Finite Group Databasebeta

← #37571464

Subgroups — #37571464

Order 24448.

320 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 49 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#27133251C211yesmaximal, characteristic, complemented
2#27132814C211yesmaximal, characteristic, complemented
3#27132797C211yesmaximal, characteristic, complemented
4#27132399C211yesmaximal, characteristic, complemented
5#27132396C211yesmaximal, characteristic, complemented
6#27132387C211yesmaximal, characteristic, complemented
7#392×C191C211yesmaximal, characteristic, complemented
8C2×#15690275—21no—
9C2×#15690272—21no—
10#20259315—21no—
11#20259308—21no—
12#20259304—21no—
13#20259301C2211yescharacteristic
14#20259296C2211yescharacteristic
15#20259295—21no—
16#20259291—21no—
17C4×C22⋊D191—21no—
18#20259243—21no—
19#20259243—21no—
20C22⋊(C191⋊C8)C2211yescharacteristic
21C22⋊(C191⋊C8)C2211yescharacteristic
22C22×D4×C191—21no—
23C22×D4×C191C2211yescharacteristic
24C2×C22⋊C4×C191—21no—
25C2×C22⋊C4×C191C2211yescharacteristic
26#95×C191C2211yescharacteristic
27C2×Dic382D411yescharacteristic, complemented
28C2×D764D411yescharacteristic, complemented
29C2×C22⋊D191—21no—
30C2×C22⋊D191—21no—
31C2×C22⋊D191—21no—
32C2×C4×D191—21no—
33#15690275—42no—
34#15690272—42no—
35C2×C191⋊C8—21no—
36C2×C191⋊C8—21no—
37C4⋊Dic191D411yescharacteristic, complemented
38C4⋊Dic191D411yescharacteristic, complemented
39Dic191⋊C4—21no—
40Dic191⋊C4—21no—
41C22⋊Dic191—21no—
42C22⋊Dic191—21no—
43C22⋊Dic191—21no—
44(C2×D191)⋊C4—21no—
45(C2×D191)⋊C4—21no—
46C4×Dic191—21no—
47C24×C191—21no—
48C24×C191D411yescharacteristic
49C2×D4×C191—21no—
50C2×D4×C191—21no—
51C2×D4×C191—42no—
52C2×D4×C191—42no—
53C2×D4×C191—42no—
54C2×D4×C191—42no—
55C22×C4×C191—21no—
56C22×C4×C191D411yescharacteristic
57C22×C4×C191C2311yescharacteristic
58C22⋊C4×C191—21no—
59C22⋊C4×C191—21no—
60C22⋊C4×C191—42no—
61C22×D191—21no—
62Dic382—41no—
63D764—41no—
64C22⋊D191—41no—
65C22⋊D191—41no—
66C22⋊D191—81no—
67C22⋊D191—81no—
68C2×Dic191—21no—
69C2×Dic191—21no—
70C2×Dic191—21no—
71C4×D191—81no—
72C191⋊C8—41no—
73C191⋊C8—41no—
74C23×C191—22no—
75C23×C191—21no—
76C23×C191—21no—
77C23×C191—21no—
78C23×C191—41no—
79C23×C191—41no—
80C23×C191—41no—
81C23×C191—41no—
82C23×C191—41no—
83C23×C191—41no—
84C23×C191—41no—
85C23×C191—41no—
86C23×C191C2×D411yescharacteristic
87D4×C191—22no—
88D4×C191—22no—
89D4×C191—41no—
90D4×C191—41no—
91D4×C191—48no—
92C2×C4×C191—22no—
93C2×C4×C191—21no—
94C2×C4×C191—41no—
95C2×C4×C191—41no—
96C2×C4×C191—41no—
97C2×C4×C191—41no—
98C2×C4×C191C2×D411yescharacteristic
99C2×C4×C191C2×D411yescharacteristic
100C22×C191—21no—
101C22×C191—21no—
102C22×C191—42no—
103C22×C191—42no—
104C22×C191—42no—
105C22×C191—41no—
106C22×C191—41no—
107C22×C191—42no—
108C22×C191—41no—
109C22×C191—81no—
110C22×C191—81no—
111C22×C191—81no—
112C22×C191—81no—
113C22×C191—81no—
114C22×C191#12011yescharacteristic
115C22×C191#13511yescharacteristic
116C22×C191AGL(1,ℤ/8)11yescharacteristic
117C2×D191—41no—
118C2×D191—41no—
119C2×D191—41no—
120C4×C191—21no—
121C4×C191—21no—
122C4×C191—42no—
123C4×C191—41no—
124Dic191—41no—
