Finite Group Databasebeta

← #26103707

Subgroups — #26103707

Order 12032.

202 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 47 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#20241322C211yesmaximal, characteristic
2#20241214C211yesmaximal, characteristic
3C47×#1857C211yesmaximal, characteristic
4#15688878—21no—
5#15688849—21no—
6C47×#519C411yescharacteristic
7C47×#428—21no—
8C2×#101×C47C411yescharacteristic
9C47×#391C2211yescharacteristic
10C47×#339—21no—
11C22×C47⋊C8—41no—
12C22⋊(C47⋊C8)—41no—
13C22×Q8×C47C2×C411yescharacteristic
14C2×M16×C47—21no—
15C4×Q8×C47—41no—
16C4×Q8×C47—41no—
17#117×C47—21no—
18#117×C47—41no—
19C2×C4⋊C4×C47D411yescharacteristic
20C2×C42×C47D411yescharacteristic
21#107×C47—41no—
22#106×C47—41no—
23#101×C47—21no—
24#101×C47C812yescomplemented
25#95×C47—41no—
26C2×Q8×C47—21no—
27C2×Q8×C47—41no—
28C2×Q8×C47—41no—
29C2×Q8×C47C2×C811yescharacteristic
30C2×Q8×C47M1611yescharacteristic
31C22×C4×C47—21no—
32C22×C4×C47—21no—
33C22×C4×C47—21no—
34C22×C4×C47C22⋊C411yescharacteristic
35M16×C47—22no—
36M16×C47—41no—
37C2×C8×C47—41no—
38C4⋊C4×C47—21no—
39C4⋊C4×C47—21no—
40C4⋊C4×C47—41no—
41C4⋊C4×C47—41no—
42C4⋊C4×C47—41no—
43C22⋊C4×C47—41no—
44C42×C47—41no—
45C42×C47—41no—
46C42×C47—42no—
47C2×C47⋊C8—42no—
48C2×C47⋊C8—81no—
49C2×C47⋊C8—81no—
50C2×C47⋊C8—81no—
51C23×C47#9911yescharacteristic
52Q8×C47—41no—
53Q8×C47—41no—
54Q8×C47—42no—
55C2×C4×C47—21no—
56C2×C4×C47—21no—
57C2×C4×C47—21no—
58C2×C4×C47—21no—
59C2×C4×C47—21no—
60C2×C4×C47—21no—
61C2×C4×C47—41no—
62C2×C4×C47—41no—
63C2×C4×C47—41no—
64C2×C4×C47—41no—
65C2×C4×C47—41no—
66C2×C4×C47—41no—
67C2×C4×C47—41no—
68C2×C4×C47—81no—
69C2×C4×C47C22⋊C811yescharacteristic
70C2×C4×C47#10111yescharacteristic
71C8×C47—42no—
72C47⋊C8—82no—
73C47⋊C8—161no—
74#7376—471nomaximal, Sylow
75C22×C47—21no—
76C22×C47—21no—
77C22×C47#32411yescharacteristic
78C22×C47AΣL(1,𝔽16)11yescharacteristic
79C22×C47#35211yescharacteristic
80C4×C47—21no—
81C4×C47—21no—
82C4×C47—41no—
83C4×C47—41no—
84C4×C47—41no—
85C4×C47—41no—
86C4×C47—81no—
87C4×C47—81no—
88C4×C47—81no—
89#1857D4711yescharacteristic
90#1319—471no—
91#1272—471no—
92C2×C47—21no—
93C2×C47—21no—
94C2×C47#132611yescharacteristic
95C2×C47#140411yescharacteristic
96C2×C47#140511yescharacteristic
97#519Dic4711yescharacteristic
98#428—21no—
99C2×#101Dic4711yescharacteristic
100#391C2×D4711yescharacteristic
101#339—21no—
102#336—941no—
103#324—941no—
104C47#737611yescharacteristic, complemented, Sylow
105C22×Q8C2×Dic4711yescharacteristic
106C2×M16—21no—
107C22×C8—1881no—
108C4×Q8—41no—
109C4×Q8—41no—
110#117—21no—
111#117—41no—
112C2×C4⋊C4C22⋊D4711yescharacteristic
113C2×C42C22⋊D4711yescharacteristic
114#107—41no—
115#106—41no—
116#101—21no—
117#101C47⋊C812yescomplemented
118C22⋊C8—1881no—
119#95—41no—
120C2×Q8—21no—
121C2×Q8—41no—
122C2×Q8—41no—
123C2×Q8C2×C47⋊C811yescharacteristic
124C2×Q8#1062532911yescharacteristic
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4C22⋊Dic4711yescharacteristic
129M16—22no—
130M16—41no—
131C2×C8—41no—
132C2×C8—3761no—
133C2×C8—3761no—
134C2×C8—3761no—
135C2×C8—1882no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C22⋊C4—41no—
142C42—41no—
143C42—41no—
144C42—42no—
145C23#1309393911yescharacteristic
146Q8—41no—
147Q8—41no—
148Q8—42no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—81no—
163C2×C4C22⋊(C47⋊C8)11yescharacteristic
164C2×C4#1309394311yescharacteristic
165C8—42no—
166C8—3762no—
167C8—7521no—
168C22—21no—
169C22—21no—
170C22#1568884911yescharacteristic
171C22#1568891011yescharacteristic
172C22#1568891311yescharacteristic
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—41no—
178C4—41no—
179C4—81no—
180C4—81no—
181C4—81no—
182C2—21no—
183C2—21no—
184C2#2024133611yescharacteristic
185C2#2024153911yescharacteristic
186C2#2024154211yescharacteristic
subgroupG/H
centerC22#15688910
commutatorC2×Q8×C47C2×C8
Frattini#391C2×D47
FittingC47×#1857C2
socleC22×C47AΣL(1,𝔽16)
prime radicalC23×C47#99
cosocleC47×#391C22
hypercenter#1857D47
hypocenterC47#7376
LCSG ▷ C2×Q8×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ #1857

Complements

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

NH ≅ G/Ntype
24. #101×C47C8semidirect
104. C47#7376semidirect
117. #101C47⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202413222
#202412142
C47×#18572
#737647