Finite Group Databasebeta

← #30674241

Subgroups — #30674241

Order 16768.

1224 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C24×Dic131C211yesmaximal, characteristic, complemented
2C2×#16862218C213yesmaximal, complemented
3C2×#16862215C211yesmaximal, characteristic, complemented
4#22553192C213yesmaximal, complemented
5#22553189C213yesmaximal, complemented
6#22553170C213yesmaximal, complemented
7C2×#120×C131C211yesmaximal, characteristic, complemented
8C22×C22⋊D131C2213yes—
9C22×C22⋊D131C2211yescharacteristic, complemented
10C23×Dic131—212no—
11C23×Dic131C2213yes—
12C23×Dic131C2213yes—
13#16862218—212no—
14#16862215—24no—
15C2×Dic131⋊C4C2213yescomplemented
16C2×C22⋊Dic131C2213yes—
17C2×C22⋊Dic131C2213yescomplemented
18C2×(C2×D131)⋊C4C2213yescomplemented
19C131⋊#95C2216yescomplemented
20C25×C131C2211yescharacteristic
21C22×D4×C131C2213yes—
22#120×C131—24no—
23C2×C22⋊C4×C131C2213yes—
24C23×D131C2311yescharacteristic
25C2×C22⋊D131—26no—
26C2×C22⋊D131—26no—
27C2×C22⋊D131—26no—
28C2×C22⋊D131—26no—
29C22×Dic131—232no—
30C22×Dic131—212no—
31C22×Dic131—212no—
32C22×Dic131—26no—
33C22×Dic131D4112yescomplemented
34C22×Dic131C2313yes—
35C22×Dic131C2313yes—
36C22×Dic131C2311yescharacteristic
37Dic131⋊C4—26no—
38C22⋊Dic131—26no—
39C22⋊Dic131—26no—
40(C2×D131)⋊C4—26no—
41C24×C131—26no—
42C24×C131—26no—
43C24×C131D414yescomplemented
44C24×C131C2311yescharacteristic
45C24×C131C2313yes—
46C2×D4×C131—26no—
47C2×D4×C131—26no—
48C2×D4×C131—26no—
49C22×C4×C131C2313yes—
50C22⋊C4×C131—26no—
51C22×D131—23no—
52C22×D131—21no—
53C22×D131—23no—
54C22⋊D131—412no—
55C22⋊D131—44no—
56C2×Dic131—224no—
57C2×Dic131—26no—
58C2×Dic131—224no—
59C2×Dic131—23no—
60C2×Dic131—23no—
61C2×Dic131—23no—
62C2×Dic131C2×D416yes—
63C2×Dic131C2×D416yes—
64C23×C131—224no—
65C23×C131—28no—
66C23×C131—224no—
67C23×C131—26no—
68C23×C131—26no—
69C23×C131—21no—
70C23×C131—23no—
71C23×C131—23no—
72C23×C131C2×D416yes—
73C23×C131C2×D416yes—
74C23×C131#4116yes—
75C23×C131C2411yescharacteristic
76D4×C131—412no—
77C2×C4×C131—23no—
78C2×C4×C131—23no—
79C2×C4×C131—23no—
80C22×C131—232no—
81C22×C131—26no—
82C22×C131—212no—
83C22×C131—26no—
84C22×C131—212no—
85C22×C131—41no—
86C22×C131—43no—
87C22×C131—43no—
88C22×C131#121112yescomplemented
89C22×C131C22×D413yes—
90C22×C131C22×D411yescharacteristic
91C22×C131C2×#4113yes—
92C2×D131—41no—
93C2×D131—43no—
94C2×D131—43no—
95C4×C131—43no—
96Dic131—26no—
97Dic131—43no—
98Dic131#12014yescomplemented
99C2×C131—212no—
100C2×C131—81no—
101C2×C131C2×#12011yescharacteristic
102C2×C131C2×#12113yes—
103C2×C131#52513yes—
104D131—81no—
105C131#284711yescharacteristic, complemented, Sylow
106#2847—1311nomaximal, Sylow
107C24×C4—1311no—
108C2×#121—1313no—
109C2×#120—1311no—
