Finite Group Databasebeta

← #30664293

Subgroups — #30664293

Order 16768.

252 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 73 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×#16862203C211yesmaximal, characteristic
2#22553369C211yesmaximal, characteristic, complemented
3#22553037C212yesmaximal, complemented
4#22553035C212yesmaximal
5#432×C131C211yesmaximal, characteristic, complemented
6C22×C4×D131C411yescharacteristic
7#16862203—21no—
8#16862203—21no—
9#16862203—21no—
10#16862203—21no—
11C2×Dic131⋊C4C411yescharacteristic
12C2×Dic131⋊C4C2211yescharacteristic
13C2×(C2×D131)⋊C4C411yescharacteristic
14C2×(C2×D131)⋊C4C2211yescharacteristic
15C2×C4×Dic131C411yescharacteristic
16C4⋊(C131⋊C8)—21no—
17C22⋊(C131⋊C8)C2212yes—
18#16862157—21no—
19C2×C42×C131C2211yescharacteristic
20C4⋊C8×C131—21no—
21C22⋊C8×C131C2212yes—
22#97×C131—21no—
23C23×D131C2×C411yescharacteristic
24C22×Dic131C2×C411yescharacteristic
25C22×Dic131C2×C411yescharacteristic
26C22×Dic131C2×C411yescharacteristic
27C2×C4×D131—22no—
28C2×C4×D131—22no—
29C2×C4×D131—21no—
30C2×C4×D131—21no—
31C2×C131⋊C8—22no—
32Dic131⋊C4—21no—
33Dic131⋊C4—21no—
34Dic131⋊C4—21no—
35Dic131⋊C4—21no—
36(C2×D131)⋊C4—21no—
37(C2×D131)⋊C4—21no—
38(C2×D131)⋊C4—21no—
39(C2×D131)⋊C4—21no—
40C4×Dic131—21no—
41C4×Dic131—21no—
42C22×C4×C131C2×C411yescharacteristic
43C22×C4×C131C2×C411yescharacteristic
44C22×C4×C131C2311yescharacteristic
45C2×C8×C131—22no—
46C42×C131D411yescharacteristic
47C42×C131D411yescharacteristic
48C42×C131D411yescharacteristic
49C42×C131D411yescharacteristic
50C22×D131—21no—
51C22×D131—21no—
52C22×D131—41no—
53C22×D131—41no—
54C22×D131M1612yes—
55C2×Dic131—21no—
56C2×Dic131—21no—
57C2×Dic131—21no—
58C2×Dic131—21no—
59C2×Dic131—41no—
60C2×Dic131—41no—
61C2×Dic131M1612yes—
62C4×D131—24no—
63C4×D131—24no—
64C131⋊C8—42no—
65C23×C131C22×C411yescharacteristic
66C2×C4×C131—21no—
67C2×C4×C131—21no—
68C2×C4×C131—41no—
69C2×C4×C131—41no—
70C2×C4×C131C22⋊C411yescharacteristic
71C2×C4×C131C22⋊C411yescharacteristic
72C2×C4×C131C22⋊C411yescharacteristic
73C2×C4×C131C22⋊C411yescharacteristic
74C2×C4×C131C2×D411yescharacteristic
75C2×C4×C131C2×D411yescharacteristic
76C8×C131—42no—
77C22×C131—21no—
78C22×C131—21no—
79C22×C131C2×C22⋊C411yescharacteristic
80C22×C131C2×M1611yescharacteristic
81C22×C131C2×M1611yescharacteristic
82C2×D131—22no—
83C2×D131—42no—
84C2×D131—41no—
85C2×D131—41no—
86C2×D131—42no—
87C4×C131—21no—
88C4×C131—21no—
89C4×C131—21no—
90C4×C131—21no—
91C4×C131—21no—
92C4×C131—21no—
93Dic131—22no—
94Dic131—41no—
95Dic131—41no—
96C2×C131—21no—
97C2×C131—21no—
98C2×C131#40711yescharacteristic
99C2×C131#41011yescharacteristic
100C2×C131#41311yescharacteristic
