Finite Group Databasebeta

← #26137154

Subgroups — #26137154

Order 12032.

184 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C2×#15688962C211yesmaximal, characteristic, complemented
2#20245681C211yesmaximal, characteristic, complemented
3#20245514C211yesmaximal, characteristic, complemented
4#20245509C211yesmaximal, characteristic
5#20241580C211yesmaximal, characteristic, complemented
6#20241577C211yesmaximal, characteristic, complemented
7C47×#1425C211yesmaximal, characteristic, complemented
8C2×Dic376C2211yescharacteristic
9C2×D752C2211yescharacteristic
10#15689539C2211yescharacteristic
11#15689399—21no—
12#15689394—21no—
13#15688962—21no—
14#15688962—21no—
15#15688957C2211yescharacteristic
16#15688956—21no—
17C32×D47—21no—
18#15688938—21no—
19#15688935—21no—
20C2×C32×C47C2211yescharacteristic
21C47×#368C2211yescharacteristic
22C47×#368C2211yescharacteristic
23#13094057—41no—
24#13094052—41no—
25C2×Dic188—21no—
26C2×D376—21no—
27#13094030D411yescharacteristic
28Dic376—21no—
29Dic376—21no—
30D752—21no—
31D752—21no—
32#13093964D411yescharacteristic
33#13093963—21no—
34C16×D47—21no—
35#13093942—41no—
36#13093940—41no—
37C47⋊C32—21no—
38C2×C16×C47C2311yescharacteristic
39#108×C47—21no—
40#108×C47—21no—
41C32×C47D411yescharacteristic
42C32×C47D411yescharacteristic
43#10625350D811yescharacteristic
44C2×D188—41no—
45C2×Dic94—41no—
46M16×C47—41no—
47M16×C47—41no—
48C2×C8×C47C2×D411yescharacteristic
49#10625329D811yescharacteristic
50Dic188—41no—
51Dic188—41no—
52Dic188—81no—
53D376—41no—
54D376—41no—
55D376—81no—
56#10625325—81no—
57#10625325—81no—
58#10625324—21no—
59C8×D47—21no—
60C16×C47C2×D411yescharacteristic
61C16×C47#4111yescharacteristic
62C47⋊C16—21no—
63C22×D47—81no—
64C2×C4×C47C2×D811yescharacteristic
65C22⋊D47—21no—
66C2×Dic47—81no—
67D188—81no—
68D188—81no—
69D188D1611yescharacteristic
70C4×D47—21no—
71Dic94—81no—
72Dic94—81no—
73Dic94D1611yescharacteristic
74C8×C47—81no—
75C8×C47—81no—
76C8×C47#12111yescharacteristic
77C8×C47#13511yescharacteristic
78C47⋊C8—21no—
79#13702—471nomaximal, Sylow
80C22×C47#50811yescharacteristic
81C2×D47—21no—
82C2×D47—161no—
83C2×D47—321no—
84C4×C47#46611yescharacteristic
85C4×C47C2×D1611yescharacteristic
86Dic47—21no—
87Dic47—161no—
88C2×SD64—471no—
89#2258—471no—
90#2222—471no—
91#2221—471no—
92#1425D4711yescharacteristic, complemented
93#1420—471no—
94#1419—471no—
95C2×C47—21no—
96C2×C47#221511yescharacteristic
97D47—41no—
98D47—321no—
99C2×Q32—471no—
100C2×D16—471no—
101#504—471no—
102#473—941no—
103#472—941no—
104SD64—941no—
105SD64—941no—
106M64—471no—
107M64—941no—
108C2×C32—941no—
109C2×C32C2×D4711yescharacteristic
110#368C2×D4711yescharacteristic
111#368C2×D4711yescharacteristic
112#362—941no—
113#361—941no—
114C47#1370211yescharacteristic, complemented, Sylow
115#137—1881no—
116AGL(1,ℤ/8)—1881no—
117C2×Q16—941no—
118C2×D8—941no—
119#131—471no—
120Q32—941no—
121Q32—941no—
122D16—941no—
123D16—941no—
124M32—471no—
125M32—941no—
126C2×C16—941no—
127C2×C16C22×D4711yescharacteristic
128#108—21no—
129#108—21no—
130#101—1881no—
131#100—1881no—
132C32—941no—
133C32D18811yescharacteristic
134C32D18811yescharacteristic
135#41—471no—
136C2×Q8—1881no—
137C2×D4—1881no—
138Q16—3761no—
139Q16—1881no—
140Q16—1881no—
141SD16—3761no—
142SD16—3761no—
143D8—3761no—
144D8—1881no—
145D8—1881no—
146M16—41no—
147M16—41no—
148M16—471no—
149M16—941no—
150C2×C8—941no—
151C2×C8D4×D4711yescharacteristic
152C16—941no—
153C16C2×D18811yescharacteristic
154C16#1062535411yescharacteristic
155C23—3761no—
156Q8—471no—
157Q8—3761no—
158Q8—3761no—
159D4—471no—
160D4—941no—
161D4—3761no—
162D4—3761no—
163C2×C4—941no—
164C2×C4—3761no—
165C2×C4D8×D4711yescharacteristic
166C8—81no—
167C8—81no—
168C8—941no—
169C8#1309399911yescharacteristic
170C8#1309404711yescharacteristic
171C22—941no—
172C22—15041no—
173C22—7521no—
174C22#1568955611yescharacteristic
175C4—941no—
176C4—7521no—
177C4#1568936011yescharacteristic
178C4D16×D4711yescharacteristic
179C2—21no—
180C2—1881no—
181C2—15041no—
182C2#2024547511yescharacteristic
subgroupG/H
centerC2#20245475
commutator, cosocleC2×C16×C47C23
FrattiniC2×C16C22×D47
FittingC47×#1425C2
socleC2×C47#2215
prime radical#20245509C2
hypercenter#1425D47
hypocenterC47#13702
LCSG ▷ C2×C16×C47 ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1425

Complements

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

NH ≅ G/Ntype
1. C2×#15688962C2semidirect
2. #20245681C2semidirect
3. #20245514C2semidirect
5. #20241580C2semidirect
6. #20241577C2semidirect
7. C47×#1425C2semidirect
92. #1425D47semidirect
114. C47#13702semidirect

Maximal subgroups

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

subgroupindex
C2×#156889622
#202456812
#202455142
#202455092
#202415802
#202415772
C47×#14252
#1370247