Finite Group Databasebeta

← #38768026

Subgroups — #38768026

Order 25472.

308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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#27164653C211yesmaximal, characteristic
2#27164295C211yesmaximal, characteristic, complemented
3#27164220C211yesmaximal, characteristic, complemented
4#27163762C211yesmaximal, characteristic, complemented
5#27163747C211yesmaximal, characteristic, complemented
6#27163747C211yesmaximal, characteristic, complemented
7#486×C199C211yesmaximal, characteristic, complemented
8C2×D4×D199—21no—
9C2×D4×D199—21no—
10C2×#15691873—21no—
11#20263926—21no—
12#20263926—21no—
13#20263923—22no—
14C4×D796C2211yescharacteristic
15C4×D796C2211yescharacteristic
16#20263888—21no—
17C4⋊(C199⋊C8)C2211yescharacteristic, complemented
18C4⋊(C199⋊C8)C2211yescharacteristic, complemented
19#20263879—21no—
20#20263875—21no—
21C4⋊Q8×C199C2211yescharacteristic, complemented
22#127×C199C2211yescharacteristic
23#102×C199—22no—
24C4×C8×C199C2211yescharacteristic, complemented
25C23×D199—22no—
26C2×D796D411yescharacteristic
27C2×D796D411yescharacteristic
28C2×C22⋊D199—22no—
29C2×C22⋊D199—22no—
30D4×D199—42no—
31D4×D199—42no—
32D4×D199—42no—
33D4×D199—42no—
34C2×C4×D199—21no—
35C2×C4×D199—21no—
36#15691873—42no—
37C2×C199⋊C8—21no—
38C2×C199⋊C8—21no—
39C4⋊Dic199D411yescharacteristic
40C4⋊Dic199D411yescharacteristic
41C22⋊Dic199—22no—
42(C2×D199)⋊C4—21no—
43(C2×D199)⋊C4—21no—
44C2×Q8×C199—21no—
45C2×D4×C199—21no—
46C2×D4×C199—21no—
47C2×D4×C199D412yes—
48C2×C8×C199—21no—
49C4⋊C4×C199—21no—
50C4⋊C4×C199—21no—
51C42×C199C2311yescharacteristic
52C22×D199—21no—
53C22×D199—21no—
54C22×D199—44no—
55C22×D199—44no—
56C22×D199—82no—
57D796—21no—
58D796—21no—
59D796D812yes—
60D796SD1612yescomplemented
61C22⋊D199—44no—
62C22⋊D199—44no—
63C2×Dic199—21no—
64C2×Dic199—21no—
65C4×D199—41no—
66C4×D199—41no—
67C4×D199—41no—
68C4×D199—41no—
69C199⋊C8—41no—
70C199⋊C8—41no—
71C23×C199—22no—
72Q8×C199—41no—
73Q8×C199—41no—
74D4×C199—42no—
75D4×C199—41no—
76D4×C199—41no—
77D4×C199—41no—
78D4×C199—41no—
79C2×C4×C199—41no—
80C2×C4×C199C2×D411yescharacteristic
81C2×C4×C199C2×D411yescharacteristic
82C2×C4×C199C2×D411yescharacteristic
83C8×C199—41no—
84C22×C199—42no—
85C22×C199—42no—
86C22×C199—42no—
87C22×C199#12011yescharacteristic
88C2×D199—22no—
89C2×D199—22no—
90C2×D199—41no—
91C2×D199—41no—
92C2×D199—41no—
93C2×D199—41no—
94C2×D199—88no—
95C4×C199—21no—
96C4×C199—81no—
97C4×C199C2×D811yescharacteristic
98C4×C199C2×SD1611yescharacteristic
99C4×C199AGL(1,ℤ/8)11yescharacteristic
100C4×C199AGL(1,ℤ/8)11yescharacteristic
101Dic199—41no—
102Dic199—41no—
103C2×C199—82no—
104C2×C199#44711yescharacteristic
105C2×C199#45011yescharacteristic
106C2×C199#45511yescharacteristic
107D199—42no—
108D199—42no—
