Finite Group Databasebeta

← #26127797

Subgroups — #26127797

Order 12032.

268 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 69 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#20254044C211yesmaximal, characteristic
2#20245203C211yesmaximal, characteristic
3#20245152C211yesmaximal, characteristic
4#20241450C211yesmaximal, characteristic, complemented
5#20241448C211yesmaximal, characteristic, complemented
6#20241427C211yesmaximal, characteristic, complemented
7C47×#1358C211yesmaximal, characteristic, complemented
8#15689734C412yes—
9M32×D47—21no—
10#15689525C2211yescharacteristic
11#15689491C412yes—
12#15689378—21no—
13#15689361—21no—
14#15689230—21no—
15Q16×Dic47—21no—
16#15689077C2211yescharacteristic
17#15688944C2211yescharacteristic
18#15688944C2211yescharacteristic
19#15688943—21no—
20#15688904—21no—
21#15688900—21no—
22C47×#494C2211yescharacteristic
23C47×#364C2211yescharacteristic
24C47×#364C2211yescharacteristic
25C2×#10625324D411yescharacteristic
26C2×C8×D47D411yescharacteristic
27#13094019—42no—
28#13094016C2×C412yes—
29#13094014—42no—
30#13094000—22no—
31Q8×Dic47—22no—
32#13093981C2×C411yescharacteristic
33#13093964—21no—
34#13093964—21no—
35#13093963—41no—
36C16×D47—41no—
37#13093958—21no—
38#13093956—21no—
39#13093948—22no—
40#13093948—22no—
41#13093935C2×C411yescharacteristic
42#13093934D411yescharacteristic
43C8×Dic47D411yescharacteristic
44C2×Q16×C47—21no—
45C2×Q16×C47—21no—
46C4⋊Q8×C47C2×C412yes—
47M32×C47—21no—
48M32×C47—21no—
49C4×C8×C47C2311yescharacteristic
50C2×Q8×C47—22no—
51C2×C4×D47C22⋊C411yescharacteristic
52Q16×C47—41no—
53Q16×C47—41no—
54Q16×C47—41no—
55Q16×C47—41no—
56C2×C8×C47C2×D411yescharacteristic
57C2×C8×C47C2×D411yescharacteristic
58C4⋊C4×C47—42no—
59C42×C47C22×C411yescharacteristic
60(C2×D47)⋊C4—21no—
61(C2×D47)⋊C4—42no—
62C4⋊Dic47—22no—
63C4⋊Dic47—42no—
64Dic47⋊C4—21no—
65Dic47⋊C4—42no—
66Dic47⋊C4—42no—
67C4×Dic47—42no—
68C4×Dic47C22⋊C411yescharacteristic
69C2×C47⋊C8C22⋊C411yescharacteristic
70C2×C47⋊C8C22⋊C411yescharacteristic
71#10625324—21no—
72#10625324—21no—
73C8×D47—21no—
74C8×D47—21no—
75C16×C47—41no—
76C16×C47—41no—
77C47⋊C16—41no—
78C47⋊C16—41no—
79C22×D47#10011yescharacteristic
80Q8×C47—42no—
81Q8×C47—42no—
82C2×C4×C47—21no—
83C2×C4×C47—42no—
84C2×C4×C47C2×C22⋊C411yescharacteristic
85C2×Dic47—21no—
86C2×Dic47—42no—
87C2×Dic47#10011yescharacteristic
88C4×D47—21no—
89C4×D47—21no—
90C8×C47#13511yescharacteristic
91C8×C47#13511yescharacteristic
92C8×C47AGL(1,ℤ/8)11yescharacteristic
93C8×C47#13711yescharacteristic
94C47⋊C8—21no—
95C47⋊C8—21no—
96#11990—471nomaximal, Sylow
97C22×C47C2×#10011yescharacteristic
98C2×D47—21no—
99C2×D47—41no—
100C4×C47—41no—
101C4×C47—82no—
102C4×C47#41611yescharacteristic
103C4×C47#41711yescharacteristic
104Dic47—21no—
105Dic47—41no—
