Finite Group Databasebeta

← #38767796

Subgroups — #38767796

Order 25472.

320 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 53 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#27164168C211yesmaximal, characteristic, complemented
2#27164151C211yesmaximal, characteristic, complemented
3#27163927C211yesmaximal, characteristic, complemented
4#27163755C211yesmaximal, characteristic, complemented
5#27163752C211yesmaximal, characteristic, complemented
6#27163742C211yesmaximal, characteristic, complemented
7#538×C199C211yesmaximal, characteristic, complemented
8C22×D796C2211yescharacteristic
9C22×C22⋊D199—21no—
10C2×#15691873—21no—
11C2×#15691871—21no—
12C2×(C2×D199)⋊C4—21no—
13#20263875—21no—
14#20263875—21no—
15C22⋊(C199⋊C8)C2211yescharacteristic
16C22⋊(C199⋊C8)C2211yescharacteristic
17C199⋊#95C2211yescharacteristic
18#126×C199—21no—
19#124×C199—21no—
20#123×C199—21no—
21C22⋊Q8×C199C2211yescharacteristic
22#121×C199—21no—
23#121×C199C2211yescharacteristic
24#120×C199—21no—
25C4×D4×C199—21no—
26C2×C22⋊C4×C199C2211yescharacteristic
27C23×D199—21no—
28C2×D796—21no—
29C2×D796—21no—
30C2×D796—42no—
31C2×C22⋊D199—42no—
32C2×C22⋊D199—42no—
33C2×C22⋊D199—42no—
34C22×Dic199—21no—
35#15691873—42no—
36#15691871—42no—
37C2×C199⋊C8—21no—
38C2×C199⋊C8—21no—
39(C2×D199)⋊C4—42no—
40C24×C199D411yescharacteristic
41C2×Q8×C199D411yescharacteristic, complemented
42C2×D4×C199—21no—
43C2×D4×C199—21no—
44C2×D4×C199—21no—
45C2×D4×C199D411yescharacteristic, complemented
46C22×C4×C199—21no—
47C22×C4×C199D411yescharacteristic
48C22×C4×C199C2311yescharacteristic
49C4⋊C4×C199—21no—
50C4⋊C4×C199—21no—
51C4⋊C4×C199D411yescharacteristic, complemented
52C4⋊C4×C199D411yescharacteristic, complemented
53C22⋊C4×C199—21no—
54C22⋊C4×C199—21no—
55C22⋊C4×C199—21no—
56C22⋊C4×C199—21no—
57C22⋊C4×C199—21no—
58C22⋊C4×C199—21no—
59C22⋊C4×C199—21no—
60C42×C199—21no—
61C22×D199—22no—
62C22×D199—41no—
63C22×D199—41no—
64C22×D199—41no—
65C22×D199—41no—
66C22×D199—41no—
67C22×D199—41no—
68D796—22no—
69D796—22no—
70D796—41no—
71D796—41no—
72C22⋊D199—48no—
73C2×Dic199—22no—
74C2×Dic199—41no—
75C2×Dic199—41no—
76C199⋊C8—41no—
77C199⋊C8—41no—
78C23×C199—21no—
79C23×C199—21no—
80C23×C199—21no—
81C23×C199—21no—
82C23×C199—41no—
83C23×C199—41no—
84C23×C199C2×D411yescharacteristic
85Q8×C199—41no—
86D4×C199—41no—
87D4×C199—41no—
88D4×C199—41no—
89D4×C199—81no—
90D4×C199—81no—
91C2×C4×C199—21no—
92C2×C4×C199—21no—
93C2×C4×C199—21no—
94C2×C4×C199—21no—
95C2×C4×C199—41no—
96C2×C4×C199—41no—
97C2×C4×C199—81no—
98C2×C4×C199C2×D411yescharacteristic
99C2×C4×C199C2×D411yescharacteristic
100C22×C199—21no—
101C22×C199—21no—
102C22×C199—42no—
103C22×C199—41no—
104C22×C199—41no—
105C22×C199—41no—
106C22×C199—41no—
107C22×C199—41no—
108C22×C199—41no—
109C22×C199—81no—
110C22×C199#12011yescharacteristic
111C22×C199#13511yescharacteristic
112C22×C199AGL(1,ℤ/8)11yescharacteristic
113C2×D199—42no—
114C2×D199—42no—
115C2×D199—42no—
116C2×D199—81no—
117C2×D199—81no—
118C2×D199—81no—
119C2×D199—81no—
120C4×C199—21no—
121C4×C199—21no—
122C4×C199—41no—
123C4×C199—41no—
124C4×C199—41no—
125C4×C199—41no—
126Dic199—42no—
127C2×C199—21no—
128C2×C199—21no—
129C2×C199—81no—
130C2×C199—81no—
131C2×C199#44911yescharacteristic