110C2×#120D13111yescharacteristic, complemented
111#386—1313no—
112#386—1313no—
113#380—1313no—
114C25C2×D13111yescharacteristic
115C22×D4—1313no—
116C22×D4—1311no—
117C22×D4C2×D13113yes—
118C23×C4—26212no—
119C23×C4—1313no—
120C23×C4—1313no—
121#121—26212no—
122#120—24no—
123#120—2624no—
124C2×C4⋊C4—1313no—
125C2×C22⋊C4—1313no—
126C2×C22⋊C4—1313no—
127C2×C22⋊C4—1313no—
128C2×C22⋊C4C2×D13113yes—
129#95—1316no—
130C24—26no—
131C24—26no—
132C24—1311no—
133C24C22⋊D13114yescomplemented
134C24C22×D13111yescharacteristic
135C24C22×D13113yes—
136C2×D4—26no—
137C2×D4—26no—
138C2×D4—26no—
139C2×D4—2626no—
140C2×D4—2626no—
141C2×D4—2626no—
142C2×D4—2626no—
143C22×C4—26232no—
144C22×C4—2626no—
145C22×C4—26212no—
146C22×C4—26212no—
147C22×C4—1313no—
148C22×C4—13112no—
149C22×C4—1313no—
150C22×C4—1311no—
151C22×C4C22×D13113yes—
152C4⋊C4—2626no—
153C22⋊C4—26no—
154C22⋊C4—2626no—
155C22⋊C4—2626no—
156C22⋊C4—2626no—
157C23—28no—
158C23—224no—
159C23—224no—
160C23—26no—
161C23—26no—
162C23—23no—
163C23—21no—
164C23—23no—
165C23—2623no—
166C23—2621no—
167C23—2623no—
168C23D4×D13116yes—
169C23C2×C22⋊D13116yes—
170C23#1452351216yes—
171C23C23×D13111yescharacteristic
172D4—412no—
173D4—52412no—
174D4—5244no—
175C2×C4—23no—
176C2×C4—23no—
177C2×C4—23no—
178C2×C4—26224no—
179C2×C4—2626no—
180C2×C4—26224no—
181C2×C4—2623no—
182C2×C4—2623no—
183C2×C4—2623no—
184C2×C4—1316no—
185C2×C4—1316no—
186C22—232no—
187C22—26no—
188C22—212no—
189C22—26no—
190C22—212no—
191C22—41no—
192C22—43no—
193C22—43no—
194C22—5241no—
195C22—5243no—
196C22—5243no—
197C22#16862218112yescomplemented
198C22C2×D4×D13113yes—
199C22C22×C22⋊D13111yescharacteristic
200C22C2×#1452351213yes—
201C4—43no—
202C4—2626no—
203C4—5243no—
204C4—1314no—
205C2—212no—
206C2—81no—
207C2—10481no—
208C2#120×D13111yescharacteristic
209C2C2×#1686221813yes—
210C2#2255381813yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC2×#120×C131C2
hypercenterC2×#120D131
hypocenterC131#2847
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C24×Dic131C2semidirect
2. C2×#16862218C2semidirect
3. C2×#16862215C2semidirect
4. #22553192C2semidirect
5. #22553189C2semidirect
6. #22553170C2semidirect
7. C2×#120×C131C2semidirect
9. C22×C22⋊D131C22semidirect
15. C2×Dic131⋊C4C22semidirect
17. C2×C22⋊Dic131C22semidirect
18. C2×(C2×D131)⋊C4C22semidirect
19. C131⋊#95C22semidirect
33. C22×Dic131D4semidirect
43. C24×C131D4semidirect
88. C22×C131#121semidirect
98. Dic131#120semidirect
105. C131#2847semidirect
110. C2×#120D131semidirect
133. C24C22⋊D131semidirect
197. C22#16862218semidirect

Maximal subgroups

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

subgroupindex
C24×Dic1312
C2×#168622182
C2×#168622152
#225531922
#225531892
#225531702
C2×#120×C1312
#2847131