Finite Group Databasebeta

← #26102975

Subgroups — #26102975

Order 12032.

316 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 47 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#20241312C211yesmaximal, characteristic, complemented
2#20241265C211yesmaximal, characteristic, complemented
3C47×#1999C211yesmaximal, characteristic
4#15689087—21no—
5C22⋊(C47⋊C16)—21no—
6C22×D8×C47C411yescharacteristic
7C2×#121×C47—21no—
8C2×#102×C47—21no—
9C2×C22⋊C8×C47C2211yescharacteristic
10C47×#340C411yescharacteristic
11C2×#10625329—41no—
12C2×C47⋊C16—41no—
13C22⋊(C47⋊C8)—22no—
14C22⋊(C47⋊C8)—41no—
15C22×D4×C47—21no—
16C22×D4×C47—41no—
17C23×C4×C47D411yescharacteristic
18C2×D8×C47—41no—
19C2×D8×C47—42no—
20C22×C8×C47C2×C411yescharacteristic
21C22×C8×C47D411yescharacteristic
22#121×C47—42no—
23#121×C47—42no—
24C2×C4⋊C4×C47—21no—
25C2×C22⋊C4×C47—41no—
26#102×C47—42no—
27C22⋊C8×C47—41no—
28C24×C47—41no—
29C24×C47SD1611yescharacteristic
30C2×D4×C47—22no—
31C2×D4×C47—41no—
32C2×D4×C47—41no—
33C2×D4×C47—41no—
34C2×D4×C47—44no—
35C2×D4×C47—41no—
36C2×D4×C47—41no—
37C2×D4×C47—82no—
38C2×D4×C47—82no—
39C22×C4×C47—21no—
40C22×C4×C47—21no—
41C22×C4×C47—41no—
42C22×C4×C47—41no—
43C22×C4×C47—41no—
44C22×C4×C47C22⋊C411yescharacteristic
45C22×C4×C47D811yescharacteristic
46D8×C47—42no—
47D8×C47—42no—
48C2×C8×C47—21no—
49C2×C8×C47—21no—
50C2×C8×C47—41no—
51C2×C8×C47—41no—
52C4⋊C4×C47—22no—
53C4⋊C4×C47—41no—
54C22⋊C4×C47—44no—
55#10625329—81no—
56#10625329—81no—
57C2×C47⋊C8—42no—
58C47⋊C16—81no—
59C23×C47—21no—
60C23×C47—21no—
61C23×C47—21no—
62C23×C47—41no—
63C23×C47—41no—
64C23×C47—42no—
65C23×C47—81no—
66C23×C47—81no—
67C23×C47—81no—
68C23×C47—81no—
69C23×C47—81no—
70C23×C47—81no—
71C23×C47#10211yescharacteristic
72D4×C47—42no—
73D4×C47—42no—
74D4×C47—81no—
75D4×C47—88no—
76D4×C47—81no—
77C2×C4×C47—21no—
78C2×C4×C47—21no—
79C2×C4×C47—21no—
80C2×C4×C47—41no—
81C2×C4×C47—41no—
82C2×C4×C47—41no—
83C2×C4×C47—42no—
84C2×C4×C47—81no—
85C2×C4×C47—81no—
86C2×C4×C47—81no—
87C2×C4×C47#9911yescharacteristic
88C2×C4×C47#10411yescharacteristic
89C8×C47—41no—
90C8×C47—41no—
91C47⋊C8—82no—
92#7084—471nomaximal, Sylow
93C22×C47—21no—
94C22×C47—21no—
95C22×C47—42no—
96C22×C47—41no—
97C22×C47—41no—
98C22×C47—41no—
99C22×C47—81no—
100C22×C47—82no—
101C22×C47—82no—
102C22×C47—82no—
103C22×C47—161no—
104C22×C47—161no—
105C22×C47—161no—
106C22×C47—161no—
107C22×C47#32711yescharacteristic
108C22×C47#35911yescharacteristic
109C22×C47#36111yescharacteristic
110C4×C47—21no—
111C4×C47—41no—
112C4×C47—82no—
113C4×C47AΣL(1,𝔽16)11yescharacteristic
114C4×C47#35311yescharacteristic
115#1999D4711yescharacteristic
116#1314—471no—
117#1293—471no—
118C2×C47—21no—
119C2×C47—21no—
120C2×C47—81no—
121C2×C47—162no—
122C2×C47#134111yescharacteristic
123C2×C47#134311yescharacteristic
124C2×C47#134911yescharacteristic
125C22×D8Dic4711yescharacteristic
126C2×#121—21no—
127C2×#102—21no—
128#407—941no—
129C2×C22⋊C8C2×D4711yescharacteristic
130C22⋊C16—941no—
131#340Dic4711yescharacteristic
132C47#708411yescharacteristic, complemented, Sylow
133C22×D4—21no—
134C22×D4—41no—
135C23×C4C22⋊D4711yescharacteristic
136C2×D8—41no—
137C2×D8—42no—
138C2×M16—1881no—
139C22×C8C2×Dic4711yescharacteristic
140C22×C8C22⋊D4711yescharacteristic
141#121—42no—
142#121—42no—
143C2×C4⋊C4—21no—
144C2×C22⋊C4—41no—
145C2×C16—1881no—
146#102—42no—
147C22⋊C8—41no—
148C22⋊C8—942no—
149C22⋊C8—1881no—
150C24—41no—
151C24#1062533511yescharacteristic
152C2×D4—22no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—44no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—82no—
160C2×D4—82no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4#1062533411yescharacteristic
167C22×C4C22⋊Dic4711yescharacteristic
168D8—42no—
169D8—42no—
170M16—3761no—
171M16—3761no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—1882no—
177C4⋊C4—22no—
178C4⋊C4—41no—
179C22⋊C4—44no—
180C16—3761no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—41no—
185C23—41no—
186C23—42no—
187C23—81no—
188C23—81no—
189C23—81no—
190C23—81no—
191C23—81no—
192C23—81no—
193C23#1309394511yescharacteristic
194D4—42no—
195D4—42no—
196D4—81no—
197D4—88no—
198D4—81no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4#1309393911yescharacteristic
210C2×C4#1309395111yescharacteristic
211C8—41no—
212C8—41no—
213C8—3762no—
214C22—21no—
215C22—21no—
216C22—42no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—82no—
222C22—82no—
223C22—82no—
224C22—161no—
225C22—161no—
226C22—161no—
227C22—161no—
228C22#1568885711yescharacteristic
229C22#1568893011yescharacteristic
230C22#1568893611yescharacteristic
231C4—21no—
232C4—41no—
233C4—82no—
234C4#1568891011yescharacteristic
235C4#1568891611yescharacteristic
236C2—21no—
237C2—21no—
238C2—81no—
239C2—162no—
240C2#2024137911yescharacteristic
241C2#2024138511yescharacteristic
242C2#2024140311yescharacteristic
subgroupG/H
centerC22#15688857
commutatorC22×C8×C47C2×C4
FrattiniC2×C22⋊C8C2×D47
Fitting, prime radicalC47×#1999C2
socleC22×C47#327
cosocleC2×C22⋊C8×C47C22
hypercenter#1999D47
hypocenterC47#7084
LCSG ▷ C22×C8×C47 ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1999

Complements

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

NH ≅ G/Ntype
1. #20241312C2semidirect
2. #20241265C2semidirect
132. C47#7084semidirect

Maximal subgroups

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

subgroupindex
#202413122
#202412652
C47×#19992
#708447