Finite Group Databasebeta

← #30664682

Subgroups — #30664682

Order 16768.

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

normal subgroups only

This group has 246 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#22553863C211yesmaximal, characteristic
2#22553368C211yesmaximal, characteristic, complemented
3#22553334C211yesmaximal, characteristic, complemented
4#22553058C211yesmaximal, characteristic, complemented
5#22553055C211yesmaximal, characteristic, complemented
6#22553055C211yesmaximal, characteristic, complemented
7#332×C131C211yesmaximal, characteristic, complemented
8C2×#14523514—21no—
9C2×#14523496—21no—
10#16862243C2211yescharacteristic
11#16862235—21no—
12#16862221—21no—
13#16862221C412yes—
14#16862208—21no—
15#16862208C412yes—
16C42×D131C2211yescharacteristic
17C4⋊(C131⋊C8)C2211yescharacteristic
18C4⋊(C131⋊C8)C2211yescharacteristic
19Dic131⋊C8—21no—
20(C2×D131)⋊C8—21no—
21#16862156—21no—
22C4⋊Q8×C131C2211yescharacteristic
23C4⋊C8×C131C2211yescharacteristic
24C4⋊C8×C131C2211yescharacteristic
25#14523514—42no—
26#14523514—42no—
27C2×D524—22no—
28C2×D524—21no—
29C2×D524—21no—
30C2×D524C2×C412yes—
31C2×C4×D131—22no—
32C2×C4×D131C2×C411yescharacteristic
33C2×C4×D131D411yescharacteristic
34C2×C4×D131D411yescharacteristic
35#14523496—42no—
36C2×C131⋊C8—21no—
37C2×C131⋊C8—21no—
38(C2×D131)⋊C4—22no—
39(C2×D131)⋊C4—22no—
40C4×Dic131C2×C411yescharacteristic
41C4×Dic131D411yescharacteristic
42C4×Dic131D411yescharacteristic
43C2×Q8×C131—21no—
44C2×C8×C131—21no—
45C2×C8×C131—21no—
46C4⋊C4×C131—21no—
47C4⋊C4×C131C2×C412yes—
48C42×C131C2311yescharacteristic
49C22×D131—22no—
50C22×D131C22⋊C411yescharacteristic
51D524—42no—
52D524—44no—
53D524—41no—
54D524—41no—
55D524—41no—
56D524—41no—
57C2×Dic131C22⋊C411yescharacteristic
58C2×Dic131C22⋊C411yescharacteristic
59C2×Dic131C22⋊C411yescharacteristic
60C4×D131—21no—
61C4×D131—21no—
62C4×D131—21no—
63C4×D131—21no—
64C4×D131—41no—
65C4×D131—44no—
66C131⋊C8—41no—
67C131⋊C8—41no—
68Q8×C131—41no—
69Q8×C131—41no—
70C2×C4×C131—22no—
71C2×C4×C131C22×C411yescharacteristic
72C2×C4×C131C2×D411yescharacteristic
73C2×C4×C131C2×D411yescharacteristic
74C8×C131—41no—
75C8×C131—41no—
76C22×C131C2×C22⋊C411yescharacteristic
77C2×D131—21no—
78C2×D131—21no—
79C2×D131—21no—
80C2×D131—42no—
81C2×D131—42no—
82C2×D131—42no—
83C4×C131—21no—
84C4×C131—42no—
85C4×C131AGL(1,ℤ/8)11yescharacteristic
86C4×C131AGL(1,ℤ/8)11yescharacteristic
87C4×C131AGL(1,ℤ/8)11yescharacteristic
88C4×C131#13711yescharacteristic
89Dic131—21no—
90Dic131—21no—
91Dic131—21no—
92C2×C131#41311yescharacteristic
93C2×C131#41711yescharacteristic
94C2×C131#41811yescharacteristic
95D131—41no—
96D131—82no—
97C131#155711yescharacteristic, complemented, Sylow
98#1557—1311nomaximal, Sylow
99#532—1311no—
100#432—1311no—
101#423—1311no—
102#332—1311no—
103#332D13111yescharacteristic, complemented
104#331—1311no—
105#331—1311no—
106C2×#41—2621no—
107C2×M16—2621no—
108C4⋊Q8C2×D13111yescharacteristic
109#127—1311no—
110#124—2621no—
111#121—2621no—
112#121—1312no—
113C4×D4—2621no—
114C4×D4—1312no—
115C2×C42—1311no—
116C4⋊C8—2621no—
117C4⋊C8—1311no—
118C4⋊C8—1311no—
119C4⋊C8C2×D13111yescharacteristic
120C4⋊C8C2×D13111yescharacteristic
121C22⋊C8—2621no—
122#97—2621no—
123#41—5242no—
124#41—5242no—
125C2×Q8—21no—
126C2×D4—2622no—
127C2×D4—2621no—
128C2×D4—2621no—
129C2×D4—1312no—
130C22×C4—2622no—
131C22×C4—1311no—
132C22×C4—1311no—
133C22×C4—1311no—
134M16—5242no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—2621no—
138C2×C8—2621no—
139C4⋊C4—21no—
140C4⋊C4C4×D13112yes—
141C22⋊C4—2622no—
142C22⋊C4—2622no—
143C42—1311no—
144C42—1311no—
145C42—1311no—
146C42C22×D13111yescharacteristic
147C23—2622no—
148C23—1311no—
149Q8—41no—
150Q8—41no—
151D4—5242no—
152D4—5244no—
153D4—5241no—
154D4—5241no—
155D4—5241no—
156D4—5241no—
157C2×C4—22no—
158C2×C4—2621no—
159C2×C4—2621no—
160C2×C4—2621no—
161C2×C4—2621no—
162C2×C4—5241no—
163C2×C4—5244no—
164C2×C4—1311no—
165C2×C4—1311no—
166C2×C4—1311no—
167C2×C4C2×C4×D13111yescharacteristic
168C2×C4D4×D13111yescharacteristic
169C2×C4D4×D13111yescharacteristic
170C8—41no—
171C8—41no—
172C8—5241no—
173C8—5241no—
174C22—2621no—
175C22—2621no—
176C22—2621no—
177C22—5242no—
178C22—5242no—
179C22—5242no—
180C22C22⋊C4×D13111yescharacteristic
181C4—21no—
182C4—42no—
183C4—2621no—
184C4—2621no—
185C4—2621no—
186C4#1686227011yescharacteristic
187C4#1686227311yescharacteristic
188C4#1686227311yescharacteristic
189C4#1686227811yescharacteristic
190C2—5241no—
191C2—10482no—
192C2#2255329511yescharacteristic
193C2#2255331211yescharacteristic
194C2#2255331511yescharacteristic
subgroupG/H
centerC22C22⋊C4×D131
commutatorC2×C4×C131C22×C4
FrattiniC42C22×D131
Fitting#332×C131C2
socleC22×C131C2×C22⋊C4
prime radical#22553863C2
cosocleC42×C131C23
hypercenter#332D131
hypocenterC131#1557
LCSG ▷ C2×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #22553368C2semidirect
3. #22553334C2semidirect
4. #22553058C2semidirect
5. #22553055C2semidirect
6. #22553055C2semidirect
7. #332×C131C2semidirect
97. C131#1557semidirect
103. #332D131semidirect

Maximal subgroups

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

subgroupindex
#225538632
#225533682
#225533342
#225530582
#225530552
#332×C1312
#1557131