Finite Group Databasebeta

← #25009423

Subgroups — #25009423

Order 11008.

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

normal subgroups only

This group has 392 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#19142541C211yesmaximal, characteristic
2#19135751C211yesmaximal, characteristic
3#19133639C212yesmaximal, complemented
4#19132623C212yesmaximal, complemented
5C43×#1318C211yesmaximal, characteristic
6#14583530C2211yescharacteristic
7#14583235—22no—
8#14582904—21no—
9#14582838C2212yes—
10#14582811—22no—
11#14582805—21no—
12#14582779—21no—
13#14582774—22no—
14#14582668C2211yescharacteristic
15#14582659—21no—
16#14582637—22no—
17C43×#390C2211yescharacteristic
18C43×#324C2212yes—
19C22×Dic86—21no—
20C23×Dic43—21no—
21C2×#10621877—21no—
22#13063324—41no—
23#13063316—41no—
24#13063304—22no—
25#13063284—21no—
26C2×C4⋊Dic43—21no—
27C2×C4⋊Dic43D412yes—
28C2×C22⋊Dic43—21no—
29C2×C22⋊Dic43—22no—
30#13063259—42no—
31#13063250—42no—
32#13063246—41no—
33#13063246—41no—
34C43⋊#95—22no—
35C43⋊#95—22no—
36C43⋊#95—22no—
37C43⋊#95D412yes—
38C22×D4×C43D411yescharacteristic
39C2×C4⋊C4×C43C2311yescharacteristic
40C2×C22⋊C4×C43—22no—
41C2×C42×C43D411yescharacteristic
42C22⋊C8×C43—22no—
43C24×C43—21no—
44C2×D4×C43—21no—
45C2×D4×C43—41no—
46C2×D4×C43—41no—
47C2×D4×C43Q1612yes—
48C22×C4×C43—21no—
49C22×C4×C43C2×D411yescharacteristic
50C22×C4×C43C2×D412yes—
51C22×Dic43—22no—
52C22×Dic43—22no—
53C22×Dic43—82no—
54C22×Dic43—81no—
55C2×Dic86—22no—
56C2×Dic86—42no—
57C2×Dic86—81no—
58C2×Dic86—81no—
59C2×C8×C43—42no—
60C4⋊C4×C43—41no—
61C22⋊C4×C43—44no—
62C42×C43—41no—
63C22⋊Dic43—41no—
64C22⋊Dic43—82no—
65C22⋊Dic43—81no—
66C4⋊Dic43—21no—
67C4⋊Dic43—21no—
68C4⋊Dic43—22no—
69C4⋊Dic43—42no—
70C4⋊Dic43—81no—
71Dic43⋊C4—42no—
72C4×Dic43—41no—
73#10621877—41no—
74#10621877—41no—
75C2×C43⋊C8—41no—
76C23×C43—22no—
77C23×C43—42no—
78C23×C43—42no—
79C23×C43—41no—
80C23×C43—41no—
81C23×C43#12011yescharacteristic
82D4×C43—42no—
83D4×C43—81no—
84C2×C4×C43—21no—
85C2×C4×C43—21no—
86C2×C4×C43—22no—
87C2×C4×C43—22no—
88C2×C4×C43—41no—
89C2×C4×C43—41no—
90C2×C4×C43—42no—
91C2×C4×C43C2×Q1611yescharacteristic
92C2×C4×C43#13711yescharacteristic
93C2×Dic43—42no—
94C2×Dic43—42no—
95C2×Dic43—42no—
96C2×Dic43—42no—
97C2×Dic43—42no—
98C2×Dic43—41no—
99C2×Dic43—41no—
100C2×Dic43—42no—
101C2×Dic43—84no—
102Dic86—84no—
103Dic86—82no—
104C8×C43—82no—
105C43⋊C8—81no—
106#12221—431nomaximal, Sylow
107C22×C43—21no—
108C22×C43—21no—
109C22×C43—44no—
110C22×C43—42no—
111C22×C43—82no—
112C22×C43—81no—
113C22×C43—81no—
114C22×C43#45111yescharacteristic
115C22×C43#45511yescharacteristic
116C22×C43B(4,𝔽2)11yescharacteristic
117C4×C43—21no—
118C4×C43—21no—
119C4×C43—42no—
120C4×C43—42no—
121Dic43—82no—
122Dic43—81no—
123Dic43—81no—
124Dic43—82no—
125#2804—431no—
126#2027—431no—
127#1617—432no—
128#1353—432no—
129#1318D4311yescharacteristic
130C2×C43—21no—
131C2×C43—21no—