101D131—42no—
102C131#146411yescharacteristic, complemented, Sylow
103#1464—1311nomaximal, Sylow
104C2×#117—1311no—
105#432—1311no—
106#432D13111yescharacteristic, complemented
107#324—1312no—
108#323—1312no—
109C23×C4—1311no—
110#117—2621no—
111#117—2621no—
112#117—2621no—
113#117—2621no—
114C2×C4⋊C4—1311no—
115C2×C4⋊C4—1311no—
116C2×C22⋊C4—1311no—
117C2×C22⋊C4—1311no—
118C2×C42—1311no—
119C2×C42C2×D13111yescharacteristic
120C4⋊C8—21no—
121C4⋊C8—2621no—
122C22⋊C8—1312no—
123C22⋊C8C2×D13112yes—
124#97—21no—
125#97—2621no—
126C24—1311no—
127C22×C4—2622no—
128C22×C4—2622no—
129C22×C4—2621no—
130C22×C4—2621no—
131C22×C4—1311no—
132C22×C4—1311no—
133C22×C4—1311no—
134C22×C4C4×D13111yescharacteristic
135C22×C4C4×D13111yescharacteristic
136C22×C4C22×D13111yescharacteristic
137C2×C8—22no—
138C2×C8—2622no—
139C4⋊C4—2621no—
140C4⋊C4—2621no—
141C4⋊C4—2621no—
142C4⋊C4—2621no—
143C22⋊C4—2621no—
144C22⋊C4—2621no—
145C22⋊C4—2621no—
146C22⋊C4—2621no—
147C42—2621no—
148C42—2621no—
149C42C22⋊D13111yescharacteristic
150C42C22⋊D13111yescharacteristic
151C42D52411yescharacteristic
152C42D52411yescharacteristic
153C23—2621no—
154C23—2621no—
155C23—5241no—
156C23—5241no—
157C23—1312no—
158C23C2×C4×D13111yescharacteristic
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—2624no—
164C2×C4—2624no—
165C2×C4—2621no—
166C2×C4—2621no—
167C2×C4—2621no—
168C2×C4—2621no—
169C2×C4—5241no—
170C2×C4—5241no—
171C2×C4—1312no—
172C2×C4(C2×D131)⋊C411yescharacteristic
173C2×C4(C2×D131)⋊C411yescharacteristic
174C2×C4(C2×D131)⋊C411yescharacteristic
175C2×C4(C2×D131)⋊C411yescharacteristic
176C2×C4C2×C22⋊D13111yescharacteristic
177C2×C4C2×D52411yescharacteristic
178C8—42no—
179C8—5242no—
180C22—21no—
181C22—21no—
182C22—2622no—
183C22—5242no—
184C22—5242no—
185C22—5241no—
186C22—5241no—
187C22C2×(C2×D131)⋊C411yescharacteristic
188C22M16×D13111yescharacteristic
189C22M16×D13111yescharacteristic
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—21no—
196C4—2622no—
197C4—5241no—
198C4—5241no—
199C2—21no—
200C2—21no—
201C2—5242no—
202C2#2255327511yescharacteristic
203C2#2255328511yescharacteristic
204C2#2255329411yescharacteristic
subgroupG/H
centerC22C2×(C2×D131)⋊C4
commutatorC23×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#432×C131C2
socleC22×C131C2×C22⋊C4
prime radicalC23×D131C2×C4
cosocleC22×C4×C131C23
hypercenter#432D131
hypocenterC131#1464
LCSG ▷ C23×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #22553369C2semidirect
3. #22553037C2semidirect
5. #432×C131C2semidirect
102. C131#1464semidirect
106. #432D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622032
#225533692
#225530372
#225530352
#432×C1312
#1464131