062no—
144#97—1061no—
145C4×C8—532no—
146C4×C8—532no—
147C4×C8—1061no—
148#41—2121no—
149#41—2121no—
150C2×Q8—531no—
151C2×D4—531no—
152C2×D4—1061no—
153C22×C4—532no—
154C22×C4—1062no—
155C22×C4—1061no—
156C22×C4C2×C4×D5311yescharacteristic
157M16—41no—
158M16—2121no—
159C2×C8—21no—
160C2×C8—41no—
161C2×C8—532no—
162C2×C8—1061no—
163C2×C8—2121no—
164C2×C8D4×D5312yes—
165C4⋊C4—1062no—
166C4⋊C4C8×D5312yes—
167C22⋊C4—21no—
168C22⋊C4—1062no—
169C42—21no—
170C42—531no—
171C42—531no—
172C42—532no—
173C42—1062no—
174C16—42no—
175C16—2122no—
176C23—1061no—
177C23M16×D5311yescharacteristic
178Q8—2121no—
179D4—2121no—
180D4—2122no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—532no—
185C2×C4—532no—
186C2×C4—1061no—
187C2×C4—2124no—
188C2×C4—2122no—
189C2×C4—2121no—
190C2×C4—2121no—
191C2×C4C22⋊C4×D5311yescharacteristic
192C2×C4C2×C8×D5311yescharacteristic
193C8—22no—
194C8—41no—
195C8—1062no—
196C8—2121no—
197C22—22no—
198C22—21no—
199C22—2122no—
200C22—2121no—
201C22C22⋊C8×D5311yescharacteristic
202C4—21no—
203C4—42no—
204C4—1062no—
205C4—1062no—
206C4—2121no—
207C4#1681059511yescharacteristic
208C4#1681060611yescharacteristic
209C2—41no—
210C2—4241no—
211C2#2027302811yescharacteristic
212C2#2027634112yes—
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#1323C2
socleC22×C53C2×C22⋊C8
prime radicalC2×#11719945C2×C4
cosocleC22×C8×C53C23
hypercenter#1323D53
hypocenterC53#11463
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
2. #20276515C2semidirect
3. #20272657C2semidirect
5. C53×#1323C2semidirect
106. #1323D53semidirect
125. C53#11463semidirect

Maximal subgroups

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

subgroupindex
#202833582
#202765152
#202726572
#202726402
C53×#13232
#1146353