Finite Group Databasebeta

← #38770839

Subgroups — #38770839

Order 25472.

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

normal subgroups only

This group has 258 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#27164698C211yesmaximal, characteristic, complemented
2#27164208C211yesmaximal, characteristic, complemented
3#27164205C211yesmaximal, characteristic
4#27163813C211yesmaximal, characteristic, complemented
5#27163809C211yesmaximal, characteristic, complemented
6#27163806C211yesmaximal, characteristic, complemented
7#455×C199C211yesmaximal, characteristic, complemented
8#20263948—21no—
9#20263937—21no—
10#20263936—21no—
11#20263936C2211yescharacteristic
12#20263931—21no—
13#20263923—21no—
14#20263923C2211yescharacteristic
15(C22⋊D199)⋊C4—21no—
16#20263912—21no—
17C22⋊C4×D199—21no—
18#20263870—21no—
19#20263870—21no—
20#20263870C2211yescharacteristic
21#20263870C2211yescharacteristic
22H(5,𝔽2)×C199C2211yescharacteristic
23#137×C199—21no—
24#124×C199C2211yescharacteristic
25#104×C199—21no—
26#100×C199C2211yescharacteristic
27C23×D199D411yescharacteristic
28C2×C22⋊D199—21no—
29C2×C22⋊D199—41no—
30C22×Dic199D411yescharacteristic, complemented
31C2×C4×D199—21no—
32C2×C4×D199—21no—
33C4⋊Dic199—21no—
34Dic199⋊C4—21no—
35Dic199⋊C4—21no—
36Dic199⋊C4—21no—
37Dic199⋊C4—41no—
38C22⋊Dic199—21no—
39C22⋊Dic199—21no—
40C22⋊Dic199—21no—
41C22⋊Dic199—41no—
42C22⋊Dic199D411yescharacteristic
43C22⋊Dic199D411yescharacteristic, complemented
44(C2×D199)⋊C4—21no—
45(C2×D199)⋊C4—21no—
46(C2×D199)⋊C4—21no—
47(C2×D199)⋊C4—41no—
48C4×Dic199—21no—
49#41×C199—21no—
50#41×C199—41no—
51C2×Q8×C199D411yescharacteristic, complemented
52C2×D4×C199—21no—
53C2×D4×C199—21no—
54C2×D4×C199—41no—
55C2×D4×C199C2311yescharacteristic
56Q16×C199—41no—
57SD16×C199—41no—
58M16×C199—21no—
59C22⋊C4×C199—21no—
60C22⋊C4×C199—21no—
61C42×C199D411yescharacteristic, complemented
62C22×D199—21no—
63C22×D199—42no—
64C22×D199—41no—
65C22⋊D199—42no—
66C22⋊D199—82no—
67C2×Dic199—21no—
68C2×Dic199—21no—
69C2×Dic199—21no—
70C2×Dic199—41no—
71C2×Dic199—41no—
72C2×Dic199—41no—
73C4×D199—42no—
74C4×D199—81no—
75C23×C199—21no—
76C23×C199—21no—
77C23×C199C2×D411yescharacteristic
78C23×C199C2×D411yescharacteristic
79Q8×C199—21no—
80Q8×C199—41no—
81D4×C199—21no—
82D4×C199—41no—
83D4×C199—41no—
84D4×C199—41no—
85D4×C199—41no—
86C2×C4×C199—21no—
87C2×C4×C199—21no—
88C2×C4×C199—21no—
89C2×C4×C199—21no—
90C2×C4×C199—41no—
91C2×C4×C199C2×D411yescharacteristic
92C8×C199—41no—
93C22×C199—21no—
94C22×C199—21no—
95C22×C199—41no—
96C22×C199—41no—
97C22×C199—41no—
98C22×C199—81no—
99C22×C199—81no—
100C22×C199#12011yescharacteristic
101C2×D199—22no—
102C2×D199—41no—
103C2×D199—41no—
104C2×D199—42no—
105C2×D199—81no—
106C4×C199—21no—
107C4×C199—41no—
108C4×C199—41no—
109C4×C199—41no—
110Dic199—41no—
111Dic199—41no—
112Dic199—41no—
113Dic199—81no—
114Dic199—81no—
115C2×C199—21no—
116C2×C199—41no—
117C2×C199—41no—
