Finite Group Databasebeta

← #20274928

Subgroups — #20274928

Order 6784.

272 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 83 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#16811100C211yesmaximal, characteristic
2#16810663C211yesmaximal, characteristic
3C2×#13133135C211yesmaximal, characteristic
4#16810528C211yesmaximal, characteristic
5#16810384C211yesmaximal, characteristic
6#16810382C211yesmaximal, characteristic
7C53×#428C211yesmaximal, characteristic
8C22×Dic106C2211yescharacteristic
9C2×#11719917C2211yescharacteristic
10C4×Dic106—21no—
11C4×Dic106—21no—
12#13133182—21no—
13#13133182—21no—
14#13133176C2211yescharacteristic
15#13133173—22no—
16C2×C4×Dic53C2211yescharacteristic
17#13133150—21no—
18#13133148—21no—
19#13133135C414yescomplemented
20#13133124—22no—
21C2×M16×C53C2211yescharacteristic
22#117×C53C2211yescharacteristic
23C2×C4⋊C4×C53C2211yescharacteristic
24#107×C53—21no—
25#106×C53—21no—
26C22×C4×C53—21no—
27C22×C4×C53C2311yescharacteristic
28C22×Dic53—21no—
29C22×Dic53D412yes—
30C2×Dic106—21no—
31C2×Dic106—21no—
32C2×Dic106—21no—
33C2×Dic106—21no—
34C2×Dic106C2×C412yes—
35C2×Dic106Q812yes—
36M16×C53—21no—
37M16×C53C2×C412yes—
38C2×C8×C53—21no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53—21no—
41C4⋊C4×C53D411yescharacteristic
42C4⋊C4×C53D411yescharacteristic
43C22⋊C4×C53—21no—
44C42×C53—21no—
45C22⋊Dic53—21no—
46C4⋊Dic53D411yescharacteristic
47C4⋊Dic53D411yescharacteristic
48#11719919—22no—
49#11719919—22no—
50C4×Dic53—22no—
51C4×Dic53—22no—
52C4×Dic53—21no—
53C4×Dic53—21no—
54C4×Dic53—21no—
55#11719917—21no—
56#11719917C2×C412yes—
57C2×#88322—21no—
58C23×C53C2×D411yescharacteristic
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—42no—
62C2×C4×C53—41no—
63C2×C4×C53C22×C411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C2×C4×C53C2×Q811yescharacteristic
67C2×C4×C53#4111yescharacteristic
68C2×C4×C53#4111yescharacteristic
69C2×Dic53—22no—
70C2×Dic53—22no—
71C2×Dic53—22no—
72C2×Dic53—21no—
73C2×Dic53—21no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76C2×Dic53C4⋊C414yes—
77Dic106—22no—
78Dic106—22no—
79Dic106—22no—
80Dic106—22no—
81C8×C53—22no—
82#88322—22no—
83C22×C53—21no—
84C22×C53—21no—
85C22×C53C2×C4⋊C411yescharacteristic
86C22×C53#12011yescharacteristic
87C22×C53#12311yescharacteristic
88C4×C53—41no—
89C4×C53—41no—
90C4×C53—81no—
91C4×C53C4×D411yescharacteristic
92C4×C53C4×D411yescharacteristic
93C4×C53C22⋊Q811yescharacteristic
94C4×C53C22⋊Q811yescharacteristic
95Dic53—24no—
96Dic53—42no—
97Dic53—41no—
98Dic53—41no—
99#1857—531nomaximal, Sylow
100C2×C53—21no—
101C2×C53—21no—
102C2×C53#38511yescharacteristic
103C2×C53#45411yescharacteristic
104C2×C53#45611yescharacteristic
105#519—531no—
106#428—531no—
107#428D5311yescharacteristic
108C2×#101—531no—
109#391—531no—
110#339—531no—
111#339—531no—
112C53#185711yescharacteristic, complemented, Sylow
113C22×Q8—531no—
114C2×M16—531no—
115C2×M16C2×D5311yescharacteristic
116C4×Q8—1061no—
117C4×Q8—1061no—
118C4×Q8—1061no—
119C4×Q8—1061no—
120#117—531no—
121#117—1062no—
122#117C2×D5311yescharacteristic
123C2×C4⋊C4C2×D5311yescharacteristic
124C2×C42—531no—
125#107—21no—
126#107—1061no—
127#106—21no—
128#106—1061no—
129#101—534no—
130#95—1062no—
131C2×Q8—532no—
132C2×Q8—532no—
133C2×Q8—1061no—
134C2×Q8—1061no—
135C2×Q8—1061no—
136C2×Q8—1061no—
137C22×C4—21no—
138C22×C4—532no—
139C22×C4—1061no—
140C22×C4C22×D5311yescharacteristic
141M16—21no—
142M16—532no—
143M16—1061no—
144M16C4×D5312yes—
145C2×C8—21no—
146C2×C8—1061no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—531no—
150C4⋊C4—531no—
151C4⋊C4—1062no—
152C4⋊C4—1062no—
153C4⋊C4C22⋊D5311yescharacteristic
154C4⋊C4C22⋊D5311yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—1061no—
157C42—21no—
158C42—1062no—
159C42—1062no—
160C42—1061no—
161C42—1061no—
162C42—1061no—
163C23D4×D5311yescharacteristic
164Q8—1062no—
165Q8—1062no—
166Q8—1062no—
167Q8—1062no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—534no—
173C2×C4—1062no—
174C2×C4—1062no—
175C2×C4—1062no—
176C2×C4—1061no—
177C2×C4—1061no—
178C2×C4—2121no—
179C2×C4—2121no—
180C2×C4C2×C4×D5311yescharacteristic
181C2×C4#1171994511yescharacteristic
182C2×C4D4×D5311yescharacteristic
183C2×C4Q8×D5311yescharacteristic
184C2×C4#1171994911yescharacteristic
185C2×C4C2×C22⋊D5311yescharacteristic
186C8—22no—
187C8—1062no—
188C22—21no—
189C22—21no—
190C22C4⋊C4×D5311yescharacteristic
191C22#1313318511yescharacteristic
192C22#1313320211yescharacteristic
193C4—41no—
194C4—41no—
195C4—81no—
196C4—1064no—
197C4—2122no—
198C4—2121no—
199C4—2121no—
200C4C4×C22⋊D5311yescharacteristic
201C4#1313317911yescharacteristic
202C4#1313319311yescharacteristic
203C4#1313319511yescharacteristic
204C2—21no—
205C2—21no—
206C2#1681049411yescharacteristic
207C2#1681079411yescharacteristic
208C2#1681080411yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#428C2
socleC22×C53#120
hypercenter#428D53
hypocenterC53#1857
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
19. #13133135C4semidirect
112. C53#1857semidirect