Finite Group Databasebeta

← #20275998

Subgroups — #20275998

Order 6784.

254 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 63 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×#13133250C211yesmaximal, characteristic
2#16810671C211yesmaximal, characteristic, complemented
3#16810641C211yesmaximal, characteristic
4#16810631C211yesmaximal, characteristic, complemented
5#16810605C211yesmaximal, characteristic
6C2×#13133135C211yesmaximal, characteristic, complemented
7C53×#344C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9C22×Dic106C2211yescharacteristic
10#13133250—22no—
11#13133250—22no—
12C2×Dic212—21no—
13C2×#11719913—21no—
14C2×#11719917C2211yescharacteristic
15C2×#11719917C2211yescharacteristic
16#13133152—22no—
17#13133145—22no—
18#13133140—22no—
19#13133135—22no—
20#13133135—21no—
21#13133133—21no—
22C2×M16×C53C2211yescharacteristic
23C2×M16×C53C2211yescharacteristic
24#117×C53C2211yescharacteristic
25C22×C4×C53C2311yescharacteristic
26C2×C22⋊D53—21no—
27C22×Dic53—21no—
28#11719945—21no—
29#11719945—21no—
30#11719945—21no—
31#11719945—21no—
32C2×D212D411yescharacteristic
33C2×C4×D53—21no—
34C2×Dic106—21no—
35C2×Dic106—21no—
36C2×Dic106—21no—
37C2×Dic106—21no—
38C2×Dic106—21no—
39C2×Dic106D411yescharacteristic
40C2×Dic106D412yes—
41M16×C53—21no—
42M16×C53—21no—
43M16×C53—21no—
44M16×C53D412yes—
45C2×C8×C53—21no—
46C2×C8×C53—21no—
47C4⋊C4×C53—21no—
48C22⋊C4×C53—21no—
49C42×C53—21no—
50#11719917—21no—
51#11719917—21no—
52#11719917—21no—
53#11719917D412yes—
54C2×#88322—21no—
55C2×#88322—21no—
56Dic212—24no—
57#11719913—24no—
58C23×C53#4111yescharacteristic
59C22×D53—21no—
60C2×C4×C53—41no—
61C2×C4×C53C2×D411yescharacteristic
62C2×C4×C53C2×D411yescharacteristic
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53#4111yescharacteristic
66C2×C4×C53#4111yescharacteristic
67C22⋊D53—41no—
68C22⋊D53—41no—
69C2×Dic53—22no—
70C2×Dic53—21no—
71C2×Dic53—41no—
72C2×Dic53—41no—
73D212—21no—
74D212—21no—
75C4×D53—41no—
76C4×D53—41no—
77Dic106—22no—
78Dic106—21no—
79Dic106—21no—
80Dic106—22no—
81Dic106—41no—
82Dic106—41no—
83C8×C53—22no—
84C8×C53—41no—
85#88322—22no—
86#88322—41no—
87C22×C53—41no—
88C22×C53#12111yescharacteristic
89C22×C53#12311yescharacteristic
90C22×C53#12411yescharacteristic
91C2×D53—41no—
92C2×D53—81no—
93C4×C53—81no—
94C4×C53#12011yescharacteristic
95C4×C53#12111yescharacteristic
96C4×C53#12111yescharacteristic
97C4×C53#12311yescharacteristic
98Dic53—42no—
99Dic53—41no—
100#2049—531nomaximal, Sylow
101C2×C53—21no—
102C2×C53—21no—
103C2×C53—21no—
104C2×C53#39411yescharacteristic
105D53—81no—
106C2×#137—531no—
107#430—531no—
108#421—531no—
109#419—531no—
110#413—531no—
111C2×#101—531no—
112#344D5311yescharacteristic, complemented
113C53#204911yescharacteristic, complemented, Sylow
114C2×#41—531no—
115C22×Q8—531no—
116#137—1062no—
117#137—1062no—
118C2×Q16—1061no—
119C2×SD16—1061no—
120C2×M16—531no—
121C2×M16—531no—
122C2×M16C2×D5311yescharacteristic
123C2×M16C2×D5311yescharacteristic
124#117C2×D5311yescharacteristic
125#108—1062no—
126#104—1062no—
127#103—1062no—
128#101—1062no—
129#101—1061no—
130#100—1061no—
131#41—1061no—
132#41—1061no—
133#41—1061no—
134#41—1061no—
135C2×Q8—531no—
136C2×Q8—532no—
137C2×Q8—1061no—
138C2×Q8—1061no—
139C2×Q8—1061no—
140C2×Q8—1061no—
141C2×Q8—1061no—
142C2×D4—531no—
143C2×D4—1061no—
144C22×C4—1061no—
145C22×C4—1061no—
146C22×C4C22×D5311yescharacteristic
147Q16—1064no—
148SD16—1064no—
149M16—21no—
150M16—21no—
151M16—21no—
152M16—532no—
153M16—1061no—
154M16—1061no—
155M16—1061no—
156M16D21212yes—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—1061no—
160C2×C8—1061no—
161C4⋊C4—21no—
162C22⋊C4—21no—
163C42—21no—
164C23—1061no—
165C23#1171994511yescharacteristic
166Q8—1062no—
167Q8—1061no—
168Q8—1061no—
169Q8—1062no—
170Q8—2121no—
171Q8—2121no—
172D4—1061no—
173D4—1061no—
174D4—2121no—
175D4—2121no—
176C2×C4—41no—
177C2×C4—1062no—
178C2×C4—1061no—
179C2×C4—2121no—
180C2×C4—2121no—
181C2×C4—2121no—
182C2×C4—2121no—
183C2×C4C2×D21211yescharacteristic
184C2×C4#1171994511yescharacteristic
185C2×C4D4×D5311yescharacteristic
186C2×C4D4×D5311yescharacteristic
187C2×C4D4×D5311yescharacteristic
188C2×C4#1171994911yescharacteristic
189C8—22no—
190C8—41no—
191C8—1062no—
192C8—2121no—
193C22—41no—
194C22—4241no—
195C22—2121no—
196C22#1313319011yescharacteristic
197C22#1313320211yescharacteristic
198C22#1313320511yescharacteristic
199C4—81no—
200C4—2121no—
201C4—2122no—
202C4#1313318711yescharacteristic
203C4#1313319011yescharacteristic
204C4#1313319211yescharacteristic
205C4#1313320211yescharacteristic
206C2—21no—
207C2—21no—
208C2—21no—
209C2—4241no—
210C2#1681054111yescharacteristic
subgroupG/H
centerC2#16810541
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#344C2
socleC2×C53#394
hypercenter#344D53
hypocenterC53#2049
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #16810671C2semidirect
4. #16810631C2semidirect
6. C2×#13133135C2semidirect
7. C53×#344C2semidirect
112. #344D53semidirect
113. C53#2049semidirect