Finite Group Databasebeta

← #30670693

Subgroups — #30670693

Order 16768.

420 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 129 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#22553240C212yesmaximal
2#22553237C211yesmaximal, characteristic
3#22553224C211yesmaximal, characteristic
4#22553216C212yesmaximal, complemented
5#22553198C212yesmaximal, complemented
6#22553193C212yesmaximal, complemented
7#22553179C212yesmaximal, complemented
8C4×C22⋊Dic131C212yesmaximal
9#385×C131C211yesmaximal, characteristic
10C22×Dic262C2211yescharacteristic
11C2×C4⋊Dic131C2211yescharacteristic
12C2×Dic131⋊C4C2212yes—
13C2×C22⋊Dic131C2212yes—
14C2×C22⋊Dic131C2212yes—
15C2×C4×Dic131C2212yes—
16C2×C4×Dic131C2212yes—
17C131⋊#95C2212yes—
18C131⋊#95C2212yes—
19C131⋊#95C2211yescharacteristic
20C131⋊#95C2212yes—
21C131⋊#95C2212yes—
22C131⋊#95C2212yes—
23C131⋊#95C2211yescharacteristic
24C131⋊#95C2212yes—
25C131⋊#95C2212yes—
26C23×C4×C131C2211yescharacteristic
27C2×C4⋊C4×C131C2212yes—
28C2×C22⋊C4×C131C2211yescharacteristic
29C2×C22⋊C4×C131C2211yescharacteristic
30#95×C131C2212yes—
31C2×Dic262—24no—
32C2×Dic262—41no—
33C22×Dic131C2312yes—
34C22×Dic131C2312yes—
35C22×Dic131C2312yes—
36C22×Dic131C2312yes—
37C4⋊Dic131—41no—
38Dic131⋊C4—24no—
39C22⋊Dic131—24no—
40C22⋊Dic131—24no—
41C4×Dic131—24no—
42C4×Dic131—42no—
43C24×C131C2311yescharacteristic
44C22×C4×C131—24no—
45C22×C4×C131—41no—
46C22×C4×C131C2311yescharacteristic
47C22×C4×C131C2312yes—
48C22×C4×C131C2311yescharacteristic
49C22×C4×C131C2312yes—
50C4⋊C4×C131—24no—
51C22⋊C4×C131—22no—
52C22⋊C4×C131—41no—
53Dic262—44no—
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×Dic131—22no—
63C2×Dic131—22no—
64C2×Dic131—22no—
65C2×Dic131#4114yes—
66C23×C131—22no—
67C23×C131—22no—
68C23×C131—21no—
69C23×C131—21no—
70C23×C131#4112yes—
71C23×C131C2411yescharacteristic
72C2×C4×C131—22no—
73C2×C4×C131—22no—
74C2×C4×C131—22no—
75C2×C4×C131—22no—
76C2×C4×C131—22no—
77C2×C4×C131—22no—
78C2×C4×C131—22no—
79C2×C4×C131—21no—
80C2×C4×C131—21no—
81C2×C4×C131—44no—
82C2×C4×C131#4114yes—
83C22×C131—24no—
84C22×C131—22no—
85C22×C131—41no—
86C22×C131—41no—
87C22×C131—42no—
88C22×C131C2×#4112yes—
89C22×C131C2×#4112yes—
90C22×C131C2×#4111yescharacteristic
91C22×C131H(5,𝔽2)11yescharacteristic
92C22×C1312−1+411yescharacteristic
93C4×C131—24no—
94C4×C131—42no—
95C4×C131—41no—
96C4×C131—41no—
97Dic131—24no—
98Dic131—42no—
99Dic131—42no—
100Dic131—42no—
101C2×C131—42no—
102C2×C131#52912yes—
103C2×C131#53911yescharacteristic
104C2×C131#55111yescharacteristic
105C2×C131#55111yescharacteristic
106C2×C131#55612yes—
107C131#252311yescharacteristic, complemented, Sylow
108#2523—1311nomaximal, Sylow
109#397—1311no—
110#397—1312no—
111#393—1311no—
112#391—1312no—
113#387—1312no—
114#387—1312no—
115#385D13111yescharacteristic
116#383—1312no—
117C4×C22⋊C4—1312no—
118C22×Q8—1311no—
119C23×C4C2×D13111yescharacteristic
120C2×C4⋊C4—1312no—
121C2×C4⋊C4—1311no—
122C2×C4⋊C4C2×D13112yes—
123C2×C22⋊C4—1312no—
124C2×C22⋊C4—1312no—
125C2×C22⋊C4C2×D13111yescharacteristic
126C2×C22⋊C4C2×D13111yescharacteristic
127C2×C42—1312no—
128C2×C42—1312no—
129#95—1312no—
130#95—1312no—
131#95—1311no—
132#95—1312no—
133#95—1312no—
134#95—1312no—
135#95—1312no—
136#95—1311no—
137#95—1312no—
138#95C2×D13112yes—
139C24C22×D13111yescharacteristic
140C2×Q8—2624no—
141C2×Q8—5241no—
142C22×C4—24no—
143C22×C4—41no—
144C22×C4—1312no—
145C22×C4—1312no—
146C22×C4—1312no—
147C22×C4—1312no—
148C22×C4C22×D13111yescharacteristic
149C22×C4C22×D13112yes—
150C22×C4C22×D13111yescharacteristic
151C22×C4C22×D13112yes—
152C4⋊C4—24no—
153C4⋊C4—2624no—
154C4⋊C4—5241no—
155C22⋊C4—22no—
156C22⋊C4—41no—
157C22⋊C4—2624no—
158C22⋊C4—2624no—
159C42—2624no—
160C42—5242no—
161C23—22no—
162C23—22no—
163C23—21no—
164C23—21no—
165C23#1452351312yes—
166C23C23×D13111yescharacteristic
167Q8—5244no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—44no—
178C2×C4—2622no—
179C2×C4—2622no—
180C2×C4—2622no—
181C2×C4—2622no—
182C2×C4—2622no—
183C2×C4—2622no—
184C2×C4—2622no—
185C2×C4—2622no—
186C2×C4—2622no—
187C2×C4—2622no—
188C2×C4—2622no—
189C2×C4—1314no—
190C2×C4#1452351214yes—
191C22—22no—
192C22—24no—
193C22—41no—
194C22—41no—
195C22—42no—
196C22#41×D13112yes—
197C22C2×#1452351212yes—
198C22C2×#1452351311yescharacteristic
199C22#1686229111yescharacteristic
200C22#1686229411yescharacteristic
201C4—24no—
202C4—41no—
203C4—41no—
204C4—42no—
205C4—2624no—
206C4—5242no—
207C4—5242no—
208C4—5242no—
209C2—42no—
210C2#2255384212yes—
211C2#2255390911yescharacteristic
212C2#2255399011yescharacteristic
213C2#2255399311yescharacteristic
214C2#2255403312yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#385×C131C2
prime radicalC24×C131C23
hypercenter#385D131
hypocenterC131#2523
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
4. #22553216C2semidirect
5. #22553198C2semidirect
6. #22553193C2semidirect
7. #22553179C2semidirect
107. C131#2523semidirect

Maximal subgroups

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

subgroupindex
#225532402
#225532372
#225532242
#225532162
#225531982
#225531932
#225531792
C4×C22⋊Dic1312
#385×C1312
#2523131