Finite Group Databasebeta

← #26111243

Subgroups — #26111243

Order 12032.

398 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 103 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#20252494C211yesmaximal, characteristic, complemented
2#20242053C211yesmaximal, characteristic, complemented
3#20241857C212yesmaximal
4#20241287C212yesmaximal, complemented
5C47×#3135C211yesmaximal, characteristic
6#15689254—24no—
7#15689184—22no—
8#15689180C2212yes—
9#15689176C2212yes—
10#15688869—22no—
11D4⋊(C47⋊C8)—24no—
12C47×#533C2211yescharacteristic
13C47×#532C2211yescharacteristic
14C47×#491C412yes—
15C47×#490C412yes—
16C47×#489—22no—
17C47×#416—22no—
18C2×#97×C47C2211yescharacteristic
19C4⋊(C47⋊C8)—24no—
20C4⋊(C47⋊C8)—22no—
21C4⋊(C47⋊C8)D414yes—
22C22⋊(C47⋊C8)—22no—
23C22⋊(C47⋊C8)—22no—
24#13093935—22no—
25#13093935—22no—
26C4×C47⋊C8—22no—
27C4×C47⋊C8—22no—
28C2×#41×C47—21no—
29C22×C8×C47—21no—
30C4⋊Q8×C47C2×C411yescharacteristic
31C4⋊Q8×C47D412yes—
32#127×C47C2×C411yescharacteristic
33#125×C47C2×C412yes—
34#124×C47—21no—
35C22⋊Q8×C47—22no—
36#121×C47—22no—
37C4×Q8×C47—22no—
38C4×D4×C47—22no—
39#117×C47D412yes—
40C2×C42×C47C2311yescharacteristic
41#103×C47—24no—
42#102×C47—24no—
43#97×C47—21no—
44#97×C47C2×C412yes—
45#41×C47—44no—
46C2×Q8×C47—22no—
47C2×Q8×C47—21no—
48C2×D4×C47—21no—
49C2×D4×C47—22no—
50C2×D4×C47—41no—
51C22×C4×C47—41no—
52C22×C4×C47C2×D411yescharacteristic
53C22×C4×C47#4112yes—
54C2×C8×C47—22no—
55C2×C8×C47—42no—
56C4⋊C4×C47—24no—
57C4⋊C4×C47—22no—
58C4⋊C4×C47—22no—
59C4⋊C4×C47C22⋊C414yes—
60C22⋊C4×C47—22no—
61C22⋊C4×C47—42no—
62C42×C47—22no—
63C42×C47C22×C411yescharacteristic
64C42×C47C2×D411yescharacteristic
65C42×C47C2×D412yes—
66C2×C47⋊C8—24no—
67C2×C47⋊C8—42no—
68C23×C47—41no—
69C23×C47#12311yescharacteristic
70Q8×C47—42no—
71Q8×C47—82no—
72D4×C47—44no—
73D4×C47—42no—
74D4×C47—82no—
75C2×C4×C47—22no—
76C2×C4×C47—24no—
77C2×C4×C47—42no—
78C2×C4×C47—41no—
79C2×C4×C47—82no—
80C2×C4×C47C2×C22⋊C411yescharacteristic
81C2×C4×C47C4×D412yes—
82C2×C4×C47#12011yescharacteristic
83C2×C4×C47#12112yes—
84C8×C47—42no—
85C47⋊C8—44no—
86C47⋊C8—82no—
87#9008—471nomaximal, Sylow
88C22×C47—22no—
89C22×C47—21no—
90C22×C47—82no—
91C22×C47—81no—
92C22×C47#38611yescharacteristic
93C4×C47—22no—
94C4×C47—22no—
95C4×C47—21no—
96C4×C47—44no—
97C4×C47—81no—
98C4×C47#45412yes—
99#3135D4711yescharacteristic
100#2987—471no—
101#1544—471no—
102#1496—472no—
103#1302—472no—
104C2×C47—41no—
105C2×C47—161no—
106C2×C47#188112yes—
107C2×C47#188811yescharacteristic
108#533C2×D4711yescharacteristic
109#532C2×D4711yescharacteristic
110#491Dic4712yes—
111#490Dic4712yes—
112#489—22no—
113#446—944no—
114#433—942no—
115#432—472no—
116#431—472no—
117#416—22no—
118C2×#97C2×D4711yescharacteristic
119#332—942no—
120D4⋊C8—944no—
121C47#900811yescharacteristic, complemented, Sylow
122C2×#41—21no—
123C22×C8—21no—
124C4⋊Q8C2×Dic4711yescharacteristic
125C4⋊Q8C22⋊D4712yes—
126#127C2×Dic4711yescharacteristic
127#125C2×Dic4712yes—
128#124—21no—
129C22⋊Q8—22no—
130#121—22no—
131C4×Q8—22no—
132C4×D4—22no—
133#117C22⋊D4712yes—
134C2×C42C22×D4711yescharacteristic
135C4⋊C8—474no—
136C4⋊C8—944no—
137C4⋊C8—942no—
138#103—24no—
139#102—24no—
140C22⋊C8—942no—
141C22⋊C8—942no—
142#97—21no—
143#97—942no—
144#97—942no—
145#97C2×Dic4712yes—
146C4×C8—942no—
147C4×C8—942no—
148#41—44no—
149C2×Q8—22no—
150C2×Q8—21no—
151C2×D4—21no—
152C2×D4—22no—
153C2×D4—41no—
154C22×C4—41no—
155C22×C4#1062535212yes—
156C22×C4C2×C22⋊D4711yescharacteristic
157C2×C8—22no—
158C2×C8—42no—
159C2×C8—944no—
160C2×C8—1882no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4C22⋊Dic4714yes—
165C22⋊C4—22no—
166C22⋊C4—42no—
167C42—22no—
168C42D4×D4712yes—
169C42C22×Dic4711yescharacteristic
170C42C2×C22⋊D4711yescharacteristic
171C23—41no—
172C23#1309400511yescharacteristic
173Q8—42no—
174Q8—82no—
175D4—44no—
176D4—42no—
177D4—82no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—82no—
183C2×C4C2×C22⋊Dic4711yescharacteristic
184C2×C4D4×Dic4712yes—
185C2×C4#1309399211yescharacteristic
186C2×C4#1309399612yes—
187C8—42no—
188C8—3762no—
189C8—1884no—
190C22—22no—
191C22—21no—
192C22—82no—
193C22—81no—
194C22#1568900311yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—44no—
199C4—81no—
200C4#1568929512yes—
201C2—41no—
202C2—161no—
203C2#2024375212yes—
204C2#2024379211yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3135C2
socleC22×C47#386
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#3135D47
hypocenterC47#9008
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3135

Complements

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

NH ≅ G/Ntype
1. #20252494C2semidirect
2. #20242053C2semidirect
4. #20241287C2semidirect
121. C47#9008semidirect

Maximal subgroups

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

subgroupindex
#202524942
#202420532
#202418572
#202412872
C47×#31352
#900847