Finite Group Databasebeta

← #37574530

Subgroups — #37574530

Order 24448.

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

normal subgroups only

This group has 260 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#27133400C211yesmaximal, characteristic, complemented
2#27132854C211yesmaximal, characteristic, complemented
3#27132851C211yesmaximal, characteristic, complemented
4#27132830C211yesmaximal, characteristic, complemented
5#27132452C211yesmaximal, characteristic, complemented
6#27132449C211yesmaximal, characteristic, complemented
7#353×C191C211yesmaximal, characteristic, complemented
8#20259367C2211yescharacteristic
9#20259348—21no—
10#20259316—21no—
11#20259306—21no—
12#20259303—21no—
13#20259303C2211yescharacteristic
14#20259294—21no—
15#20259294—21no—
16#20259294—21no—
17#20259292—21no—
18#20259290—21no—
19#20259290C2211yescharacteristic
20#20259250—21no—
21#20259239C2211yescharacteristic
22#20259236—21no—
23#20259236—21no—
24#127×C191C2211yescharacteristic
25#99×C191C2211yescharacteristic
26#99×C191C2211yescharacteristic
27C23×D191D411yescharacteristic
28#15690287—21no—
29#15690286—41no—
30C2×D764—41no—
31C2×C22⋊D191—21no—
32C2×C22⋊D191—21no—
33C2×C22⋊D191—21no—
34C2×C22⋊D191—21no—
35C2×C22⋊D191—21no—
36D4×D191—41no—
37C22×Dic191—21no—
38C22×Dic191D411yescharacteristic, complemented
39#15690275—41no—
40#15690272—41no—
41#15690271—21no—
42C4⋊Dic191—41no—
43Dic191⋊C4—21no—
44Dic191⋊C4—21no—
45Dic191⋊C4—21no—
46C22⋊Dic191—21no—
47C22⋊Dic191—21no—
48C22⋊Dic191D411yescharacteristic
49C22⋊Dic191D411yescharacteristic, complemented
50(C2×D191)⋊C4—21no—
51(C2×D191)⋊C4—21no—
52(C2×D191)⋊C4—21no—
53(C2×D191)⋊C4—41no—
54(C2×D191)⋊C4—41no—
55C2×D4×C191—21no—
56C2×D4×C191—21no—
57C2×D4×C191D411yescharacteristic, complemented
58C2×D4×C191C2311yescharacteristic
59C22⋊C4×C191—21no—
60C22⋊C4×C191—21no—
61C42×C191D411yescharacteristic, complemented
62C22×D191—21no—
63C22×D191—21no—
64C22×D191—21no—
65C22×D191—41no—
66C22×D191—81no—
67Dic382—21no—
68D764—21no—
69D764—82no—
70C22⋊D191—41no—
71C22⋊D191—41no—
72C22⋊D191—41no—
73C22⋊D191—81no—
74C22⋊D191—81no—
75C22⋊D191—81no—
76C2×Dic191—21no—
77C2×Dic191—21no—
78C2×Dic191—21no—
79C2×Dic191—21no—
80C2×Dic191—21no—
81C2×Dic191—41no—
82C2×Dic191—81no—
83C4×D191—41no—
84C191⋊C8—41no—
85C23×C191—21no—
86C23×C191C2×D411yescharacteristic
87C23×C191C2×D411yescharacteristic
88D4×C191—41no—
89D4×C191—41no—
90D4×C191—81no—
91D4×C191—81no—
92C2×C4×C191—21no—
93C2×C4×C191—41no—
94C2×C4×C191—41no—
95C2×C4×C191C2×D411yescharacteristic
96C22×C191—21no—
97C22×C191—21no—
98C22×C191—41no—
99C22×C191—41no—
100C22×C191—41no—
101C22×C191—41no—
102C22×C191—41no—
103C22×C191#12011yescharacteristic
104C2×D191—41no—
105C2×D191—41no—
106C2×D191—42no—
107C2×D191—41no—
108C2×D191—41no—
109C2×D191—81no—
110C2×D191—81no—
111C2×D191—81no—
112C4×C191—21no—
113C4×C191—41no—
114C4×C191—81no—
115C4×C191—81no—
116Dic191—41no—
117Dic191—41no—
118Dic191—41no—
119Dic191—41no—
120C2×C191—21no—
121C2×C191—41no—
122C2×C191—41no—
123C2×C191—81no—
124C2×C191B(4,𝔽2)11yescharacteristic
125D191—81no—
