Finite Group Databasebeta

← #30664913

Subgroups — #30664913

Order 16768.

308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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
1#22553450C212yesmaximal, complemented
2#22553224C211yesmaximal, characteristic, complemented
3#22553046C212yesmaximal, complemented
4#22553036C211yesmaximal, characteristic, complemented
5#534×C131C211yesmaximal, characteristic
6C22×Dic262C2211yescharacteristic, complemented
7C2×#14523501—22no—
8C2×C22⋊Dic131—21no—
9#16862169—22no—
10C22⋊(C131⋊C8)C2212yescomplemented
11C131⋊#95—21no—
12C131⋊#95C2211yescharacteristic, complemented
13C22×D4×C131—21no—
14#123×C131—22no—
15#121×C131—21no—
16#121×C131C2212yes—
17#120×C131—22no—
18C2×C22⋊C4×C131C2211yescharacteristic
19C2×Dic262—22no—
20C2×Dic262—42no—
21C22×Dic131—21no—
22C22×Dic131—21no—
23#14523501—44no—
24C2×C131⋊C8—22no—
25C22⋊Dic131—42no—
26C24×C131—21no—
27C24×C131D411yescharacteristic
28C2×D4×C131—22no—
29C2×D4×C131—22no—
30C2×D4×C131—21no—
31C2×D4×C131—42no—
32C2×D4×C131—42no—
33C2×D4×C131D412yes—
34C22×C4×C131—21no—
35C22×C4×C131D411yescharacteristic
36C22×C4×C131C2311yescharacteristic
37C4⋊C4×C131—21no—
38C4⋊C4×C131D412yes—
39C22⋊C4×C131—22no—
40C22⋊C4×C131—22no—
41C22⋊C4×C131—22no—
42Dic262—24no—
43Dic262—42no—
44C2×Dic131—22no—
45C2×Dic131—41no—
46C2×Dic131—41no—
47C2×Dic131—41no—
48C2×Dic131—41no—
49C2×Dic131—41no—
50C131⋊C8—42no—
51C23×C131—21no—
52C23×C131—21no—
53C23×C131—22no—
54C23×C131—41no—
55C23×C131—41no—
56C23×C131—42no—
57C23×C131—81no—
58C23×C131—81no—
59C23×C131SD1612yescomplemented
60C23×C131C2×D411yescharacteristic
61D4×C131—42no—
62D4×C131—44no—
63D4×C131—42no—
64D4×C131—42no—
65D4×C131—81no—
66C2×C4×C131—21no—
67C2×C4×C131—22no—
68C2×C4×C131—41no—
69C2×C4×C131—42no—
70C2×C4×C131—41no—
71C2×C4×C131C2×D412yes—
72C22×C131—21no—
73C22×C131—21no—
74C22×C131—22no—
75C22×C131—42no—
76C22×C131—41no—
77C22×C131—41no—
78C22×C131—42no—
79C22×C131—42no—
80C22×C131—42no—
81C22×C131—81no—
82C22×C131—84no—
83C22×C131#12011yescharacteristic
84C22×C131C2×SD1611yescharacteristic
85C22×C131AGL(1,ℤ/8)11yescharacteristic
86C4×C131—22no—
87C4×C131—42no—
88C4×C131—41no—
89Dic131—42no—
90Dic131—81no—
91C2×C131—21no—
92C2×C131—21no—
93C2×C131—42no—
94C2×C131—82no—
95C2×C131#45011yescharacteristic
96C2×C131#45511yescharacteristic
97C2×C131B(4,𝔽2)11yescharacteristic
98C131#160111yescharacteristic, complemented, Sylow
99#1601—1311nomaximal, Sylow
100#534D13111yescharacteristic
101#448—1312no—
102#393—1311no—
103#327—1312no—
104#323—1311no—
105C22×Q8—1311no—
106C22×D4—21no—
107C2×SD16—2622no—
108#123—22no—
109#121—21no—
110#121C2×D13112yes—
111#120—22no—
112C2×C22⋊C4—2621no—
113C2×C22⋊C4C2×D13111yescharacteristic
114#103—2622no—
115C22⋊C8—1312no—
116#95—2621no—
117#95—1311no—
118C24—21no—
119C24C22⋊D13111yescharacteristic
120C2×Q8—2622no—
121C2×Q8—5242no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4C22⋊D13112yes—
128C22×C4—21no—
129C22×C4—2621no—
130C22×C4—2621no—
131C22×C4C22⋊D13111yescharacteristic
132C22×C4C22×D13111yescharacteristic
133SD16—5244no—
134C2×C8—2622no—
135C4⋊C4—21no—
136C4⋊C4C22⋊D13112yes—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—5242no—
141C23—21no—
142C23—21no—
143C23—22no—
144C23—41no—
145C23—41no—
146C23—42no—
147C23—81no—
148C23—81no—
149C23#1452350112yescomplemented
150C23C2×C22⋊D13111yescharacteristic
151Q8—2624no—
152Q8—5242no—
153D4—42no—
154D4—44no—
155D4—42no—
156D4—42no—
157D4—81no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—2622no—
164C2×C4—5241no—
165C2×C4—5241no—
166C2×C4—5241no—
167C2×C4—5241no—
168C2×C4—5241no—
169C2×C4C2×C22⋊D13112yes—
170C8—5242no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—81no—
181C22—84no—
182C22#1686221511yescharacteristic
183C22C2×#1452350111yescharacteristic
184C22#1686227211yescharacteristic
185C4—22no—
186C4—42no—
187C4—41no—
188C4—5242no—
189C4—10481no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—82no—
194C2#2255346311yescharacteristic
195C2#2255349011yescharacteristic
196C2#2255349911yescharacteristic
subgroupG/H
centerC22#16862215
commutator, cosocleC22×C4×C131C23
FrattiniC22×C4C22×D131
Fitting, prime radical#534×C131C2
socleC22×C131#120
hypercenter#534D131
hypocenterC131#1601
LCSG ▷ C22×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #22553450C2semidirect
2. #22553224C2semidirect
3. #22553046C2semidirect
4. #22553036C2semidirect
6. C22×Dic262C22semidirect
10. C22⋊(C131⋊C8)C22semidirect
12. C131⋊#95C22semidirect
59. C23×C131SD16semidirect
98. C131#1601semidirect
149. C23#14523501semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225534502
#225532242
#225530462
#225530362
#534×C1312
#1601131