Finite Group Databasebeta

← #20287523

Subgroups — #20287523

Order 6784.

326 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 115 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#16811348C211yesmaximal, characteristic
2#16811348C211yesmaximal, characteristic
3#16810990C212yesmaximal
4#16810932C212yesmaximal
5#16810905C211yesmaximal, characteristic
6#16810832C212yesmaximal
7#16810830C211yesmaximal, characteristic
8#16810753C211yesmaximal, characteristic
9#16810727C212yesmaximal
10C4×Dic212C211yesmaximal, characteristic
11C53×#487C211yesmaximal, characteristic
12C2×Dic212C2211yescharacteristic
13C2×#11719925C2212yes—
14#13133215—21no—
15#13133215C2212yes—
16#13133208—21no—
17#13133208C2212yes—
18#13133207—22no—
19#13133207—22no—
20Q8×Dic53—21no—
21Q8×Dic53C2212yes—
22C4×Dic106C2211yescharacteristic
23C4×Dic106C2211yescharacteristic
24C4×Dic106C2211yescharacteristic
25#13133182—21no—
26#13133182C2212yes—
27#13133151C2211yescharacteristic
28#13133148C2212yes—
29C4⋊#88322C2211yescharacteristic
30#13133146C2212yes—
31#13133142C2211yescharacteristic
32#13133142C2211yescharacteristic
33#13133141C2212yes—
34#13133140C2212yes—
35#13133127C2212yes—
36C4⋊Q8×C53C2211yescharacteristic
37C4⋊Q8×C53C2211yescharacteristic
38#103×C53C2212yes—
39#103×C53C2212yes—
40C4×C8×C53C2211yescharacteristic
41C2×Q8×C53—21no—
42C2×Q8×C53C2312yes—
43C2×Dic106D411yescharacteristic
44C2×Dic106C2311yescharacteristic
45C2×Dic106C2311yescharacteristic
46C2×C8×C53C2311yescharacteristic
47C2×C8×C53C2311yescharacteristic
48C4⋊C4×C53—21no—
49C4⋊C4×C53—21no—
50C4⋊C4×C53—21no—
51C4⋊C4×C53C2312yes—
52C42×C53C2311yescharacteristic
53#11719925—24no—
54C4⋊Dic53—22no—
55C4⋊Dic53—22no—
56C4⋊Dic53D411yescharacteristic
57C4⋊Dic53C2311yescharacteristic
58C4⋊Dic53C2311yescharacteristic
59#11719919—21no—
60#11719919—21no—
61#11719919—22no—
62#11719919—22no—
63#11719919—22no—
64#11719919—22no—
65#11719919D412yes—
66C4×Dic53—21no—
67C4×Dic53—21no—
68C4×Dic53—22no—
69C4×Dic53—22no—
70C4×Dic53C2312yes—
71C2×#88322C2312yes—
72Dic212—22no—
73Q8×C53—22no—
74Q8×C53—22no—
75Q8×C53—42no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53C2×D411yescharacteristic
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2411yescharacteristic
81C2×Dic53—21no—
82C2×Dic53—21no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53C2×D412yes—
86C2×Dic53C2×D412yes—
87Dic106—22no—
88Dic106—21no—
89Dic106—21no—
90Dic106—21no—
91Dic106#4112yes—
92C8×C53—21no—
93C8×C53—21no—
94#88322—22no—
95C22×C53C22×D411yescharacteristic
96C4×C53—21no—
97C4×C53—42no—
98C4×C53—42no—
99C4×C53#13712yes—
100C4×C53C2×#4111yescharacteristic
101C4×C532−1+411yescharacteristic
102Dic53—22no—
103Dic53—22no—
104Dic53—22no—
105Dic53—41no—
106Dic53—41no—
107Dic53—41no—
108#3374—531nomaximal, Sylow
109C2×C53#54811yescharacteristic
110C2×C53C2×#13711yescharacteristic
111C2×C53#57811yescharacteristic
112#557—531no—
113#557—531no—
114#491—532no—
115#487D5311yescharacteristic
116#479—532no—
117#475—531no—
118#462—531no—
119#462—532no—
120#446—531no—
121#441—532no—
122C4×Q16—531no—
123C53#337411yescharacteristic, complemented, Sylow
124C2×Q16—532no—
125C2×Q16—531no—
126C4⋊Q8—532no—
127C4⋊Q8—1061no—
128C4⋊Q8C2×D5311yescharacteristic
129C4⋊Q8C2×D5311yescharacteristic
130#125—532no—
131#125—1062no—
132#125—1062no—
133#125—1061no—
134C4×Q8—531no—
135C4×Q8—532no—
136C4×Q8—532no—
137C4×Q8—531no—
138C4×Q8—531no—
139C4×Q8—1061no—
140C4×Q8—1061no—
141#107—531no—
142#106—532no—
143C4⋊C8—532no—
144C4⋊C8—531no—
145#103—532no—
146#103—531no—
147#103—531no—
148#103—532no—
149#103C2×D5312yes—
150#103C2×D5312yes—
151#97—532no—
152C4×C8C2×D5311yescharacteristic
153C2×Q8—21no—
154C2×Q8—531no—
155C2×Q8—531no—
156C2×Q8—531no—
157C2×Q8C22×D5312yes—
158Q16—1062no—
159Q16—1064no—
160C2×C8—532no—
161C2×C8C22×D5311yescharacteristic
162C2×C8C22×D5311yescharacteristic
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—531no—
167C4⋊C4—532no—
168C4⋊C4—531no—
169C4⋊C4—531no—
170C4⋊C4—1062no—
171C4⋊C4—1062no—
172C4⋊C4—1062no—
173C4⋊C4—1062no—
174C4⋊C4—1062no—
175C4⋊C4—1062no—
176C4⋊C4—1061no—
177C4⋊C4—1061no—
178C4⋊C4C22×D5312yes—
179C42—532no—
180C42—1062no—
181C42—1062no—
182C42—1061no—
183C42—1061no—
184C42C22×D5311yescharacteristic
185Q8—22no—
186Q8—22no—
187Q8—42no—
188Q8—532no—
189Q8—1062no—
190Q8—1061no—
191Q8—1061no—
192Q8—1061no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—532no—
196C2×C4—532no—
197C2×C4—1061no—
198C2×C4—1061no—
199C2×C4—1061no—
200C2×C4—1061no—
201C2×C4D4×D5311yescharacteristic
202C2×C4D4×D5311yescharacteristic
203C2×C4C23×D5311yescharacteristic
204C8—21no—
205C8—21no—
206C8—1062no—
207C22C2×D4×D5311yescharacteristic
208C4—21no—
209C4—42no—
210C4—42no—
211C4—1062no—
212C4—1062no—
213C4—1062no—
214C4—2121no—
215C4—2121no—
216C4—2121no—
217C4#1313324612yes—
218C4#41×D5311yescharacteristic
219C4#1313326411yescharacteristic
220C2#1681128111yescharacteristic
221C2C2×#1313324611yescharacteristic
222C2#1681143611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#487C2
socleC22×C53C22×D4
hypercenter#487D53
hypocenterC53#3374
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
123. C53#3374semidirect