Finite Group Databasebeta

← #27300372

Subgroups — #27300372

Order 13568.

342 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 79 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#20274820C212yesmaximal
3#20272845C212yesmaximal
4#20272836C211yesmaximal, characteristic
5C53×#1392C211yesmaximal, characteristic
6#41×Dic53C2211yescharacteristic
7#16810681—22no—
8#16810673—22no—
9#16810651C411yescharacteristic, complemented
10#16810649C411yescharacteristic, complemented
11#16810597C2212yes—
12#16810590C2211yescharacteristic
13M16×Dic53C411yescharacteristic, complemented
14C2×C8×Dic53C411yescharacteristic, complemented
15#16810384—22no—
16#16810380—22no—
17C53×#413C2212yes—
18C53×#408C2211yescharacteristic
19C2×#11719917C2×C411yescharacteristic
20C22×#88322C2×C411yescharacteristic
21Q8×Dic53—21no—
22D4×Dic53—22no—
23D4×Dic53—21no—
24#13133176—22no—
25#13133176C2×C411yescharacteristic
26#13133176D412yes—
27C2×C4×Dic53C2×C411yescharacteristic
28C2×C4×Dic53D412yes—
29#13133146—22no—
30#13133146—22no—
31#13133132—22no—
32#13133132—22no—
33C22⋊#88322—22no—
34#13133127—21no—
35C8×Dic53—21no—
36C8×Dic53—21no—
37C8×Dic53—21no—
38C2×#41×C53C2311yescharacteristic
39C2×M16×C53—22no—
40C2×M16×C53C2×C411yescharacteristic
41C22×C8×C53C2×C411yescharacteristic
42#101×C53—22no—
43#100×C53—22no—
44C22⋊C8×C53—22no—
45#41×C53—41no—
46#41×C53—41no—
47C2×Q8×C53#4111yescharacteristic
48C2×D4×C53#4111yescharacteristic
49C2×D4×C53#4112yes—
50C22×C4×C53C22×C411yescharacteristic
51C22×C4×C53C2×D412yes—
52C22×Dic53—21no—
53C22×Dic53—22no—
54M16×C53—41no—
55M16×C53—42no—
56M16×C53—42no—
57C2×C8×C53—21no—
58C2×C8×C53—21no—
59C2×C8×C53—41no—
60C2×C8×C53—42no—
61C22⋊Dic53—21no—
62C22⋊Dic53—22no—
63C22⋊Dic53—42no—
64C4⋊Dic53—21no—
65C4⋊Dic53—22no—
66C4⋊Dic53—42no—
67C4×Dic53—21no—
68C4×Dic53—22no—
69C4×Dic53—42no—
70C4×Dic53C22⋊C411yescharacteristic
71C4×Dic53C22⋊C411yescharacteristic
72C4×Dic53C22⋊C412yes—
73#11719917—41no—
74C2×#88322—21no—
75C2×#88322—21no—
76C2×#88322—41no—
77C23×C53#11711yescharacteristic
78C23×C53#12312yes—
79Q8×C53—41no—
80D4×C53—42no—
81D4×C53—41no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53C2×C22⋊C411yescharacteristic
87C2×C4×C53#11711yescharacteristic
88C2×C4×C53#12312yes—
89C2×Dic53—21no—
90C2×Dic53—21no—
91C2×Dic53—22no—
92C2×Dic53—42no—
93C2×Dic53—44no—
94C2×Dic53—82no—
95C8×C53—41no—
96C8×C53—41no—
97C8×C53—82no—
98#88322—41no—
99#88322—41no—
100#12864—531nomaximal, Sylow
101C22×C53—22no—
102C22×C53—21no—
103C22×C53—41no—
104C22×C53—42no—
105C22×C53#38111yescharacteristic
106C4×C53—22no—
107C4×C53—21no—
108C4×C53#41011yescharacteristic
109C4×C53#41011yescharacteristic
110Dic53—22no—
111Dic53—41no—
112Dic53—42no—
113Dic53—162no—
114#2922—531no—
115#1827—532no—
116#1395—532no—
117#1392D5311yescharacteristic
