Finite Group Databasebeta

← #37574512

Subgroups — #37574512

Order 24448.

258 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 41 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#27133344C211yesmaximal, characteristic, complemented
2#27132854C211yesmaximal, characteristic, complemented
3#27132849C211yesmaximal, characteristic
4#27132843C211yesmaximal, characteristic, complemented
5#27132457C211yesmaximal, characteristic, complemented
6#27132454C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C191C211yesmaximal, characteristic, complemented
8#20259367C2211yescharacteristic
9#20259355—21no—
10#20259315—21no—
11#20259309C2211yescharacteristic
12#20259306—21no—
13#20259303—21no—
14#20259303C2211yescharacteristic
15#20259299—21no—
16#20259291—21no—
17D4×Dic191—21no—
18#20259278—21no—
19C2×C22⋊Dic191—21no—
20#20259248—21no—
21#20259237C2211yescharacteristic
22#20259236—21no—
23#20259236—21no—
24#120×C191C2211yescharacteristic
25#100×C191C2211yescharacteristic
26#99×C191C2211yescharacteristic
27#15690287—21no—
28#15690286—41no—
29C2×Dic382D411yescharacteristic, complemented
30C2×C22⋊D191—21no—
31C2×C22⋊D191—21no—
32C2×C22⋊D191—41no—
33D4×D191—41no—
34C22×Dic191—21no—
35C22×Dic191—21no—
36C22×Dic191D411yescharacteristic, complemented
37Dic764—41no—
38#15690276—41no—
39C4⋊Dic191—21no—
40C4⋊Dic191—41no—
41Dic191⋊C4—21no—
42Dic191⋊C4—21no—
43Dic191⋊C4—21no—
44C22⋊Dic191—21no—
45C22⋊Dic191—21no—
46C22⋊Dic191—21no—
47C22⋊Dic191—21no—
48C22⋊Dic191—21no—
49C22⋊Dic191—21no—
50C22⋊Dic191—41no—
51C22⋊Dic191D411yescharacteristic, complemented
52(C2×D191)⋊C4—41no—
53C4×Dic191—21no—
54C4×Dic191D411yescharacteristic, complemented
55C24×C191D411yescharacteristic
56C2×D4×C191—21no—
57C2×D4×C191C2311yescharacteristic
58M16×C191—21no—
59C22⋊C4×C191—21no—
60C22⋊C4×C191—21no—
61C22⋊C4×C191D411yescharacteristic
62C22×D191—21no—
63C22×D191—21no—
64Dic382—21no—
65Dic382—41no—
66D764—21no—
67C22⋊D191—41no—
68C22⋊D191—41no—
69C22⋊D191—41no—
70C22⋊D191—82no—
71C2×Dic191—21no—
72C2×Dic191—21no—
73C2×Dic191—21no—
74C2×Dic191—21no—
75C2×Dic191—21no—
76C2×Dic191—21no—
77C2×Dic191—42no—
78C2×Dic191—41no—
79C2×Dic191—41no—
80C2×Dic191—81no—
81C4×D191—41no—
82C23×C191—21no—
83C23×C191—42no—
84C23×C191—41no—
85C23×C191C2×D411yescharacteristic
86C23×C191C2×D411yescharacteristic
87D4×C191—41no—
88D4×C191—42no—
89C2×C4×C191—21no—
90C2×C4×C191—41no—
91C2×C4×C191C2×D411yescharacteristic
92C8×C191—41no—
93C22×C191—22no—
94C22×C191—21no—
95C22×C191—21no—
96C22×C191—41no—
97C22×C191—41no—
98C22×C191—41no—
99C22×C191—42no—
100C22×C191—41no—
101C22×C191—81no—
102C22×C191#12011yescharacteristic
103C2×D191—41no—
104C2×D191—81no—
105C2×D191—81no—
106C4×C191—21no—
107C4×C191—41no—
108C4×C191—81no—
109Dic191—41no—
110Dic191—41no—
111Dic191—41no—
112Dic191—41no—
113Dic191—41no—
114Dic191—81no—
115C2×C191—21no—
116C2×C191—41no—
117C2×C191—41no—
118C2×C191—42no—
119C2×C191B(4,𝔽2)11yescharacteristic
