Finite Group Databasebeta

← #20290487

Subgroups — #20290487

Order 6784.

324 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 100 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×#13133280C212yesmaximal
2#16811582C211yesmaximal, characteristic
3C4×C4⋊#6157C211yesmaximal, characteristic
4#16811496C212yesmaximal
5C4×Q8×D53C211yesmaximal, characteristic
6C2×D53⋊C8C2212yes—
7C2×C4⋊#6157C2212yes—
8C2×C4×#6157—21no—
9#13133280C418yescomplemented
10#13133272—21no—
11#13133272—21no—
12(C2×#6157)⋊C4—21no—
13C2×Q8×D53C2211yescharacteristic
14Q8×Dic53C411yescharacteristic, complemented
15C4×Dic106—21no—
16C4×Dic106C411yescharacteristic, complemented
17#13133182—21no—
18#13133182C411yescharacteristic, complemented
19C4⋊C4×D53—21no—
20C4⋊C4×D53C2211yescharacteristic
21C42×D53—21no—
22C42×D53C2211yescharacteristic
23C4×Q8×C53C411yescharacteristic, complemented
24C22×#6157—22no—
25C2×Q8×C53C2×C411yescharacteristic
26Q8×D53—22no—
27Q8×D53C2×C414yes—
28C2×C4×D53—21no—
29C2×C4×D53—21no—
30C2×C4×D53D411yescharacteristic
31C2×C4×D53D411yescharacteristic
32C2×C4×D53C2311yescharacteristic
33C2×Dic106—21no—
34C2×Dic106C2×C411yescharacteristic
35C2×#88324—22no—
36C4⋊#6157—22no—
37C4⋊#6157C2×C414yes—
38C4×#6157—42no—
39D53⋊C8—22no—
40D53⋊C8C2×C414yes—
41C4⋊C4×C53—21no—
42C4⋊C4×C53C2×C411yescharacteristic
43C42×C53—21no—
44C42×C53C2×C411yescharacteristic
45C4⋊Dic53—21no—
46C4⋊Dic53C2×C411yescharacteristic
47#11719919—21no—
48#11719919—21no—
49#11719919—21no—
50C4×Dic53—21no—
51C4×Dic53—21no—
52C4×Dic53—21no—
53C4×Dic53C2×C411yescharacteristic
54C4×Dic53D411yescharacteristic
55C4×Dic53D411yescharacteristic
56C22×D53C2×D411yescharacteristic
57C2×#6157—24no—
58C2×#6157—42no—
59C2×#6157—42no—
60Q8×C53—21no—
61Q8×C53C4212yescomplemented
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53C22⋊C411yescharacteristic
65C2×C4×C53C22⋊C411yescharacteristic
66C2×C4×C53C22×C411yescharacteristic
67C2×Dic53—21no—
68C2×Dic53—21no—
69C2×Dic53C22⋊C411yescharacteristic
70C2×Dic53C22⋊C411yescharacteristic
71C2×Dic53C2×D411yescharacteristic
72C4×D53—24no—
73C4×D53—21no—
74C4×D53—21no—
75C4×D53—22no—
76C4×D53—41no—
77C4×D53—41no—
78C4×D53C22×C412yes—
79C4×D53#4112yes—
80Dic106—21no—
81Dic106—24no—
82Dic106C4212yescomplemented
83#88324—24no—
84C22×C53C2×C22⋊C411yescharacteristic
85C2×D53C4×D412yes—
86C2×D53AGL(1,ℤ/8)11yescharacteristic
87C2×D53#13712yes—
88C2×D53#13711yescharacteristic
89#6157—44no—
90C4×C53—22no—
91C4×C53—21no—
92C4×C53—21no—
93C4×C53—41no—
94C4×C53C2×C4211yescharacteristic
95C4×C53#11711yescharacteristic
96Dic53—22no—
97Dic53—21no—
98Dic53—21no—
99Dic53—41no—
100Dic53C4×D412yes—
101#1763—531nomaximal, Sylow