132C2×C43—82no—
133C2×C43#160211yescharacteristic
134C2×C43#219811yescharacteristic
135C2×C43#220211yescharacteristic
136#544—431no—
137#484—862no—
138#418—861no—
139#400—432no—
140#393—862no—
141#391—861no—
142#390C2×D4311yescharacteristic
143#386—861no—
144#385—862no—
145#343—431no—
146#339—861no—
147#328—862no—
148#324C2×D4312yes—
149C43#1222111yescharacteristic, complemented, Sylow
150C22×Q8—861no—
151C22×D4D17211yescharacteristic
152C23×C4—861no—
153C2×M16—861no—
154C4⋊Q8—1721no—
155#125—1721no—
156C22⋊Q8—862no—
157#117—861no—
158C2×C4⋊C4—432no—
159C2×C4⋊C4—861no—
160C2×C4⋊C4C22×D4311yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—862no—
163C2×C22⋊C4—861no—
164C2×C42D17211yescharacteristic
165#107—1722no—
166#103—1722no—
167#102—1721no—
168#102—1721no—
169C22⋊C8—22no—
170#95—432no—
171#95—862no—
172#95—862no—
173#95—862no—
174C24—21no—
175C2×Q8—862no—
176C2×Q8—3441no—
177C2×Q8—3441no—
178C2×Q8—1722no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4Dic17212yes—
183C22×C4—21no—
184C22×C4—862no—
185C22×C4—862no—
186C22×C4—3442no—
187C22×C4—3441no—
188C22×C4C2×D17211yescharacteristic
189C22×C4D4×D4312yes—
190M16—1721no—
191M16—1721no—
192C2×C8—42no—
193C2×C8—1721no—
194C4⋊C4—41no—
195C4⋊C4—861no—
196C4⋊C4—861no—
197C4⋊C4—862no—
198C4⋊C4—3441no—
199C4⋊C4—1722no—
200C4⋊C4—1722no—
201C22⋊C4—44no—
202C22⋊C4—3442no—
203C22⋊C4—3441no—
204C22⋊C4—1721no—
205C42—41no—
206C42—1721no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23#1306329511yescharacteristic
213Q8—3442no—
214Q8—3444no—
215D4—42no—
216D4—81no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—3444no—
225C2×C4—1722no—
226C2×C4—1722no—
227C2×C4—1722no—
228C2×C4—1722no—
229C2×C4—1722no—
230C2×C4—1721no—
231C2×C4—1721no—
232C2×C4—1722no—
233C2×C4C2×Dic17211yescharacteristic
234C2×C4#1306335811yescharacteristic
235C8—82no—
236C8—3441no—
237C22—21no—
238C22—21no—
239C22—44no—
240C22—42no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22#1458305711yescharacteristic
245C22#1458307911yescharacteristic
246C22#1458308711yescharacteristic
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—3442no—
252C4—3441no—
253C4—3442no—
254C4—3441no—
255C2—21no—
256C2—21no—
257C2—82no—
258C2#1913355411yescharacteristic
259C2#1913659311yescharacteristic
260C2#1913661311yescharacteristic
subgroupG/H
centerC22#14583087
commutator, cosocleC2×C4⋊C4×C43C23
FrattiniC2×C4⋊C4C22×D43
FittingC43×#1318C2
socleC22×C43B(4,𝔽2)
prime radicalC22×D4×C43D4
hypercenter#1318D43
hypocenterC43#12221
LCSG ▷ C2×C4⋊C4×C43 ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1318

Complements

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

NH ≅ G/Ntype
3. #19133639C2semidirect
4. #19132623C2semidirect
149. C43#12221semidirect

Maximal subgroups

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

subgroupindex
#191425412
#191357512
#191336392
#191326232
C43×#13182
#1222143