Finite Group Databasebeta

← #30668762

Subgroups — #30668762

Order 16768.

428 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 177 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
1C2×#16862238C211yesmaximal, characteristic
2#22553209C212yesmaximal
3#22553199C212yesmaximal
4#22553197C212yesmaximal
5#22553195C212yesmaximal
6#22553194C212yesmaximal
7C4×C4⋊Dic131C211yesmaximal, characteristic
8#22553163C212yesmaximal
9#383×C131C211yesmaximal, characteristic
10#16862238C418yescomplemented
11C2×C4⋊Dic131C2212yes—
12C2×C4⋊Dic131C2211yescharacteristic
13C2×C4⋊Dic131C2211yescharacteristic
14C2×Dic131⋊C4C2212yes—
15C2×Dic131⋊C4C2212yes—
16C2×Dic131⋊C4C2212yes—
17C2×Dic131⋊C4C2212yes—
18C2×C4×Dic131C2212yes—
19C2×C4×Dic131C2211yescharacteristic
20C2×C4×Dic131C2212yes—
21C2×C4×Dic131C2211yescharacteristic
22C131⋊#95C2212yes—
23C131⋊#95C2212yes—
24C131⋊#95C2212yes—
25C131⋊#95C2212yes—
26C131⋊#95C2212yes—
27C2×C42×C131C2211yescharacteristic
28#95×C131C2212yes—
29#95×C131C2212yes—
30#95×C131C2211yescharacteristic
31#95×C131C2211yescharacteristic
32C22×Dic131C2312yes—
33C22×Dic131C2312yes—
34C22×Dic131C2312yes—
35C22×Dic131C2312yes—
36C4⋊Dic131—24no—
37C4⋊Dic131C2×C414yes—
38C4⋊Dic131C2×C414yes—
39Dic131⋊C4—24no—
40Dic131⋊C4—42no—
41Dic131⋊C4C2×C418yes—
42Dic131⋊C4C2×C418yes—
43C4×Dic131—24no—
44C4×Dic131—41no—
45C4×Dic131—41no—
46C4×Dic131—42no—
47C22×C4×C131C2312yes—
48C22×C4×C131C2312yes—
49C22×C4×C131C2311yescharacteristic
50C22×C4×C131C2311yescharacteristic
51C22×C4×C131C2311yescharacteristic
52C42×C131C2×C414yes—
53C2×Dic131—22no—
54C2×Dic131—22no—
55C2×Dic131—22no—
56C2×Dic131—22no—
57C2×Dic131—22no—
58C2×Dic131—22no—
59C2×Dic131—22no—
60C2×Dic131—22no—
61C2×Dic131—22no—
62C2×Dic131C22×C414yes—
63C2×Dic131C22×C414yes—
64C2×Dic131#4114yes—
65C23×C131C2411yescharacteristic
66C2×C4×C131—22no—
67C2×C4×C131—21no—
68C2×C4×C131—22no—
69C2×C4×C131—21no—
70C2×C4×C131—22no—
71C2×C4×C131—22no—
72C2×C4×C131—22no—
73C2×C4×C131—22no—
74C2×C4×C131—22no—
75C2×C4×C131—22no—
76C2×C4×C131C22×C412yes—
77C2×C4×C131C22×C412yes—
78C2×C4×C131C22×C412yes—
79C22×C131C23×C411yescharacteristic
80C22×C131C2×#4112yes—
81C22×C131H(5,𝔽2)11yescharacteristic
82C22×C1312−1+412yes—
83C22×C1312−1+411yescharacteristic
84C4×C131—22no—
85C4×C131—22no—
86C4×C131—22no—
87C4×C131—42no—
88C4×C131—42no—
89Dic131—24no—
90Dic131—24no—
91Dic131—24no—
92Dic131—42no—
93C2×C131C4×#4111yescharacteristic
94C2×C131#51911yescharacteristic
95C2×C131#52011yescharacteristic
96C2×C131#53912yes—
97C2×C131#54112yes—
98C131#233611yescharacteristic, complemented, Sylow
99#2336—1311nomaximal, Sylow
100C2×#125—1311no—
101#389—1312no—
102#387—1312no—
103#387—1312no—
104#387—1312no—
105#387—1312no—
106#383D13111yescharacteristic
107C4×C4⋊C4—1311no—
108C4×C4⋊C4—1312no—
109#125—1318no—
110C2×C4⋊C4—1312no—
111C2×C4⋊C4—1312no—
112C2×C4⋊C4—1312no—
113C2×C4⋊C4—1312no—
114C2×C4⋊C4—1311no—
115C2×C4⋊C4—1311no—
116C2×C4⋊C4—1312no—
117C2×C42—1312no—
118C2×C42—1311no—
119C2×C42—1311no—
120C2×C42—1312no—
121C2×C42C2×D13111yescharacteristic
122#95—1312no—
123#95—1312no—
124#95—1312no—
125#95—1312no—
126#95—1312no—
127#95C2×D13112yes—
128#95C2×D13111yescharacteristic
129#95C2×D13112yes—
130#95C2×D13111yescharacteristic
131C22×C4—1312no—
132C22×C4—1312no—
133C22×C4—1312no—
134C22×C4—1312no—
135C22×C4C22×D13112yes—
136C22×C4C22×D13112yes—
137C22×C4C22×D13111yescharacteristic
138C22×C4C22×D13111yescharacteristic
139C22×C4C22×D13111yescharacteristic
140C4⋊C4—2624no—
141C4⋊C4—2624no—
142C4⋊C4—5242no—
143C4⋊C4—1318no—
144C4⋊C4—1318no—
145C4⋊C4—1314no—
146C4⋊C4—1314no—
147C42—2624no—
148C42—5241no—
149C42—5241no—
150C42—5242no—
151C42C4×D13114yes—
152C23C23×D13111yescharacteristic
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—2622no—
164C2×C4—2622no—
165C2×C4—2622no—
166C2×C4—2622no—
167C2×C4—2622no—
168C2×C4—2622no—
169C2×C4—2622no—
170C2×C4—2622no—
171C2×C4—2622no—
172C2×C4—1314no—
173C2×C4—1314no—
174C2×C4—1314no—
175C2×C4C2×C4×D13112yes—
176C2×C4C2×C4×D13112yes—
177C2×C4C2×C4×D13112yes—
178C22C22×C4×D13111yescharacteristic
179C22#41×D13112yes—
180C22#1686229211yescharacteristic
181C22#1686229312yes—
182C22#1686229311yescharacteristic
183C4—22no—
184C4—22no—
185C4—22no—
186C4—42no—
187C4—42no—
188C4—2624no—
189C4—2624no—
190C4—2624no—
191C4—5242no—
192C2#2255378011yescharacteristic
193C2#2255378911yescharacteristic
194C2#2255379211yescharacteristic
195C2#2255390312yes—
196C2#2255391912yes—
subgroupG/H
center, FrattiniC23C23×D131
commutatorC22×C131C23×C4
Fitting#383×C131C2
socle, prime radical, cosocleC23×C131C24
hypercenter#383D131
hypocenterC131#2336
LCSG ▷ C22×C131 ▷ C131
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
10. #16862238C4semidirect
98. C131#2336semidirect

Maximal subgroups

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

subgroupindex
C2×#168622382
#225532092
#225531992
#225531972
#225531952
#225531942
C4×C4⋊Dic1312
#225531632
#383×C1312
#2336131