118#1391—531no—
119C2×C53—21no—
120C2×C53—42no—
121C2×C53—41no—
122C2×C53#182011yescharacteristic
123C4×#41—531no—
124#433—1062no—
125#431—1062no—
126#424—531no—
127#424—531no—
128#413C2×D5312yes—
129#410—532no—
130#408—531no—
131#408C2×D5311yescharacteristic
132C4×M16—531no—
133C2×C4×C8—531no—
134#339—1062no—
135H(3,ℤ/4)—1062no—
136C53#1286411yescharacteristic, complemented, Sylow
137C2×#41C22×D5311yescharacteristic
138C2×M16—22no—
139C2×M16—531no—
140C2×M16C4×D5311yescharacteristic
141C22×C8—531no—
142C22×C8C4×D5311yescharacteristic
143C4×Q8—1061no—
144C4×D4—1062no—
145C4×D4—1061no—
146#117—531no—
147#117—532no—
148#117—1062no—
149C2×C42—531no—
150C2×C42—532no—
151C4⋊C8—1062no—
152C4⋊C8—1062no—
153#101—22no—
154#100—22no—
155#99—1062no—
156#99—1062no—
157C22⋊C8—22no—
158C22⋊C8—1062no—
159#97—1061no—
160C4×C8—1061no—
161C4×C8—1061no—
162C4×C8—1061no—
163#41—41no—
164#41—41no—
165C2×Q8#1171994711yescharacteristic
166C2×D4#1171994512yes—
167C2×D4#1171994911yescharacteristic
168C22×C4—1061no—
169C22×C4—1062no—
170C22×C4C2×C4×D5311yescharacteristic
171C22×C4D4×D5312yes—
172M16—41no—
173M16—42no—
174M16—42no—
175M16—2121no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—41no—
179C2×C8—42no—
180C2×C8—1061no—
181C2×C8—1061no—
182C2×C8—2121no—
183C4⋊C4—1061no—
184C4⋊C4—1062no—
185C4⋊C4—2122no—
186C22⋊C4—1061no—
187C22⋊C4—1062no—
188C22⋊C4—2122no—
189C42—531no—
190C42—531no—
191C42—532no—
192C42—1061no—
193C42—1062no—
194C42—2122no—
195C23#1313317411yescharacteristic
196C23#1313320212yes—
197Q8—41no—
198D4—42no—
199D4—41no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—1061no—
205C2×C4—1061no—
206C2×C4—1062no—
207C2×C4—4242no—
208C2×C4—2122no—
209C2×C4—2124no—
210C2×C4C22⋊C4×D5311yescharacteristic
211C2×C4#1313317511yescharacteristic
212C2×C4#1313319912yes—
213C8—41no—
214C8—41no—
215C8—82no—
216C8—2121no—
217C8—2121no—
218C22—22no—
219C22—21no—
220C22—41no—
221C22—42no—
222C22#1681048511yescharacteristic
223C4—22no—
224C4—21no—
225C4—8482no—
226C4—1062no—
227C4—2121no—
228C4—2122no—
229C4#1681059511yescharacteristic
230C4#1681059511yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2#2027479111yescharacteristic
subgroupG/H
centerC4#16810595
commutatorC22×C4×C53C22×C4
FrattiniC2×#41C22×D53
FittingC53×#1392C2
socleC2×C53#1820
prime radical, cosocleC2×#41×C53C23
hypercenter#1392D53
hypocenterC53#12864
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1392

Complements

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

NH ≅ G/Ntype
9. #16810651C4semidirect
10. #16810649C4semidirect
13. M16×Dic53C4semidirect
14. C2×C8×Dic53C4semidirect
136. C53#12864semidirect

Maximal subgroups

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

subgroupindex
#202833552
#202748202
#202728452
#202728362
C53×#13922
#1286453