Finite Group Databasebeta

← #20251415

Subgroups — #20251415

Order 6016.

528 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13093998C211yesmaximal, characteristic
2#15689058C212yesmaximal, complemented
3#15689053C212yesmaximal, complemented
4#15689040C212yesmaximal, complemented
5#15689038C214yesmaximal, complemented
6#15689034C212yesmaximal, complemented
7#15688986C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C47C211yesmaximal, characteristic, complemented
9C22×D188C2211yescharacteristic
10C22×C22⋊D47C2212yes—
11#13093998—24no—
12C2×C4⋊Dic47C2211yescharacteristic
13C2×Dic47⋊C4C2214yes—
14C2×C22⋊Dic47C2212yes—
15C2×(C2×D47)⋊C4C2212yes—
16C2×(C2×D47)⋊C4C2214yes—
17C2×(C2×D47)⋊C4C2214yes—
18C47⋊#95C2212yes—
19C47⋊#95C2212yes—
20C47⋊#95C2214yes—
21C22×Q8×C47C2211yescharacteristic
22C23×C4×C47C2211yescharacteristic
23C22⋊Q8×C47—24no—
24C2×C4⋊C4×C47C2212yes—
25C2×C4⋊C4×C47C2211yescharacteristic
26C2×C22⋊C4×C47C2212yes—
27C24×C47C2311yescharacteristic
28C23×D47C2312yes—
29C2×Q8×C47—22no—
30C2×Q8×C47—42no—
31C22×C4×C47—42no—
32C22×C4×C47D414yes—
33C22×C4×C47C2312yes—
34C22×C4×C47C2312yes—
35C22×C4×C47C2311yescharacteristic
36C22×C4×C47C2311yescharacteristic
37C2×C22⋊D47—24no—
38C2×C22⋊D47—44no—
39C22×Dic47C2314yes—
40C22×Dic47C2312yes—
41C2×D188—22no—
42C2×D188—42no—
43C4⋊C4×C47—22no—
44C4⋊C4×C47—24no—
45C22⋊C4×C47—24no—
46C22⋊Dic47—42no—
47(C2×D47)⋊C4—24no—
48(C2×D47)⋊C4—44no—
49(C2×D47)⋊C4—44no—
50C4⋊Dic47—22no—
51Dic47⋊C4—44no—
52C23×C47—22no—
53C23×C47—22no—
54C23×C47—22no—
55C23×C47C2×D412yes—
56C23×C47C2411yescharacteristic
57C22×D47—22no—
58C22×D47—24no—
59C22×D47—24no—
60C22×D47—24no—
61Q8×C47—44no—
62C2×C4×C47—22no—
63C2×C4×C47—22no—
64C2×C4×C47—22no—
65C2×C4×C47—22no—
66C2×C4×C47—22no—
67C2×C4×C47—22no—
68C2×C4×C47—24no—
69C2×C4×C47—44no—
70C2×C4×C47C2×D412yes—
71C2×C4×C47C2×D412yes—
72C22⋊D47—48no—
73C2×Dic47—22no—
74C2×Dic47—24no—
75C2×Dic47—24no—
76C2×Dic47—24no—
77C2×Dic47—24no—
78D188—44no—
79C22×C47—22no—
80C22×C47—22no—
81C22×C47—22no—
82C22×C47—44no—
83C22×C47C22×D411yescharacteristic
84C22×C47H(5,𝔽2)12yes—
85C22×C47H(5,𝔽2)12yes—
86C22×C472−1+412yes—
87C2×D47—42no—
88C2×D47—42no—
89C2×D47—42no—
90C2×D47—48no—
91C4×C47—22no—
92C4×C47—22no—
93C4×C47—42no—
94C4×C47—42no—
95Dic47—42no—
96Dic47—44no—
97#2816—471nomaximal, Sylow
98C2×C47—42no—
99C2×C47#53411yescharacteristic
100C2×C47#53511yescharacteristic
101C2×C47#53611yescharacteristic
102C2×C47#56214yes—
103D47—82no—
104C2×C22⋊Q8D4711yescharacteristic, complemented
105C2×#121—471no—
106#399—472no—
107#398—472no—
108#394—472no—
109#394—474no—
110#393—472no—
111#381—471no—
112C47#281611yescharacteristic, complemented, Sylow
113C22×Q8C2×D4711yescharacteristic
114C22×D4—472no—
115C22×D4—471no—
116C23×C4C2×D4711yescharacteristic
117C22⋊Q8—24no—
118#121—944no—
119C2×C4⋊C4—471no—
120C2×C4⋊C4—474no—
121C2×C4⋊C4C2×D4712yes—
122C2×C4⋊C4C2×D4711yescharacteristic
123C2×C22⋊C4—472no—
124C2×C22⋊C4—472no—
125C2×C22⋊C4—474no—
126C2×C22⋊C4—474no—
127C2×C22⋊C4C2×D4712yes—
128#95—472no—
129#95—472no—
130#95—474no—
131C24—472no—
132C24C22×D4711yescharacteristic
133C2×Q8—22no—
134C2×Q8—42no—
135C2×D4—944no—
136C2×D4—942no—
137C2×D4—1882no—
138C2×D4—1884no—
139C22×C4—42no—
140C22×C4—474no—
141C22×C4—472no—
142C22×C4C22⋊D4714yes—
143C22×C4C22×D4712yes—
144C22×C4C22×D4712yes—
145C22×C4C22×D4711yescharacteristic
146C22×C4C22×D4711yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—24no—
149C4⋊C4—942no—
150C4⋊C4—1884no—
151C22⋊C4—24no—
152C22⋊C4—944no—
153C22⋊C4—1884no—
154C22⋊C4—1884no—
155C22⋊C4—1882no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—942no—
160C23—944no—
161C23—944no—
162C23—944no—
163C23C2×C22⋊D4712yes—
164C23C23×D4711yescharacteristic
165Q8—44no—
166D4—1884no—
167D4—1888no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—44no—
176C2×C4—942no—
177C2×C4—944no—
178C2×C4—944no—
179C2×C4—944no—
180C2×C4—944no—
181C2×C4C2×C22⋊D4712yes—
182C2×C4C2×C22⋊D4712yes—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—44no—
187C22—1888no—
188C22—1882no—
189C22—1882no—
190C22—1882no—
191C22C22×C22⋊D4711yescharacteristic
192C22#1309406912yes—
193C22#1309407012yes—
194C22#1309407212yes—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—1884no—
200C4—1882no—
201C2—42no—
202C2—3762no—
203C2#1568968211yescharacteristic
204C2#1568968711yescharacteristic
205C2#1568969711yescharacteristic
206C2#1568986914yes—
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC2×C22⋊Q8×C47C2
hypercenterC2×C22⋊Q8D47
hypocenterC47#2816
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #15689058C2semidirect
3. #15689053C2semidirect
4. #15689040C2semidirect
5. #15689038C2semidirect
6. #15689034C2semidirect
7. #15688986C2semidirect
8. C2×C22⋊Q8×C47C2semidirect
104. C2×C22⋊Q8D47semidirect
112. C47#2816semidirect