Finite Group Databasebeta

← #20290625

Subgroups — #20290625

Order 6784.

262 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 62 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#16811609C211yesmaximal, characteristic
2#16811602C211yesmaximal, characteristic
3#16811494C212yesmaximal, complemented
4#16811490C212yesmaximal, complemented
5#16811091C211yesmaximal, characteristic
6D53⋊M16C2212yes—
7#13133296C2212yes—
8#13133278—21no—
9#13133277—21no—
10#13133276—21no—
11#13133275—21no—
12C2×#11719945C2211yescharacteristic
13#13133182—21no—
14#13133179—21no—
15#13133177—21no—
16#13133177—21no—
17#13133174C411yescharacteristic, complemented
18#13133174C2211yescharacteristic
19C2×C4×Dic53C411yescharacteristic, complemented
20C42×D53C411yescharacteristic, complemented
21C42×D53C2211yescharacteristic
22#117×C53C411yescharacteristic, complemented
23C22×C4×C53C2×C411yescharacteristic
24C2×C22⋊D53D411yescharacteristic
25C2×C22⋊D53D411yescharacteristic
26C22×Dic53—21no—
27#11719945—22no—
28#11719945—22no—
29C2×D212D411yescharacteristic
30C2×C4×D53—21no—
31C2×C4×D53—21no—
32C2×C4×D53C2×C411yescharacteristic
33C2×C4×D53C2311yescharacteristic
34C2×Dic106D411yescharacteristic
35C22⋊#6157—22no—
36#11719940—22no—
37C4⋊#6157—22no—
38C4×#6157—22no—
39#11719936—22no—
40D53⋊C8—22no—
41C4⋊C4×C53—21no—
42C22⋊C4×C53—21no—
43C42×C53C2×C411yescharacteristic
44C42×C53C2×C411yescharacteristic
45#11719921—21no—
46#11719921—21no—
47#11719919—21no—
48#11719919—21no—
49C4×Dic53—21no—
50C4×Dic53—21no—
51C4×Dic53—21no—
52C4×Dic53C2×C411yescharacteristic
53C4×Dic53C2×C411yescharacteristic
54C23×C53C22⋊C411yescharacteristic
55C22×D53C22⋊C411yescharacteristic
56C22×D53C2×D411yescharacteristic
57C2×#6157—42no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53C22⋊C411yescharacteristic
63C2×C4×C53C22×C411yescharacteristic
64C22⋊D53—22no—
65C22⋊D53—22no—
66C2×Dic53—21no—
67C2×Dic53—21no—
68C2×Dic53—42no—
69C2×Dic53C22⋊C411yescharacteristic
70C2×Dic53C2×D411yescharacteristic
71D212—22no—
72C4×D53—21no—
73C4×D53—21no—
74C4×D53—42no—
75C4×D53—42no—
76C4×D53#4112yes—
77C4×D53#4112yes—
78Dic106—22no—
79#88324—42no—
80C22×C53—21no—
81C22×C53—41no—
82C22×C53C2×C22⋊C411yescharacteristic
83C2×D53—21no—
84C2×D53—41no—
85C2×D53—41no—
86C2×D53#12312yes—
87#6157—82no—
88C4×C53—21no—
89C4×C53—41no—
90C4×C53—41no—
91C4×C53#11711yescharacteristic
92C4×C53#11711yescharacteristic
93Dic53—22no—
94Dic53—21no—
95Dic53—41no—
96Dic53#12312yes—
97#1827—531nomaximal, Sylow
98C2×C53—21no—
99C2×C53—41no—
100C2×C53#38111yescharacteristic
101D53—22no—
102D53—41no—
103C4×#41—531no—
104#413—531no—
105#410—531no—
106#339—532no—
107H(3,ℤ/4)—532no—
108C53#182711yescharacteristic, complemented, Sylow
109C2×#41—531no—
110C2×M16—532no—
111C4×Q8—1061no—
112C4×D4—1061no—
113C4×D4—1061no—
114C4×D4—1061no—
115#117—531no—
116#117—531no—
117#117—532no—
118#117#615711yescharacteristic, complemented
119C2×C42—531no—
120C2×C42—531no—
121C2×C42—531no—
122#101—1061no—
123#100—1061no—
124#99—1061no—
125#99—1061no—
126#41—1062no—
127#41—1062no—
128C2×Q8—531no—
129C2×D4—531no—
130C2×D4—531no—
131C2×D4—531no—
132C22×C4—531no—
133C22×C4—531no—
134C22×C4—1061no—
135C22×C4—1061no—
136C22×C4—1061no—
137C22×C4C2×#615711yescharacteristic
138M16—1062no—
139M16—1062no—
140C2×C8—1062no—
141C4⋊C4—21no—
142C4⋊C4—1061no—
143C4⋊C4—1061no—
144C4⋊C4—1062no—
145C22⋊C4—21no—
146C22⋊C4—1061no—
147C22⋊C4—1061no—
148C22⋊C4—1062no—
149C42—531no—
150C42—531no—
151C42—1061no—
152C42—1061no—
153C42—1062no—
154C42—1061no—
155C42C2×#615711yescharacteristic
156C42C2×#615711yescharacteristic
157C23—531no—
158C23—531no—
159C23C22⋊#615711yescharacteristic
160Q8—1062no—
161D4—1062no—
162D4—1062no—
163D4—1062no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—531no—
169C2×C4—532no—
170C2×C4—532no—
171C2×C4—531no—
172C2×C4—1061no—
173C2×C4—1061no—
174C2×C4—1061no—
175C2×C4—1061no—
176C2×C4—2122no—
177C2×C4—2122no—
178C2×C4—2122no—
179C2×C4—2122no—
180C2×C4C22⋊#615711yescharacteristic
181C2×C4C22×#615711yescharacteristic
182C8—2122no—
183C22—21no—
184C22—41no—
185C22—532no—
186C22—1061no—
187C22—2121no—
188C22—2121no—
189C22C2×C22⋊#615711yescharacteristic
190C4—21no—
191C4—41no—
192C4—41no—
193C4—532no—
194C4—1062no—
195C4—1061no—
196C4—4242no—
197C4—2121no—
198C4#1313329611yescharacteristic
199C4#1313329611yescharacteristic
200C2—21no—
201C2—41no—
202C2—1062no—
203C2—2121no—
204C2#1681156611yescharacteristic
subgroupG/H
centerC4#13133296
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#117×C53C4
socleC2×C53#381
hypercenter#117#6157
hypocenterC53#1827
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C42 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #16811494C2semidirect
4. #16811490C2semidirect
17. #13133174C4semidirect
19. C2×C4×Dic53C4semidirect
20. C42×D53C4semidirect
22. #117×C53C4semidirect
108. C53#1827semidirect
118. #117#6157semidirect