Finite Group Databasebeta

← #26129686

Subgroups — #26129686

Order 12032.

356 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 67 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#20245469C212yesmaximal, complemented
2#20243004C211yesmaximal, characteristic, complemented
3#20241427C211yesmaximal, characteristic, complemented
4#20241361C212yesmaximal, complemented
5C47×#3293C211yesmaximal, characteristic
6#15689563—22no—
7#15689526—22no—
8#15689503C2211yescharacteristic
9#15688944C2212yes—
10#15688938—22no—
11C8×C47⋊C8C2211yescharacteristic
12C47×#578—22no—
13C47×#494C2211yescharacteristic
14C47×#473—22no—
15C47×#456—22no—
16C47×#443C2212yes—
17C2×Dic188—22no—
18#13094024—21no—
19#13093970—42no—
20#13093968—42no—
21#13093964—22no—
22#13093957—44no—
23C4×C47⋊C8—21no—
242−1+4×C47—22no—
25#137×C47—24no—
26#137×C47—44no—
27#135×C47—42no—
28#135×C47D412yes—
29C2×Q16×C47—22no—
30C2×Q16×C47—42no—
31#131×C47—22no—
32C4⋊Q8×C47D412yes—
33#108×C47D412yes—
34#104×C47—22no—
35#101×C47—22no—
36C4×C8×C47C2311yescharacteristic
37#41×C47—22no—
38#41×C47—44no—
39#41×C47—82no—
40C2×Q8×C47—22no—
41C2×Q8×C47—44no—
42C2×Dic94—41no—
43Q16×C47—44no—
44Q16×C47—42no—
45Q16×C47—42no—
46Q16×C47D812yes—
47SD16×C47—22no—
48SD16×C47—44no—
49D8×C47D812yes—
50M16×C47—22no—
51M16×C47—42no—
52C2×C8×C47—42no—
53C2×C8×C47C2×D412yes—
54C4⋊C4×C47—42no—
55C42×C47C2×D411yescharacteristic
56C4⋊Dic47—42no—
57C2×C47⋊C8—22no—
58Dic188—42no—
59Dic188—42no—
60C47⋊C16—42no—
61Q8×C47—22no—
62Q8×C47—44no—
63Q8×C47—82no—
64D4×C47—22no—
65D4×C47—42no—
66D4×C47—82no—
67C2×C4×C47—21no—
68C2×C4×C47—42no—
69C2×C4×C47—42no—
70C2×C4×C47—84no—
71C2×C4×C47#12011yescharacteristic
72C2×Dic47—81no—
73Dic94—82no—
74C8×C47—42no—
75C8×C47C2×D812yes—
76C8×C47AGL(1,ℤ/8)12yes—
77C47⋊C8—24no—
78#12313—471nomaximal, Sylow
79C22×C47—42no—
80C22×C47#45311yescharacteristic
81C4×C47—22no—
82C4×C47—42no—
83C4×C47—82no—
84C4×C47#44712yes—
85Dic47—161no—
86#3293D4711yescharacteristic
87#2214—472no—
88#1713—471no—
89#1358—471no—
90#1335—472no—
91C2×C47—21no—
92C2×C47—82no—
93C2×C47#161911yescharacteristic
94#578—22no—
95#510—942no—
96#500—942no—
97#494—471no—
98#494C2×D4711yescharacteristic
99#473—22no—
100#456—22no—
101#443C2×D4712yes—
102#364—472no—
103#362—942no—
104C82—471no—
105C47#1231311yescharacteristic, complemented, Sylow
1062−1+4—22no—
107#137—24no—
108#137—44no—
109#135—42no—
110#135C22⋊D4712yes—
111C2×Q16—22no—
112C2×Q16—42no—
113C2×Q16—942no—
114#131—22no—
115C4⋊Q8—941no—
116C4⋊Q8C22⋊D4712yes—
117Q32—1882no—
118SD32—1882no—
119M32—942no—
120#108C22⋊D4712yes—
121#107—1884no—
122#104—22no—
123#101—22no—
124C4×C8—941no—
125C4×C8C22×D4711yescharacteristic
126#41—22no—
127#41—44no—
128#41—82no—
129C2×Q8—22no—
130C2×Q8—44no—
131C2×Q8—1881no—
132Q16—44no—
133Q16—42no—
134Q16—42no—
135Q16—1882no—
136Q16—1882no—
137Q16#1062533412yes—
138SD16—22no—
139SD16—44no—
140D8#1062533412yes—
141M16—22no—
142M16—42no—
143C2×C8—42no—
144C2×C8—942no—
145C2×C8C2×C22⋊D4712yes—
146C4⋊C4—42no—
147C4⋊C4—1882no—
148C42C2×C22⋊D4711yescharacteristic
149C16—1882no—
150Q8—22no—
151Q8—44no—
152Q8—82no—
153Q8—3762no—
154D4—22no—
155D4—42no—
156D4—82no—
157C2×C4—21no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—84no—
161C2×C4—3761no—
162C2×C4#1309399311yescharacteristic
163C8—42no—
164C8—944no—
165C8C2×#1062533412yes—
166C8#1309405012yes—
167C22—42no—
168C22#1568929111yescharacteristic
169C4—22no—
170C4—42no—
171C4—82no—
172C4—7521no—
173C4#1568925712yes—
174C2—21no—
175C2—82no—
176C2#2024244211yescharacteristic
subgroupG/H
centerC2#20242442
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
Fitting, prime radicalC47×#3293C2
socleC2×C47#1619
hypercenter#3293D47
hypocenterC47#12313
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3293

Complements

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

NH ≅ G/Ntype
1. #20245469C2semidirect
2. #20243004C2semidirect
3. #20241427C2semidirect
4. #20241361C2semidirect
105. C47#12313semidirect

Maximal subgroups

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

subgroupindex
#202454692
#202430042
#202414272
#202413612
C47×#32932
#1231347