Finite Group Databasebeta

← #37573571

Subgroups — #37573571

Order 24448.

380 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 71 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#27133461C211yesmaximal, characteristic, complemented
2#27133230C211yesmaximal, characteristic, complemented
3#27132679C211yesmaximal, characteristic, complemented
4#27132668C211yesmaximal, characteristic, complemented
5#27132637C211yesmaximal, characteristic, complemented
6#27132435C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C191C211yesmaximal, characteristic, complemented
8C2×#15690287C2211yescharacteristic
9#20259353—21no—
10#20259352—21no—
11#20259348—21no—
12#20259347—21no—
13#20259342—22no—
14C2×#15690271C2211yescharacteristic
15#20259318—21no—
16#20259309—21no—
17#20259293—22no—
18#20259291—22no—
19#20259281C2211yescharacteristic
20#20259281C2211yescharacteristic
21#20259248—22no—
22#20259243—22no—
23#20259237—22no—
24C2×#41×C191C2211yescharacteristic
25C22×D4×C191C2211yescharacteristic
26AGL(1,ℤ/8)×C191—22no—
27AGL(1,ℤ/8)×C191—22no—
28C2×SD16×C191—21no—
29C2×D8×C191—21no—
30C2×M16×C191C2211yescharacteristic
31#15690287—21no—
32#15690287—21no—
33#15690287—21no—
34#15690287—21no—
35C2×Dic382D411yescharacteristic
36C2×D764D411yescharacteristic
37C2×C22⋊D191—24no—
38C2×C22⋊D191—21no—
39C2×C4×D191—21no—
40#15690277—22no—
41#15690275—22no—
42#15690274—22no—
43#15690272—22no—
44#15690271—21no—
45#15690271—21no—
46C2×C191⋊C8D412yescomplemented
47C4⋊Dic191—21no—
48C4⋊Dic191—21no—
49C22⋊Dic191—24no—
50C22⋊Dic191—21no—
51C22⋊Dic191D412yes—
52C4×Dic191—21no—
53C4×Dic191—21no—
54C24×C191—21no—
55#41×C191—21no—
56#41×C191—21no—
57#41×C191—21no—
58#41×C191—21no—
59C2×Q8×C191D411yescharacteristic, complemented
60C2×D4×C191—21no—
61C2×D4×C191—21no—
62C2×D4×C191—21no—
63C2×D4×C191—21no—
64C2×D4×C191—21no—
65C2×D4×C191—21no—
66C2×D4×C191D412yes—
67C2×D4×C191D411yescharacteristic, complemented
68C22×C4×C191—21no—
69C22×C4×C191C2311yescharacteristic
70SD16×C191—24no—
71D8×C191—24no—
72M16×C191—21no—
73M16×C191D412yescomplemented
74C2×C8×C191—21no—
75C22×D191—21no—
76Dic382—21no—
77Dic382—21no—
78D764—21no—
79D764—21no—
80C22⋊D191—41no—
81C22⋊D191—41no—
82C22⋊D191—84no—
83C2×Dic191—22no—
84C2×Dic191—21no—
85C2×Dic191—41no—
86C4×D191—41no—
87C4×D191—41no—
88C191⋊C8—22no—
89C23×C191—22no—
90C23×C191—21no—
91C23×C191—41no—
92C23×C191—41no—
93C23×C191—82no—
94C23×C191C2×D411yescharacteristic
95Q8×C191—21no—
96Q8×C191—21no—
97D4×C191—21no—
98D4×C191—22no—
99D4×C191—21no—
100D4×C191—22no—
101D4×C191—41no—
102D4×C191—41no—
103D4×C191—41no—
104D4×C191—41no—
105C2×C4×C191—21no—
106C2×C4×C191—41no—
107C2×C4×C191—41no—
108C2×C4×C191C2×D411yescharacteristic
109C2×C4×C191C2×D411yescharacteristic
110C2×C4×C191C2×D411yescharacteristic
111C2×C4×C191C2×D411yescharacteristic
112C2×C4×C191C2×D411yescharacteristic
113C2×C4×C191#4111yescharacteristic
114C8×C191—22no—
115C22×C191—41no—
116C22×C191—44no—
117C22×C191—41no—
118C22×C191—42no—
119C22×C191—81no—
120C22×C191—82no—
121C22×C191—82no—
122C22×C191#12011yescharacteristic
123C22×C191#12111yescharacteristic
124C22×C191#12711yescharacteristic
125C2×D191—42no—
126C2×D191—41no—
127C4×C191—41no—
128C4×C191#12011yescharacteristic
129C4×C191#12011yescharacteristic
130C4×C191#12111yescharacteristic
131C4×C191#12111yescharacteristic
132Dic191—42no—
133Dic191—41no—
134Dic191—81no—
135C2×C191—21no—
136C2×C191—21no—
137C2×C191—21no—
138C2×C191—82no—
139C2×C191—81no—
140C2×C191#39211yescharacteristic
141D191—81no—
