Finite Group Databasebeta

← #26128183

Subgroups — #26128183

Order 12032.

226 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C4×#13093967C211yesmaximal, characteristic
2#20242285C211yesmaximal, characteristic, complemented
3#20242182C211yesmaximal, characteristic
4#20241461C211yesmaximal, characteristic, complemented
5#20241350C211yesmaximal, characteristic
6#20241307C211yesmaximal, characteristic, complemented
7C47×#1332C211yesmaximal, characteristic, complemented
8C2×#13093967C411yescharacteristic
9#15689254—21no—
10C4×D376C2211yescharacteristic
11#15689188—21no—
12#15688950C411yescharacteristic
13#15688927C411yescharacteristic
14#15688923C411yescharacteristic
15C4×C47⋊C16C2211yescharacteristic
16#15688876C2211yescharacteristic
17Q8⋊(C47⋊C8)—21no—
18#15688850—21no—
19C8×C47⋊C8C2211yescharacteristic
20C4×Q16×C47C2211yescharacteristic
21C4×C16×C47C2211yescharacteristic
22C47×#335C2211yescharacteristic
23C2×D376C2×C411yescharacteristic
24C2×C8×D47—41no—
25C4×D188—21no—
26#13093967—22no—
27C2×C47⋊C16C2×C411yescharacteristic
28C2×C47⋊C16C2×C411yescharacteristic
29#13093958C2×C411yescharacteristic
30C4⋊(C47⋊C8)—21no—
31C4⋊(C47⋊C8)—41no—
32#13093946—21no—
33(C2×D47)⋊C8—41no—
34#13093935—41no—
35C4×C47⋊C8D411yescharacteristic
36C4×C47⋊C8D411yescharacteristic
37C2×Q16×C47C2×C411yescharacteristic
38C4×Q8×C47—21no—
39C2×C16×C47—21no—
40#107×C47C2×C411yescharacteristic
41C4⋊C8×C47—21no—
42#103×C47—21no—
43C4×C8×C47C2311yescharacteristic
44C2×Q8×C47—21no—
45C2×D188—21no—
46C2×C4×D47—41no—
47Q16×C47—21no—
48Q16×C47—21no—
49C2×C8×C47—41no—
50C2×C8×C47C22×C411yescharacteristic
51C2×C8×C47#4111yescharacteristic
52C4⋊C4×C47—21no—
53C4⋊C4×C47—41no—
54C42×C47—41no—
55C42×C47C2×D411yescharacteristic
56(C2×D47)⋊C4—41no—
57C4⋊Dic47—21no—
58C2×C47⋊C8—21no—
59C2×C47⋊C8—41no—
60C2×C47⋊C8D811yescharacteristic
61C2×C47⋊C8D811yescharacteristic
62D376—21no—
63D376M1612yes—
64C8×D47—82no—
65C16×C47—41no—
66C47⋊C16—21no—
67C47⋊C16—21no—
68C22×D47—41no—
69Q8×C47—41no—
70Q8×C47—41no—
71C2×C4×C47—41no—
72C2×C4×C47—41no—
73C2×C4×C47C4×D411yescharacteristic
74C2×C4×C47C2×D811yescharacteristic
75C2×C4×C47#13511yescharacteristic
76C2×Dic47—41no—
77D188—22no—
78D188—41no—
79C4×D47—81no—
80C4×D47—81no—
81C8×C47—21no—
82C8×C47—81no—
83C8×C47C2×M1611yescharacteristic
84C8×C47#13111yescharacteristic
85C47⋊C8—21no—
86C47⋊C8—41no—
87C47⋊C8—81no—
88C47⋊C8SD3212yescomplemented
89#12070—471nomaximal, Sylow
90C22×C47C4×D811yescharacteristic
91C2×D47—42no—
92C2×D47—81no—
93C2×D47—81no—
94C4×C47—21no—
95C4×C47—81no—
96C4×C47—81no—
97C4×C47#43511yescharacteristic
98C4×C47#44411yescharacteristic
99C4×C47C2×SD3211yescharacteristic
100C4×C47#50811yescharacteristic
101Dic47—81no—
102C4×SD32—471no—
103#1591—471no—
104C8×D8—471no—
105#1372—471no—
106#1332D4711yescharacteristic, complemented
