Finite Group Databasebeta

← #30664499

Subgroups — #30664499

Order 16768.

222 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 77 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#22553369C211yesmaximal, characteristic, complemented
2#22553178C211yesmaximal, characteristic, complemented
3#22553049C212yesmaximal, complemented
4#22553046C212yesmaximal
5#529×C131C211yesmaximal, characteristic
6C2×C4⋊Dic131—21no—
7C4⋊(C131⋊C8)—21no—
8C22⋊(C131⋊C8)C2212yes—
9#16862157—21no—
10C131⋊#95—21no—
11C131⋊#95C2212yes—
12#126×C131—21no—
13#125×C131C411yescharacteristic, complemented
14#124×C131C411yescharacteristic, complemented
15#123×C131—21no—
16C22⋊Q8×C131C2211yescharacteristic
17#121×C131C2211yescharacteristic
18C4×Q8×C131C411yescharacteristic, complemented
19C4×D4×C131—21no—
20C4×D4×C131C411yescharacteristic, complemented
21#117×C131—21no—
22C2×C42×C131C2211yescharacteristic
23C22×Dic131—22no—
24C2×C131⋊C8—22no—
25C4⋊Dic131—42no—
26C2×Q8×C131C2×C411yescharacteristic
27C2×D4×C131—21no—
28C2×D4×C131C2×C411yescharacteristic
29C22×C4×C131—21no—
30C22×C4×C131D411yescharacteristic
31C22×C4×C131D411yescharacteristic
32C22×C4×C131C2311yescharacteristic
33C4⋊C4×C131—21no—
34C4⋊C4×C131—21no—
35C4⋊C4×C131—21no—
36C4⋊C4×C131—21no—
37C4⋊C4×C131C2×C411yescharacteristic
38C4⋊C4×C131C2×C411yescharacteristic
39C22⋊C4×C131—21no—
40C22⋊C4×C131—21no—
41C22⋊C4×C131—21no—
42C22⋊C4×C131—21no—
43C22⋊C4×C131—21no—
44C42×C131—21no—
45C42×C131C2×C411yescharacteristic
46C42×C131C2×C411yescharacteristic
47C42×C131D411yescharacteristic
48C42×C131D411yescharacteristic
49C2×Dic131—42no—
50C2×Dic131—42no—
51C2×Dic131—42no—
52C131⋊C8—42no—
53C23×C131—21no—
54C23×C131C2×D411yescharacteristic
55Q8×C131—41no—
56D4×C131—41no—
57D4×C131—41no—
58D4×C131—41no—
59C2×C4×C131—21no—
60C2×C4×C131—21no—
61C2×C4×C131—21no—
62C2×C4×C131—21no—
63C2×C4×C131—21no—
64C2×C4×C131—41no—
65C2×C4×C131—41no—
66C2×C4×C131—42no—
67C2×C4×C131C22⋊C411yescharacteristic
68C2×C4×C131C22⋊C411yescharacteristic
69C2×C4×C131C22⋊C411yescharacteristic
70C2×C4×C131C22⋊C411yescharacteristic
71C2×C4×C131C22×C411yescharacteristic
72C2×C4×C131C2×D411yescharacteristic
73C22×C131—21no—
74C22×C131—21no—
75C22×C131—41no—
76C22×C131—41no—
77C22×C131—41no—
78C22×C131C2×C22⋊C411yescharacteristic
79C22×C131AGL(1,ℤ/8)11yescharacteristic
80C22×C131#13711yescharacteristic
81C4×C131—21no—
82C4×C131—21no—
83C4×C131—21no—
84C4×C131—21no—
85C4×C131—21no—
86C4×C131—21no—
87C4×C131—41no—
88C4×C131—41no—
89C4×C131—41no—
90Dic131—82no—
91C2×C131—21no—
92C2×C131—21no—
93C2×C131—81no—
94C2×C131#41011yescharacteristic
95C2×C131#41711yescharacteristic
96C2×C131#42111yescharacteristic
97C131#151211yescharacteristic, complemented, Sylow
98#1512—1311nomaximal, Sylow
99#529D13111yescharacteristic
100#432—1311no—
101#382—1311no—
102#328—1312no—
103#327—1312no—
104#126—21no—
105#125Dic13111yescharacteristic, complemented
106#124Dic13111yescharacteristic, complemented
107#123—21no—
108C22⋊Q8C2×D13111yescharacteristic
109#121C2×D13111yescharacteristic
110C4×Q8Dic13111yescharacteristic, complemented
111C4×D4—21no—
112C4×D4Dic13111yescharacteristic, complemented
113#117—21no—
114C2×C4⋊C4—2621no—
115C2×C42C2×D13111yescharacteristic
116C4⋊C8—2621no—
117C22⋊C8—1312no—
118#97—2621no—
119#95—2621no—
120#95—1312no—
121C2×Q8C2×Dic13111yescharacteristic
122C2×D4—21no—
123C2×D4C2×Dic13111yescharacteristic
124C22×C4—21no—
125C22×C4—2622no—
126C22×C4C22⋊D13111yescharacteristic
127C22×C4C22⋊D13111yescharacteristic
128C22×C4C22×D13111yescharacteristic
129C2×C8—2622no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—5242no—
135C4⋊C4C2×Dic13111yescharacteristic
136C4⋊C4C2×Dic13111yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C2×Dic13111yescharacteristic
144C42C2×Dic13111yescharacteristic
145C42C22⋊D13111yescharacteristic
146C42C22⋊D13111yescharacteristic
147C23—21no—
148C23C2×C22⋊D13111yescharacteristic
149Q8—41no—
150D4—41no—
151D4—41no—
152D4—41no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—5242no—
162C2×C4—5242no—
163C2×C4—5242no—
164C2×C4C22⋊Dic13111yescharacteristic
165C2×C4C22⋊Dic13111yescharacteristic
166C2×C4C22⋊Dic13111yescharacteristic
167C2×C4C22⋊Dic13111yescharacteristic
168C2×C4C22×Dic13111yescharacteristic
169C2×C4C2×C22⋊D13111yescharacteristic
170C8—5242no—
171C22—21no—
172C22—21no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22C2×C22⋊Dic13111yescharacteristic
177C22#1686227111yescharacteristic
178C22#1686227611yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—10482no—
189C2—21no—
190C2—21no—
191C2—81no—
192C2#2255328611yescharacteristic
193C2#2255331311yescharacteristic
194C2#2255332911yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic131
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#529×C131C2
socleC22×C131C2×C22⋊C4
prime radical#121×C131C22
cosocleC22×C4×C131C23
hypercenter#529D131
hypocenterC131#1512
LCSG ▷ C2×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #22553369C2semidirect
2. #22553178C2semidirect
3. #22553049C2semidirect
13. #125×C131C4semidirect
14. #124×C131C4semidirect
18. C4×Q8×C131C4semidirect
20. C4×D4×C131C4semidirect
97. C131#1512semidirect
105. #125Dic131semidirect
106. #124Dic131semidirect
110. C4×Q8Dic131semidirect
112. C4×D4Dic131semidirect

Maximal subgroups

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

subgroupindex
#225533692
#225531782
#225530492
#225530462
#529×C1312
#1512131