126D191—81no—
127C191#220011yescharacteristic, complemented, Sylow
128#2200—1911nomaximal, Sylow
129#560—1911no—
130#458—1911no—
131B(4,𝔽2)—1911no—
132#453—1911no—
133#353D19111yescharacteristic, complemented
134#352—1911no—
135AΣL(1,𝔽16)—1911no—
136H(5,𝔽2)—1911no—
137AGL(1,ℤ/8)—3821no—
138#127C2×D19111yescharacteristic
139#126—3821no—
140#123—3821no—
141#123—3821no—
142#123—1911no—
143#121—3821no—
144#121—3821no—
145#121—3821no—
146#120—3821no—
147#120—3821no—
148#120—1911no—
149#104—3821no—
150#100—1911no—
151#99—3821no—
152#99—3821no—
153#99C2×D19111yescharacteristic
154#99C2×D19111yescharacteristic
155C24—1911no—
156#41—3821no—
157#41—7641no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—3821no—
161C2×D4—3821no—
162C2×D4—3821no—
163C2×D4—3821no—
164C2×D4—3821no—
165C2×D4—7641no—
166C2×D4—7641no—
167C2×D4D76411yescharacteristic, complemented
168C2×D4C22×D19111yescharacteristic
169C22×C4—3821no—
170C22×C4—1911no—
171SD16—7641no—
172D8—7641no—
173M16—3821no—
174C4⋊C4—3821no—
175C4⋊C4—3821no—
176C4⋊C4—3821no—
177C4⋊C4—7641no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—3821no—
181C22⋊C4—3821no—
182C22⋊C4—3821no—
183C22⋊C4—3821no—
184C22⋊C4—3821no—
185C22⋊C4—1911no—
186C22⋊C4—1911no—
187C22⋊C4—7641no—
188C22⋊C4—7641no—
189C42D76411yescharacteristic, complemented
190C23—21no—
191C23—3821no—
192C23—3821no—
193C23—3821no—
194C23—15281no—
195C23—7641no—
196C23D4×D19111yescharacteristic
197C23D4×D19111yescharacteristic
198Q8—3821no—
199D4—41no—
200D4—41no—
201D4—81no—
202D4—81no—
203D4—3821no—
204D4—15282no—
205D4—15281no—
206D4—15281no—
207D4—15281no—
208D4—7641no—
209D4—7641no—
210D4—7641no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—3821no—
215C2×C4—3821no—
216C2×C4—3821no—
217C2×C4—3821no—
218C2×C4—3821no—
219C2×C4—15281no—
220C2×C4—7641no—
221C2×C4—7641no—
222C2×C4C2×D76411yescharacteristic
223C8—7641no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—15281no—
232C22—15281no—
233C22—15281no—
234C22—7641no—
235C22—7641no—
236C22—7642no—
237C22—7641no—
238C22—7641no—
239C22#2025929211yescharacteristic
240C4—21no—
241C4—41no—
242C4—81no—
243C4—81no—
244C4—7641no—
245C4—7641no—
246C4—7641no—
247C4—7641no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—15281no—
253C2—15281no—
254C2#2713285111yescharacteristic
subgroupG/H
centerC2#27132851
commutator, cosocleC2×D4×C191C23
FrattiniC2×D4C22×D191
Fitting#353×C191C2
socleC2×C191B(4,𝔽2)
hypercenter#353D191
hypocenterC191#2200
LCSG ▷ C2×D4×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
1. #27133400C2semidirect
2. #27132854C2semidirect
3. #27132851C2semidirect
4. #27132830C2semidirect
5. #27132452C2semidirect
6. #27132449C2semidirect
7. #353×C191C2semidirect
38. C22×Dic191D4semidirect
49. C22⋊Dic191D4semidirect
57. C2×D4×C191D4semidirect
61. C42×C191D4semidirect
127. C191#2200semidirect
133. #353D191semidirect
167. C2×D4D764semidirect
189. C42D764semidirect

Maximal subgroups

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

subgroupindex
#271334002
#271328542
#271328512
#271328302
#271324522
#271324492
#353×C1912
#2200191