118C2×C199—81no—
119C2×C199B(4,𝔽2)11yescharacteristic
120D199—42no—
121C199#219711yescharacteristic, complemented, Sylow
122#2197—1991nomaximal, Sylow
123#551—1991no—
124#458—1991no—
125B(4,𝔽2)—1991no—
126#455D19911yescharacteristic, complemented
127#354—1991no—
128#352—1991no—
129AΣL(1,𝔽16)—1991no—
130H(5,𝔽2)C2×D19911yescharacteristic
131#137—21no—
132#126—3981no—
133#124C2×D19911yescharacteristic
134#123—3981no—
135#123—3981no—
136#123—1991no—
137C22⋊Q8—3981no—
138#120—3981no—
139#120—1991no—
140C4×D4—3981no—
141#117—3981no—
142C2×C22⋊C4—3981no—
143#104—21no—
144#100C2×D19911yescharacteristic
145#99—3981no—
146#99—3981no—
147#99—1991no—
148#99—1991no—
149C24—1991no—
150#41—21no—
151#41—41no—
152C2×Q8C22⋊D19911yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—7961no—
157C2×D4—3981no—
158C2×D4C22×D19911yescharacteristic
159C22×C4—3981no—
160C22×C4—3981no—
161C22×C4—1991no—
162Q16—41no—
163SD16—41no—
164M16—21no—
165C4⋊C4—7961no—
166C4⋊C4—3981no—
167C4⋊C4—3981no—
168C4⋊C4—3981no—
169C4⋊C4—3981no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—7961no—
173C22⋊C4—7961no—
174C22⋊C4—3981no—
175C22⋊C4—3981no—
176C22⋊C4—3981no—
177C22⋊C4—3981no—
178C22⋊C4—3981no—
179C22⋊C4—3981no—
180C22⋊C4—1991no—
181C22⋊C4—1991no—
182C42—3981no—
183C42C22⋊D19911yescharacteristic, complemented
184C23—21no—
185C23—21no—
186C23—7961no—
187C23—7962no—
188C23—3981no—
189C23D4×D19911yescharacteristic
190C23D4×D19911yescharacteristic
191Q8—21no—
192Q8—41no—
193D4—21no—
194D4—41no—
195D4—41no—
196D4—41no—
197D4—41no—
198D4—7962no—
199D4—15922no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—7962no—
206C2×C4—7961no—
207C2×C4—7961no—
208C2×C4—7961no—
209C2×C4—15921no—
210C2×C4—3981no—
211C2×C4—3981no—
212C2×C4—3981no—
213C2×C4C2×C22⋊D19911yescharacteristic
214C8—41no—
215C22—21no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22—7961no—
223C22—7962no—
224C22—7961no—
225C22—15921no—
226C22—3982no—
227C22#2026392311yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—7961no—
233C4—7961no—
234C4—7961no—
235C4—15921no—
236C4—15921no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—81no—
241C2—7962no—
242C2#2716420411yescharacteristic
subgroupG/H
centerC2#27164204
commutator, cosocleC2×D4×C199C23
FrattiniC2×D4C22×D199
Fitting#455×C199C2
socleC2×C199B(4,𝔽2)
prime radical#27164205C2
hypercenter#455D199
hypocenterC199#2197
LCSG ▷ C2×D4×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #27164698C2semidirect
2. #27164208C2semidirect
4. #27163813C2semidirect
5. #27163809C2semidirect
6. #27163806C2semidirect
7. #455×C199C2semidirect
30. C22×Dic199D4semidirect
43. C22⋊Dic199D4semidirect
51. C2×Q8×C199D4semidirect
61. C42×C199D4semidirect
121. C199#2197semidirect
126. #455D199semidirect
152. C2×Q8C22⋊D199semidirect
183. C42C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
#271646982
#271642082
#271642052
#271638132
#271638092
#271638062
#455×C1992
#2197199