Finite Group Databasebeta

← #20275497

Subgroups — #20275497

Order 6784.

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

normal subgroups only

This group has 196 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
1C2×#13133152C211yesmaximal, characteristic
2#16810666C211yesmaximal, characteristic
3C2×#13133150C211yesmaximal, characteristic
4#16810579C211yesmaximal, characteristic
5#16810392C211yesmaximal, characteristic
6#16810391C211yesmaximal, characteristic
7C53×#428C211yesmaximal, characteristic
8C2×#11719917C2211yescharacteristic
9C2×#11719917C2211yescharacteristic
10C22×#88322C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12#13133152C414yescomplemented
13#13133150—21no—
14#13133150—21no—
15#13133150—21no—
16#13133148—21no—
17#13133128—21no—
18C4×#88322—21no—
19C2×M16×C53C2211yescharacteristic
20#117×C53C2211yescharacteristic
21C2×C4⋊C4×C53C2211yescharacteristic
22#107×C53—21no—
23#106×C53—21no—
24C22×C4×C53—21no—
25C22×C4×C53C2311yescharacteristic
26C22×Dic53—21no—
27M16×C53—21no—
28M16×C53C2×C412yes—
29C2×C8×C53—21no—
30C4⋊C4×C53—21no—
31C4⋊C4×C53—21no—
32C4⋊C4×C53D411yescharacteristic
33C4⋊C4×C53D411yescharacteristic
34C22⋊C4×C53—21no—
35C42×C53—21no—
36C4⋊Dic53—21no—
37C4⋊Dic53—21no—
38#11719917—21no—
39#11719917—21no—
40#11719917C2×C412yes—
41#11719917Q812yes—
42C2×#88322—21no—
43C2×#88322—21no—
44C2×#88322C2×C412yes—
45C2×#88322D411yescharacteristic
46C2×#88322D411yescharacteristic
47C2×#88322Q812yes—
48C23×C53#4111yescharacteristic
49C2×C4×C53—21no—
50C2×C4×C53—21no—
51C2×C4×C53—41no—
52C2×C4×C53—81no—
53C2×C4×C53C22×C411yescharacteristic
54C2×C4×C53C2×D411yescharacteristic
55C2×C4×C53C2×D411yescharacteristic
56C2×C4×C53C2×Q811yescharacteristic
57C2×C4×C53C2×Q811yescharacteristic
58C2×C4×C53#4111yescharacteristic
59C2×Dic53—41no—
60C2×Dic53—81no—
61C8×C53—22no—
62#88322—22no—
63#88322—22no—
64#88322C4⋊C412yes—
65#88322C4⋊C412yes—
66C22×C53—21no—
67C22×C53—21no—
68C22×C53C4×Q811yescharacteristic
69C22×C53#12111yescharacteristic
70C22×C53#12511yescharacteristic
71C4×C53—41no—
72C4×C53—41no—
73C4×C53—81no—
74C4×C53C2×C4⋊C411yescharacteristic
75C4×C53C4×D411yescharacteristic
76C4×C53C22⋊Q811yescharacteristic
77C4×C53C4⋊Q811yescharacteristic
78Dic53—81no—
79#1952—531nomaximal, Sylow
80C2×C53—21no—
81C2×C53—21no—
82C2×C53#38911yescharacteristic
83C2×C53#47211yescharacteristic
84C2×C53#47311yescharacteristic
85C2×#108—531no—
86#428—531no—
87#428D5311yescharacteristic
88C2×#107—531no—
89#405—531no—
90#343—531no—
91#343—531no—
92C53#195211yescharacteristic, complemented, Sylow
93C2×M16—531no—
94C2×M16—531no—
95C2×M16C2×D5311yescharacteristic
96C22×C8—531no—
97#117C2×D5311yescharacteristic
98C2×C4⋊C4—531no—
99C2×C4⋊C4C2×D5311yescharacteristic
100#108—534no—
101#107—21no—
102#107—1061no—
103#107—1061no—
104#107—1061no—
105#106—21no—
106#106—1061no—
107#97—1061no—
108C4×C8—1061no—
109C22×C4—21no—
110C22×C4—1061no—
111C22×C4C22×D5311yescharacteristic
112M16—21no—
113M16—532no—
114M16—532no—
115M16—1061no—
116M16—1061no—
117M16C4×D5312yes—
118C2×C8—21no—
119C2×C8—532no—
120C2×C8—531no—
121C2×C8—531no—
122C2×C8—532no—
123C2×C8—1061no—
124C2×C8—1061no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—1061no—
128C4⋊C4—1061no—
129C4⋊C4C22⋊D5311yescharacteristic
130C4⋊C4C22⋊D5311yescharacteristic
131C22⋊C4—21no—
132C42—21no—
133C23#1171994711yescharacteristic
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—81no—
138C2×C4—4241no—
139C2×C4—2121no—
140C2×C4C2×C4×D5311yescharacteristic
141C2×C4#1171994511yescharacteristic
142C2×C4D4×D5311yescharacteristic
143C2×C4Q8×D5311yescharacteristic
144C2×C4Q8×D5311yescharacteristic
145C2×C4C2×C22⋊D5311yescharacteristic
146C8—22no—
147C8—532no—
148C8—532no—
149C8—1062no—
150C8—1062no—
151C22—21no—
152C22—21no—
153C22#1313318211yescharacteristic
154C22#1313318911yescharacteristic
155C22#1313320711yescharacteristic
156C4—41no—
157C4—41no—
158C4—81no—
159C4—4241no—
160C4C4⋊C4×D5311yescharacteristic
161C4C4×C22⋊D5311yescharacteristic
162C4#1313319511yescharacteristic
163C4#1313321511yescharacteristic
164C2—21no—
165C2—21no—
166C2#1681051811yescharacteristic
167C2#1681089111yescharacteristic
168C2#1681089611yescharacteristic
subgroupG/H
centerC22#13133189
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#428C2
socleC22×C53#121
hypercenter#428D53
hypocenterC53#1952
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
12. #13133152C4semidirect
92. C53#1952semidirect