120D191—81no—
121C191#219711yescharacteristic, complemented, Sylow
122#2197—1911nomaximal, Sylow
123#551—1911no—
124#458—1911no—
125B(4,𝔽2)—1911no—
126#455—1911no—
127#354—1911no—
128#352—1911no—
129AΣL(1,𝔽16)D19111yescharacteristic, complemented
130H(5,𝔽2)—1911no—
131#137—3821no—
132#126—3821no—
133#124—1911no—
134#123—3821no—
135#123—3821no—
136#123—1911no—
137C22⋊Q8—3821no—
138#120—3821no—
139#120C2×D19111yescharacteristic
140C4×D4—3821no—
141#117—3821no—
142C2×C22⋊C4—3821no—
143#104—3821no—
144#100C2×D19111yescharacteristic
145#99—3821no—
146#99—3821no—
147#99—1911no—
148#99C2×D19111yescharacteristic
149C24D76411yescharacteristic
150#41—3821no—
151#41—7641no—
152C2×Q8—1911no—
153C2×D4—21no—
154C2×D4—3821no—
155C2×D4—3821no—
156C2×D4—7641no—
157C2×D4—7641no—
158C2×D4C22×D19111yescharacteristic
159C22×C4—3821no—
160C22×C4—3821no—
161C22×C4—1911no—
162Q16—7641no—
163SD16—7641no—
164M16—21no—
165C4⋊C4—3821no—
166C4⋊C4—3821no—
167C4⋊C4—3821no—
168C4⋊C4—3821no—
169C4⋊C4—7641no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—3821no—
173C22⋊C4—3821no—
174C22⋊C4—3821no—
175C22⋊C4—3821no—
176C22⋊C4—3821no—
177C22⋊C4—3821no—
178C22⋊C4—1911no—
179C22⋊C4—7641no—
180C22⋊C4—7641no—
181C22⋊C4D76411yescharacteristic
182C42—3821no—
183C42—1911no—
184C23—21no—
185C23—41no—
186C23—42no—
187C23—3821no—
188C23—3821no—
189C23D4×D19111yescharacteristic
190C23C2×D76411yescharacteristic
191Q8—3821no—
192Q8—7641no—
193D4—41no—
194D4—42no—
195D4—3821no—
196D4—15282no—
197D4—7641no—
198D4—7641no—
199D4—7641no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—3821no—
203C2×C4—3821no—
204C2×C4—3821no—
205C2×C4—3821no—
206C2×C4—3821no—
207C2×C4—3821no—
208C2×C4—15281no—
209C2×C4—7641no—
210C2×C4—7641no—
211C2×C4—7642no—
212C2×C4—7641no—
213C2×C4D4×D19111yescharacteristic
214C8—41no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22—15281no—
225C22—15281no—
226C22—7641no—
227C22#2025929211yescharacteristic
228C4—21no—
229C4—41no—
230C4—81no—
231C4—15281no—
232C4—7641no—
233C4—7641no—
234C4—7641no—
235C4—7641no—
236C4—7641no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—15281no—
242C2#2713285111yescharacteristic
subgroupG/H
centerC2#27132851
commutator, cosocleC2×D4×C191C23
FrattiniC2×D4C22×D191
FittingAΣL(1,𝔽16)×C191C2
socleC2×C191B(4,𝔽2)
prime radical#27132849C2
hypercenterAΣL(1,𝔽16)D191
hypocenterC191#2197
LCSG ▷ C2×D4×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
1. #27133344C2semidirect
2. #27132854C2semidirect
4. #27132843C2semidirect
5. #27132457C2semidirect
6. #27132454C2semidirect
7. AΣL(1,𝔽16)×C191C2semidirect
29. C2×Dic382D4semidirect
36. C22×Dic191D4semidirect
51. C22⋊Dic191D4semidirect
54. C4×Dic191D4semidirect
121. C191#2197semidirect
129. AΣL(1,𝔽16)D191semidirect

Maximal subgroups

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

subgroupindex
#271333442
#271328542
#271328492
#271328432
#271324572
#271324542
AΣL(1,𝔽16)×C1912
#2197191