Finite Group Databasebeta

← #38767851

Subgroups — #38767851

Order 25472.

254 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 51 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#27164595C211yesmaximal, characteristic
2#27164331C211yesmaximal, characteristic, complemented
3#27163949C211yesmaximal, characteristic, complemented
4#27163753C211yesmaximal, characteristic, complemented
5#27163752C211yesmaximal, characteristic, complemented
6#27163743C211yesmaximal, characteristic, complemented
7#480×C199C211yesmaximal, characteristic, complemented
8C2×#15691885—21no—
9#20263933—21no—
10#20263932—21no—
11#20263929C2211yescharacteristic
12#20263928—21no—
13#20263927—21no—
14#20263927—21no—
15C2×Dic199⋊C4—21no—
16C2×Dic199⋊C4C2211yescharacteristic
17#20263888—21no—
18#20263875—21no—
19C22⋊(C199⋊C8)C2211yescharacteristic
20C199⋊#95—21no—
21C199⋊#95C2211yescharacteristic
22#121×C199C2211yescharacteristic
23C2×C4⋊C4×C199C2211yescharacteristic
24#107×C199—21no—
25#102×C199—21no—
26C22⋊C8×C199C2211yescharacteristic
27#15691885—42no—
28#15691885—42no—
29C2×Dic398—21no—
30C2×D796D411yescharacteristic
31C2×C22⋊D199—21no—
32C2×C22⋊D199—21no—
33C2×C22⋊D199—21no—
34C22×Dic199—21no—
35C22×Dic199—21no—
36C22×Dic199D411yescharacteristic
37C22×Dic199D411yescharacteristic
38C2×C4×D199—21no—
39C2×C199⋊C8—21no—
40C4⋊Dic199D411yescharacteristic
41Dic199⋊C4—21no—
42Dic199⋊C4—21no—
43Dic199⋊C4—21no—
44Dic199⋊C4—42no—
45C22⋊Dic199—21no—
46(C2×D199)⋊C4—21no—
47(C2×D199)⋊C4—21no—
48C2×D4×C199—21no—
49C2×D4×C199D411yescharacteristic
50C22×C4×C199—21no—
51C22×C4×C199C2311yescharacteristic
52C2×C8×C199—21no—
53C4⋊C4×C199—21no—
54C4⋊C4×C199—21no—
55C4⋊C4×C199D411yescharacteristic
56C22⋊C4×C199—21no—
57C22×D199—21no—
58Dic398—42no—
59D796—41no—
60C22⋊D199—42no—
61C22⋊D199—41no—
62C22⋊D199—41no—
63C22⋊D199—81no—
64C2×Dic199—21no—
65C2×Dic199—21no—
66C2×Dic199—22no—
67C2×Dic199—21no—
68C2×Dic199—21no—
69C2×Dic199—42no—
70C2×Dic199—41no—
71C2×Dic199—41no—
72C2×Dic199—41no—
73C2×Dic199D812yes—
74C4×D199—42no—
75C199⋊C8—41no—
76C23×C199—21no—
77C23×C199C2×D411yescharacteristic
78D4×C199—41no—
79D4×C199—41no—
80D4×C199—41no—
81C2×C4×C199—21no—
82C2×C4×C199—41no—
83C2×C4×C199—41no—
84C2×C4×C199—41no—
85C2×C4×C199—41no—
86C2×C4×C199C2×D411yescharacteristic
87C2×C4×C199C2×D411yescharacteristic
88C8×C199—41no—
89C22×C199—21no—
90C22×C199—21no—
91C22×C199—41no—
92C22×C199—41no—
93C22×C199—41no—
94C22×C199#12011yescharacteristic
95C22×C199C2×D811yescharacteristic
96C22×C199AGL(1,ℤ/8)11yescharacteristic
97C2×D199—41no—
98C2×D199—41no—
99C2×D199—41no—
100C4×C199—21no—
101C4×C199—21no—
102C4×C199—41no—
103C4×C199—81no—
104Dic199—22no—
105Dic199—42no—
106Dic199—41no—
107Dic199—41no—
108C2×C199—21no—
109C2×C199—21no—
110C2×C199—81no—
111C2×C199#44711yescharacteristic
112C2×C199#45611yescharacteristic
