Finite Group Databasebeta

← #27303485

Subgroups — #27303485

Order 13568.

436 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#16811003C211yesmaximal, characteristic
2C2×#16810440C212yesmaximal
3#20276436C212yesmaximal
4#20275463C211yesmaximal, characteristic
5C53×#1376C211yesmaximal, characteristic
6C22×Dic212—21no—
7C22×Dic212C2212yes—
8C2×#13133217—21no—
9#16811003—24no—
10C2×#13133153C2211yescharacteristic
11#16810661C2211yescharacteristic
12#16810649—21no—
13#16810440—24no—
14#16810428—24no—
15#16810383—21no—
16C2×M32×C53C2212yes—
17C2×C4×C8×C53C2211yescharacteristic
18C22×Dic106—22no—
19C2×Dic212—24no—
20C2×Dic212—22no—
21C2×Dic212—22no—
22C2×Dic212—22no—
23C2×Dic212—22no—
24C2×Dic212—22no—
25C2×Dic212—42no—
26C2×Dic212—42no—
27C2×Dic212D414yes—
28C2×#11719917—21no—
29#13133217—24no—
30#13133217—42no—
31#13133176—21no—
32C2×C4⋊Dic53—42no—
33#13133153—22no—
34#13133151D412yes—
35#13133149—21no—
36#13133146—42no—
37C22×C8×C53D411yescharacteristic
38C22×C8×C53C2311yescharacteristic
39C2×C42×C53D411yescharacteristic
40M32×C53—22no—
41M32×C53—22no—
42C2×C16×C53—22no—
43C4×C8×C53—22no—
44C22×C4×C53—21no—
45C22×C4×C53C2×D411yescharacteristic
46C22×Dic53—42no—
47C2×Dic106—24no—
48C2×Dic106—42no—
49C2×Dic106—42no—
50C2×Dic106—42no—
51C2×Dic106—42no—
52C2×Dic106—42no—
53C2×C8×C53—21no—
54C2×C8×C53—21no—
55C2×C8×C53D812yes—
56C2×C8×C53C2×D411yescharacteristic
57C2×C8×C53C2×D412yes—
58C2×C8×C53#4112yes—
59C2×C8×C53#4111yescharacteristic
60C42×C53—21no—
61C42×C53SD1612yes—
62C22⋊Dic53—41no—
63C4⋊Dic53—22no—
64C4⋊Dic53—48no—
65C4×Dic53—41no—
66#11719917—41no—
67#11719917—41no—
68C2×#88322—41no—
69Dic212—24no—
70Dic212—24no—
71Dic212—48no—
72Dic212—42no—
73Dic212—42no—
74C16×C53—42no—
75C23×C53AGL(1,ℤ/8)11yescharacteristic
76C2×C4×C53—24no—
77C2×C4×C53—22no—
78C2×C4×C53#12011yescharacteristic
79C2×C4×C53#12111yescharacteristic
80C2×C4×C53#12411yescharacteristic
81C2×C4×C53C2×D811yescharacteristic
82C2×C4×C53C2×SD1611yescharacteristic
83C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
84C2×Dic53—44no—
85C2×Dic53—42no—
86C2×Dic53—82no—
87C2×Dic53—82no—
88Dic106—44no—
89Dic106—44no—
90Dic106—82no—
91Dic106—82no—
92C8×C53—22no—
93C8×C53#12112yes—
94C8×C53#12312yes—
95#88322—81no—
96#13415—531nomaximal, Sylow
97C22×C53—22no—
98C22×C53#45011yescharacteristic
99C22×C53#45911yescharacteristic
100C22×C53#49011yescharacteristic
101C4×C53—24no—
102C4×C53#39411yescharacteristic
103C4×C53#44711yescharacteristic
104C4×C53#46011yescharacteristic
105C4×C53#48611yescharacteristic
106Dic53—82no—
107Dic53—84no—
108C2×#494—531no—
109C2×#362—532no—
110#2141—532no—
111#1945—531no—
112#1376D5311yescharacteristic
113C2×C53—21no—
114C2×C53—21no—
115C2×C53#204111yescharacteristic
116C2×C53#221412yes—
