Finite Group Databasebeta

← #30666399

Subgroups — #30666399

Order 16768.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#16862220C211yesmaximal, characteristic
2C2×#16862168C212yesmaximal, complemented
3#22553172C211yesmaximal, characteristic, complemented
4#22553075C212yesmaximal, complemented
5C22×M16×C131C211yesmaximal, characteristic, complemented
6C22×D524C2211yescharacteristic
7C22×C22⋊D131—21no—
8#16862220—22no—
9#16862220C414yescomplemented
10C2×C4⋊Dic131C2212yes—
11C2×C4⋊Dic131C2211yescharacteristic
12C2×C22⋊Dic131—21no—
13C2×(C2×D131)⋊C4—21no—
14#16862168—24no—
15C131⋊#95—21no—
16C23×C4×C131C2211yescharacteristic
17C2×M16×C131—22no—
18C2×M16×C131—24no—
19C22×C8×C131C2212yes—
20C23×D131—21no—
21C2×D524—22no—
22C2×D524—21no—
23C2×D524—22no—
24C2×D524C2×C412yes—
25C2×C22⋊D131—24no—
26C2×C22⋊D131—44no—
27C22×Dic131—21no—
28C22×Dic131—22no—
29C4⋊Dic131—21no—
30C4⋊Dic131—22no—
31C4⋊Dic131—22no—
32C4⋊Dic131C2×C412yes—
33C22⋊Dic131—42no—
34(C2×D131)⋊C4—24no—
35C24×C131D411yescharacteristic
36C22×C4×C131—22no—
37C22×C4×C131—22no—
38C22×C4×C131C2×C412yes—
39C22×C4×C131D412yes—
40C22×C4×C131D411yescharacteristic
41C22×C4×C131C2311yescharacteristic
42M16×C131—28no—
43C2×C8×C131—22no—
44C2×C8×C131—22no—
45C2×C8×C131D414yescomplemented
46C22×D131—22no—
47C22×D131—42no—
48C22×D131—42no—
49C22×D131—41no—
50C22×D131—41no—
51D524—24no—
52D524—41no—
53D524—41no—
54C22⋊D131—48no—
55C2×Dic131—22no—
56C2×Dic131—42no—
57C2×Dic131—42no—
58C2×Dic131—42no—
59C2×Dic131—42no—
60C23×C131—22no—
61C23×C131—21no—
62C23×C131—22no—
63C23×C131—21no—
64C23×C131C22⋊C412yes—
65C23×C131C2×D411yescharacteristic
66C2×C4×C131—22no—
67C2×C4×C131—24no—
68C2×C4×C131—22no—
69C2×C4×C131—22no—
70C2×C4×C131C22⋊C412yes—
71C2×C4×C131C22×C411yescharacteristic
72C2×C4×C131C2×D412yes—
73C2×C4×C131C2×D411yescharacteristic
74C2×C4×C131#4112yes—
75C8×C131—24no—
76C22×C131—22no—
77C22×C131—22no—
78C22×C131—22no—
79C22×C131—42no—
80C22×C131—42no—
81C22×C131C2×C22⋊C411yescharacteristic
82C22×C131#12112yes—
83C22×C131AGL(1,ℤ/8)11yescharacteristic
84C22×C131AGL(1,ℤ/8)12yes—
85C22×C131#13711yescharacteristic
86C2×D131—42no—
87C2×D131—42no—
88C2×D131—42no—
89C2×D131—82no—
90C2×D131—82no—
91C4×C131—22no—
92C4×C131C4×D412yes—
93C4×C131#12011yescharacteristic
94C4×C131#12311yescharacteristic
95Dic131—42no—
96Dic131—82no—
97C2×C131—42no—
98C2×C131#38611yescharacteristic
99C2×C131#41711yescharacteristic
100C2×C131#41811yescharacteristic
101C2×C131#46812yes—
102C2×C131#46912yes—
103D131—82no—
104C131#189511yescharacteristic, complemented, Sylow
105#1895—1311nomaximal, Sylow
106C22×M16D13111yescharacteristic, complemented
107C2×#121—1311no—
108C2×#102—1312no—
109#380—1311no—
110#340—1312no—
111C22×D4—2621no—
112C22×D4—1311no—
113C23×C4C2×D13111yescharacteristic
114C2×M16—22no—
115C2×M16—24no—
116C22×C8C2×D13112yes—
117#121—2622no—
118#121—1314no—
119C2×C4⋊C4—1312no—
120C2×C4⋊C4—1311no—
121C2×C22⋊C4—2621no—
122C2×C22⋊C4—2621no—
123#102—2624no—
124#95—2621no—
125C24—2621no—
126C24D52411yescharacteristic
127C2×D4—2624no—
128C2×D4—2622no—
129C2×D4—2621no—
130C2×D4—2622no—
131C2×D4—5244no—
132C2×D4—1312no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—2621no—
136C22×C4—2622no—
137C22×C4C4×D13112yes—
138C22×C4C22⋊D13112yes—
139C22×C4D52411yescharacteristic
140C22×C4C22×D13111yescharacteristic
141M16—28no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8C22⋊D13114yescomplemented
145C4⋊C4—2621no—
146C4⋊C4—2622no—
147C4⋊C4—2622no—
148C4⋊C4—1312no—
149C22⋊C4—2624no—
150C22⋊C4—5242no—
151C23—22no—
152C23—21no—
153C23—22no—
154C23—21no—
155C23—2622no—
156C23—5242no—
157C23—5242no—
158C23—5241no—
159C23—5241no—
160C23(C2×D131)⋊C412yes—
161C23C2×D52411yescharacteristic
162D4—2624no—
163D4—5248no—
164D4—5241no—
165D4—5241no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—2622no—
171C2×C4—5242no—
172C2×C4—5242no—
173C2×C4—5242no—
174C2×C4—5242no—
175C2×C4(C2×D131)⋊C412yes—
176C2×C4C2×C4×D13111yescharacteristic
177C2×C4C2×C22⋊D13111yescharacteristic
178C2×C4C2×C22⋊D13112yes—
179C2×C4#1452351312yes—
180C8—24no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—5242no—
187C22—5242no—
188C22—5242no—
189C22—10482no—
190C22—10482no—
191C22C2×(C2×D131)⋊C411yescharacteristic
192C22#1686222012yes—
193C22#1686227412yes—
194C22#1686227411yescharacteristic
195C22#1686227911yescharacteristic
196C4—22no—
197C4—5242no—
198C4—10482no—
199C4C4×C22⋊D13112yes—
200C4#1686221511yescharacteristic
201C4#1686222911yescharacteristic
202C2—42no—
203C2—10482no—
204C2#2255319111yescharacteristic
205C2#2255331411yescharacteristic
206C2#2255331811yescharacteristic
207C2#2255355812yes—
208C2#2255356512yes—
subgroupG/H
centerC23C2×D524
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
FittingC22×M16×C131C2
socleC23×C131C2×D4
prime radicalC2×#16862220C2
cosocleC22×C4×C131C23
hypercenterC22×M16D131
hypocenterC131#1895
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#16862168C2semidirect
3. #22553172C2semidirect
4. #22553075C2semidirect
5. C22×M16×C131C2semidirect
9. #16862220C4semidirect
45. C2×C8×C131D4semidirect
104. C131#1895semidirect
106. C22×M16D131semidirect
144. C2×C8C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622202
C2×#168621682
#225531722
#225530752
C22×M16×C1312
#1895131