142C191#200811yescharacteristic, complemented, Sylow
143#2008—1911nomaximal, Sylow
144#575—1911no—
145C2×AGL(1,ℤ/8)D19111yescharacteristic, complemented
146#535—1911no—
147#421—1911no—
148#418—1911no—
149#410—1911no—
150#344—1911no—
151C2×#41—1911no—
152C2×#41C2×D19111yescharacteristic
153C22×D4C2×D19111yescharacteristic
154#137—3821no—
155#137—3821no—
156AGL(1,ℤ/8)—22no—
157AGL(1,ℤ/8)—22no—
158AGL(1,ℤ/8)—3821no—
159AGL(1,ℤ/8)—3821no—
160#135—3822no—
161C2×SD16—21no—
162C2×D8—21no—
163C2×M16—1911no—
164C2×M16C2×D19111yescharacteristic
165#127—3821no—
166#124—3821no—
167#121—3822no—
168#120—3822no—
169#117—1911no—
170#117—1911no—
171#104—3822no—
172#102—3822no—
173#99—3822no—
174C24—21no—
175#41—21no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—3821no—
180#41—3821no—
181#41—3821no—
182#41—3821no—
183C2×Q8—1911no—
184C2×Q8C22⋊D19111yescharacteristic, complemented
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—3824no—
192C2×D4—3821no—
193C2×D4—1911no—
194C2×D4C22⋊D19112yes—
195C2×D4C22⋊D19111yescharacteristic, complemented
196C22×C4—21no—
197C22×C4—3821no—
198C22×C4C22×D19111yescharacteristic
199Q16—3822no—
200SD16—24no—
201SD16—3822no—
202SD16—3822no—
203D8—24no—
204D8—3822no—
205M16—21no—
206M16—3821no—
207M16—3821no—
208M16C22⋊D19112yescomplemented
209C2×C8—21no—
210C2×C8—1912no—
211C4⋊C4—3821no—
212C4⋊C4—3821no—
213C22⋊C4—3824no—
214C22⋊C4—3821no—
215C22⋊C4—1912no—
216C42—3821no—
217C42—3821no—
218C23—22no—
219C23—21no—
220C23—41no—
221C23—41no—
222C23—82no—
223C23—3821no—
224C23D4×D19111yescharacteristic
225Q8—21no—
226Q8—21no—
227Q8—3821no—
228Q8—3821no—
229D4—21no—
230D4—21no—
231D4—22no—
232D4—22no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—3821no—
238D4—3821no—
239D4—15284no—
240D4—7641no—
241D4—7641no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—3822no—
246C2×C4—3821no—
247C2×C4—7641no—
248C2×C4—7641no—
249C2×C4—7641no—
250C2×C4D4×D19111yescharacteristic
251C2×C4D4×D19111yescharacteristic
252C2×C4C2×C22⋊D19111yescharacteristic
253C2×C4C2×C22⋊D19111yescharacteristic
254C2×C4C2×C22⋊D19111yescharacteristic
255C2×C4#1569028611yescharacteristic
256C8—22no—
257C8—3822no—
258C22—41no—
259C22—44no—
260C22—41no—
261C22—42no—
262C22—81no—
263C22—82no—
264C22—82no—
265C22—7642no—
266C22—7641no—
267C22#2025929011yescharacteristic
268C22#2025929411yescharacteristic
269C22#2025931811yescharacteristic
270C4—41no—
271C4—15281no—
272C4—7642no—
273C4—7641no—
274C4#2025929011yescharacteristic
275C4#2025929111yescharacteristic
276C4#2025929311yescharacteristic
277C4#2025929411yescharacteristic
278C2—21no—
279C2—21no—
280C2—21no—
281C2—82no—
282C2—81no—
283C2—15281no—
284C2#2713257111yescharacteristic
subgroupG/H
centerC2#27132571
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
FittingC2×AGL(1,ℤ/8)×C191C2
socleC2×C191#392
hypercenterC2×AGL(1,ℤ/8)D191
hypocenterC191#2008
LCSG ▷ C22×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27133461C2semidirect
2. #27133230C2semidirect
3. #27132679C2semidirect
4. #27132668C2semidirect
5. #27132637C2semidirect
6. #27132435C2semidirect
7. C2×AGL(1,ℤ/8)×C191C2semidirect
46. C2×C191⋊C8D4semidirect
59. C2×Q8×C191D4semidirect
67. C2×D4×C191D4semidirect
73. M16×C191D4semidirect
142. C191#2008semidirect
145. C2×AGL(1,ℤ/8)D191semidirect
184. C2×Q8C22⋊D191semidirect
195. C2×D4C22⋊D191semidirect
208. M16C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
#271334612
#271332302
#271326792
#271326682
#271326372
#271324352
C2×AGL(1,ℤ/8)×C1912
#2008191