102C2×C53C4×C22⋊C411yescharacteristic
103C2×C53#41711yescharacteristic
104C2×C53#41911yescharacteristic
105D53#44012yes—
106D53#44112yes—
107C2×C4×Q8—531no—
108C2×#103—532no—
109C2×#97—531no—
110C4×C4⋊C4—531no—
111#340—532no—
112C53#176311yescharacteristic, complemented, Sylow
113C22×Q8—531no—
114C22×C8—532no—
115C4×Q8—531no—
116C4×Q8—531no—
117C4×Q8—531no—
118C4×Q8—1061no—
119C4×Q8—1061no—
120C4×Q8#615711yescharacteristic, complemented
121C2×C4⋊C4—532no—
122C2×C4⋊C4—531no—
123C2×C4⋊C4—1061no—
124C2×C42—531no—
125C2×C42—1061no—
126C2×C42—1061no—
127#103—538no—
128#97—1061no—
129#97—1061no—
130#95—1061no—
131C2×Q8—534no—
132C2×Q8—531no—
133C2×Q8—1061no—
134C2×Q8—1062no—
135C2×Q8C2×#615711yescharacteristic
136C22×C4—531no—
137C22×C4—531no—
138C22×C4—531no—
139C22×C4—1061no—
140C22×C4—1061no—
141C22×C4—1062no—
142C2×C8—534no—
143C2×C8—1062no—
144C2×C8—1062no—
145C4⋊C4—21no—
146C4⋊C4—534no—
147C4⋊C4—531no—
148C4⋊C4—1061no—
149C4⋊C4—1061no—
150C4⋊C4—1061no—
151C4⋊C4—1062no—
152C4⋊C4—1061no—
153C4⋊C4C2×#615711yescharacteristic
154C42—21no—
155C42—531no—
156C42—531no—
157C42—531no—
158C42—1061no—
159C42—1061no—
160C42—1061no—
161C42—2122no—
162C42C2×#615711yescharacteristic
163C23—531no—
164Q8—21no—
165Q8—532no—
166Q8—1064no—
167Q8—1061no—
168Q8C4×#615712yescomplemented
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—531no—
172C2×C4—532no—
173C2×C4—532no—
174C2×C4—531no—
175C2×C4—531no—
176C2×C4—1061no—
177C2×C4—1061no—
178C2×C4—1062no—
179C2×C4—1064no—
180C2×C4—1061no—
181C2×C4—1064no—
182C2×C4—1061no—
183C2×C4—2121no—
184C2×C4—2121no—
185C2×C4—2122no—
186C2×C4—2122no—
187C2×C4C22⋊#615711yescharacteristic
188C2×C4C22⋊#615711yescharacteristic
189C2×C4C22×#615711yescharacteristic
190C8—1064no—
191C22—532no—
192C22—531no—
193C22—531no—
194C22—532no—
195C22C2×C22⋊#615711yescharacteristic
196C4—22no—
197C4—21no—
198C4—21no—
199C4—41no—
200C4—532no—
201C4—1062no—
202C4—1061no—
203C4—1061no—
204C4—2124no—
205C4—2121no—
206C4C2×C4×#615711yescharacteristic
207C4#1313329611yescharacteristic
208C2—532no—
209C2—532no—
210C2C4×C22⋊#615711yescharacteristic
211C2#1681161411yescharacteristic
212C2#1681161611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC4×C53C2×C42
FrattiniC2×C4C22×#6157
FittingC4×Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC4×Q8#6157
hypocenterC53#1763
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
9. #13133280C4semidirect
14. Q8×Dic53C4semidirect
16. C4×Dic106C4semidirect
18. #13133182C4semidirect
23. C4×Q8×C53C4semidirect
61. Q8×C53C42semidirect
82. Dic106C42semidirect
112. C53#1763semidirect
120. C4×Q8#6157semidirect
168. Q8C4×#6157semidirect