Finite Group Databasebeta

← #38769865

Subgroups — #38769865

Order 25472.

478 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#20263981C211yesmaximal, characteristic, complemented
2#27164585C211yesmaximal, characteristic, complemented
3C2×#20263883C211yesmaximal, characteristic, complemented
4#27164023C211yesmaximal, characteristic, complemented
5C2×#20263872C211yesmaximal, characteristic, complemented
6#27163778C211yesmaximal, characteristic, complemented
7C2×#127×C199C211yesmaximal, characteristic, complemented
8C2×#15691886C2211yescharacteristic
9#20263981—22no—
10#20263981—22no—
11C2×#15691874—21no—
12C2×#15691871—21no—
13C2×#15691870C2211yescharacteristic
14C2×#15691870C2211yescharacteristic
15#20263951—21no—
16#20263942—21no—
17#20263926—22no—
18#20263924—22no—
19#20263914C2211yescharacteristic
20#20263883—22no—
21#20263876—22no—
22#20263872—22no—
23C22×D4×C199—22no—
24C22×D4×C199C2211yescharacteristic
25C22×D4×C199C2211yescharacteristic
26#127×C199—22no—
27#127×C199—22no—
28C2×C42×C199C2211yescharacteristic
29#15691886—21no—
30#15691886—21no—
31#15691886—21no—
32#15691886—21no—
33C2×Dic398D411yescharacteristic, complemented
34C2×D796D411yescharacteristic, complemented
35C2×C22⋊D199—24no—
36C2×C22⋊D199—21no—
37C2×C4×D199—21no—
38#15691874—24no—
39#15691871—24no—
40#15691870—21no—
41#15691870—21no—
42#15691870—21no—
43#15691870D412yescomplemented
44C2×C199⋊C8—21no—
45C2×C199⋊C8—21no—
46C4⋊Dic199D412yescomplemented
47C22⋊Dic199—24no—
48C22⋊Dic199—21no—
49C4×Dic199—21no—
50C24×C199—21no—
51C24×C199D412yes—
52C2×D4×C199—28no—
53C2×D4×C199—21no—
54C2×D4×C199—21no—
55C2×D4×C199—21no—
56C2×D4×C199—21no—
57C2×D4×C199—21no—
58C2×D4×C199—21no—
59C2×D4×C199—21no—
60C2×D4×C199—21no—
61C2×D4×C199—21no—
62C2×D4×C199—21no—
63C2×D4×C199—21no—
64C2×D4×C199—48no—
65C2×D4×C199D412yescomplemented
66C22×C4×C199—21no—
67C22×C4×C199C2311yescharacteristic
68C42×C199—21no—
69C42×C199D412yescomplemented
70C22×D199—21no—
71Dic398—21no—
72Dic398—21no—
73D796—21no—
74D796—21no—
75C22⋊D199—41no—
76C22⋊D199—41no—
77C22⋊D199—84no—
78C2×Dic199—22no—
79C2×Dic199—21no—
80C4×D199—41no—
81C4×D199—41no—
82C199⋊C8—22no—
83C199⋊C8—41no—
84C23×C199—22no—
85C23×C199—22no—
86C23×C199—22no—
87C23×C199—22no—
88C23×C199—44no—
89C23×C199—44no—
90C23×C199—41no—
91C23×C199—41no—
92C23×C199C2×D411yescharacteristic
93D4×C199—22no—
94D4×C199—22no—
95D4×C199—416no—
96D4×C199—41no—
97D4×C199—41no—
98D4×C199—41no—
99D4×C199—41no—
100D4×C199—41no—
101D4×C199—41no—
102C2×C4×C199—24no—
103C2×C4×C199—22no—
104C2×C4×C199C2×D411yescharacteristic
105C2×C4×C199C2×D411yescharacteristic
106C2×C4×C199C2×D411yescharacteristic
107C2×C4×C199C2×D411yescharacteristic
108C2×C4×C199C2×D411yescharacteristic
109C2×C4×C199#4111yescharacteristic
110C22×C199—22no—
111C22×C199—44no—
112C22×C199—44no—
113C22×C199—42no—
114C22×C199—42no—
115C22×C199—44no—
116C22×C199—82no—
117C22×C199—82no—
118C22×C199—82no—
119C22×C199#12011yescharacteristic
120C22×C199#12011yescharacteristic
121C22×C199#12111yescharacteristic
122C2×D199—41no—
123C2×D199—42no—
124C4×C199—24no—
125C4×C199#12011yescharacteristic
126C4×C199#12111yescharacteristic
127C4×C199#12111yescharacteristic
128C4×C199#12711yescharacteristic
129Dic199—41no—
130Dic199—42no—
131C2×C199—21no—
132C2×C199—21no—
133C2×C199—82no—
134C2×C199—82no—
135C2×C199#39211yescharacteristic
