Finite Group Databasebeta

← #26111602

Subgroups — #26111602

Order 12032.

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

normal subgroups only

This group has 426 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#20253713C211yesmaximal, characteristic
2#20252490C211yesmaximal, characteristic, complemented
3#20241806C212yesmaximal
4#20241226C212yesmaximal, complemented
5C47×#1526C211yesmaximal, characteristic, complemented
6#15689673C2211yescharacteristic
7#15689480—22no—
8#15689475C412yes—
9#15689471C412yes—
10#15689254—22no—
11Q8×C47⋊C8—22no—
12#15689184—22no—
13#15689176C2212yes—
14#15689117—22no—
15C2×C4×C47⋊C8C2211yescharacteristic
16#15688852—22no—
17#15688852—22no—
18C47×#533C2211yescharacteristic
19C4×M16×C47C2212yes—
20C47×#330—22no—
21C2×#10625350—21no—
22C22×C47⋊C8—21no—
23#13094024C2×C411yescharacteristic
24#13094021C2×C411yescharacteristic
25#13094012—21no—
26#13093998—22no—
27C4×D188—22no—
28C4⋊(C47⋊C8)—24no—
29C4⋊(C47⋊C8)D412yes—
30C4⋊(C47⋊C8)D412yes—
31#13093947—24no—
32#13093944—24no—
33C22⋊(C47⋊C8)—22no—
34#13093935—22no—
35C4×C47⋊C8—22no—
36C4×C47⋊C8—21no—
37C4×C47⋊C8—22no—
38C4×C47⋊C8C2×C412yes—
39C2×M16×C47—22no—
40C4⋊Q8×C47C2×C412yes—
41#125×C47D412yes—
42C22⋊Q8×C47—22no—
43C4×Q8×C47—22no—
44#117×C47D412yes—
45C2×C42×C47C2311yescharacteristic
46#97×C47—22no—
47C4×C8×C47—22no—
48C2×Q8×C47—22no—
49C22×C4×C47C2×D411yescharacteristic
50C22×C4×C47#4112yes—
51C2×C22⋊D47—41no—
52#10625350—44no—
53C2×D188—21no—
54C2×D188—22no—
55C2×C4×D47—41no—
56C2×Dic94—21no—
57M16×C47—24no—
58M16×C47—42no—
59C2×C8×C47—42no—
60C4⋊C4×C47—22no—
61C4⋊C4×C47—24no—
62C4⋊C4×C47C22⋊C414yes—
63C22⋊C4×C47—22no—
64C42×C47—22no—
65C42×C47C22×C411yescharacteristic
66C42×C47C2×D411yescharacteristic
67C42×C47C2×D412yes—
68(C2×D47)⋊C4—42no—
69C4⋊Dic47—22no—
70C2×C47⋊C8—22no—
71C2×C47⋊C8—22no—
72C2×C47⋊C8—22no—
73C2×C47⋊C8—42no—
74C23×C47#12311yescharacteristic
75C22×D47—41no—
76Q8×C47—42no—
77Q8×C47—42no—
78C2×C4×C47—22no—
79C2×C4×C47—22no—
80C2×C4×C47—22no—
81C2×C4×C47—24no—
82C2×C4×C47C2×C22⋊C411yescharacteristic
83C2×C4×C47C4×D412yes—
84C2×C4×C47#12011yescharacteristic
85C2×C4×C47#12112yes—
86C22⋊D47—82no—
87C2×Dic47—41no—
88D188—42no—
89D188—42no—
90D188—42no—
91C4×D47—82no—
92Dic94—42no—
93C8×C47—44no—
94C47⋊C8—42no—
95C47⋊C8—42no—
96C47⋊C8—42no—
97#9070—471nomaximal, Sylow
98C22×C47—22no—
99C22×C47—21no—
100C22×C47#38611yescharacteristic
101C2×D47—82no—
102C2×D47—81no—
103C4×C47—22no—
104C4×C47—21no—
105C4×C47—44no—
106C4×C47#44312yes—
107C4×C47#44412yes—
108C4×C47#45412yes—
109Dic47—81no—
110#3131—471no—
111#2986—471no—
112#1526D4711yescharacteristic, complemented
113#1484—472no—
114#1278—472no—
