Finite Group Databasebeta

← #26104033

Subgroups — #26104033

Order 12032.

274 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 59 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#20245053C211yesmaximal, characteristic
2#20243772C211yesmaximal, characteristic
3C47×#1543C211yesmaximal, characteristic
4#15689890C411yescharacteristic, complemented
5#15689140—21no—
6#15689097—21no—
7#15689097C411yescharacteristic, complemented
8#15688918C412yescomplemented
9#15688916—21no—
10#15688883—21no—
11C47×#532C2211yescharacteristic
12C47×#431—21no—
13C47×#333C412yescomplemented
14C47×#331—21no—
15#13094031—42no—
16C2×#10625329—21no—
17C2×#10625329—41no—
18#13093983—21no—
19#13093983—41no—
20#13093951—42no—
21#13093939—22no—
22#13093939—41no—
23#13093939—41no—
24C2×#41×C47C2×C411yescharacteristic
25C2×#41×C47D411yescharacteristic
26C4⋊Q8×C47C2×C411yescharacteristic
27#127×C47D411yescharacteristic
28#124×C47C2×C411yescharacteristic
29#124×C47Q811yescharacteristic
30#121×C47—41no—
31C4×D4×C47—41no—
32C2×C42×C47D411yescharacteristic
33C4⋊C8×C47—22no—
34C22⋊C8×C47—41no—
35C4×C8×C47—41no—
36#41×C47—41no—
37#41×C47—41no—
38#41×C47—41no—
39#41×C47—41no—
40C2×Q8×C47C4211yescharacteristic
41C2×Q8×C47C4⋊C411yescharacteristic
42C2×D4×C47—21no—
43C2×D4×C47—21no—
44C2×D4×C47—41no—
45C2×D4×C47C22⋊C411yescharacteristic
46C2×D4×C47C4⋊C411yescharacteristic
47C22×C4×C47—21no—
48C22×C4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47C22⋊C411yescharacteristic
51C2×C8×C47—42no—
52C4⋊C4×C47—41no—
53C22⋊C4×C47—41no—
54C22⋊C4×C47—41no—
55C42×C47—21no—
56C42×C47C22⋊C411yescharacteristic
57C42×C47C4⋊C411yescharacteristic
58C22⋊Dic47—42no—
59C22⋊Dic47—41no—
60C4⋊Dic47—41no—
61C4⋊Dic47—81no—
62C4×Dic47—41no—
63C4×Dic47—81no—
64#10625329—22no—
65#10625329—41no—
66#10625329—42no—
67C2×C47⋊C8—41no—
68C2×C47⋊C8—42no—
69C23×C47—21no—
70C23×C47—21no—
71C23×C47#9911yescharacteristic
72Q8×C47—41no—
73Q8×C47—41no—
74D4×C47—41no—
75D4×C47—41no—
76D4×C47—42no—
77D4×C47—82no—
78D4×C47—81no—
79C2×C4×C47—21no—
80C2×C4×C47—21no—
81C2×C4×C47—21no—
82C2×C4×C47—21no—
83C2×C4×C47—21no—
84C2×C4×C47—21no—
85C2×C4×C47—42no—
86C2×C4×C47—41no—
87C2×C4×C47—41no—
88C2×C4×C47—41no—
89C2×C4×C47—41no—
90C2×C4×C47#9511yescharacteristic
91C2×C4×C47#9911yescharacteristic
92C2×Dic47—81no—
93C2×Dic47—82no—
94C8×C47—82no—
95C47⋊C8—42no—
96#7490—471nomaximal, Sylow
97C22×C47—21no—
98C22×C47—41no—
99C22×C47—41no—
100C22×C47—41no—
101C22×C47—81no—
102C22×C47—81no—
103C22×C47#34211yescharacteristic
104C4×C47—22no—
105C4×C47—21no—
106C4×C47—41no—
107C4×C47—41no—
108C4×C47—41no—
109C4×C47AΣL(1,𝔽16)11yescharacteristic
110C4×C47#35211yescharacteristic
111Dic47—161no—
112Dic47—162no—
113#2126—471no—
114#1884—471no—
115#1543D4711yescharacteristic
116C2×C47—21no—
117C2×C47—41no—
118C2×C47—81no—
119C2×C47—81no—
120C2×C47#139011yescharacteristic
121#568—471no—
122#532C2×D4711yescharacteristic
123#431—21no—
124#421—941no—
125#410—471no—
126#410—941no—
127#354—472no—
128#353—941no—
129#339—941no—
130#333Dic4712yescomplemented
131#331—21no—
132C47#749011yescharacteristic, complemented, Sylow
133C2×#41C4×D4711yescharacteristic
134C2×#41D18811yescharacteristic
135#131—1882no—
136C2×M16—941no—
137C2×M16—1881no—
138C4⋊Q8C2×Dic4711yescharacteristic
139#127C22⋊D4711yescharacteristic
140#124Dic9411yescharacteristic
141#124C4×D4711yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—941no—
145#117—1881no—
146C2×C42C22⋊D4711yescharacteristic
147C4⋊C8—22no—
148#104—1882no—
149#99—942no—
150#99—1881no—
151#99—1881no—
152C22⋊C8—41no—
153C4×C8—41no—
154#41—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158C2×Q8C4×Dic4711yescharacteristic
159C2×Q8C4⋊Dic4711yescharacteristic
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4Dic47⋊C411yescharacteristic
164C2×D4(C2×D47)⋊C411yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4(C2×D47)⋊C411yescharacteristic
169M16—942no—
170M16—1881no—
171M16—1882no—
172C2×C8—42no—
173C2×C8—1881no—
174C2×C8—1882no—
175C4⋊C4—41no—
176C4⋊C4—3761no—
177C4⋊C4—1881no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—1882no—
181C22⋊C4—1881no—
182C42—21no—
183C42—3761no—
184C42—1881no—
185C42Dic47⋊C411yescharacteristic
186C42C22⋊Dic4711yescharacteristic
187C23—21no—
188C23—21no—
189C23#1309393811yescharacteristic
190Q8—41no—
191Q8—41no—
192D4—41no—
193D4—41no—
194D4—42no—
195D4—82no—
196D4—81no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—3761no—
209C2×C4—3762no—
210C2×C4C47⋊#9511yescharacteristic
211C2×C4#1309393811yescharacteristic
212C8—82no—
213C8—1882no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—81no—
219C22—81no—
220C22#1568888911yescharacteristic
221C4—22no—
222C4—21no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—7521no—
227C4—7522no—
228C4#1568891111yescharacteristic
229C4#1568891411yescharacteristic
230C2—21no—
231C2—41no—
232C2—81no—
233C2—81no—
234C2#2024150711yescharacteristic
subgroupG/H
centerC4#15688911
commutatorC2×Q8×C47C42
Frattini#532C2×D47
FittingC47×#1543C2
socleC2×C47#1390
prime radical, cosocleC47×#532C22
hypercenter#1543D47
hypocenterC47#7490
LCSG ▷ C2×Q8×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1543

Complements

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

NH ≅ G/Ntype
4. #15689890C4semidirect
7. #15689097C4semidirect
8. #15688918C4semidirect
13. C47×#333C4semidirect
130. #333Dic47semidirect
132. C47#7490semidirect

Maximal subgroups

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

subgroupindex
#202450532
#202437722
C47×#15432
#749047