106Dic47—82no—
107#3177—471no—
108#2166—471no—
109#2153—471no—
110#1366—471no—
111#1366—471no—
112#1358—471no—
113#1358D4711yescharacteristic, complemented
114C2×C47—21no—
115C2×C47#155111yescharacteristic
116D47—41no—
117#541—472no—
118C2×M32—941no—
119#499—471no—
120#494C2×D4711yescharacteristic
121#491—472no—
122#470—941no—
123#467—941no—
124#441—941no—
125C4×Q16—941no—
126#405—471no—
127#364—471no—
128#364—471no—
129#364—941no—
130#364C2×D4711yescharacteristic
131#364C2×D4711yescharacteristic
132#349—941no—
133#347—941no—
134C47#1199011yescharacteristic, complemented, Sylow
135C2×Q16—21no—
136C2×Q16—21no—
137C2×M16—471no—
138C22×C8—471no—
139C4⋊Q8C4×D4712yes—
140#126—1882no—
141#125—472no—
142#125—1882no—
143C22⋊Q8—942no—
144C4×Q8—942no—
145#117—471no—
146M32—21no—
147M32—21no—
148M32—941no—
149M32—941no—
150M32—1881no—
151C2×C16—1881no—
152#107—941no—
153#106—941no—
154#103—942no—
155#103—942no—
156#97—471no—
157#97—471no—
158C4×C8—471no—
159C4×C8C22×D4711yescharacteristic
160C2×Q8—22no—
161C22×C4—471no—
162Q16—41no—
163Q16—41no—
164Q16—41no—
165Q16—41no—
166M16—941no—
167M16—941no—
168C2×C8—471no—
169C2×C8—471no—
170C2×C8—941no—
171C2×C8—941no—
172C2×C8D4×D4711yescharacteristic
173C2×C8D4×D4711yescharacteristic
174C4⋊C4—42no—
175C4⋊C4—941no—
176C4⋊C4—942no—
177C4⋊C4—1882no—
178C4⋊C4—1882no—
179C4⋊C4—1882no—
180C22⋊C4—941no—
181C22⋊C4—1882no—
182C42—471no—
183C42—1882no—
184C42C2×C4×D4711yescharacteristic
185C16—41no—
186C16—41no—
187C16—1881no—
188C16—1881no—
189C23—471no—
190Q8—42no—
191Q8—42no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—471no—
195C2×C4—941no—
196C2×C4—941no—
197C2×C4—941no—
198C2×C4—1882no—
199C2×C4C22⋊C4×D4711yescharacteristic
200C8—941no—
201C8—941no—
202C8#1309404311yescharacteristic
203C8#1309404511yescharacteristic
204C8#1309404811yescharacteristic
205C8#1309405311yescharacteristic
206C22—941no—
207C22—1881no—
208C22#100×D4711yescharacteristic
209C4—41no—
210C4—82no—
211C4—941no—
212C4—3762no—
213C4—1881no—
214C4#1568911611yescharacteristic
215C4#1568912111yescharacteristic
216C2—21no—
217C2—1881no—
218C2#2024208011yescharacteristic
subgroupG/H
centerC2#20242080
commutatorC42×C47C22×C4
FrattiniC4×C8C22×D47
FittingC47×#1358C2
socleC2×C47#1551
prime radicalC22×D47#100
cosocleC4×C8×C47C23
hypercenter#1358D47
hypocenterC47#11990
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1358

Complements

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

NH ≅ G/Ntype
4. #20241450C2semidirect
5. #20241448C2semidirect
6. #20241427C2semidirect
7. C47×#1358C2semidirect
113. #1358D47semidirect
134. C47#11990semidirect

Maximal subgroups

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

subgroupindex
#202540442
#202452032
#202451522
#202414502
#202414482
#202414272
C47×#13582
#1199047