109C199#163911yescharacteristic, complemented, Sylow
110#1639—1991nomaximal, Sylow
111D42—1991no—
112#486D19911yescharacteristic, complemented
113#474—1991no—
114#461—1991no—
115#331—1991no—
116D4⋊C8—1991no—
117D4⋊C8—1991no—
118C22×D4—3981no—
119C22×D4—3981no—
120C2×SD16—3981no—
121C4⋊Q8C2×D19911yescharacteristic, complemented
122#127C2×D19911yescharacteristic
123#121—3981no—
124#121—3981no—
125#120—3982no—
126C4×D4—1991no—
127C4×D4—1991no—
128#107—3981no—
129C4⋊C8—1991no—
130C4⋊C8—1991no—
131#103—3981no—
132#102—22no—
133#102—3981no—
134C4×C8C2×D19911yescharacteristic, complemented
135C24—3982no—
136C2×Q8—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—7962no—
140C2×D4—7962no—
141C2×D4—7962no—
142C2×D4—7962no—
143C2×D4—3982no—
144C2×D4—3982no—
145C2×D4—1991no—
146C2×D4—1991no—
147C2×D4C22⋊D19912yes—
148C22×C4—3981no—
149C22×C4—3981no—
150SD16—7962no—
151C2×C8—21no—
152C2×C8—3981no—
153C2×C8—3981no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—1991no—
157C4⋊C4—1991no—
158C22⋊C4—3982no—
159C22⋊C4—3981no—
160C22⋊C4—3981no—
161C42C22×D19911yescharacteristic
162C23—22no—
163C23—7964no—
164C23—7964no—
165C23—15922no—
166C23—3981no—
167C23—3981no—
168Q8—41no—
169Q8—41no—
170D4—42no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—7964no—
176D4—7964no—
177D4—3981no—
178D4—3981no—
179D4—1992no—
180D4—1992no—
181C2×C4—41no—
182C2×C4—7961no—
183C2×C4—7961no—
184C2×C4—7961no—
185C2×C4—7961no—
186C2×C4—3981no—
187C2×C4—3981no—
188C2×C4D4×D19911yescharacteristic
189C2×C4D4×D19911yescharacteristic
190C2×C4C2×C22⋊D19911yescharacteristic
191C8—41no—
192C8—7961no—
193C8—7961no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—7961no—
198C22—7961no—
199C22—7961no—
200C22—7961no—
201C22—15928no—
202C22—3982no—
203C22—3982no—
204C22#2026392311yescharacteristic
205C4—21no—
206C4—81no—
207C4—7961no—
208C4—7961no—
209C4D8×D19911yescharacteristic
210C4SD16×D19911yescharacteristic
211C4#2026397911yescharacteristic
212C4#2026398211yescharacteristic
213C2—82no—
214C2—7962no—
215C2—7962no—
216C2#2716415011yescharacteristic
217C2#2716416711yescharacteristic
218C2#2716419511yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#486×C199C2
socleC22×C199#120
prime radical#27164653C2
hypercenter#486D199
hypocenterC199#1639
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #27164295C2semidirect
3. #27164220C2semidirect
4. #27163762C2semidirect
5. #27163747C2semidirect
6. #27163747C2semidirect
7. #486×C199C2semidirect
17. C4⋊(C199⋊C8)C22semidirect
18. C4⋊(C199⋊C8)C22semidirect
21. C4⋊Q8×C199C22semidirect
24. C4×C8×C199C22semidirect
60. D796SD16semidirect
109. C199#1639semidirect
112. #486D199semidirect
121. C4⋊Q8C2×D199semidirect
134. C4×C8C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271646532
#271642952
#271642202
#271637622
#271637472
#486×C1992
#1639199