Finite Group Databasebeta

← #28311353

Subgroups — #28311353

Order 13568.

334 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 68 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#20290625C212yesmaximal
2#20289816C212yesmaximal
3#20289806C211yesmaximal, characteristic
4#20289803C211yesmaximal, characteristic
5#20283365C211yesmaximal, characteristic
6#16811609C2212yes—
7#16811602C2212yes—
8#16811494—22no—
9#16811490—22no—
10#16811091C2211yescharacteristic
11#16810683—22no—
12#16810679—22no—
13#16810650C411yescharacteristic, complemented
14#16810591C2211yescharacteristic
15#16810589C2211yescharacteristic
16M16×Dic53C411yescharacteristic, complemented
17C2×#13133127C411yescharacteristic, complemented
18C53×#424C411yescharacteristic, complemented
19D53⋊M16—22no—
20#13133296—22no—
21#13133278—22no—
22#13133277—22no—
23#13133276—22no—
24#13133275—22no—
25C2×#11719945C2311yescharacteristic
26C2×#11719917C2×C411yescharacteristic
27C22×#88322C2×C411yescharacteristic
28#13133182—21no—
29#13133179—21no—
30#13133177—22no—
31#13133174D412yes—
32C2×C4×Dic53C2×C411yescharacteristic
33C42×D53D412yes—
34C4⋊#88322—22no—
35#13133129—22no—
36#13133129—22no—
37#13133127—21no—
38#13133127—21no—
39#13133127—21no—
40C8×Dic53—21no—
41C2×M16×C53C2×C411yescharacteristic
42C22×C8×C53C2×C411yescharacteristic
43#117×C53C2×C411yescharacteristic
44C4⋊C8×C53—22no—
45C22×C4×C53C22×C411yescharacteristic
46C2×C22⋊D53#4112yes—
47C22×Dic53—21no—
48#11719945—41no—
49#11719945—41no—
50C2×D212#4111yescharacteristic
51C2×C4×D53—22no—
52C2×C4×D53C2×D412yes—
53C2×Dic106#4111yescharacteristic
54C22⋊#6157—42no—
55#11719940—42no—
56C4⋊#6157—42no—
57C4×#6157—42no—
58#11719936—42no—
59D53⋊C8—42no—
60M16×C53—41no—
61C2×C8×C53—21no—
62C2×C8×C53—21no—
63C2×C8×C53—41no—
64C4⋊C4×C53—21no—
65C22⋊C4×C53—21no—
66C42×C53C22⋊C412yes—
67#11719921—22no—
68#11719919—22no—
69C4×Dic53—21no—
70C4×Dic53—22no—
71C4×Dic53C22⋊C411yescharacteristic
72C4×Dic53C22⋊C411yescharacteristic
73#11719917—41no—
74C2×#88322—21no—
75C2×#88322—21no—
76C2×#88322—41no—
77C23×C53#11711yescharacteristic
78C22×D53#12312yes—
79C2×#6157—82no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53C2×C22⋊C411yescharacteristic
84C2×C4×C53#11711yescharacteristic
85C22⋊D53—42no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—81no—
89C2×Dic53#12312yes—
90D212—41no—
91C4×D53—22no—
92C4×D53—22no—
93C4×D53—82no—
94Dic106—41no—
95#88324—82no—
96C8×C53—41no—
97C8×C53—41no—
98#88322—41no—
99#88322—41no—
100#12865—531nomaximal, Sylow
101C22×C53—21no—
102C22×C53—41no—
103C22×C53#38111yescharacteristic
104C2×D53—22no—
105C2×D53—42no—
106#6157—162no—
107C4×C53—21no—
108C4×C53—42no—
109C4×C53#41011yescharacteristic
110C4×C53#41011yescharacteristic
111Dic53—22no—
112Dic53—41no—
113Dic53—41no—
114#2923—531no—
115#1827—532no—
116#1395—532no—
117#1392—531no—
118#1391—531no—
119C2×C53—21no—
120C2×C53—41no—
121C2×C53#182011yescharacteristic
122D53—42no—
123C4×#41—531no—
124#433—1062no—
125#432—1062no—
126#424—531no—
127#424#615711yescharacteristic, complemented
128#413—532no—
129#410—532no—
130#408—531no—
131#408—531no—
132C4×M16—531no—
133C2×#97—531no—
134#339—1062no—
135H(3,ℤ/4)—1062no—
136C53#1286511yescharacteristic, complemented, Sylow
137C2×#41—531no—
138C2×M16—531no—
139C2×M16—1062no—
140C2×M16C2×#615711yescharacteristic
141C22×C8—531no—
142C22×C8C2×#615711yescharacteristic
143C4×Q8—1061no—
144C4×D4—1062no—
145C4×D4—1061no—
146#117—532no—
147#117—1062no—
148#117C2×#615711yescharacteristic
149C2×C42—531no—
150C2×C42—532no—
151C4⋊C8—22no—
152C4⋊C8—1062no—
153#101—1062no—
154#100—1062no—
155#99—1062no—
156#99—1062no—
157C22⋊C8—1062no—
158C22⋊C8—1062no—
159#97—1061no—
160#97—1061no—
161#97—1061no—
162C4×C8—1061no—
163#41—2121no—
164#41—2121no—
165C2×Q8—531no—
166C2×D4—532no—
167C2×D4—531no—
168C22×C4—532no—
169C22×C4—1062no—
170C22×C4—1061no—
171C22×C4C22×#615711yescharacteristic
172M16—41no—
173M16—2122no—
174M16—2122no—
175M16—2121no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—41no—
179C2×C8—1061no—
180C2×C8—1061no—
181C2×C8—2122no—
182C2×C8—2121no—
183C4⋊C4—21no—
184C4⋊C4—1062no—
185C4⋊C4—2122no—
186C22⋊C4—21no—
187C22⋊C4—1062no—
188C22⋊C4—2122no—
189C42—531no—
190C42—531no—
191C42—1062no—
192C42—1061no—
193C42—2122no—
194C42C22⋊#615712yes—
195C23—532no—
196C23#1313329611yescharacteristic
197Q8—2121no—
198D4—2121no—
199D4—2122no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—532no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1061no—
207C2×C4—1061no—
208C2×C4—4242no—
209C2×C4—4242no—
210C2×C4—4241no—
211C2×C4C2×C22⋊#615711yescharacteristic
212C2×C4#1313329611yescharacteristic
213C8—41no—
214C8—41no—
215C8—4242no—
216C8—2121no—
217C8—2121no—
218C22—21no—
219C22—41no—
220C22—1062no—
221C22—2122no—
222C22#1681156611yescharacteristic
223C4—21no—
224C4—42no—
225C4—8482no—
226C4—1062no—
227C4—2121no—
228C4—2121no—
229C4#1681160311yescharacteristic
230C4#1681160411yescharacteristic
231C2—21no—
232C2—41no—
233C2—2122no—
234C2#2029061311yescharacteristic
subgroupG/H
centerC4#16811603
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#424C4
socleC2×C53#1820
prime radical, cosocleC2×#11719945C23
hypercenter#424#6157
hypocenterC53#12865
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
13. #16810650C4semidirect
16. M16×Dic53C4semidirect
17. C2×#13133127C4semidirect
18. C53×#424C4semidirect
127. #424#6157semidirect
136. C53#12865semidirect

Maximal subgroups

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

subgroupindex
#202906252
#202898162
#202898062
#202898032
#202833652
#1286553