Finite Group Databasebeta

← #20287595

Subgroups — #20287595

Order 6784.

338 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 125 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, complemented
2#16811334C211yesmaximal, characteristic
3#16811023C212yesmaximal, complemented
4#16810933C212yesmaximal, complemented
5#16810922C212yesmaximal
6#16810915C211yesmaximal, characteristic, complemented
7#16810900C211yesmaximal, characteristic
8C8×Dic106C211yesmaximal, characteristic, complemented
9C4×#11719913C211yesmaximal, characteristic
10#16810710C212yesmaximal
11C53×#498C211yesmaximal, characteristic, complemented
12C2×#11719913C2211yescharacteristic
13#13133215—21no—
14#13133208C2212yes—
15#13133208C2212yes—
16#13133207—22no—
17#13133197—21no—
18#13133193—22no—
19Q8×Dic53—21no—
20C4×Dic106C2211yescharacteristic
21C4×Dic106C2211yescharacteristic
22#13133182C2212yes—
23C4×D212C2211yescharacteristic
24#13133179C2212yes—
25C4⋊C4×D53—21no—
26#13133150C2212yes—
27#13133150C2212yes—
28#13133149C2211yescharacteristic
29C4⋊#88322C2211yescharacteristic
30#13133146C2212yes—
31#13133142C2211yescharacteristic
32#13133140C2212yes—
33#13133139C2211yescharacteristic
34#13133137C2212yes—
35C8×Dic53C2212yes—
36C4⋊Q8×C53C2211yescharacteristic
37C4⋊Q8×C53C2211yescharacteristic
38#106×C53C2212yes—
39#106×C53C2212yes—
40C4×C8×C53C2211yescharacteristic
41C2×Q8×C53—21no—
42C2×Q8×C53—21no—
43C2×D212C2311yescharacteristic
44C2×C4×D53—21no—
45C2×C4×D53—22no—
46C2×Dic106D411yescharacteristic
47C2×Dic106C2311yescharacteristic
48C2×C8×C53C2311yescharacteristic
49C2×C8×C53C2311yescharacteristic
50C4⋊C4×C53—21no—
51C4⋊C4×C53—21no—
52C4⋊C4×C53C2312yes—
53C4⋊C4×C53C2312yes—
54C42×C53C2311yescharacteristic
55#11719921—21no—
56#11719921—22no—
57C4⋊Dic53—22no—
58C4⋊Dic53—22no—
59C4⋊Dic53D411yescharacteristic
60C4⋊Dic53C2311yescharacteristic
61C4⋊Dic53C2311yescharacteristic
62#11719919—21no—
63#11719919—22no—
64#11719919—22no—
65#11719919—22no—
66#11719919D412yes—
67C4×Dic53—21no—
68C4×Dic53—22no—
69C4×Dic53C2312yes—
70C2×#88322C2312yes—
71#11719913Q814yescomplemented
72C22×D53—21no—
73Q8×C53—42no—
74Q8×C53—81no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53C2×D411yescharacteristic
78C2×C4×C53C2×D411yescharacteristic
79C2×C4×C53C2411yescharacteristic
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—21no—
83C2×Dic53C2×D412yes—
84C2×Dic53C2×D412yes—
85D212—21no—
86D212C2×Q812yes—
87C4×D53—42no—
88C4×D53—82no—
89Dic106—22no—
90Dic106—21no—
91Dic106C2×Q812yes—
92C8×C53—21no—
93C8×C53C2×Q812yes—
94#88322—22no—
95C22×C53C22×D411yescharacteristic
96C2×D53—22no—
97C2×D53—41no—
98C2×D53—41no—
99C4×C53—21no—
100C4×C53—42no—
101C4×C53—42no—
102C4×C53#13512yes—
103C4×C53C22×Q811yescharacteristic
104C4×C532−1+411yescharacteristic
105Dic53—22no—