125Dic191—41no—
126Dic191—41no—
127C2×C191—21no—
128C2×C191—21no—
129C2×C191—81no—
130C2×C191—82no—
131C2×C191#44911yescharacteristic
132C2×C191#45311yescharacteristic
133C2×C191B(4,𝔽2)11yescharacteristic
134D191—81no—
135C191#160011yescharacteristic, complemented, Sylow
136#1600—1911nomaximal, Sylow
137#538—1911no—
138#450—1911no—
139#447—1911no—
140#392D19111yescharacteristic, complemented
141#328—1911no—
142#327—1911no—
143#323—1911no—
144C22×D4—21no—
145C22×D4C2×D19111yescharacteristic
146C2×SD16—3821no—
147C2×D8—3821no—
148#126—3821no—
149#124—3821no—
150#123—3821no—
151C22⋊Q8—1911no—
152#121—3821no—
153#121—1911no—
154#120—3821no—
155C4×D4—3821no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4C2×D19111yescharacteristic
158#102—3821no—
159#102—3821no—
160C22⋊C8—1911no—
161C22⋊C8—1911no—
162#95C2×D19111yescharacteristic
163C24—21no—
164C24C22⋊D19111yescharacteristic
165C2×Q8—1911no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—3821no—
173C2×D4—3821no—
174C2×D4—3821no—
175C2×D4—1911no—
176C22×C4—21no—
177C22×C4—3821no—
178C22×C4C22⋊D19111yescharacteristic
179C22×C4C22×D19111yescharacteristic
180SD16—7642no—
181D8—7642no—
182C2×C8—3821no—
183C2×C8—3821no—
184C4⋊C4—3821no—
185C4⋊C4—3821no—
186C4⋊C4—1911no—
187C4⋊C4—1911no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—42no—
191C22⋊C4—3821no—
192C22⋊C4—3821no—
193C22⋊C4—3821no—
194C22⋊C4—3821no—
195C22⋊C4—3821no—
196C42—3821no—
197C23—22no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—3821no—
210C23C2×C22⋊D19111yescharacteristic
211Q8—7641no—
212D4—22no—
213D4—22no—
214D4—41no—
215D4—41no—
216D4—48no—
217D4—15281no—
218D4—15281no—
219D4—7641no—
220D4—7641no—
221D4—7641no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—3821no—
229C2×C4—3821no—
230C2×C4—3821no—
231C2×C4—15281no—
232C2×C4D4×D19111yescharacteristic
233C2×C4D4×D19111yescharacteristic
234C8—7641no—
235C8—7641no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—42no—
244C22—41no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—7641no—
251C22—7641no—
252C22—7641no—
253C22#2025929011yescharacteristic
254C22#2025934211yescharacteristic
255C22#2025934811yescharacteristic
256C4—21no—
257C4—21no—
258C4—42no—
259C4—41no—
260C4—7641no—
261C4—7641no—
262C4—7641no—
263C2—21no—
264C2—21no—
265C2—81no—
266C2—82no—
267C2—15281no—
268C2#2713280611yescharacteristic
269C2#2713283011yescharacteristic
270C2#2713284911yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#392×C191C2
socleC22×C191#120
hypercenter#392D191
hypocenterC191#1600
LCSG ▷ C22×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #27133251C2semidirect
2. #27132814C2semidirect
3. #27132797C2semidirect
4. #27132399C2semidirect
5. #27132396C2semidirect
6. #27132387C2semidirect
7. #392×C191C2semidirect
27. C2×Dic382D4semidirect
28. C2×D764D4semidirect
37. C4⋊Dic191D4semidirect
38. C4⋊Dic191D4semidirect
135. C191#1600semidirect
140. #392D191semidirect

Maximal subgroups

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

subgroupindex
#271332512
#271328142
#271327972
#271323992
#271323962
#271323872
#392×C1912
#1600191