Finite Group Databasebeta

← #26422701

Subgroups — #26422701

Order 12032.

1778 conjugacy classes of subgroups; only the 539 normal subgroups are shown.

normal subgroups only

This group has 1778 conjugacy classes of subgroups — too many to draw. Only its 539 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#20251652C213yesmaximal
2#20251648C2112yesmaximal, complemented
3#20246655C2112yesmaximal, complemented
4#20245766C213yesmaximal
5C47×#2840C211yesmaximal, characteristic
6C24×Dic47C2211yescharacteristic
7C2×Q8×Dic47C2214yescomplemented
8C2×#13093980C2216yes—
9#15689063C22112yescomplemented
10#15689059C22112yes—
11#15689057C22124yescomplemented
12#15689046C22124yes—
13#15689042C22112yescomplemented
14#15688984C2216yes—
15#15688981C22112yes—
16#15688979C2213yes—
17#15688974C22112yescomplemented
18C22⋊C4×Dic47C22112yes—
19C47×#393C22112yes—
20C47×#379C2213yes—
21C23×Dic47—44no—
22C23×Dic47Q818yescomplemented
23C23×Dic47C2313yes—
24C23×Dic47C2313yes—
25Q8×Dic47—48no—
26#13093980—412no—
27C2×C4⋊Dic47C23112yes—
28C2×Dic47⋊C4C23124yes—
29C2×Dic47⋊C4C23112yes—
30C2×C22⋊Dic47C23112yes—
31C2×C22⋊Dic47C2316yes—
32C2×C4×Dic47C23112yes—
33C47⋊#95C23124yes—
34C47⋊#95C23112yes—
35C25×C47C2311yescharacteristic
36C22×Q8×C47C2314yes—
37C2×C22⋊C4×C47C23112yes—
38C2×C22⋊C4×C47C2316yes—
39#95×C47C23112yes—
40C24×C47—44no—
41C24×C47—42no—
42C24×C47C2×Q814yes—
43C24×C47C2413yes—
44C2×Q8×C47—48no—
45C2×Q8×C47—44no—
46C22×C4×C47C24112yes—
47C22×Dic47—212no—
48C22×Dic47—416no—
49C22×Dic47—46no—
50C22×Dic47C2×Q8112yes—
51C22×Dic47C2×Q8112yes—
52C22×Dic47C24112yes—
53C22×Dic47C2413yes—
54C22×Dic47C2411yescharacteristic
55C22⋊C4×C47—412no—
56C22⋊C4×C47—46no—
57C22⋊Dic47—412no—
58C22⋊Dic47—46no—
59C4⋊Dic47—412no—
60Dic47⋊C4—424no—
61Dic47⋊C4—412no—
62C4×Dic47—224no—
63C23×C47—26no—
64C23×C47—212no—
65C23×C47—44no—
66C23×C47—48no—
67C23×C47—416no—
68C23×C47C22×Q816yes—
69C23×C47C2511yescharacteristic
70Q8×C47—416no—
71C2×C4×C47—212no—
72C2×C4×C47—224no—
73C2×Dic47—26no—
74C2×Dic47—224no—
75C2×Dic47—212no—
76C2×Dic47—224no—
77C2×Dic47—412no—
78C2×Dic47C22×Q816yes—
79C2×Dic47C22×Q812yes—
80C2×Dic47H(5,𝔽2)14yes—
81#36674—471nomaximal, Sylow
82C22×C47—26no—
83C22×C47—212no—
84C22×C47—416no—
85C22×C47—412no—
86C22×C47C23×Q811yescharacteristic
87C22×C47C2×H(5,𝔽2)12yes—
88C22×C47#58514yes—
89C4×C47—412no—
90Dic47—26no—
91Dic47—412no—
92Dic47#54314yes—
93#2842—473no—
94#2841—4712no—
95#2840D4711yescharacteristic
96#2396—4712no—
97#2280—473no—
98C2×C47—46no—
99C2×C47C2×#54311yescharacteristic
100C2×C47#345812yes—
101C2×C47#352914yes—
102C24×C4—471no—
103C2×C4×Q8—474no—
104C2×#117—476no—
105#400—4712no—
106#399—4712no—
107#398—4724no—
108#396—4724no—
109#395—4712no—
110#393C2×D47112yes—
111#381—476no—
112#380—4712no—
113#379—473no—
114#379C2×D4713yes—
115C4×C4⋊C4—4712no—
116C4×C22⋊C4—4712no—
117C47#3667411yescharacteristic, complemented, Sylow
118C25C22×D4711yescharacteristic
119C22×Q8C22×D4714yes—
120C23×C4—473no—
121C23×C4—478no—
122C23×C4—473no—
123C23×C4—1884no—
124C4×Q8—1888no—
125#117—18812no—
126C2×C4⋊C4—4712no—
127C2×C4⋊C4—4712no—
128C2×C4⋊C4—4724no—
129C2×C22⋊C4—476no—
130C2×C22⋊C4—4712no—
131C2×C22⋊C4C22×D4716yes—
132C2×C22⋊C4C22×D47112yes—
133C2×C42—4712no—
134#95—4724no—
135#95—4712no—
136#95C22×D47112yes—
137C24—42no—
138C24—44no—
139C24Q8×D4714yes—
140C24C23×D4713yes—
141C2×Q8—44no—
142C2×Q8—48no—
143C22×C4—471no—
144C22×C4—4712no—
145C22×C4—473no—
146C22×C4—4712no—
147C22×C4—4712no—
148C22×C4—9412no—
149C22×C4—18816no—
150C22×C4—1886no—
151C22×C4C23×D47112yes—
152C4⋊C4—18824no—
153C4⋊C4—18812no—
154C4⋊C4—18812no—
155C22⋊C4—412no—
156C22⋊C4—46no—
157C22⋊C4—18812no—
158C22⋊C4—1886no—
159C42—9424no—
160C23—26no—
161C23—212no—
162C23—44no—
163C23—48no—
164C23—416no—
165C23C2×Q8×D4716yes—
166C23C24×D4711yescharacteristic
167Q8—416no—
168C2×C4—212no—
169C2×C4—224no—
170C2×C4—472no—
171C2×C4—474no—
172C2×C4—476no—
173C2×C4—9424no—
174C2×C4—946no—
175C2×C4—9412no—
176C2×C4—9424no—
177C2×C4—18812no—
178C22—26no—
179C22—212no—
180C22—416no—
181C22—412no—
182C22C22×Q8×D4711yescharacteristic
183C22H(5,𝔽2)×D4712yes—
184C22#1568995814yes—
185C4—412no—
186C4—474no—
187C4—946no—
188C4—18812no—
189C2—46no—
190C2#543×D4711yescharacteristic
191C2#2025679012yes—
192C2#2025752814yes—
subgroupG/H
center, FrattiniC23C24×D47
commutator, socle, cosocleC23×C47C25
FittingC47×#2840C2
prime radicalC25×C47C23
hypercenter#2840D47
hypocenterC47#36674
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2840

Complements

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

NH ≅ G/Ntype
2. #20251648C2semidirect
3. #20246655C2semidirect
7. C2×Q8×Dic47C22semidirect
9. #15689063C22semidirect
11. #15689057C22semidirect
13. #15689042C22semidirect
17. #15688974C22semidirect
22. C23×Dic47Q8semidirect
117. C47#36674semidirect

Maximal subgroups

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

subgroupindex
#202516522
#202516482
#202466552
#202457662
C47×#28402
#3667447