Finite Group Databasebeta

← #26119371

Subgroups — #26119371

Order 12032.

416 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 99 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#20251766C211yesmaximal, characteristic
2#20241952C212yesmaximal
3#20241248C212yesmaximal
4#20241221C211yesmaximal, characteristic
5C47×#1994C211yesmaximal, characteristic
6C2×#13094024—21no—
7#15689176C2212yes—
8#15689029—24no—
9#15688989—21no—
10#15688989C2212yes—
11#15688860—24no—
12C47×#533C2211yescharacteristic
13C47×#423C2211yescharacteristic
14C43×C47C2211yescharacteristic
15C47×#339—21no—
16C47×#339—21no—
17C22×Dic94—22no—
18#13094024—42no—
19#13094024—42no—
20C2×C4⋊Dic47—22no—
21C2×C4⋊Dic47—22no—
22C4⋊(C47⋊C8)—22no—
23C22⋊(C47⋊C8)—22no—
24C4×C47⋊C8—22no—
25C47⋊#95—24no—
26C47⋊#95—22no—
27C47⋊#95D414yes—
28C2×M16×C47—21no—
29C4⋊Q8×C47Q812yescomplemented
30#125×C47—21no—
31C22⋊Q8×C47—22no—
32C4×Q8×C47—22no—
33#117×C47—21no—
34C2×C42×C47—22no—
35C2×C42×C47D411yescharacteristic
36C2×C42×C47D411yescharacteristic
37C2×C42×C47C2311yescharacteristic
38C4⋊C8×C47—22no—
39C2×Q8×C47—22no—
40C22×C4×C47—21no—
41C22×C4×C47—21no—
42C22×C4×C47C2×D412yes—
43C22×C4×C47C2×D411yescharacteristic
44C22×Dic47—24no—
45C2×Dic94—44no—
46C2×Dic94—48no—
47M16×C47—41no—
48M16×C47—41no—
49C2×C8×C47—41no—
50C4⋊C4×C47—22no—
51C4⋊C4×C47—22no—
52C4⋊C4×C47—42no—
53C22⋊C4×C47—41no—
54C42×C47—21no—
55C42×C47—21no—
56C42×C47—44no—
57C42×C47—41no—
58C42×C47SD1612yes—
59C42×C47Q1612yes—
60C42×C47C2×Q811yescharacteristic
61C42×C47#4111yescharacteristic
62C42×C47#4112yes—
63C4⋊Dic47—44no—
64C4⋊Dic47—42no—
65C4⋊Dic47—42no—
66C2×C47⋊C8—42no—
67C23×C47#12011yescharacteristic
68Q8×C47—82no—
69C2×C4×C47—24no—
70C2×C4×C47—22no—
71C2×C4×C47—22no—
72C2×C4×C47—42no—
73C2×C4×C47—42no—
74C2×C4×C47—42no—
75C2×C4×C47C22⋊Q812yes—
76C2×C4×C47C22⋊Q811yescharacteristic
77C2×C4×C47#12412yes—
78C2×C4×C47#12411yescharacteristic
79C2×C4×C47C2×SD1611yescharacteristic
80C2×C4×C47C2×Q1611yescharacteristic
81C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
82C2×C4×C47#13711yescharacteristic
83C2×Dic47—48no—
84C2×Dic47—44no—
85Dic94—88no—
86C8×C47—81no—
87C47⋊C8—82no—
88#10363—471nomaximal, Sylow
89C22×C47—22no—
90C22×C47#39311yescharacteristic
91C22×C47#45011yescharacteristic
92C22×C47#45111yescharacteristic
93C4×C47—24no—
94C4×C47—22no—
95C4×C47—22no—
96C4×C47—42no—
97C4×C47—82no—
98C4×C47#47511yescharacteristic
99C4×C47#47711yescharacteristic
100C4×C47#48711yescharacteristic
101C4×C47#48911yescharacteristic
102Dic47—84no—
103#2862—471no—
104#1994D4711yescharacteristic
105#1518—472no—
106#1286—472no—
107#1276—471no—
108C2×C47—21no—
109C2×C47—21no—
110C2×C47#202411yescharacteristic
111C2×C47#203112yes—
112#533C2×D4711yescharacteristic
113C2×C4⋊Q8—941no—
114#431—472no—
115#423C2×D4711yescharacteristic
116#391—944no—
117#382—472no—
118#382—941no—
119C43C2×D4711yescharacteristic
120#339—21no—
121#339—21no—
122#328—944no—
123C47#1036311yescharacteristic, complemented, Sylow
124C22×Q8—942no—
125C2×M16—21no—
126C4⋊Q8—1882no—
127C4⋊Q8—1882no—
128C4⋊Q8Dic9412yescomplemented
129#125—21no—
130C22⋊Q8—22no—
131C4×Q8—22no—
132#117—21no—
133C2×C4⋊C4—942no—
134C2×C4⋊C4—942no—
135C2×C42—22no—
136C2×C42C22⋊D4711yescharacteristic
137C2×C42C22⋊D4711yescharacteristic
138C2×C42C22×D4711yescharacteristic
139C4⋊C8—22no—
140C4⋊C8—942no—
141C22⋊C8—942no—
142C4×C8—942no—
143#95—474no—
144#95—944no—
145#95—942no—
146C2×Q8—22no—
147C2×Q8—1884no—
148C2×Q8—1888no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—944no—
152C22×C4D4×D4712yes—
153C22×C4C2×C22⋊D4711yescharacteristic
154M16—41no—
155M16—41no—
156C2×C8—41no—
157C2×C8—1882no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—42no—
161C4⋊C4—1884no—
162C4⋊C4—1882no—
163C4⋊C4—1882no—
164C22⋊C4—41no—
165C42—21no—
166C42—21no—
167C42—44no—
168C42—41no—
169C42#1062533612yes—
170C42#1062533712yes—
171C42C2×Dic9411yescharacteristic
172C42#1062535011yescharacteristic
173C42#1062535212yes—
174C23#1309399211yescharacteristic
175Q8—82no—
176Q8—3768no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—1888no—
184C2×C4—1884no—
185C2×C4#1309400112yes—
186C2×C4#1309400311yescharacteristic
187C2×C4#1309401012yes—
188C2×C4#1309401111yescharacteristic
189C2×C4C2×#1062533611yescharacteristic
190C2×C4C2×#1062533711yescharacteristic
191C2×C4#1309404911yescharacteristic
192C2×C4#1309405411yescharacteristic
193C8—81no—
194C8—3762no—
195C22—22no—
196C22#1568903511yescharacteristic
197C22#1568927311yescharacteristic
198C22#1568927811yescharacteristic
199C4—24no—
200C4—22no—
201C4—22no—
202C4—42no—
203C4—82no—
204C4—3764no—
205C4#1568940811yescharacteristic
206C4#1568941811yescharacteristic
207C4#1568947311yescharacteristic
208C4#1568948211yescharacteristic
209C2—21no—
210C2—21no—
211C2#2024452911yescharacteristic
212C2#2024456012yes—
subgroupG/H
centerC22#15689035
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1994C2
socleC22×C47#393
prime radicalC23×C47#120
hypercenter#1994D47
hypocenterC47#10363
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
29. C4⋊Q8×C47Q8semidirect
123. C47#10363semidirect
128. C4⋊Q8Dic94semidirect

Maximal subgroups

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

subgroupindex
#202517662
#202419522
#202412482
#202412212
C47×#19942
#1036347