106Dic53—22no—
107Dic53—22no—
108Dic53—41no—
109Dic53—41no—
110#3381—531nomaximal, Sylow
111C2×C53D4×Q811yescharacteristic
112C2×C53C2×#13511yescharacteristic
113C2×C53#57711yescharacteristic
114D53—42no—
115#557—531no—
116#554—531no—
117#499—532no—
118#498D5311yescharacteristic, complemented
119#479—532no—
120#478—532no—
121#477—531no—
122#474—531no—
123C8×Q8—531no—
124C4×SD16—532no—
125C4×SD16—531no—
126C53#338111yescharacteristic, complemented, Sylow
127C2×SD16—531no—
128C4⋊Q8—1061no—
129C4⋊Q8C2×D5311yescharacteristic
130C4⋊Q8C2×D5311yescharacteristic
131#125—532no—
132#125—532no—
133#125—1062no—
134C22⋊Q8—1061no—
135C22⋊Q8—1062no—
136C4×Q8—531no—
137C4×Q8—532no—
138C4×Q8—531no—
139C4×Q8—1061no—
140C4×D4—532no—
141C4×D4—531no—
142C2×C4⋊C4—1061no—
143#107—532no—
144#107—532no—
145#106—531no—
146#106C2×D5312yes—
147#106C2×D5312yes—
148C4⋊C8—532no—
149C4⋊C8—531no—
150#103—531no—
151#103—532no—
152#102—531no—
153#102—532no—
154C4×C8—532no—
155C4×C8C2×D5311yescharacteristic
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—531no—
159C2×Q8—531no—
160C2×D4—531no—
161C22×C4—1062no—
162C22×C4—1061no—
163SD16—534no—
164C2×C8—532no—
165C2×C8C22×D5311yescharacteristic
166C2×C8C22×D5311yescharacteristic
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—531no—
170C4⋊C4—532no—
171C4⋊C4—531no—
172C4⋊C4—531no—
173C4⋊C4—1062no—
174C4⋊C4—1062no—
175C4⋊C4—1062no—
176C4⋊C4—1062no—
177C4⋊C4—1062no—
178C4⋊C4—1061no—
179C4⋊C4C22×D5312yes—
180C4⋊C4C22×D5312yes—
181C22⋊C4—1062no—
182C22⋊C4—1061no—
183C42—532no—
184C42—1062no—
185C42—1061no—
186C42C22×D5311yescharacteristic
187C23—1061no—
188Q8—42no—
189Q8—81no—
190Q8—532no—
191Q8—1062no—
192Q8—1061no—
193D4—532no—
194D4—1061no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—532no—
198C2×C4—532no—
199C2×C4—1061no—
200C2×C4—1061no—
201C2×C4—1061no—
202C2×C4—4242no—
203C2×C4—2122no—
204C2×C4D4×D5311yescharacteristic
205C2×C4D4×D5311yescharacteristic
206C2×C4C23×D5311yescharacteristic
207C8—21no—
208C8—1062no—
209C8Q8×D5312yes—
210C22—1062no—
211C22—2121no—
212C22—2121no—
213C22C2×D4×D5311yescharacteristic
214C4—21no—
215C4—42no—
216C4—42no—
217C4—1062no—
218C4—1062no—
219C4—1062no—
220C4—2121no—
221C4—2121no—
222C4#1313323812yes—
223C4C2×Q8×D5311yescharacteristic
224C4#1313326511yescharacteristic
225C2—2122no—
226C2#1681128811yescharacteristic
227C2C2×#1313323811yescharacteristic
228C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#498C2
socleC22×C53C22×D4
hypercenter#498D53
hypocenterC53#3381
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
1. #16811348C2semidirect
3. #16811023C2semidirect
4. #16810933C2semidirect
6. #16810915C2semidirect
8. C8×Dic106C2semidirect
11. C53×#498C2semidirect
71. #11719913Q8semidirect
118. #498D53semidirect
126. C53#3381semidirect