Finite Group Databasebeta

← #26127795

Subgroups — #26127795

Order 12032.

292 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 87 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#20245204C211yesmaximal, characteristic, complemented
2#20245154C211yesmaximal, characteristic, complemented
3#20241449C212yesmaximal, complemented
4#20241425C212yesmaximal, complemented
5C47×#3176C211yesmaximal, characteristic
6C2×#13093964—21no—
7#15688944—21no—
8#15688944C2212yes—
9#15688944C2212yes—
10#15688905—21no—
11#15688901—21no—
12C47×#540C412yes—
13C47×#499C2211yescharacteristic
14C47×#493C2211yescharacteristic
15C47×#490C412yes—
16C47×#469—21no—
17C47×#466—21no—
18C47×#442—21no—
19C4×D8×C47—21no—
20C47×#405C2211yescharacteristic
21#13093964—22no—
22#13093964—22no—
23#13093964—41no—
24C2×C47⋊C16—41no—
25C2×D8×C47—21no—
26C2×D8×C47—21no—
27C2×M16×C47D411yescharacteristic
28C22×C8×C47D411yescharacteristic
29#127×C47C2×C412yes—
30#125×C47C2×C412yes—
31#123×C47—42no—
32#121×C47—22no—
33#121×C47—42no—
34C4×D4×C47—22no—
35#117×C47C2×C411yescharacteristic
36#107×C47—21no—
37#106×C47—21no—
38#102×C47—22no—
39#102×C47—22no—
40#97×C47C2×C411yescharacteristic
41#97×C47D411yescharacteristic
42C4×C8×C47D411yescharacteristic
43C4×C8×C47C2311yescharacteristic
44C2×D4×C47—22no—
45C2×D4×C47—42no—
46C2×D4×C47—42no—
47C22×C4×C47—42no—
48C22×C4×C47C22⋊C411yescharacteristic
49D8×C47—42no—
50D8×C47—41no—
51D8×C47—41no—
52M16×C47—22no—
53C2×C8×C47—21no—
54C2×C8×C47—21no—
55C2×C8×C47C22⋊C411yescharacteristic
56C2×C8×C47C22⋊C411yescharacteristic
57C2×C8×C47C2×D411yescharacteristic
58C2×C8×C47C2×D411yescharacteristic
59C4⋊C4×C47—21no—
60C4⋊C4×C47—22no—
61C4⋊C4×C47—42no—
62C22⋊C4×C47—21no—
63C22⋊C4×C47—42no—
64C22⋊C4×C47—42no—
65C42×C47C22⋊C411yescharacteristic
66C42×C47C22×C411yescharacteristic
67C47⋊C16—42no—
68C47⋊C16—42no—
69C23×C47—42no—
70C23×C47#10011yescharacteristic
71D4×C47—42no—
72D4×C47—42no—
73D4×C47—162no—
74D4×C47—162no—
75C2×C4×C47—21no—
76C2×C4×C47—21no—
77C2×C4×C47—21no—
78C2×C4×C47—21no—
79C2×C4×C47—42no—
80C2×C4×C47—162no—
81C2×C4×C47#10011yescharacteristic
82C2×C4×C47C2×C22⋊C411yescharacteristic
83C8×C47—21no—
84C8×C47—21no—
85C8×C47#13512yes—
86C8×C47AGL(1,ℤ/8)11yescharacteristic
87C8×C47#13711yescharacteristic
88#11989—471nomaximal, Sylow
89C22×C47—21no—
90C22×C47—41no—
91C22×C47—82no—
92C22×C47—82no—
93C22×C47—82no—
94C22×C47C2×#10011yescharacteristic
95C4×C47—21no—
96C4×C47—41no—
97C4×C47—41no—
98C4×C47—82no—
99C4×C47#41611yescharacteristic
100C4×C47#41711yescharacteristic
101#3176D4711yescharacteristic
102#2166—471no—
103#2153—471no—
104#1366—472no—
105#1357—472no—
106C2×C47—21no—
107C2×C47—41no—
108C2×C47—162no—
109C2×C47#155111yescharacteristic
110#540Dic4712yes—
111C2×M32—941no—
112#499C2×D4711yescharacteristic
113#493C2×D4711yescharacteristic
114#490Dic4712yes—
115#469—21no—
116#466—21no—
117#442—21no—
118C4×D8—21no—
119#405C2×D4711yescharacteristic
120#364—472no—
121#364—472no—
122#364—941no—
123#349—941no—
124#347—941no—
125C47#1198911yescharacteristic, complemented, Sylow
126C2×D8—21no—
127C2×D8—21no—
128C2×M16C22⋊D4711yescharacteristic
129C22×C8C22⋊D4711yescharacteristic
130#127C2×Dic4712yes—
131#125C2×Dic4712yes—
132#123—42no—
133#121—22no—
134#121—42no—
135C4×D4—22no—
136#117C2×Dic4711yescharacteristic
137M32—942no—
138M32—942no—
139M32—1881no—
140C2×C16—1881no—
141#107—21no—
142#106—21no—
143#102—22no—
144#102—22no—
145#97C2×Dic4711yescharacteristic
146#97C22⋊D4711yescharacteristic
147C4×C8C22⋊D4711yescharacteristic
148C4×C8C22×D4711yescharacteristic
149C2×D4—22no—
150C2×D4—42no—
151C2×D4—42no—
152C22×C4—42no—
153C22×C4C22⋊Dic4711yescharacteristic
154D8—42no—
155D8—41no—
156D8—41no—
157M16—22no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8C22⋊Dic4711yescharacteristic
161C2×C8C22⋊Dic4711yescharacteristic
162C2×C8C2×C22⋊D4711yescharacteristic
163C2×C8C2×C22⋊D4711yescharacteristic
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C22⋊C4—21no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C42C22⋊Dic4711yescharacteristic
171C42C22×Dic4711yescharacteristic
172C16—1882no—
173C16—1882no—
174C23—42no—
175C23#1309394111yescharacteristic
176D4—42no—
177D4—42no—
178D4—162no—
179D4—162no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—42no—
185C2×C4—162no—
186C2×C4#1309394111yescharacteristic
187C2×C4C2×C22⋊Dic4711yescharacteristic
188C8—21no—
189C8—21no—
190C8#1309404412yes—
191C8#1309404911yescharacteristic
192C8#1309405411yescharacteristic
193C22—21no—
194C22—41no—
195C22—82no—
196C22—82no—
197C22—82no—
198C22C2×#1309394111yescharacteristic
199C4—21no—
200C4—41no—
201C4—41no—
202C4—82no—
203C4#1568911911yescharacteristic
204C4#1568912411yescharacteristic
205C2—21no—
206C2—41no—
207C2—162no—
208C2#2024208311yescharacteristic
subgroupG/H
centerC2#20242083
commutatorC42×C47C22×C4
FrattiniC4×C8C22×D47
Fitting, prime radicalC47×#3176C2
socleC2×C47#1551
cosocleC4×C8×C47C23
hypercenter#3176D47
hypocenterC47#11989
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #3176

Complements

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

NH ≅ G/Ntype
1. #20245204C2semidirect
2. #20245154C2semidirect
3. #20241449C2semidirect
4. #20241425C2semidirect
125. C47#11989semidirect

Maximal subgroups

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

subgroupindex
#202452042
#202451542
#202414492
#202414252
C47×#31762
#1198947