107#1331—471no—
108#1312—471no—
109C2×C47#158111yescharacteristic
110C2×C47C4×SD3211yescharacteristic
111C2×C47#217911yescharacteristic
112D47—82no—
113C2×SD32—471no—
114#446—941no—
115C4×Q16C2×D4711yescharacteristic
116C4×D8—471no—
117C8×D4—941no—
118#367—471no—
119#358—471no—
120#357—471no—
121C4×C16—471no—
122C4×C16C2×D4711yescharacteristic
123#335—471no—
124#335C2×D4711yescharacteristic
125Q8⋊C8—941no—
126D4⋊C8—941no—
127C82—471no—
128C47#1207011yescharacteristic, complemented, Sylow
129C2×Q16C4×D4711yescharacteristic
130C2×D8—471no—
131C22×C8—1881no—
132C4×Q8—21no—
133C4×D4—941no—
134SD32—942no—
135C2×C16—21no—
136C2×C16—471no—
137C2×C16—471no—
138#107—471no—
139#107C4×D4711yescharacteristic
140C4⋊C8—21no—
141C4⋊C8—941no—
142C4⋊C8—1881no—
143#103—21no—
144#102—941no—
145C22⋊C8—1881no—
146#97—1881no—
147C4×C8—471no—
148C4×C8—471no—
149C4×C8C22×D4711yescharacteristic
150C2×Q8—21no—
151C2×D4—941no—
152C22×C4—1881no—
153Q16—21no—
154Q16—21no—
155D8—472no—
156D8—941no—
157C2×C8—41no—
158C2×C8—471no—
159C2×C8—471no—
160C2×C8—941no—
161C2×C8—3762no—
162C2×C8—1881no—
163C2×C8C2×C4×D4711yescharacteristic
164C2×C8#1062535211yescharacteristic
165C4⋊C4—21no—
166C4⋊C4—41no—
167C4⋊C4—941no—
168C22⋊C4—1881no—
169C42—41no—
170C42D4×D4711yescharacteristic
171C16—41no—
172C16—941no—
173C16—941no—
174C23—1881no—
175Q8—41no—
176Q8—41no—
177D4—942no—
178D4—1881no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—3761no—
182C2×C4—3761no—
183C2×C4—1881no—
184C2×C4(C22⋊D47)⋊C411yescharacteristic
185C2×C4D8×D4711yescharacteristic
186C2×C4#1309404511yescharacteristic
187C8—21no—
188C8—81no—
189C8—472no—
190C8—941no—
191C8—3761no—
192C8—1881no—
193C8M16×D4711yescharacteristic
194C8#1309403011yescharacteristic
195C22—3761no—
196C22—3761no—
197C22—1882no—
198C22#1568920311yescharacteristic
199C4—21no—
200C4—81no—
201C4—81no—
202C4—3761no—
203C4#1568919711yescharacteristic
204C4#1568924211yescharacteristic
205C4SD32×D4711yescharacteristic
206C4#1568955611yescharacteristic
207C2—3762no—
208C2#2024222011yescharacteristic
209C2#2024523711yescharacteristic
210C2#2024527111yescharacteristic
subgroupG/H
centerC2×C4D8×D47
commutatorC2×C8×C47C22×C4
FrattiniC4×C8C22×D47
FittingC47×#1332C2
socleC22×C47C4×D8
prime radicalC2×D376C2×C4
cosocleC4×C8×C47C23
hypercenter#1332D47
hypocenterC47#12070
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ #1332

Complements

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

NH ≅ G/Ntype
2. #20242285C2semidirect
4. #20241461C2semidirect
6. #20241307C2semidirect
7. C47×#1332C2semidirect
88. C47⋊C8SD32semidirect
106. #1332D47semidirect
128. C47#12070semidirect

Maximal subgroups

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

subgroupindex
C4×#130939672
#202422852
#202421822
#202414612
#202413502
#202413072
C47×#13322
#1207047