Finite Group Databasebeta

← #30664296

Subgroups — #30664296

Order 16768.

316 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 69 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×(C22⋊D131)⋊C4C211yesmaximal, characteristic
2#22553274C212yesmaximal, complemented
3#22553037C212yesmaximal, complemented
4#22553036C211yesmaximal, characteristic, complemented
5#323×C131C211yesmaximal, characteristic, complemented
6C22×C22⋊D131C411yescharacteristic
7C23×Dic131C411yescharacteristic
8C22×C4×D131C2211yescharacteristic
9C2×#14523496—22no—
10(C22⋊D131)⋊C4—22no—
11(C22⋊D131)⋊C4—22no—
12C2×Dic131⋊C4C2211yescharacteristic
13C2×(C2×D131)⋊C4C411yescharacteristic
14C2×C4×Dic131C411yescharacteristic
15C22⋊(C131⋊C8)C2212yes—
16(C2×D131)⋊C8—22no—
17C2×C22⋊C4×C131C2211yescharacteristic
18C22⋊C8×C131C2212yes—
19C23×D131C2×C411yescharacteristic
20C2×C22⋊D131—22no—
21C2×C22⋊D131—22no—
22C2×C22⋊D131—22no—
23C22×Dic131—21no—
24C22×Dic131—21no—
25C22×Dic131—42no—
26C22×Dic131C2×C411yescharacteristic
27C22×Dic131C2×C411yescharacteristic
28C22×Dic131C2×C411yescharacteristic
29C2×C4×D131—42no—
30C2×C4×D131D412yes—
31C2×C4×D131D412yes—
32#14523496—44no—
33C2×C131⋊C8—22no—
34Dic131⋊C4—22no—
35(C2×D131)⋊C4—22no—
36C4×Dic131—21no—
37C4×Dic131—21no—
38C24×C131C2×C411yescharacteristic
39C22×C4×C131C2×C411yescharacteristic
40C22×C4×C131C2311yescharacteristic
41C2×C8×C131—22no—
42C22⋊C4×C131—22no—
43C22×D131—21no—
44C22×D131—21no—
45C22×D131—41no—
46C22×D131—41no—
47C22×D131C22⋊C412yes—
48C22⋊D131—28no—
49C2×Dic131—21no—
50C2×Dic131—21no—
51C2×Dic131—22no—
52C2×Dic131—21no—
53C2×Dic131—42no—
54C2×Dic131—42no—
55C2×Dic131—41no—
56C2×Dic131C22⋊C411yescharacteristic
57C2×Dic131C22⋊C411yescharacteristic
58C2×Dic131M1612yescomplemented
59C4×D131—24no—
60C4×D131—42no—
61C131⋊C8—42no—
62C23×C131—21no—
63C23×C131—21no—
64C23×C131—41no—
65C23×C131—41no—
66C23×C131M1612yescomplemented
67C23×C131C22×C411yescharacteristic
68C2×C4×C131—21no—
69C2×C4×C131—41no—
70C2×C4×C131—41no—
71C2×C4×C131C2×D412yes—
72C8×C131—42no—
73C22×C131—21no—
74C22×C131—21no—
75C22×C131—22no—
76C22×C131—42no—
77C22×C131—42no—
78C22×C131—41no—
79C22×C131—41no—
80C22×C131C2×C22⋊C411yescharacteristic
81C22×C131C2×M1611yescharacteristic
82C22×C131C2×M1611yescharacteristic
83C2×D131—22no—
84C2×D131—22no—
85C2×D131—22no—
86C2×D131—42no—
87C2×D131—42no—
88C4×C131—22no—
89C4×C131—41no—
90Dic131—22no—
91Dic131—21no—
92Dic131—41no—
93Dic131#9912yes—
94C2×C131—21no—
95C2×C131—21no—
96C2×C131—42no—
97C2×C131#40711yescharacteristic
98C2×C131C2×#9911yescharacteristic
99C2×C131#41311yescharacteristic
100D131—42no—
101C131#146511yescharacteristic, complemented, Sylow
102#1465—1311nomaximal, Sylow
103C2×C4×D4—1311no—
104#407—1312no—
105#324—1312no—
106#323—1311no—
107#323D13111yescharacteristic, complemented
108C22×D4—1311no—
109C23×C4—1311no—
110C23×C4—1311no—
111C2×M16—2622no—
112C4×D4—2622no—
113C4×D4—2622no—
114C2×C4⋊C4—1311no—
115C2×C22⋊C4—1311no—
116C2×C22⋊C4C2×D13111yescharacteristic
117C2×C42—1311no—
118C22⋊C8—2622no—
119C22⋊C8—1312no—
120C22⋊C8C2×D13112yes—
121C24—1311no—
122C24C4×D13111yescharacteristic
123C2×D4—2622no—
124C2×D4—2622no—
125C2×D4—2622no—
126C22×C4—2621no—
127C22×C4—2621no—
128C22×C4—5242no—
129C22×C4—5242no—
130C22×C4—1311no—
131C22×C4—1311no—
132C22×C4—1312no—
133C22×C4—1311no—
134C22×C4—1312no—
135C22×C4C4×D13111yescharacteristic
136C22×C4C22×D13111yescharacteristic
137M16—5244no—
138C2×C8—22no—
139C2×C8—2622no—
140C4⋊C4—2622no—
141C22⋊C4—22no—
142C22⋊C4—2622no—
143C42—2621no—
144C42—2621no—
145C23—21no—
146C23—21no—
147C23—41no—
148C23—41no—
149C23—2621no—
150C23—2621no—
151C23—5241no—
152C23—5241no—
153C23—1312no—
154C23#1452349612yescomplemented
155C23C2×C4×D13111yescharacteristic
156D4—2628no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—2624no—
161C2×C4—2621no—
162C2×C4—2621no—
163C2×C4—2622no—
164C2×C4—2621no—
165C2×C4—5242no—
166C2×C4—5242no—
167C2×C4—5242no—
168C2×C4—5241no—
169C2×C4—1312no—
170C2×C4—1311no—
171C2×C4—1311no—
172C2×C4D4×D13112yes—
173C8—42no—
174C8—5242no—
175C22—21no—
176C22—21no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—2622no—
183C22—2622no—
184C22—2622no—
185C22—5242no—
186C22—5242no—
187C22C22⋊C4×D13111yescharacteristic
188C22M16×D13111yescharacteristic
189C22C2×#1452349611yescharacteristic
190C4—22no—
191C4—41no—
192C4—2622no—
193C4—2621no—
194C4—5241no—
195C4—1312no—
196C2—21no—
197C2—21no—
198C2—42no—
199C2—5242no—
200C2#2255327411yescharacteristic
201C2#99×D13111yescharacteristic
202C2#2255329311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D131
commutatorC23×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#323×C131C2
socleC22×C131C2×C22⋊C4
prime radicalC22×C22⋊D131C4
cosocleC22×C4×C131C23
hypercenter#323D131
hypocenterC131#1465
LCSG ▷ C23×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22553274C2semidirect
3. #22553037C2semidirect
4. #22553036C2semidirect
5. #323×C131C2semidirect
58. C2×Dic131M16semidirect
66. C23×C131M16semidirect
101. C131#1465semidirect
107. #323D131semidirect
154. C23#14523496semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D131)⋊C42
#225532742
#225530372
#225530362
#323×C1312
#1465131