Finite Group Databasebeta

← #38768816

Subgroups — #38768816

Order 25472.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#27164854C211yesmaximal, characteristic
2#27164036C212yesmaximal, complemented
3#27164012C212yesmaximal, complemented
4#27163985C211yesmaximal, characteristic, complemented
5#410×C199C211yesmaximal, characteristic, complemented
6#20264003C412yescomplemented
7#20264002—22no—
8#20264001—22no—
9#20264000—22no—
10#20264000C412yescomplemented
11C2×#15691886—22no—
12C2×#15691886C2212yes—
13#41×D199—24no—
14C2×#15691870C2211yescharacteristic, complemented
15C22×C199⋊C8C2211yescharacteristic, complemented
16#20263883—24no—
17#20263878—22no—
18#20263875—22no—
19C22⋊(C199⋊C8)—22no—
20C2×#41×C199C2211yescharacteristic
21#117×C199C2212yescomplemented
22#99×C199—22no—
23#15691887—24no—
24#15691886—24no—
25#15691886—24no—
26#15691886—24no—
27#15691886—22no—
28#15691886—22no—
29#15691886C2×C414yes—
30#15691886D414yes—
31#15691885—28no—
32#15691885—24no—
33C2×Dic398—22no—
34C2×Dic398D412yes—
35Q8×D199—24no—
36C2×D796—22no—
37C2×D796D412yes—
38C2×C22⋊D199—24no—
39C2×C22⋊D199—22no—
40D4×D199—28no—
41D4×D199—24no—
42C2×C4×D199—24no—
43C2×C4×D199—22no—
44#15691870—22no—
45C2×C199⋊C8—21no—
46C2×C199⋊C8—22no—
47C2×C199⋊C8—21no—
48#41×C199—22no—
49#41×C199—22no—
50C2×Q8×C199C2×C411yescharacteristic
51C2×D4×C199C2×C411yescharacteristic
52C2×D4×C199D412yes—
53C22×C4×C199D412yes—
54C22×C4×C199C2311yescharacteristic
55C4⋊C4×C199—22no—
56C22⋊C4×C199—22no—
57C42×C199—22no—
58C22×D199—24no—
59C22×D199—22no—
60Dic398—24no—
61Dic398—22no—
62Dic398C22⋊C414yes—
63D796—24no—
64D796—22no—
65D796C22⋊C414yes—
66C22⋊D199—28no—
67C22⋊D199—24no—
68C22⋊D199—24no—
69C2×Dic199—24no—
70C2×Dic199—22no—
71C4×D199—28no—
72C4×D199—24no—
73C4×D199—24no—
74C199⋊C8—41no—
75C199⋊C8—41no—
76C23×C199C22⋊C412yes—
77C23×C199C2×D411yescharacteristic
78Q8×C199—22no—
79D4×C199—24no—
80D4×C199—22no—
81C2×C4×C199—22no—
82C2×C4×C199—22no—
83C2×C4×C199—42no—
84C2×C4×C199C22⋊C412yes—
85C2×C4×C199C22×C411yescharacteristic
86C2×C4×C199C2×D412yes—
87C2×C4×C199C2×D412yes—
88C2×C4×C199C2×D411yescharacteristic
89C22×C199—22no—
90C22×C199—41no—
91C22×C199—42no—
92C22×C199C2×C22⋊C411yescharacteristic
93C22×C199#12012yes—
94C2×D199—24no—
95C2×D199—82no—
96C2×D199—84no—
97C4×C199—22no—
98C4×C199—82no—
99C4×C199C2×C22⋊C412yes—
100C4×C199#12011yescharacteristic
101C4×C199#12011yescharacteristic
102Dic199—24no—
103C2×C199—22no—
104C2×C199—21no—
105C2×C199—42no—
106C2×C199#37911yescharacteristic
107D199—44no—
108C199#179411yescharacteristic, complemented, Sylow
109#1794—1991nomaximal, Sylow
110#585—1991no—
111#421—1992no—
112#416—1992no—
113#410D19911yescharacteristic, complemented
114#408—1991no—
1152−1+4—3982no—
1162−1+4—1992no—
117H(5,𝔽2)—3982no—
118H(5,𝔽2)—3982no—
119H(5,𝔽2)—1992no—
120C2×#41—3984no—
121C2×#41—3982no—
122C2×#41—1992no—
123C2×#41C2×D19911yescharacteristic
124C2×M16—1991no—
125C22×C8—1991no—
126#117C2×D19912yescomplemented
127#104—3984no—
128#103—3982no—
129#102—3982no—
130#99—22no—
131C22⋊C8—3982no—
132#41—22no—
133#41—22no—
134#41—3984no—
135#41—3984no—
136#41—3988no—
137#41—3984no—
138#41—3984no—
139#41—3984no—
140#41—3982no—
141#41—3982no—
142#41—1994no—
143#41—1994no—
144C2×Q8—3982no—
145C2×Q8—3984no—
146C2×Q8—1992no—
147C2×Q8C4×D19911yescharacteristic
148C2×D4—3982no—
149C2×D4—3988no—
150C2×D4—3984no—
151C2×D4—3984no—
152C2×D4—3982no—
153C2×D4—1992no—
154C2×D4C4×D19911yescharacteristic
155C2×D4C22⋊D19912yes—
156C22×C4—3984no—
157C22×C4—3982no—
158C22×C4D79612yes—
159C22×C4C22×D19911yescharacteristic
160M16—3982no—
161C2×C8—3982no—
162C2×C8—3981no—
163C2×C8—3981no—
164C4⋊C4—22no—
165C22⋊C4—22no—
166C42—22no—
167C23—3984no—
168C23—3982no—
169C23(C2×D199)⋊C412yes—
170C23C2×C22⋊D19911yescharacteristic
171Q8—22no—
172Q8—3984no—
173Q8—3982no—
174Q8—1994no—
175D4—24no—
176D4—22no—
177D4—3984no—
178D4—3988no—
179D4—3984no—
180D4—3984no—
181D4—3982no—
182D4—1994no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—3988no—
187C2×C4—3984no—
188C2×C4—3982no—
189C2×C4—3984no—
190C2×C4—3984no—
191C2×C4(C2×D199)⋊C412yes—
192C2×C4C2×C4×D19911yescharacteristic
193C2×C4D4×D19912yes—
194C2×C4D4×D19912yes—
195C2×C4C2×D79611yescharacteristic
196C8—7961no—
197C8—7961no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—15922no—
202C22—15924no—
203C22—3984no—
204C22C2×(C2×D199)⋊C411yescharacteristic
205C22#2026392312yes—
206C4—22no—
207C4—82no—
208C4—3984no—
209C4C22⋊C4×D19912yes—
210C4#2026392511yescharacteristic
211C4#2026392511yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—7964no—
216C2#2716387311yescharacteristic
subgroupG/H
centerC4#20263925
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#410×C199C2
socleC2×C199#379
prime radical#27164854C2
cosocleC22×C4×C199C23
hypercenter#410D199
hypocenterC199#1794
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #27164036C2semidirect
3. #27164012C2semidirect
4. #27163985C2semidirect
5. #410×C199C2semidirect
6. #20264003C4semidirect
10. #20264000C4semidirect
14. C2×#15691870C22semidirect
15. C22×C199⋊C8C22semidirect
21. #117×C199C22semidirect
108. C199#1794semidirect
113. #410D199semidirect
126. #117C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271648542
#271640362
#271640122
#271639852
#410×C1992
#1794199