Finite Group Databasebeta

← #30666407

Subgroups — #30666407

Order 16768.

358 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 89 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#22553823C211yesmaximal, characteristic
2C2×#16862205C211yesmaximal, characteristic, complemented
3C2×#16862172C212yesmaximal, complemented
4#22553073C212yesmaximal, complemented
5#407×C131C211yesmaximal, characteristic, complemented
6C2×#14523513C2211yescharacteristic
7#16862229—21no—
8#16862225C412yescomplemented
9#16862220C412yescomplemented
10#16862215—21no—
11C4×C22⋊D131—22no—
12#16862205—24no—
13#16862205C2211yescharacteristic
14C2×C4⋊Dic131—21no—
15C2×C22⋊Dic131—21no—
16C2×C4×Dic131C2212yes—
17#16862172—24no—
18C23×C4×C131C2211yescharacteristic
19C2×M16×C131C2212yes—
20C22⋊C8×C131—22no—
21#14523513—22no—
22#14523513—22no—
23C2×Dic262C2×C411yescharacteristic
24C2×D524C2×C411yescharacteristic
25C2×C22⋊D131—22no—
26C2×C22⋊D131—21no—
27C22×Dic131—22no—
28C2×C4×D131—21no—
29C4⋊Dic131—24no—
30C4⋊Dic131C2×C412yes—
31Dic131⋊C4—22no—
32C22⋊Dic131—24no—
33C22⋊Dic131—22no—
34C22⋊Dic131—21no—
35(C2×D131)⋊C4—22no—
36C4×Dic131—22no—
37C4×Dic131—21no—
38C4×Dic131D414yescomplemented
39C24×C131D411yescharacteristic
40C22×C4×C131—22no—
41C22×C4×C131—22no—
42C22×C4×C131C2×C412yes—
43C22×C4×C131D412yes—
44C22×C4×C131D411yescharacteristic
45C22×C4×C131C2311yescharacteristic
46M16×C131—22no—
47M16×C131—22no—
48C2×C8×C131—22no—
49C22×D131—21no—
50Dic262—22no—
51D524—22no—
52C22⋊D131—44no—
53C22⋊D131—41no—
54C22⋊D131—41no—
55C2×Dic131—22no—
56C2×Dic131—21no—
57C2×Dic131—24no—
58C2×Dic131—44no—
59C4×D131—42no—
60C23×C131—21no—
61C23×C131—21no—
62C23×C131—42no—
63C23×C131C22⋊C412yes—
64C23×C131C2×D411yescharacteristic
65C2×C4×C131—21no—
66C2×C4×C131—21no—
67C2×C4×C131—24no—
68C2×C4×C131—24no—
69C2×C4×C131C22⋊C412yes—
70C2×C4×C131C22×C411yescharacteristic
71C2×C4×C131C2×D412yes—
72C2×C4×C131C2×D411yescharacteristic
73C2×C4×C131#4112yes—
74C8×C131—42no—
75C22×C131—22no—
76C22×C131—22no—
77C22×C131—22no—
78C22×C131—22no—
79C22×C131—42no—
80C22×C131—42no—
81C22×C131C2×C22⋊C411yescharacteristic
82C22×C131#12011yescharacteristic
83C22×C131#12311yescharacteristic
84C2×D131—41no—
85C2×D131—42no—
86C4×C131—22no—
87C4×C131C4×D412yes—
88C4×C131#12112yes—
89Dic131—44no—
90Dic131—41no—
91Dic131—42no—
92C2×C131—21no—
93C2×C131—21no—
94C2×C131—42no—
95C2×C131#38611yescharacteristic
96C2×C131#42112yes—
97D131—81no—
98C131#189711yescharacteristic, complemented, Sylow
99#1897—1311nomaximal, Sylow
100#525—1311no—
101C2×#117—1311no—
102C2×#104—1312no—
103#407D13111yescharacteristic, complemented
104#339—1312no—
105C2×#41—1311no—
106C23×C4C2×D13111yescharacteristic
107C2×M16C2×D13112yes—
108#123—2621no—
109C22⋊Q8—1312no—
110#121—1312no—
111#120—2621no—
112C4×D4—2622no—
113#117—2624no—
114#117—1311no—
115C2×C4⋊C4—2621no—
116C2×C22⋊C4—2621no—
117C2×C42—1312no—
118#104—2624no—
119C22⋊C8—22no—
120C24D52411yescharacteristic
121#41—2622no—
122#41—2622no—
123C2×Q8—1311no—
124C2×D4—2622no—
125C2×D4—2621no—
126C2×D4—1311no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—2622no—
130C22×C4—2621no—
131C22×C4C4×D13112yes—
132C22×C4C22⋊D13112yes—
133C22×C4D52411yescharacteristic
134C22×C4C22×D13111yescharacteristic
135M16—22no—
136M16—22no—
137C2×C8—22no—
138C4⋊C4—2624no—
139C4⋊C4—2622no—
140C4⋊C4—1312no—
141C22⋊C4—2624no—
142C22⋊C4—2622no—
143C22⋊C4—2621no—
144C22⋊C4—2622no—
145C42—2622no—
146C42—2621no—
147C42—1314no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—2621no—
152C23(C2×D131)⋊C412yes—
153C23C2×D52411yescharacteristic
154Q8—2622no—
155D4—2622no—
156D4—5244no—
157D4—5241no—
158D4—5241no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—2624no—
164C2×C4—2622no—
165C2×C4—2621no—
166C2×C4—5244no—
167C2×C4—5242no—
168C2×C4(C2×D131)⋊C412yes—
169C2×C4C2×C4×D13111yescharacteristic
170C2×C4D4×D13112yes—
171C2×C4C2×C22⋊D13111yescharacteristic
172C2×C4#1452351212yes—
173C8—42no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—5241no—
181C22—5242no—
182C22C2×(C2×D131)⋊C411yescharacteristic
183C22#1686221611yescharacteristic
184C22#1686223011yescharacteristic
185C4—22no—
186C4—5244no—
187C4—5241no—
188C4—5242no—
189C4(C22⋊D131)⋊C412yes—
190C4#1686221812yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—10481no—
195C2#2255318911yescharacteristic
196C2#2255332812yes—
subgroupG/H
centerC2×C4C2×C22⋊D131
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#407×C131C2
socleC22×C131C2×C22⋊C4
prime radical#22553823C2
cosocleC22×C4×C131C23
hypercenter#407D131
hypocenterC131#1897
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#16862205C2semidirect
3. C2×#16862172C2semidirect
4. #22553073C2semidirect
5. #407×C131C2semidirect
8. #16862225C4semidirect
9. #16862220C4semidirect
38. C4×Dic131D4semidirect
98. C131#1897semidirect
103. #407D131semidirect

Maximal subgroups

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

subgroupindex
#225538232
C2×#168622052
C2×#168621722
#225530732
#407×C1312
#1897131