Finite Group Databasebeta

← #38767813

Subgroups — #38767813

Order 25472.

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

normal subgroups only

This group has 244 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#27164173C211yesmaximal, characteristic, complemented
2#27164156C211yesmaximal, characteristic
3#27163938C211yesmaximal, characteristic, complemented
4#27163755C211yesmaximal, characteristic, complemented
5#27163752C211yesmaximal, characteristic
6#27163744C211yesmaximal, characteristic, complemented
7#539×C199C211yesmaximal, characteristic
8C22×Dic398C2211yescharacteristic
9C2×#15691876—21no—
10C2×#15691874—21no—
11C2×Dic199⋊C4—21no—
12#20263878—21no—
13#20263878—21no—
14C22⋊(C199⋊C8)C2211yescharacteristic
15C22⋊(C199⋊C8)C2211yescharacteristic
16C199⋊#95—21no—
17C199⋊#95C2211yescharacteristic
18#126×C199—21no—
19#125×C199—21no—
20#123×C199—21no—
21#123×C199—21no—
22C22⋊Q8×C199—21no—
23C22⋊Q8×C199C2211yescharacteristic
24#121×C199C2211yescharacteristic
25C4×D4×C199—21no—
26C2×C4⋊C4×C199C2211yescharacteristic
27C2×Dic398—21no—
28C2×Dic398—21no—
29C2×Dic398—42no—
30C22×Dic199—21no—
31C22×Dic199—21no—
32#15691876—42no—
33#15691874—42no—
34C2×C199⋊C8—21no—
35C2×C199⋊C8—21no—
36Dic199⋊C4—42no—
37C2×Q8×C199D411yescharacteristic
38C2×D4×C199—21no—
39C2×D4×C199D411yescharacteristic
40C22×C4×C199—21no—
41C22×C4×C199D411yescharacteristic
42C22×C4×C199D411yescharacteristic
43C22×C4×C199C2311yescharacteristic
44C4⋊C4×C199—21no—
45C4⋊C4×C199—21no—
46C4⋊C4×C199—21no—
47C4⋊C4×C199—21no—
48C4⋊C4×C199—21no—
49C4⋊C4×C199—21no—
50C4⋊C4×C199D411yescharacteristic
51C4⋊C4×C199D411yescharacteristic
52C22⋊C4×C199—21no—
53C22⋊C4×C199—21no—
54C22⋊C4×C199—21no—
55C22⋊C4×C199—21no—
56C22⋊C4×C199—21no—
57C42×C199—21no—
58Dic398—22no—
59Dic398—22no—
60Dic398—41no—
61Dic398—41no—
62C2×Dic199—22no—
63C2×Dic199—22no—
64C2×Dic199—41no—
65C2×Dic199—41no—
66C2×Dic199—41no—
67C2×Dic199—41no—
68C199⋊C8—41no—
69C199⋊C8—41no—
70C23×C199—21no—
71C23×C199C2×D411yescharacteristic
72Q8×C199—41no—
73D4×C199—41no—
74D4×C199—41no—
75D4×C199—41no—
76C2×C4×C199—21no—
77C2×C4×C199—21no—
78C2×C4×C199—21no—
79C2×C4×C199—21no—
80C2×C4×C199—21no—
81C2×C4×C199—21no—
82C2×C4×C199—21no—
83C2×C4×C199—41no—
84C2×C4×C199—41no—
85C2×C4×C199—81no—
86C2×C4×C199C2×D411yescharacteristic
87C2×C4×C199C2×D411yescharacteristic
88C22×C199—21no—
89C22×C199—21no—
90C22×C199—41no—
91C22×C199—41no—
92C22×C199—41no—
93C22×C199#12011yescharacteristic
94C22×C199#13511yescharacteristic
95C22×C199#13711yescharacteristic
96C4×C199—21no—
97C4×C199—21no—
98C4×C199—41no—
99C4×C199—41no—
100C4×C199—41no—
101C4×C199—41no—
102C4×C199—41no—
103Dic199—42no—
104Dic199—42no—
105C2×C199—21no—
106C2×C199—21no—
107C2×C199—81no—
108C2×C199#45211yescharacteristic
109C2×C199#45611yescharacteristic
110C2×C199B(4,𝔽2)11yescharacteristic
111C199#160311yescharacteristic, complemented, Sylow
112#1603—1991nomaximal, Sylow
113#539D19911yescharacteristic
114#451—1991no—
115#448—1991no—
116#395—1991no—
117#328—1991no—
118#327—1991no—
119#324—1991no—
120C22×Q8—1991no—
121C2×Q16—3981no—
122C2×SD16—3981no—
123#126—21no—
124#125—21no—
125#123—21no—
126#123—21no—
127C22⋊Q8—21no—
128C22⋊Q8C2×D19911yescharacteristic
129#121C2×D19911yescharacteristic
130C4×D4—21no—
131C2×C4⋊C4—3981no—
132C2×C4⋊C4C2×D19911yescharacteristic
133#103—3981no—
134#103—3981no—
135C22⋊C8—1991no—
136C22⋊C8—1991no—
137#95—3981no—
138#95—1991no—
139C2×Q8—7962no—
140C2×Q8—3981no—
141C2×Q8—3981no—
142C2×Q8C22⋊D19911yescharacteristic
143C2×D4—21no—
144C2×D4C22⋊D19911yescharacteristic
145C22×C4—21no—
146C22×C4—3981no—
147C22×C4—3981no—
148C22×C4C22⋊D19911yescharacteristic
149C22×C4C22⋊D19911yescharacteristic
150C22×C4C22×D19911yescharacteristic
151Q16—7962no—
152SD16—7962no—
153C2×C8—3981no—
154C2×C8—3981no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—7962no—
162C4⋊C4C22⋊D19911yescharacteristic
163C4⋊C4C22⋊D19911yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C42—21no—
170C23—21no—
171C23C2×C22⋊D19911yescharacteristic
172Q8—41no—
173Q8—7961no—
174Q8—7961no—
175Q8—3982no—
176Q8—3982no—
177D4—41no—
178D4—41no—
179D4—41no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—7961no—
191C2×C4—7961no—
192C2×C4—7961no—
193C2×C4—7961no—
194C2×C4—3982no—
195C2×C4—3982no—
196C2×C4C2×C22⋊D19911yescharacteristic
197C2×C4C2×C22⋊D19911yescharacteristic
198C8—7961no—
199C8—7961no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22#2026392411yescharacteristic
206C22#2026397511yescharacteristic
207C22#2026398511yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—41no—
213C4—41no—
214C4—41no—
215C4—7962no—
216C4—7962no—
217C2—21no—
218C2—21no—
219C2—81no—
220C2#2716418011yescharacteristic
221C2#2716420111yescharacteristic
222C2#2716420511yescharacteristic
subgroupG/H
centerC22#20263924
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#539×C199C2
socleC22×C199#120
prime radical#121×C199C22
hypercenter#539D199
hypocenterC199#1603
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. #27164173C2semidirect
3. #27163938C2semidirect
4. #27163755C2semidirect
6. #27163744C2semidirect
111. C199#1603semidirect

Maximal subgroups

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

subgroupindex
#271641732
#271641562
#271639382
#271637552
#271637522
#271637442
#539×C1992
#1603199