Finite Group Databasebeta

← #38767778

Subgroups — #38767778

Order 25472.

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

normal subgroups only

This group has 272 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#27164617C211yesmaximal, characteristic, complemented
2#27164152C211yesmaximal, characteristic
3#27163932C211yesmaximal, characteristic, complemented
4#27163755C211yesmaximal, characteristic, complemented
5#27163753C211yesmaximal, characteristic, complemented
6#27163743C211yesmaximal, characteristic, complemented
7#450×C199C211yesmaximal, characteristic, complemented
8C2×#15691871—21no—
9#20263950—21no—
10#20263946—21no—
11#20263940—21no—
12#20263939—21no—
13#20263931—21no—
14#20263929C2211yescharacteristic
15(C22⋊D199)⋊C4—21no—
16C2×Dic199⋊C4C2211yescharacteristic
17C2×C22⋊Dic199—21no—
18C2×C22⋊Dic199—21no—
19#20263876—21no—
20C22⋊(C199⋊C8)C2211yescharacteristic
21C199⋊#95C2211yescharacteristic
22C22×D4×C199C2211yescharacteristic
23C2×SD16×C199—21no—
24C22⋊Q8×C199C2211yescharacteristic
25#102×C199—21no—
26C22⋊C8×C199C2211yescharacteristic
27C2×D796D411yescharacteristic
28C2×C22⋊D199—21no—
29C22×Dic199—21no—
30C22×Dic199D411yescharacteristic
31C22×Dic199D411yescharacteristic
32C2×C4×D199—21no—
33#15691871—42no—
34C2×C199⋊C8—21no—
35C4⋊Dic199—21no—
36C4⋊Dic199—21no—
37C4⋊Dic199D411yescharacteristic
38Dic199⋊C4—21no—
39Dic199⋊C4—21no—
40Dic199⋊C4—21no—
41C22⋊Dic199—42no—
42C22⋊Dic199—42no—
43(C2×D199)⋊C4—21no—
44(C2×D199)⋊C4—21no—
45(C2×D199)⋊C4—21no—
46(C2×D199)⋊C4—21no—
47C4×Dic199—21no—
48C24×C199—21no—
49C2×Q8×C199D411yescharacteristic
50C2×D4×C199—21no—
51C2×D4×C199—21no—
52C2×D4×C199—42no—
53C22×C4×C199C2311yescharacteristic
54SD16×C199—42no—
55C2×C8×C199—21no—
56C4⋊C4×C199—21no—
57C4⋊C4×C199D411yescharacteristic
58C22⋊C4×C199—21no—
59C22×D199—21no—
60D796—41no—
61C22⋊D199—41no—
62C22⋊D199—41no—
63C2×Dic199—21no—
64C2×Dic199—21no—
65C2×Dic199—21no—
66C2×Dic199—21no—
67C2×Dic199—21no—
68C2×Dic199—41no—
69C2×Dic199—41no—
70C2×Dic199—41no—
71C2×Dic199—41no—
72C4×D199—81no—
73C199⋊C8—41no—
74C23×C199—22no—
75C23×C199—41no—
76C23×C199—41no—
77C23×C199—41no—
78C23×C199—41no—
79C23×C199—41no—
80C23×C199—41no—
81C23×C199C2×D411yescharacteristic
82Q8×C199—41no—
83D4×C199—22no—
84D4×C199—22no—
85D4×C199—41no—
86D4×C199—41no—
87C2×C4×C199—21no—
88C2×C4×C199—21no—
89C2×C4×C199—41no—
90C2×C4×C199C2×D411yescharacteristic
91C2×C4×C199C2×D411yescharacteristic
92C8×C199—41no—
93C22×C199—21no—
94C22×C199—21no—
95C22×C199—42no—
96C22×C199—42no—
97C22×C199—42no—
98C22×C199—81no—
99C22×C199—81no—
100C22×C199—81no—
101C22×C199—81no—
102C22×C199#12011yescharacteristic
103C22×C199#13511yescharacteristic
104C22×C199AGL(1,ℤ/8)11yescharacteristic
105C2×D199—41no—
106C2×D199—41no—
107C2×D199—41no—
108C4×C199—21no—
109C4×C199—21no—
110C4×C199—41no—
111C4×C199—41no—
112Dic199—41no—
113Dic199—41no—
114Dic199—41no—
115Dic199—81no—
116C2×C199—21no—
117C2×C199—21no—
118C2×C199—82no—
