Finite Group Databasebeta

← #20276107

Subgroups — #20276107

Order 6784.

248 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 89 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×#13133150C211yesmaximal, characteristic
2C2×C4⋊#88322C211yesmaximal, characteristic
3C2×#13133140C211yesmaximal, characteristic
4C2×#13133140C211yesmaximal, characteristic
5#16810529C211yesmaximal, characteristic
6#16810385C211yesmaximal, characteristic
7C53×#389C211yesmaximal, characteristic
8C22×Dic106C2211yescharacteristic
9C22×#88322C2211yescharacteristic
10C22×#88322C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12#13133150—22no—
13C4⋊#88322—22no—
14#13133140—22no—
15#13133140—22no—
16#13133124—21no—
17#13133124—21no—
18C2×C4⋊C4×C53—21no—
19C2×C4⋊C4×C53C2211yescharacteristic
20C2×C4⋊C4×C53C2211yescharacteristic
21C2×C42×C53C2211yescharacteristic
22#95×C53—21no—
23C22×C4×C53—21no—
24C22×C4×C53—21no—
25C22×C4×C53D411yescharacteristic
26C22×C4×C53D411yescharacteristic
27C22×C4×C53C2311yescharacteristic
28C22×Dic53—21no—
29C22×Dic53—21no—
30C2×Dic106—21no—
31C2×Dic106—21no—
32C2×Dic106—21no—
33C2×Dic106—21no—
34C2×Dic106—21no—
35C2×Dic106Q812yes—
36C4⋊C4×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53—42no—
40C4⋊C4×C53D412yes—
41C42×C53—21no—
42C42×C53—21no—
43C4⋊Dic53—21no—
44C4⋊Dic53—21no—
45C2×#88322—21no—
46C2×#88322—21no—
47C2×#88322—21no—
48C2×#88322—21no—
49C2×#88322—21no—
50C2×#88322D412yes—
51C23×C53C2×D411yescharacteristic
52C2×C4×C53—21no—
53C2×C4×C53—21no—
54C2×C4×C53—22no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—41no—
59C2×C4×C53—41no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53Q1612yes—
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C2×C4×C53C2×Q811yescharacteristic
67C2×C4×C53#4111yescharacteristic
68C2×C4×C53#4111yescharacteristic
69C2×C4×C53#4111yescharacteristic
70C2×Dic53—22no—
71C2×Dic53—41no—
72C2×Dic53—41no—
73C2×Dic53—41no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76Dic106—22no—
77Dic106—22no—
78Dic106—41no—
79Dic106—41no—
80#88322—22no—
81#88322—41no—
82C22×C53#12111yescharacteristic
83C22×C53#12111yescharacteristic
84C22×C53C22⋊Q811yescharacteristic
85C22×C53C2×Q1611yescharacteristic
86C22×C53#13511yescharacteristic
87C22×C53AGL(1,ℤ/8)11yescharacteristic
88C22×C53#13711yescharacteristic
89C4×C53—22no—
90C4×C53—42no—
91C4×C53—41no—
92C4×C53—81no—
93C4×C53#12111yescharacteristic
94C4×C53C22⋊Q811yescharacteristic
95C4×C53C22⋊Q811yescharacteristic
96C4×C53#12611yescharacteristic
97Dic53—42no—
98Dic53—81no—
99#2068—531nomaximal, Sylow
100C2×C53#39611yescharacteristic
101C2×C53#46211yescharacteristic
102C2×C53#46311yescharacteristic
103C2×C53#46711yescharacteristic
104C2×C53#46811yescharacteristic
105C2×C53#47711yescharacteristic
106C2×C53#47911yescharacteristic
107C2×#107—531no—
108C2×C4⋊C8—531no—
109C2×#103—531no—
110C2×#103—531no—
111#391—531no—
112#389D5311yescharacteristic
113#340—531no—
114C53#206811yescharacteristic, complemented, Sylow
115C22×Q8—531no—
116C22×C8—531no—
117C22×C8—531no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—531no—
120C2×C4⋊C4C2×D5311yescharacteristic
121C2×C4⋊C4C2×D5311yescharacteristic
122C2×C42C2×D5311yescharacteristic
123#107—1062no—
124C4⋊C8—1062no—
125#103—1062no—
126#103—1062no—
127#95—21no—
128#95—1061no—
129#95—1061no—
130C2×Q8—532no—
131C2×Q8—1061no—
132C2×Q8—1061no—
133C2×Q8—1061no—
134C2×Q8—1061no—
135C2×Q8—1061no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—1061no—
139C22×C4—1061no—
140C22×C4C22⋊D5311yescharacteristic
141C22×C4C22⋊D5311yescharacteristic
142C22×C4C22×D5311yescharacteristic
143C2×C8—532no—
144C2×C8—1061no—
145C2×C8—1061no—
146C2×C8—1061no—
147C2×C8—1061no—
148C2×C8—1061no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—42no—
153C4⋊C4—1061no—
154C4⋊C4—1061no—
155C4⋊C4D21212yes—
156C42—21no—
157C42—21no—
158C23C2×C22⋊D5311yescharacteristic
159Q8—1062no—
160Q8—1062no—
161Q8—2121no—
162Q8—2121no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—1062no—
175C2×C4—2121no—
176C2×C4—2121no—
177C2×C4—2121no—
178C2×C4—2121no—
179C2×C4—2121no—
180C2×C4#1171992512yes—
181C2×C4C2×D21211yescharacteristic
182C2×C4#1171994511yescharacteristic
183C2×C4D4×D5311yescharacteristic
184C2×C4#1171994711yescharacteristic
185C2×C4Q8×D5311yescharacteristic
186C2×C4#1171994911yescharacteristic
187C8—1062no—
188C8—2121no—
189C22#1313318911yescharacteristic
190C22#1313319111yescharacteristic
191C22#1313319311yescharacteristic
192C22C2×#1171992511yescharacteristic
193C22#1313323711yescharacteristic
194C22#1313324311yescharacteristic
195C22#1313324711yescharacteristic
196C4—22no—
197C4—42no—
198C4—41no—
199C4—81no—
200C4—4241no—
201C4—2122no—
202C4#1313319211yescharacteristic
203C4#1313319511yescharacteristic
204C4#1313319711yescharacteristic
205C4#1313321211yescharacteristic
206C2#1681054511yescharacteristic
207C2#1681083311yescharacteristic
208C2#1681083811yescharacteristic
209C2#1681086211yescharacteristic
210C2#1681086711yescharacteristic
211C2#1681091511yescharacteristic
212C2#1681092711yescharacteristic
subgroupG/H
centerC23C2×C22⋊D53
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#389C2
socleC23×C53C2×D4
hypercenter#389D53
hypocenterC53#2068
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
114. C53#2068semidirect