132C2×C199#45311yescharacteristic
133C2×C199B(4,𝔽2)11yescharacteristic
134D199—82no—
135C199#160011yescharacteristic, complemented, Sylow
136#1600—1991nomaximal, Sylow
137#538D19911yescharacteristic, complemented
138#450—1991no—
139#447—1991no—
140#392—1991no—
141#328—1991no—
142#327—1991no—
143#323—1991no—
144C22×D4—3981no—
145C22×D4—1991no—
146C2×SD16—3981no—
147C2×D8—3981no—
148#126—21no—
149#124—21no—
150#123—21no—
151C22⋊Q8C2×D19911yescharacteristic
152#121—21no—
153#121C2×D19911yescharacteristic
154#120—21no—
155C4×D4—21no—
156C2×C22⋊C4—3981no—
157C2×C22⋊C4C2×D19911yescharacteristic
158#102—3981no—
159#102—3981no—
160C22⋊C8—1991no—
161C22⋊C8—1991no—
162#95—1991no—
163C24—3981no—
164C24C22⋊D19911yescharacteristic
165C2×Q8C22⋊D19911yescharacteristic, complemented
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—7962no—
170C2×D4—7962no—
171C2×D4—7962no—
172C2×D4—7962no—
173C2×D4—3981no—
174C2×D4—3981no—
175C2×D4C22⋊D19911yescharacteristic, complemented
176C22×C4—21no—
177C22×C4—3981no—
178C22×C4C22⋊D19911yescharacteristic
179C22×C4C22×D19911yescharacteristic
180SD16—7962no—
181D8—7962no—
182C2×C8—3981no—
183C2×C8—3981no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4C22⋊D19911yescharacteristic, complemented
187C4⋊C4C22⋊D19911yescharacteristic, complemented
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—7962no—
196C42—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—41no—
203C23—7961no—
204C23—7961no—
205C23—7961no—
206C23—7961no—
207C23—7961no—
208C23—7961no—
209C23—3982no—
210C23C2×C22⋊D19911yescharacteristic
211Q8—41no—
212D4—41no—
213D4—41no—
214D4—41no—
215D4—81no—
216D4—81no—
217D4—7961no—
218D4—7961no—
219D4—7968no—
220D4—3982no—
221D4—3982no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—81no—
229C2×C4—7961no—
230C2×C4—7961no—
231C2×C4—3982no—
232C2×C4C2×C22⋊D19911yescharacteristic
233C2×C4C2×C22⋊D19911yescharacteristic
234C8—7961no—
235C8—7961no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—81no—
246C22—7962no—
247C22—7962no—
248C22—7962no—
249C22—15921no—
250C22—15921no—
251C22—15921no—
252C22—15921no—
253C22#2026392411yescharacteristic
254C22#2026397511yescharacteristic
255C22#2026398011yescharacteristic
256C4—21no—
257C4—21no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—7962no—
263C2—21no—
264C2—21no—
265C2—81no—
266C2—81no—
267C2—15922no—
268C2#2716416311yescharacteristic
269C2#2716418611yescharacteristic
270C2#2716420511yescharacteristic
subgroupG/H
centerC22#20263924
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#538×C199C2
socleC22×C199#120
hypercenter#538D199
hypocenterC199#1600
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #27164168C2semidirect
2. #27164151C2semidirect
3. #27163927C2semidirect
4. #27163755C2semidirect
5. #27163752C2semidirect
6. #27163742C2semidirect
7. #538×C199C2semidirect
41. C2×Q8×C199D4semidirect
45. C2×D4×C199D4semidirect
51. C4⋊C4×C199D4semidirect
52. C4⋊C4×C199D4semidirect
135. C199#1600semidirect
137. #538D199semidirect
165. C2×Q8C22⋊D199semidirect
175. C2×D4C22⋊D199semidirect
186. C4⋊C4C22⋊D199semidirect
187. C4⋊C4C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
#271641682
#271641512
#271639272
#271637552
#271637522
#271637422
#538×C1992
#1600199