Finite Group Databasebeta

← #20275208

Subgroups — #20275208

Order 6784.

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

normal subgroups only

This group has 310 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
1C2×#13133250C211yesmaximal, characteristic
2#16811100C211yesmaximal, characteristic, complemented
3#16810639C211yesmaximal, characteristic
4#16810633C211yesmaximal, characteristic, complemented
5#16810596C211yesmaximal, characteristic
6#16810380C211yesmaximal, characteristic, complemented
7C53×#344C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9C22×Dic106C2211yescharacteristic
10#13133250—22no—
11#13133250—22no—
12C2×Dic212C412yescomplemented
13C2×#11719913C412yescomplemented
14C4×Dic106—22no—
15#13133182—22no—
16#13133176C2211yescharacteristic
17#13133176C2211yescharacteristic
18#13133173—22no—
19#13133143—22no—
20#13133142—22no—
21#13133131—22no—
22C2×M16×C53C2211yescharacteristic
23C2×M16×C53C2211yescharacteristic
24#117×C53C2211yescharacteristic
25C22×C4×C53C2311yescharacteristic
26C2×C22⋊D53—21no—
27C22×Dic53—21no—
28#11719945—21no—
29#11719945—21no—
30#11719945—21no—
31#11719945—21no—
32C2×D212C2×C411yescharacteristic
33C2×C4×D53—21no—
34C2×Dic106—21no—
35C2×Dic106—21no—
36C2×Dic106—21no—
37C2×Dic106C2×C411yescharacteristic
38C2×Dic106C2×C412yes—
39C2×Dic106D412yes—
40C2×Dic106D412yes—
41M16×C53—21no—
42M16×C53—21no—
43M16×C53—21no—
44M16×C53—21no—
45C2×C8×C53—21no—
46C2×C8×C53C2×C412yes—
47C4⋊C4×C53—21no—
48C22⋊C4×C53D412yes—
49C42×C53—21no—
50C22⋊Dic53—21no—
51C22⋊Dic53—21no—
52C4⋊Dic53—21no—
53C4⋊Dic53—21no—
54#11719919—24no—
55C4×Dic53—24no—
56C4×Dic53—21no—
57C4×Dic53D412yes—
58Dic212—24no—
59#11719913—24no—
60C23×C53C2×D411yescharacteristic
61C22×D53—21no—
62C2×C4×C53—22no—
63C2×C4×C53C22×C411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C2×C4×C53#4111yescharacteristic
68C2×C4×C53#4111yescharacteristic
69C22⋊D53—41no—
70C22⋊D53—41no—
71C2×Dic53—22no—
72C2×Dic53—22no—
73C2×Dic53—22no—
74C2×Dic53—21no—
75C2×Dic53—22no—
76C2×Dic53—41no—
77D212—21no—
78D212—21no—
79C4×D53—41no—
80C4×D53—41no—
81Dic106—22no—
82Dic106—22no—
83Dic106—22no—
84Dic106—21no—
85Dic106—21no—
86Dic106C22⋊C414yes—
87C8×C53—22no—
88C8×C53—41no—
89C22×C53—41no—
90C22×C53C4×D411yescharacteristic
91C22×C53#12011yescharacteristic
92C22×C53#12111yescharacteristic
93C2×D53—41no—
94C2×D53—81no—
95C4×C53—42no—
96C4×C53C2×C22⋊C411yescharacteristic
97C4×C53C4×D411yescharacteristic
98C4×C53#12111yescharacteristic
99C4×C53#12311yescharacteristic
100Dic53—24no—
101Dic53—42no—
102Dic53—41no—
103Dic53—81no—
104#1905—531nomaximal, Sylow
105C2×C53—21no—
106C2×C53—21no—
107C2×C53—21no—
108C2×C53#38611yescharacteristic
109D53—81no—
110C2×#137—531no—
111#519—531no—
112#421—531no—
113#419—531no—
114#410—531no—
115#344D5311yescharacteristic, complemented
116H(3,ℤ/4)—531no—
117C53#190511yescharacteristic, complemented, Sylow
118C2×#41—531no—
119C22×Q8—531no—
120#137—1062no—
121#137—1062no—
122C2×Q16—532no—
123C2×SD16—532no—
124C2×M16C2×D5311yescharacteristic
125C2×M16C2×D5311yescharacteristic
126C4×Q8—1062no—
127C4×Q8—1062no—
128#117—531no—
129#117—531no—
130#117—1062no—
131#117C2×D5311yescharacteristic
132#104—1062no—
133#103—1062no—
134#99—1062no—
135#41—1061no—
136#41—1061no—
137#41—1061no—
138#41—1061no—
139C2×Q8—532no—
140C2×Q8—531no—
141C2×Q8—532no—
142C2×Q8—532no—
143C2×Q8—1061no—
144C2×Q8—1061no—
145C2×Q8—1061no—
146C2×D4—531no—
147C2×D4—1061no—
148C22×C4—1061no—
149C22×C4—1061no—
150C22×C4C22×D5311yescharacteristic
151Q16—1064no—
152SD16—1064no—
153M16—21no—
154M16—21no—
155M16—21no—
156M16—21no—
157C2×C8—21no—
158C2×C8C4×D5312yes—
159C4⋊C4—21no—
160C4⋊C4—1064no—
161C4⋊C4—1061no—
162C4⋊C4—1061no—
163C22⋊C4—1061no—
164C22⋊C4—1061no—
165C22⋊C4D21212yes—
166C42—21no—
167C42—532no—
168C42—1064no—
169C42—1061no—
170C23—1061no—
171C23C2×D21211yescharacteristic
172Q8—534no—
173Q8—1062no—
174Q8—1062no—
175Q8—1062no—
176Q8—1061no—
177Q8—1061no—
178D4—1061no—
179D4—1061no—
180D4—2121no—
181D4—2121no—
182C2×C4—22no—
183C2×C4—1062no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4—1061no—
187C2×C4—1062no—
188C2×C4—2121no—
189C2×C4—2121no—
190C2×C4—2121no—
191C2×C4C2×C4×D5311yescharacteristic
192C2×C4#1171994511yescharacteristic
193C2×C4D4×D5311yescharacteristic
194C2×C4D4×D5311yescharacteristic
195C2×C4D4×D5311yescharacteristic
196C2×C4#1171994911yescharacteristic
197C8—22no—
198C8—41no—
199C22—41no—
200C22—4241no—
201C22—2121no—
202C22C4×D21211yescharacteristic
203C22#1313318711yescharacteristic
204C22#1313319211yescharacteristic
205C4—42no—
206C4—1064no—
207C4—4241no—
208C4—2122no—
209C4—2121no—
210C4C22⋊C4×D5311yescharacteristic
211C4#1313317911yescharacteristic
212C4#1313319011yescharacteristic
213C4#1313320211yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—4241no—
218C2#1681050111yescharacteristic
subgroupG/H
centerC2#16810501
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#344C2
socleC2×C53#386
hypercenter#344D53
hypocenterC53#1905
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #16811100C2semidirect
4. #16810633C2semidirect
6. #16810380C2semidirect
7. C53×#344C2semidirect
12. C2×Dic212C4semidirect
13. C2×#11719913C4semidirect
115. #344D53semidirect
117. C53#1905semidirect