Finite Group Databasebeta

← #27293304

Subgroups — #27293304

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#20283355C211yesmaximal, characteristic
2#20276515C212yesmaximal, complemented
3#20272656C212yesmaximal, complemented
4#20272641C211yesmaximal, characteristic
5C53×#1315C211yesmaximal, characteristic
6#41×Dic53C411yescharacteristic
7#16810681—22no—
8#16810673—22no—
9#16810651C411yescharacteristic
10#16810649C2211yescharacteristic
11#16810590C411yescharacteristic
12M16×Dic53C411yescharacteristic
13C2×C8×Dic53C2211yescharacteristic
14#16810441—22no—
15C22⋊#11719908C2212yes—
16#16810398—22no—
17C53×#408C2211yescharacteristic
18C53×C22⋊C16C2212yes—
19C2×#11719917C2×C411yescharacteristic
20C22×#88322C2×C411yescharacteristic
21Q8×Dic53C812yes—
22D4×Dic53—22no—
23D4×Dic53C812yes—
24#13133176—21no—
25#13133176C2×C411yescharacteristic
26C2×C4×Dic53—21no—
27C2×C4×Dic53C2×C411yescharacteristic
28C2×#11719908—22no—
29#13133146—22no—
30#13133146—22no—
31C22⋊#88322—22no—
32#13133127—21no—
33C8×Dic53—21no—
34C8×Dic53D412yes—
35C8×Dic53D412yes—
36C2×#41×C53C2×C411yescharacteristic
37C2×M16×C53C2×C411yescharacteristic
38C22×C8×C53C2311yescharacteristic
39C2×C16×C53—22no—
40C22⋊C8×C53—22no—
41#41×C53—41no—
42#41×C53—41no—
43C2×Q8×C53C2×C811yescharacteristic
44C2×D4×C53—21no—
45C2×D4×C53C2×C811yescharacteristic
46C22×C4×C53—21no—
47C22×C4×C53C22×C411yescharacteristic
48C22×Dic53—22no—
49C22×Dic53M1612yes—
50M16×C53—41no—
51C2×C8×C53—21no—
52C2×C8×C53—41no—
53C2×C8×C53C2×D412yes—
54C22⋊Dic53—21no—
55C22⋊Dic53—22no—
56C4⋊Dic53—22no—
57C4⋊Dic53C2×C812yes—
58C4×Dic53—21no—
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×C53—21no—
70C23×C53C2×M1611yescharacteristic
71Q8×C53—41no—
72D4×C53—41no—
73D4×C53—42no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—41no—
78C2×C4×C53—41no—
79C2×C4×C53C2×C22⋊C411yescharacteristic
80C2×C4×C53C22×C811yescharacteristic
81C2×Dic53—22no—
82C2×Dic53—42no—
83C2×Dic53—44no—
84C2×Dic53C22⋊C812yes—
85C2×Dic53C22⋊C812yes—
86C8×C53—22no—
87C8×C53—41no—
88#88322—22no—
89#88322—41no—
90#11463—531nomaximal, Sylow
91C22×C53—22no—
92C22×C53—21no—
93C22×C53—42no—
94C22×C53—41no—
95C22×C53C2×C22⋊C811yescharacteristic
96C4×C53—21no—
97C4×C53—41no—
98C4×C53#41011yescharacteristic
99C4×C53#41311yescharacteristic
100Dic53—22no—
101Dic53—22no—
102Dic53—42no—
103#2922—531no—
104#2163—532no—
105#1323—532no—
106#1315—531no—
107#1315D5311yescharacteristic
108C2×C53—41no—
109C2×C53—81no—
110C2×C53#145811yescharacteristic
111C2×C53#211712yes—
112C4×#41—531no—
113#433—1062no—
114#431—1062no—
115#424—531no—
116#424—531no—
117#408—531no—
118#408C2×D5311yescharacteristic
119C4×M16—531no—
120C2×C4×C8—531no—
121C4⋊C16—1062no—
122C22⋊C16—532no—
123C22⋊C16C2×D5312yes—
124#346—1062no—
125C53#1146311yescharacteristic, complemented, Sylow
126C2×#41C4×D5311yescharacteristic
127C2×M16—531no—
128C2×M16C4×D5311yescharacteristic
129C22×C8—531no—
130C22×C8C22×D5311yescharacteristic
131C4×Q8—532no—
132C4×D4—532no—
133C4×D4—1062no—
134#117—531no—
135#117—1061no—
136C2×C42—531no—
137C2×C42—1061no—
138C2×C16—22no—
139C2×C16—1062no—
140C4⋊C8—1062no—
141C4⋊C8—1062no—
142C22⋊C8—22no—
143C22⋊C8—1062no—
144#97—1061no—
145C4×C8—532no—
146C4×C8—532no—
147C4×C8—1061no—
148#41—41no—
149#41—41no—
150C2×Q8C8×D5311yescharacteristic
151C2×D4—21no—
152C2×D4C8×D5311yescharacteristic
153C22×C4—21no—
154C22×C4—532no—
155C22×C4—1062no—
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—532no—
166C4⋊C4—1062no—
167C22⋊C4—1062no—
168C22⋊C4—1061no—
169C42—531no—
170C42—531no—
171C42—532no—
172C42—1062no—
173C42—1061no—
174C16—42no—
175C16—2122no—
176C23—21no—
177C23M16×D5311yescharacteristic
178Q8—41no—
179D4—41no—
180D4—42no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—532no—
187C2×C4—532no—
188C2×C4—1062no—
189C2×C4—2122no—
190C2×C4—2124no—
191C2×C4C22⋊C4×D5311yescharacteristic
192C2×C4C2×C8×D5311yescharacteristic
193C8—22no—
194C8—41no—
195C8—1062no—
196C8—2121no—
197C22—22no—
198C22—21no—
199C22—42no—
200C22—41no—
201C22C22⋊C8×D5311yescharacteristic
202C4—21no—
203C4—41no—
204C4—1062no—
205C4—1062no—
206C4—2122no—
207C4#1681059511yescharacteristic
208C4#1681060411yescharacteristic
209C2—41no—
210C2—81no—
211C2#2027302611yescharacteristic
212C2#2027634112yes—
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#1315C2
socleC22×C53C2×C22⋊C8
prime radicalC2×#41×C53C2×C4
cosocleC22×C8×C53C23
hypercenter#1315D53
hypocenterC53#11463
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #20276515C2semidirect
3. #20272656C2semidirect
125. C53#11463semidirect

Maximal subgroups

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

subgroupindex
#202833552
#202765152
#202726562
#202726412
C53×#13152
#1146353