Finite Group Databasebeta

← #20275990

Subgroups — #20275990

Order 6784.

274 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 65 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×#13133245C211yesmaximal, characteristic
2#16810639C211yesmaximal, characteristic, complemented
3#16810629C211yesmaximal, characteristic, complemented
4#16810605C211yesmaximal, characteristic, complemented
5C2×#13133133C211yesmaximal, characteristic, complemented
6#16810395C211yesmaximal, characteristic, complemented
7C53×#430C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9C22×D212C2211yescharacteristic
10#13133245—22no—
11#13133245—22no—
12C2×#11719913—21no—
13C2×D424—21no—
14C2×#11719917C2211yescharacteristic
15#13133176C2211yescharacteristic
16#13133143—22no—
17#13133139—22no—
18#13133135—21no—
19#13133133—22no—
20#13133133—21no—
21C2×M16×C53C2211yescharacteristic
22C2×M16×C53C2211yescharacteristic
23C2×M16×C53C2211yescharacteristic
24#108×C53—22no—
25C23×D53—21no—
26C22×C4×C53C2311yescharacteristic
27C2×C22⋊D53—21no—
28#11719945—21no—
29#11719945—21no—
30#11719945—21no—
31#11719945—21no—
32C2×D212—21no—
33C2×D212—21no—
34C2×D212—21no—
35C2×D212—21no—
36C2×D212—21no—
37C2×D212D411yescharacteristic
38C2×D212D412yes—
39C2×C4×D53—21no—
40C2×Dic106D411yescharacteristic
41M16×C53—21no—
42M16×C53—21no—
43M16×C53—21no—
44M16×C53—21no—
45M16×C53D412yes—
46M16×C53D412yes—
47C2×C8×C53—21no—
48C2×C8×C53—21no—
49C2×C8×C53—21no—
50C22⋊Dic53—21no—
51C4⋊Dic53—21no—
52C4×Dic53—21no—
53#11719917—21no—
54#11719917—21no—
55C2×#88322—21no—
56D424—24no—
57#11719913—24no—
58C23×C53#4111yescharacteristic
59C22×D53—22no—
60C22×D53—21no—
61C22×D53—41no—
62C22×D53—41no—
63C22×D53—82no—
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C2×C4×C53C2×D411yescharacteristic
68C2×C4×C53#4111yescharacteristic
69C2×C4×C53#4111yescharacteristic
70C22⋊D53—41no—
71C22⋊D53—41no—
72C2×Dic53—21no—
73C2×Dic53—41no—
74D212—22no—
75D212—21no—
76D212—21no—
77D212—22no—
78D212—41no—
79D212—41no—
80C4×D53—41no—
81C4×D53—41no—
82Dic106—21no—
83Dic106—21no—
84C8×C53—22no—
85C8×C53—22no—
86C8×C53—41no—
87#88322—41no—
88C22×C53—41no—
89C22×C53#12111yescharacteristic
90C22×C53#12311yescharacteristic
91C22×C53#12411yescharacteristic
92C2×D53—41no—
93C2×D53—42no—
94C2×D53—82no—
95C2×D53—82no—
96C2×D53—81no—
97C2×D53—82no—
98C4×C53#12011yescharacteristic
99C4×C53#12111yescharacteristic
100C4×C53#12111yescharacteristic
101C4×C53#12311yescharacteristic
102Dic53—41no—
103Dic53—81no—
104#2048—531nomaximal, Sylow
105C2×C53—21no—
106C2×C53—21no—
107C2×C53—21no—
108C2×C53#39411yescharacteristic
109D53—81no—
110D53—82no—
111C2×AGL(1,ℤ/8)—531no—
112#430D5311yescharacteristic, complemented
113#421—531no—
114#418—531no—
115#413—531no—
116C2×#100—531no—
117#344—531no—
118C53#204811yescharacteristic, complemented, Sylow
119C2×#41—531no—
120C22×D4—531no—
121AGL(1,ℤ/8)—1062no—
122AGL(1,ℤ/8)—1062no—
123C2×SD16—1061no—
124C2×D8—1061no—
125C2×M16—531no—
126C2×M16C2×D5311yescharacteristic
127C2×M16C2×D5311yescharacteristic
128C2×M16C2×D5311yescharacteristic
129#117—531no—
130#108—22no—
131#104—1062no—
132#102—1062no—
133#101—1061no—
134#100—1061no—
135#100—1062no—
136C24—1061no—
137#41—1061no—
138#41—1061no—
139#41—1061no—
140#41—1061no—
141C2×Q8—531no—
142C2×D4—531no—
143C2×D4—532no—
144C2×D4—1061no—
145C2×D4—1061no—
146C2×D4—1061no—
147C2×D4—1061no—
148C2×D4—1061no—
149C2×D4—1061no—
150C22×C4—1061no—
151C22×C4C22×D5311yescharacteristic
152SD16—1064no—
153D8—1064no—
154M16—21no—
155M16—21no—
156M16—21no—
157M16—21no—
158M16—1061no—
159M16—1061no—
160M16D21212yes—
161M16C22⋊D5312yes—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—1061no—
166C4⋊C4—1061no—
167C22⋊C4—1061no—
168C42—1061no—
169C23—1062no—
170C23—1061no—
171C23—4242no—
172C23—2121no—
173C23—2121no—
174C23#1171994911yescharacteristic
175Q8—1061no—
176Q8—1061no—
177D4—1061no—
178D4—1062no—
179D4—1061no—
180D4—1062no—
181D4—2121no—
182D4—2121no—
183D4—2121no—
184D4—2121no—
185C2×C4—1061no—
186C2×C4—2121no—
187C2×C4—2121no—
188C2×C4—2121no—
189C2×C4C2×D21211yescharacteristic
190C2×C4#1171994511yescharacteristic
191C2×C4D4×D5311yescharacteristic
192C2×C4D4×D5311yescharacteristic
193C2×C4#1171994911yescharacteristic
194C2×C4C2×C22⋊D5311yescharacteristic
195C8—22no—
196C8—22no—
197C8—41no—
198C8—2121no—
199C22—41no—
200C22—4242no—
201C22—4241no—
202C22—4242no—
203C22—4242no—
204C22—2121no—
205C22—2122no—
206C22#1313319211yescharacteristic
207C22#1313320211yescharacteristic
208C22#1313320611yescharacteristic
209C4—4241no—
210C4—2121no—
211C4#1313318511yescharacteristic
212C4#1313319111yescharacteristic
213C4#1313319211yescharacteristic
214C4#1313320211yescharacteristic
215C2—21no—
216C2—21no—
217C2—21no—
218C2—4242no—
219C2—4241no—
220C2#1681054011yescharacteristic
subgroupG/H
centerC2#16810540
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#430C2
socleC2×C53#394
hypercenter#430D53
hypocenterC53#2048
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #430

Complements

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

NH ≅ G/Ntype
2. #16810639C2semidirect
3. #16810629C2semidirect
4. #16810605C2semidirect
5. C2×#13133133C2semidirect
6. #16810395C2semidirect
7. C53×#430C2semidirect
112. #430D53semidirect
118. C53#2048semidirect