Finite Group Databasebeta

← #30664663

Subgroups — #30664663

Order 16768.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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#22553860C211yesmaximal, characteristic
2#22553368C211yesmaximal, characteristic, complemented
3#22553364C211yesmaximal, characteristic, complemented
4#22553059C211yesmaximal, characteristic, complemented
5#22553057C211yesmaximal, characteristic, complemented
6#22553056C211yesmaximal, characteristic, complemented
7#331×C131C211yesmaximal, characteristic, complemented
8C2×#14523512—21no—
9#16862246C2211yescharacteristic
10#16862233—21no—
11#16862217—21no—
12#16862217C412yes—
13D4×Dic131—21no—
14D4×Dic131C412yes—
15C42×D131C2211yescharacteristic
16C4⋊(C131⋊C8)C2211yescharacteristic
17C4⋊(C131⋊C8)C2211yescharacteristic
18(C2×D131)⋊C8—21no—
19(C2×D131)⋊C8—21no—
20#16862156—21no—
21C8×Dic131—21no—
22#127×C131C2211yescharacteristic
23C4⋊C8×C131C2211yescharacteristic
24C4⋊C8×C131C2211yescharacteristic
25#14523512—42no—
26#14523512—42no—
27C2×Dic262—21no—
28C2×C22⋊D131—22no—
29C22×Dic131—22no—
30C2×C4×D131C2×C411yescharacteristic
31C2×C4×D131D411yescharacteristic
32C2×C4×D131D411yescharacteristic
33C2×C131⋊C8—21no—
34C2×C131⋊C8—21no—
35C4⋊Dic131—21no—
36C4⋊Dic131C2×C412yes—
37C22⋊Dic131—22no—
38C22⋊Dic131—22no—
39C4×Dic131C2×C411yescharacteristic
40C4×Dic131D411yescharacteristic
41C4×Dic131D411yescharacteristic
42C2×D4×C131—21no—
43C2×D4×C131—21no—
44C2×D4×C131C2×C412yes—
45C2×C8×C131—21no—
46C2×C8×C131—21no—
47C42×C131C2311yescharacteristic
48C22×D131C22⋊C411yescharacteristic
49Dic262—41no—
50Dic262—41no—
51C22⋊D131—44no—
52C2×Dic131—22no—
53C2×Dic131—44no—
54C2×Dic131C22⋊C411yescharacteristic
55C2×Dic131C22⋊C411yescharacteristic
56C2×Dic131C22⋊C411yescharacteristic
57C4×D131—21no—
58C4×D131—21no—
59C4×D131—21no—
60C4×D131—21no—
61C4×D131—41no—
62C131⋊C8—41no—
63C131⋊C8—41no—
64C23×C131—22no—
65D4×C131—42no—
66D4×C131—41no—
67D4×C131—41no—
68D4×C131—41no—
69D4×C131—41no—
70C2×C4×C131C22×C411yescharacteristic
71C2×C4×C131C2×D411yescharacteristic
72C2×C4×C131C2×D411yescharacteristic
73C8×C131—41no—
74C8×C131—41no—
75C22×C131—42no—
76C22×C131—42no—
77C22×C131—42no—
78C22×C131C2×C22⋊C411yescharacteristic
79C2×D131—21no—
80C2×D131—21no—
81C2×D131—21no—
82C4×C131—21no—
83C4×C131#13511yescharacteristic
84C4×C131#13511yescharacteristic
85C4×C131AGL(1,ℤ/8)11yescharacteristic
86C4×C131AGL(1,ℤ/8)11yescharacteristic
87Dic131—21no—
88Dic131—21no—
89Dic131—21no—
90Dic131—42no—
91C2×C131—82no—
92C2×C131#41311yescharacteristic
93C2×C131#41611yescharacteristic
94C2×C131#41811yescharacteristic
95D131—41no—
96C131#155311yescharacteristic, complemented, Sylow
97#1553—1311nomaximal, Sylow
98#532—1311no—
99#432—1311no—
100#431—1311no—
101#332—1311no—
102#332—1311no—
103#331—1311no—
104#331D13111yescharacteristic, complemented
105C2×#41—2621no—
106C4⋊Q8—1311no—
107#127C2×D13111yescharacteristic
108#124—2621no—
109#121—2621no—
110#121—1312no—
111C4×D4—2621no—
112C4×D4—1312no—
113C2×C42—1311no—
114C4⋊C8—1311no—
115C4⋊C8—1311no—
116C4⋊C8C2×D13111yescharacteristic
117C4⋊C8C2×D13111yescharacteristic
118C22⋊C8—2621no—
119C22⋊C8—2621no—
120#97—2621no—
121C4×C8—2621no—
122#41—5242no—
123#41—5242no—
124C2×Q8—2621no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—2622no—
128C2×D4C4×D13112yes—
129C22×C4—2622no—
130C22×C4—1311no—
131C22×C4—1311no—
132C22×C4—1311no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—2621no—
136C2×C8—2621no—
137C4⋊C4—2621no—
138C4⋊C4—1312no—
139C22⋊C4—2622no—
140C22⋊C4—2622no—
141C42—1311no—
142C42—1311no—
143C42—1311no—
144C42C22×D13111yescharacteristic
145C23—22no—
146C23—1311no—
147Q8—5241no—
148Q8—5241no—
149D4—42no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—41no—
154D4—5244no—
155C2×C4—2621no—
156C2×C4—2621no—
157C2×C4—2621no—
158C2×C4—2621no—
159C2×C4—2622no—
160C2×C4—5241no—
161C2×C4—5244no—
162C2×C4—1311no—
163C2×C4—1311no—
164C2×C4—1311no—
165C2×C4C2×C4×D13111yescharacteristic
166C2×C4D4×D13111yescharacteristic
167C2×C4D4×D13111yescharacteristic
168C8—41no—
169C8—41no—
170C8—5241no—
171C8—5241no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22—2621no—
176C22—2621no—
177C22—2621no—
178C22C22⋊C4×D13111yescharacteristic
179C4—21no—
180C4—2621no—
181C4—2621no—
182C4—2621no—
183C4—5242no—
184C4#1686226511yescharacteristic
185C4#1686226711yescharacteristic
186C4#1686227011yescharacteristic
187C4#1686227311yescharacteristic
188C2—82no—
189C2—5241no—
190C2#2255329311yescharacteristic
191C2#2255330511yescharacteristic
192C2#2255331511yescharacteristic
subgroupG/H
centerC22C22⋊C4×D131
commutatorC2×C4×C131C22×C4
FrattiniC42C22×D131
Fitting#331×C131C2
socleC22×C131C2×C22⋊C4
prime radical#22553860C2
cosocleC42×C131C23
hypercenter#331D131
hypocenterC131#1553
LCSG ▷ C2×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #22553368C2semidirect
3. #22553364C2semidirect
4. #22553059C2semidirect
5. #22553057C2semidirect
6. #22553056C2semidirect
7. #331×C131C2semidirect
96. C131#1553semidirect
104. #331D131semidirect

Maximal subgroups

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

subgroupindex
#225538602
#225533682
#225533642
#225530592
#225530572
#225530562
#331×C1312
#1553131