136C2×C199#45312yes—
137D199—81no—
138C199#200211yescharacteristic, complemented, Sylow
139#2002—1991nomaximal, Sylow
140C2×AGL(1,ℤ/8)—1991no—
141#535—1991no—
142C2×#127D19911yescharacteristic, complemented
143C2×#104—1991no—
144#418—1991no—
145C2×#100—1991no—
146#339—1991no—
147C2×#41—1991no—
148C22×D4—22no—
149C22×D4C2×D19911yescharacteristic
150C22×D4C2×D19911yescharacteristic
151AGL(1,ℤ/8)—3982no—
152AGL(1,ℤ/8)—3982no—
153C2×SD16—3981no—
154C2×D8—3981no—
155C2×M16—1991no—
156C2×M16—1991no—
157#127—22no—
158#127—22no—
159#127—3981no—
160#124—3981no—
161#121—3982no—
162#120—3982no—
163#117—1991no—
164C2×C42C2×D19911yescharacteristic
165#104—3982no—
166#102—3982no—
167#100—3982no—
168C24—21no—
169C24C22⋊D19912yes—
170#41—3981no—
171#41—3981no—
172#41—3981no—
173#41—3981no—
174C2×Q8—1991no—
175C2×D4—28no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—48no—
188C2×D4—3981no—
189C2×D4—3984no—
190C2×D4—1991no—
191C2×D4C22⋊D19912yescomplemented
192C22×C4—21no—
193C22×C4—3981no—
194C22×C4C22×D19911yescharacteristic
195SD16—3984no—
196D8—3984no—
197M16—3981no—
198M16—3981no—
199M16—3981no—
200M16—1992no—
201C2×C8—3981no—
202C2×C8—3981no—
203C4⋊C4—1992no—
204C22⋊C4—3984no—
205C22⋊C4—3981no—
206C42—21no—
207C42—3981no—
208C42C22⋊D19912yescomplemented
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—44no—
214C23—44no—
215C23—41no—
216C23—41no—
217C23—3981no—
218C23C2×C22⋊D19911yescharacteristic
219Q8—3981no—
220Q8—3981no—
221D4—22no—
222D4—22no—
223D4—416no—
224D4—41no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—7961no—
231D4—7961no—
232D4—15924no—
233D4—3981no—
234D4—3981no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—7961no—
238C2×C4—7961no—
239C2×C4—3982no—
240C2×C4—3981no—
241C2×C4D4×D19911yescharacteristic
242C2×C4D4×D19911yescharacteristic
243C2×C4D4×D19911yescharacteristic
244C2×C4C2×C22⋊D19911yescharacteristic
245C2×C4C2×C22⋊D19911yescharacteristic
246C2×C4#1569188511yescharacteristic
247C8—7961no—
248C8—3982no—
249C22—22no—
250C22—44no—
251C22—44no—
252C22—42no—
253C22—42no—
254C22—44no—
255C22—82no—
256C22—82no—
257C22—82no—
258C22—7961no—
259C22—7962no—
260C22#2026392311yescharacteristic
261C22#2026392411yescharacteristic
262C22#2026392711yescharacteristic
263C4—24no—
264C4—7961no—
265C4—7962no—
266C4#2026392311yescharacteristic
267C4#2026392611yescharacteristic
268C4#2026392711yescharacteristic
269C4#2026395111yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2—82no—
274C2—15921no—
275C2#2716392611yescharacteristic
276C2#2716418512yes—
subgroupG/H
centerC22#20263923
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
FittingC2×#127×C199C2
socleC22×C199#120
hypercenterC2×#127D199
hypocenterC199#2002
LCSG ▷ C22×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#20263981C2semidirect
2. #27164585C2semidirect
3. C2×#20263883C2semidirect
4. #27164023C2semidirect
5. C2×#20263872C2semidirect
6. #27163778C2semidirect
7. C2×#127×C199C2semidirect
33. C2×Dic398D4semidirect
34. C2×D796D4semidirect
43. #15691870D4semidirect
46. C4⋊Dic199D4semidirect
65. C2×D4×C199D4semidirect
69. C42×C199D4semidirect
138. C199#2002semidirect
142. C2×#127D199semidirect
191. C2×D4C22⋊D199semidirect
208. C42C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
C2×#202639812
#271645852
C2×#202638832
#271640232
C2×#202638722
#271637782
C2×#127×C1992
#2002199