117C22×Q16—532no—
118C22×Q16—1061no—
119C2×C4⋊Q8—1061no—
120C2×M32C2×D5312yes—
121#494—1064no—
122C2×#108—531no—
123#427—531no—
124#424—1061no—
125C2×C4×C8C2×D5311yescharacteristic
126#362—1064no—
127#358—1064no—
128#339—1061no—
129C53#1341511yescharacteristic, complemented, Sylow
130C22×Q8—1062no—
131C2×Q16—534no—
132C2×Q16—1064no—
133C2×Q16—1062no—
134C2×Q16—1062no—
135C2×Q16—1062no—
136C2×Q16—1062no—
137C2×Q16—1062no—
138C2×Q16—2122no—
139C2×Q16—2122no—
140C2×M16—1061no—
141C22×C8D21211yescharacteristic
142C22×C8C22×D5311yescharacteristic
143C4⋊Q8—1064no—
144C4⋊Q8—2122no—
145#117—1061no—
146C2×C4⋊C4—2122no—
147C2×C42D21211yescharacteristic
148M32—22no—
149M32—22no—
150C2×C16—22no—
151#108—1062no—
152#107—532no—
153#106—1061no—
154C4⋊C8—2122no—
155C4×C8—22no—
156C2×Q8—1064no—
157C2×Q8—2122no—
158C2×Q8—2122no—
159C2×Q8—2122no—
160C2×Q8—2122no—
161C2×Q8—2122no—
162C22×C4—21no—
163C22×C4—2122no—
164C22×C4C2×D21211yescharacteristic
165Q16—1064no—
166Q16—1064no—
167Q16—2128no—
168Q16—2122no—
169Q16—2122no—
170M16—2121no—
171M16—2121no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—2121no—
175C2×C8D42412yes—
176C2×C8#1171994512yes—
177C2×C8D4×D5311yescharacteristic
178C2×C8D4×D5312yes—
179C2×C8#1171994911yescharacteristic
180C4⋊C4—1062no—
181C4⋊C4—2128no—
182C22⋊C4—2121no—
183C42—21no—
184C42—2121no—
185C42#1171991312yes—
186C16—42no—
187C23#1313324511yescharacteristic
188Q8—4242no—
189Q8—4242no—
190Q8—2124no—
191Q8—2124no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—4242no—
195C2×C4—4242no—
196C2×C4—2124no—
197C2×C4—2122no—
198C2×C4#1313318711yescharacteristic
199C2×C4#1313319211yescharacteristic
200C2×C4#1313320511yescharacteristic
201C2×C4C2×D42411yescharacteristic
202C2×C4C2×#1171991311yescharacteristic
203C2×C4#1313324511yescharacteristic
204C8—22no—
205C8—4241no—
206C8#1313319012yes—
207C8#1313320212yes—
208C22—22no—
209C22#1681077611yescharacteristic
210C22#1681081911yescharacteristic
211C22#1681098811yescharacteristic
212C4—24no—
213C4—4242no—
214C4—4244no—
215C4#1681054111yescharacteristic
216C4#1681075911yescharacteristic
217C4#1681082411yescharacteristic
218C4#1681097011yescharacteristic
219C2—21no—
220C2—21no—
221C2#2027595011yescharacteristic
222C2#2027679712yes—
subgroupG/H
centerC22#16810819
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#1376C2
socleC22×C53#459
prime radicalC23×C53AGL(1,ℤ/8)
hypercenter#1376D53
hypocenterC53#13415
LCSG ▷ C22×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1376

Complements

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

NH ≅ G/Ntype
129. C53#13415semidirect

Maximal subgroups

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

subgroupindex
C2×#168110032
C2×#168104402
#202764362
#202754632
C53×#13762
#1341553