Finite Group Databasebeta

← #38767814

Subgroups — #38767814

Order 25472.

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

normal subgroups only

This group has 244 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#27164613C211yesmaximal, characteristic
2#27164176C211yesmaximal, characteristic, complemented
3#27164156C211yesmaximal, characteristic
4#27163939C211yesmaximal, characteristic, complemented
5#27163754C211yesmaximal, characteristic, complemented
6#27163743C211yesmaximal, characteristic, complemented
7#327×C199C211yesmaximal, characteristic
8C22×Dic398C2211yescharacteristic
9C2×Dic796—21no—
10C2×#15691874—21no—
11#20263948—21no—
12#20263945—21no—
13#20263936—21no—
14#20263936—21no—
15#20263934C2211yescharacteristic
16#20263932—21no—
17D4×Dic199—21no—
18C2×C4⋊Dic199—21no—
19C2×Dic199⋊C4C2211yescharacteristic
20#20263880—21no—
21#20263878—21no—
22C22⋊(C199⋊C8)C2211yescharacteristic
23C199⋊#95—21no—
24#121×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic
26#95×C199C2211yescharacteristic
27C2×Dic398—21no—
28C2×Dic398—21no—
29C2×Dic398—42no—
30C2×Dic398D411yescharacteristic
31C22×Dic199—21no—
32C22×Dic199—21no—
33C22×Dic199D411yescharacteristic
34C22×Dic199D411yescharacteristic
35Dic796—42no—
36#15691874—42no—
37C2×C199⋊C8—21no—
38C4⋊Dic199—21no—
39C4⋊Dic199—42no—
40C4⋊Dic199D411yescharacteristic
41Dic199⋊C4—21no—
42Dic199⋊C4—21no—
43Dic199⋊C4—21no—
44Dic199⋊C4—21no—
45Dic199⋊C4—21no—
46C22⋊Dic199—21no—
47C22⋊Dic199—21no—
48C22⋊Dic199—21no—
49C22⋊Dic199—21no—
50C4×Dic199—21no—
51C2×D4×C199—21no—
52C2×D4×C199D411yescharacteristic
53C22×C4×C199—21no—
54C22×C4×C199C2311yescharacteristic
55C2×C8×C199—21no—
56C4⋊C4×C199D411yescharacteristic
57C22⋊C4×C199—21no—
58Dic398—22no—
59Dic398—22no—
60Dic398—41no—
61Dic398—41no—
62Dic398—41no—
63C2×Dic199—22no—
64C2×Dic199—21no—
65C2×Dic199—21no—
66C2×Dic199—21no—
67C2×Dic199—21no—
68C2×Dic199—21no—
69C2×Dic199—21no—
70C2×Dic199—41no—
71C2×Dic199—41no—
72C2×Dic199—41no—
73C2×Dic199—81no—
74C199⋊C8—41no—
75C23×C199—21no—
76C23×C199C2×D411yescharacteristic
77D4×C199—41no—
78D4×C199—41no—
79D4×C199—41no—
80C2×C4×C199—22no—
81C2×C4×C199—21no—
82C2×C4×C199—41no—
83C2×C4×C199—41no—
84C2×C4×C199—41no—
85C2×C4×C199C2×D411yescharacteristic
86C2×C4×C199C2×D411yescharacteristic
87C8×C199—41no—
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—42no—
99C4×C199—41no—
100Dic199—41no—
101Dic199—41no—
102Dic199—42no—
103Dic199—41no—
104Dic199—41no—
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#539—1991no—
114#451—1991no—
115#448—1991no—
116#395—1991no—
117#328—1991no—
118#327D19911yescharacteristic
119#324—1991no—
120C22×Q8—1991no—
121C2×Q16—3981no—
122C2×SD16—3981no—
123#126—3981no—
124#125—3981no—
125#123—3981no—
126#123—3981no—
127C22⋊Q8—3981no—
128C22⋊Q8—1991no—
129#121C2×D19911yescharacteristic
130C4×D4—3981no—
131C2×C4⋊C4—3981no—
132C2×C4⋊C4—1991no—
133#103—3981no—
134#103—3981no—
135C22⋊C8—1991no—
136C22⋊C8C2×D19911yescharacteristic
137#95—3981no—
138#95C2×D19911yescharacteristic
139C2×Q8—7962no—
140C2×Q8—3981no—
141C2×Q8—3981no—
142C2×Q8—1991no—
143C2×D4—21no—
144C2×D4D79611yescharacteristic
145C22×C4—21no—
146C22×C4—3981no—
147C22×C4—3981no—
148C22×C4—1991no—
149C22×C4—1991no—
150C22×C4C22×D19911yescharacteristic
151Q16—7962no—
152SD16—7962no—
153C2×C8—21no—
154C2×C8—3981no—
155C4⋊C4—7962no—
156C4⋊C4—3981no—
157C4⋊C4—3981no—
158C4⋊C4—3981no—
159C4⋊C4—3981no—
160C4⋊C4—3981no—
161C4⋊C4—3981no—
162C4⋊C4—1991no—
163C4⋊C4D79611yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—3981no—
166C22⋊C4—3981no—
167C22⋊C4—3981no—
168C22⋊C4—3981no—
169C42—3981no—
170C23—21no—
171C23D4×D19911yescharacteristic
172Q8—7961no—
173Q8—7961no—
174Q8—7961no—
175Q8—3982no—
176Q8—3982no—
177D4—41no—
178D4—41no—
179D4—41no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—7961no—
186C2×C4—7961no—
187C2×C4—7961no—
188C2×C4—15921no—
189C2×C4—3982no—
190C2×C4—3981no—
191C2×C4—3981no—
192C2×C4—3981no—
193C2×C4—3981no—
194C2×C4—3981no—
195C2×C4—3981no—
196C2×C4D4×D19911yescharacteristic
197C2×C4C2×D79611yescharacteristic
198C8—41no—
199C8—7961no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22#2026392511yescharacteristic
206C22#2026397411yescharacteristic
207C22#2026398411yescharacteristic
208C4—21no—
209C4—21no—
210C4—42no—
211C4—41no—
212C4—7961no—
213C4—7961no—
214C4—7962no—
215C4—7961no—
216C4—7961no—
217C2—21no—
218C2—21no—
219C2—81no—
220C2#2716418111yescharacteristic
221C2#2716420211yescharacteristic
222C2#2716420611yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#327×C199C2
socleC22×C199#120
prime radical#121×C199C22
hypercenter#327D199
hypocenterC199#1603
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #27164176C2semidirect
4. #27163939C2semidirect
5. #27163754C2semidirect
6. #27163743C2semidirect
111. C199#1603semidirect

Maximal subgroups

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

subgroupindex
#271646132
#271641762
#271641562
#271639392
#271637542
#271637432
#327×C1992
#1603199