119C2×C199#44911yescharacteristic
120C2×C199#45411yescharacteristic
121C2×C199B(4,𝔽2)11yescharacteristic
122D199—81no—
123C199#159711yescharacteristic, complemented, Sylow
124#1597—1991nomaximal, Sylow
125#539—1991no—
126#450D19911yescharacteristic, complemented
127#447—1991no—
128#394—1991no—
129#328—1991no—
130#327—1991no—
131#324—1991no—
132C22×D4C2×D19911yescharacteristic
133C2×SD16—21no—
134C2×D8—3981no—
135#126—3981no—
136#125—3981no—
137#123—3981no—
138#123—3981no—
139C22⋊Q8—3981no—
140C22⋊Q8C2×D19911yescharacteristic
141#121—1991no—
142C4×D4—3981no—
143C2×C4⋊C4—1991no—
144C2×C22⋊C4—3981no—
145C2×C22⋊C4—3981no—
146#102—21no—
147#102—3981no—
148C22⋊C8—1991no—
149C22⋊C8C2×D19911yescharacteristic
150#95—1991no—
151C24—21no—
152C2×Q8C22⋊D19911yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—42no—
156C2×D4—3981no—
157C2×D4—1991no—
158C22×C4—3981no—
159C22×C4—3981no—
160C22×C4—1991no—
161C22×C4—1991no—
162C22×C4C22×D19911yescharacteristic
163SD16—42no—
164D8—7962no—
165C2×C8—21no—
166C2×C8—3981no—
167C4⋊C4—21no—
168C4⋊C4—3981no—
169C4⋊C4—3981no—
170C4⋊C4—3981no—
171C4⋊C4—3981no—
172C4⋊C4—3981no—
173C4⋊C4—1991no—
174C4⋊C4C22⋊D19911yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—7962no—
177C22⋊C4—7962no—
178C22⋊C4—3981no—
179C22⋊C4—3981no—
180C22⋊C4—3981no—
181C22⋊C4—3981no—
182C42—3981no—
183C23—22no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—3981no—
191C23D4×D19911yescharacteristic
192Q8—41no—
193D4—22no—
194D4—22no—
195D4—41no—
196D4—41no—
197D4—7961no—
198D4—7961no—
199D4—7961no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—7961no—
204C2×C4—7961no—
205C2×C4—7961no—
206C2×C4—7961no—
207C2×C4—15921no—
208C2×C4—3981no—
209C2×C4—3981no—
210C2×C4—3981no—
211C2×C4—3981no—
212C2×C4—3981no—
213C2×C4D4×D19911yescharacteristic
214C2×C4C2×C22⋊D19911yescharacteristic
215C8—41no—
216C8—7961no—
217C22—21no—
218C22—21no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—81no—
223C22—81no—
224C22—81no—
225C22—81no—
226C22—7961no—
227C22—7961no—
228C22—7961no—
229C22#2026392311yescharacteristic
230C22#2026397611yescharacteristic
231C22#2026398211yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—7961no—
237C4—7961no—
238C4—7961no—
239C4—15921no—
240C2—21no—
241C2—21no—
242C2—82no—
243C2—15921no—
244C2#2716416211yescharacteristic
245C2#2716418911yescharacteristic
246C2#2716420411yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#450×C199C2
socleC22×C199#120
prime radical#27164152C2
hypercenter#450D199
hypocenterC199#1597
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #27164617C2semidirect
3. #27163932C2semidirect
4. #27163755C2semidirect
5. #27163753C2semidirect
6. #27163743C2semidirect
7. #450×C199C2semidirect
123. C199#1597semidirect
126. #450D199semidirect

Maximal subgroups

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

subgroupindex
#271646172
#271641522
#271639322
#271637552
#271637532
#271637432
#450×C1992
#1597199