Finite Group Databasebeta

← #20276266

Subgroups — #20276266

Order 6784.

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

normal subgroups only

This group has 238 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×#13133150C211yesmaximal, characteristic
2C2×#13133148C211yesmaximal, characteristic
3C2×#13133140C211yesmaximal, characteristic
4C2×#13133140C211yesmaximal, characteristic
5#16810529C211yesmaximal, characteristic
6#16810378C211yesmaximal, characteristic
7C53×#382C211yesmaximal, characteristic
8C22×Dic106C2211yescharacteristic
9C22×#88322C2211yescharacteristic
10C22×#88322C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12#13133150—22no—
13#13133148—22no—
14#13133140—22no—
15#13133140—22no—
16#13133124—22no—
17C2×C4⋊C4×C53C2211yescharacteristic
18C2×C4⋊C4×C53C2211yescharacteristic
19C2×C42×C53C2211yescharacteristic
20#95×C53—22no—
21C22×C4×C53—21no—
22C22×C4×C53—21no—
23C22×C4×C53D412yes—
24C22×C4×C53C2311yescharacteristic
25C22×Dic53—21no—
26C22×Dic53—21no—
27C2×Dic106—21no—
28C2×Dic106—21no—
29C2×Dic106—21no—
30C2×Dic106—21no—
31C2×Dic106—21no—
32C2×Dic106Q812yes—
33C4⋊C4×C53—21no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53—21no—
37C42×C53—21no—
38C42×C53—21no—
39C4⋊Dic53—21no—
40C4⋊Dic53—21no—
41C2×#88322—21no—
42C2×#88322—21no—
43C2×#88322—21no—
44C2×#88322—21no—
45C2×#88322D412yes—
46C2×#88322D412yes—
47C23×C53C2×D411yescharacteristic
48C2×C4×C53—22no—
49C2×C4×C53—22no—
50C2×C4×C53—22no—
51C2×C4×C53—41no—
52C2×C4×C53—41no—
53C2×C4×C53—42no—
54C2×C4×C53—42no—
55C2×C4×C53C2×D411yescharacteristic
56C2×C4×C53C2×D411yescharacteristic
57C2×C4×C53C2×Q811yescharacteristic
58C2×C4×C53#4111yescharacteristic
59C2×C4×C53#4111yescharacteristic
60C2×C4×C53#4111yescharacteristic
61C2×Dic53—22no—
62C2×Dic53—41no—
63C2×Dic53—41no—
64C2×Dic53—41no—
65C2×Dic53—42no—
66Dic106—22no—
67Dic106—22no—
68Dic106—41no—
69Dic106—41no—
70#88322—22no—
71#88322—22no—
72C22×C53C22⋊Q811yescharacteristic
73C22×C53#12311yescharacteristic
74C22×C53#12711yescharacteristic
75C22×C53#13512yes—
76C22×C53#13712yes—
77C4×C53—42no—
78C4×C53—81no—
79C4×C53—81no—
80C4×C53C22⋊Q811yescharacteristic
81C4×C53C22⋊Q811yescharacteristic
82C4×C53#12311yescharacteristic
83C4×C53#12311yescharacteristic
84Dic53—42no—
85Dic53—81no—
86#2097—531nomaximal, Sylow
87C2×C53#39911yescharacteristic
88C2×C53#47912yes—
89C2×C53#48512yes—
90C2×C53#49511yescharacteristic
91C2×C53#49711yescharacteristic
92C2×#107—531no—
93C2×#106—531no—
94C2×#103—531no—
95C2×#103—531no—
96#391—531no—
97#382D5311yescharacteristic
98#336—531no—
99C53#209711yescharacteristic, complemented, Sylow
100C22×Q8—531no—
101C22×C8—531no—
102C22×C8—531no—
103C2×C4⋊C4—531no—
104C2×C4⋊C4C2×D5311yescharacteristic
105C2×C4⋊C4C2×D5311yescharacteristic
106C2×C42C2×D5311yescharacteristic
107#107—1062no—
108#106—1062no—
109#103—1062no—
110#103—1062no—
111#95—22no—
112#95—1062no—
113C2×Q8—532no—
114C2×Q8—1061no—
115C2×Q8—1061no—
116C2×Q8—1061no—
117C2×Q8—1061no—
118C2×Q8—1061no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—1061no—
122C22×C4—1061no—
123C22×C4C22⋊D5312yes—
124C22×C4C22×D5311yescharacteristic
125C2×C8—532no—
126C2×C8—532no—
127C2×C8—1061no—
128C2×C8—1061no—
129C2×C8—1061no—
130C2×C8—1061no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—1061no—
136C4⋊C4—1061no—
137C42—21no—
138C42—21no—
139C23C2×C22⋊D5311yescharacteristic
140Q8—1062no—
141Q8—1062no—
142Q8—2121no—
143Q8—2121no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—1062no—
152C2×C4—2121no—
153C2×C4—2122no—
154C2×C4—2121no—
155C2×C4—2121no—
156C2×C4#1171994511yescharacteristic
157C2×C4#1171994511yescharacteristic
158C2×C4D4×D5311yescharacteristic
159C2×C4D4×D5311yescharacteristic
160C2×C4Q8×D5311yescharacteristic
161C2×C4#1171994911yescharacteristic
162C8—1062no—
163C8—1062no—
164C22#1313319311yescharacteristic
165C22#1313320011yescharacteristic
166C22#1313321311yescharacteristic
167C22#1313323712yes—
168C22#1313324712yes—
169C4—42no—
170C4—81no—
171C4—81no—
172C4—4241no—
173C4—2122no—
174C4#1313319511yescharacteristic
175C4#1313319511yescharacteristic
176C4#1313320211yescharacteristic
177C4#1313320211yescharacteristic
178C2#1681055811yescharacteristic
179C2#1681092712yes—
180C2#1681096412yes—
181C2#1681100511yescharacteristic
182C2#1681101811yescharacteristic
subgroupG/H
centerC23C2×C22⋊D53
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#382C2
socleC23×C53C2×D4
hypercenter#382D53
hypocenterC53#2097
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
99. C53#2097semidirect