113C2×C199#45811yescharacteristic
114D199—81no—
115C199#161011yescharacteristic, complemented, Sylow
116#1610—1991nomaximal, Sylow
117#537—1991no—
118#480—1991no—
119#480D19911yescharacteristic, complemented
120#398—1991no—
121#327—1991no—
122#327—1991no—
123#324—1991no—
124C2×#41—3981no—
125C22⋊Q8—3981no—
126C22⋊Q8—3981no—
127#121—3981no—
128#121—3981no—
129#121—3981no—
130#121—1991no—
131#121C2×D19911yescharacteristic
132C2×C4⋊C4—3981no—
133C2×C4⋊C4—1991no—
134C2×C4⋊C4C2×D19911yescharacteristic
135#107—21no—
136#107—3981no—
137#102—21no—
138#102—3981no—
139C22⋊C8—1991no—
140C22⋊C8C2×D19911yescharacteristic
141#95—3981no—
142#95—1991no—
143#41—7962no—
144#41—7962no—
145C2×Q8—3981no—
146C2×D4—21no—
147C2×D4—3981no—
148C2×D4—3981no—
149C2×D4—3981no—
150C2×D4—1991no—
151C2×D4C22⋊D19911yescharacteristic
152C22×C4—21no—
153C22×C4—3981no—
154C22×C4—3981no—
155C22×C4—3981no—
156C22×C4—1991no—
157C22×C4—1991no—
158C22×C4C22×D19911yescharacteristic
159C2×C8—21no—
160C2×C8—3981no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—7962no—
164C4⋊C4—3981no—
165C4⋊C4—3981no—
166C4⋊C4—3981no—
167C4⋊C4—1991no—
168C4⋊C4C22⋊D19911yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—3981no—
171C22⋊C4—3981no—
172C22⋊C4—3981no—
173C23—21no—
174C23—3981no—
175C23D4×D19911yescharacteristic
176Q8—7962no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—7961no—
181D4—7961no—
182D4—7961no—
183D4—7962no—
184D4—15921no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—7962no—
191C2×C4—7961no—
192C2×C4—7961no—
193C2×C4—7962no—
194C2×C4—7961no—
195C2×C4—3982no—
196C2×C4—3981no—
197C2×C4—3981no—
198C2×C4—3981no—
199C2×C4—3981no—
200C2×C4—1992no—
201C2×C4D4×D19911yescharacteristic
202C2×C4C2×C22⋊D19911yescharacteristic
203C8—41no—
204C8—7961no—
205C22—21no—
206C22—21no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—7961no—
211C22—7961no—
212C22—7961no—
213C22#2026392311yescharacteristic
214C22D8×D19911yescharacteristic
215C22#2026397911yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—81no—
220C4—7962no—
221C4—7961no—
222C4—7961no—
223C4—3982no—
224C2—21no—
225C2—21no—
226C2—81no—
227C2—15921no—
228C2#2716415011yescharacteristic
229C2#2716419911yescharacteristic
230C2#2716420711yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#480×C199C2
socleC22×C199#120
prime radical#27164595C2
hypercenter#480D199
hypocenterC199#1610
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #27164331C2semidirect
3. #27163949C2semidirect
4. #27163753C2semidirect
5. #27163752C2semidirect
6. #27163743C2semidirect
7. #480×C199C2semidirect
115. C199#1610semidirect
119. #480D199semidirect

Maximal subgroups

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

subgroupindex
#271645952
#271643312
#271639492
#271637532
#271637522
#271637432
#480×C1992
#1610199