115C2×C47—41no—
116C2×C47#187612yes—
117C2×C47#188811yescharacteristic
118D47—161no—
119#533C2×D4711yescharacteristic
120#532—471no—
121#488—942no—
122#487—472no—
123#486—472no—
124#446—942no—
125C8×Q8—942no—
126#433—942no—
127#431—472no—
128#416—942no—
129C4×M16C2×D4712yes—
130C2×C4×C8—471no—
131#330—22no—
132D4⋊C8—942no—
133D4⋊C8—942no—
134C47#907011yescharacteristic, complemented, Sylow
135C2×#41—941no—
136C2×M16—22no—
137C22×C8—941no—
138C4⋊Q8—471no—
139C4⋊Q8C4×D4712yes—
140#127—471no—
141#125C22⋊D4712yes—
142#124—941no—
143C22⋊Q8—22no—
144#121—942no—
145C4×Q8—22no—
146C4×D4—942no—
147#117D18812yes—
148C2×C42C22×D4711yescharacteristic
149C4⋊C8—472no—
150C4⋊C8—472no—
151C4⋊C8—944no—
152#103—944no—
153#102—944no—
154C22⋊C8—942no—
155#97—22no—
156#97—942no—
157C4×C8—22no—
158C4×C8—472no—
159C4×C8—942no—
160C4×C8—942no—
161C4×C8—941no—
162#41—1884no—
163C2×Q8—22no—
164C2×Q8—941no—
165C2×D4—941no—
166C2×D4—942no—
167C2×D4—1881no—
168C22×C4—1881no—
169C22×C4C2×D18811yescharacteristic
170C22×C4#1062535212yes—
171M16—24no—
172M16—42no—
173C2×C8—42no—
174C2×C8—942no—
175C2×C8—942no—
176C2×C8—942no—
177C2×C8—1882no—
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—942no—
181C4⋊C4(C2×D47)⋊C414yes—
182C22⋊C4—22no—
183C22⋊C4—1882no—
184C42—22no—
185C42C2×C4×D4711yescharacteristic
186C42D4×D4712yes—
187C42C2×C22⋊D4711yescharacteristic
188C23—1881no—
189C23#1309400811yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—1882no—
193D4—3762no—
194D4—1882no—
195D4—1882no—
196D4—1882no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—3762no—
202C2×C4—1881no—
203C2×C4C2×(C2×D47)⋊C411yescharacteristic
204C2×C4(C22⋊D47)⋊C412yes—
205C2×C4#1309399411yescharacteristic
206C2×C4#1309399612yes—
207C8—44no—
208C8—1882no—
209C8—1882no—
210C8—1882no—
211C22—22no—
212C22—21no—
213C22—3762no—
214C22—3761no—
215C22#1568900011yescharacteristic
216C4—22no—
217C4—21no—
218C4—44no—
219C4—3761no—
220C4#1568923712yes—
221C4#1568924212yes—
222C4#1568929812yes—
223C2—41no—
224C2—7521no—
225C2#2024371312yes—
226C2#2024378911yescharacteristic
subgroupG/H
centerC2×C4#13093994
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1526C2
socleC22×C47#386
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#1526D47
hypocenterC47#9070
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1526

Complements

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

NH ≅ G/Ntype
2. #20252490C2semidirect
4. #20241226C2semidirect
5. C47×#1526C2semidirect
112. #1526D47semidirect
134. C47#9070semidirect

Maximal subgroups

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

subgroupindex
#202537132
#202524902
#202418062
#202412262
C47×#15262
#907047