Finite Group Databasebeta

← #38769249

Subgroups — #38769249

Order 25472.

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

normal subgroups only

This group has 444 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#27164493C211yesmaximal, characteristic, complemented
2#27164023C212yesmaximal, complemented
3C2×#20263872C211yesmaximal, characteristic, complemented
4#27163778C212yesmaximal, complemented
5C2×#127×C199C211yesmaximal, characteristic
6C2×#15691870C2212yes—
7D4×Dic199—24no—
8#20263914C2212yes—
9C2×C22⋊Dic199—22no—
10#20263876—24no—
11#20263872—22no—
12C22×D4×C199—22no—
13C22×D4×C199C2211yescharacteristic
14C22×D4×C199C2211yescharacteristic
15#127×C199—22no—
16#127×C199C414yescomplemented
17C2×C42×C199C2211yescharacteristic
18C22×Dic199—24no—
19#15691870—22no—
20#15691870—22no—
21C2×C199⋊C8—22no—
22C4⋊Dic199D414yescomplemented
23C22⋊Dic199—24no—
24C22⋊Dic199—22no—
25C4×Dic199—22no—
26C24×C199—21no—
27C24×C199D412yes—
28C2×D4×C199—28no—
29C2×D4×C199—22no—
30C2×D4×C199—22no—
31C2×D4×C199—21no—
32C2×D4×C199—22no—
33C2×D4×C199—22no—
34C2×D4×C199—48no—
35C2×D4×C199C2×C412yes—
36C2×D4×C199C2×C412yescomplemented
37C2×D4×C199D412yes—
38C22×C4×C199—21no—
39C22×C4×C199C2311yescharacteristic
40C42×C199—21no—
41C42×C199C2×C412yescomplemented
42C2×Dic199—24no—
43C2×Dic199—84no—
44C199⋊C8—42no—
45C23×C199—22no—
46C23×C199—22no—
47C23×C199—22no—
48C23×C199—44no—
49C23×C199—44no—
50C23×C199—41no—
51C23×C199—41no—
52C23×C199C22⋊C414yes—
53C23×C199C2×D411yescharacteristic
54D4×C199—24no—
55D4×C199—24no—
56D4×C199—24no—
57D4×C199—416no—
58D4×C199—42no—
59C2×C4×C199—24no—
60C2×C4×C199—22no—
61C2×C4×C199C22×C411yescharacteristic
62C2×C4×C199C2×D412yes—
63C2×C4×C199C2×D411yescharacteristic
64C2×C4×C199#4112yes—
65C22×C199—22no—
66C22×C199—24no—
67C22×C199—44no—
68C22×C199—44no—
69C22×C199—44no—
70C22×C199—42no—
71C22×C199—82no—
72C22×C199—82no—
73C22×C199—82no—
74C22×C199C2×C22⋊C411yescharacteristic
75C22×C199#12011yescharacteristic
76C22×C199#12311yescharacteristic
77C4×C199—24no—
78C4×C199C4×D412yes—
79C4×C199#12112yes—
80Dic199—44no—
81C2×C199—21no—
82C2×C199—21no—
83C2×C199—44no—
84C2×C199—82no—
85C2×C199#38611yescharacteristic
86C2×C199#45312yes—
87C199#188711yescharacteristic, complemented, Sylow
88#1887—1991nomaximal, Sylow
89C2×#127D19911yescharacteristic
90#518—1991no—
91#418—1992no—
92C2×#100—1991no—
93#339—1992no—
94C22×D4—22no—
95C22×D4C2×D19911yescharacteristic
96C22×D4C2×D19911yescharacteristic
97C2×M16—1992no—
98#127—22no—
99#127Dic19914yescomplemented
100C4×D4—3984no—
101#117—1992no—
102C2×C22⋊C4—3982no—
103C2×C42C2×D19911yescharacteristic
104#102—3984no—
105#100—3982no—
106C24—21no—
107C24C22⋊D19912yes—
108C2×D4—28no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—48no—
115C2×D4C2×Dic19912yes—
116C2×D4C2×Dic19912yescomplemented
117C2×D4C22⋊D19912yes—
118C22×C4—21no—
119C22×C4—3984no—
120C22×C4C22×D19911yescharacteristic
121M16—3982no—
122M16—3982no—
123C2×C8—3982no—
124C4⋊C4—1994no—
125C22⋊C4—3984no—
126C22⋊C4—3982no—
127C42—21no—
128C42—3982no—
129C42C2×Dic19912yescomplemented
130C23—22no—
131C23—22no—
132C23—22no—
133C23—44no—
134C23—44no—
135C23—41no—
136C23—41no—
137C23C22⋊Dic19914yes—
138C23C2×C22⋊D19911yescharacteristic
139D4—24no—
140D4—24no—
141D4—24no—
142D4—416no—
143D4—42no—
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—15924no—
147C2×C4—3984no—
148C2×C4C22×Dic19911yescharacteristic
149C2×C4D4×D19912yes—
150C2×C4C2×C22⋊D19911yescharacteristic
151C2×C4#1569188512yes—
152C8—7962no—
153C22—22no—
154C22—24no—
155C22—44no—
156C22—44no—
157C22—44no—
158C22—42no—
159C22—82no—
160C22—82no—
161C22—82no—
162C22C2×C22⋊Dic19911yescharacteristic
163C22#2026392311yescharacteristic
164C22#2026393611yescharacteristic
165C4—24no—
166C4—7964no—
167C4D4×Dic19912yes—
168C4#2026392712yes—
169C2—21no—
170C2—21no—
171C2—44no—
172C2—82no—
173C2#2716389811yescharacteristic
174C2#2716418512yes—
subgroupG/H
centerC22#20263923
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting, prime radicalC2×#127×C199C2
socleC22×C199#120
cosocleC22×C4×C199C23
hypercenterC2×#127D199
hypocenterC199#1887
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #27164493C2semidirect
2. #27164023C2semidirect
3. C2×#20263872C2semidirect
4. #27163778C2semidirect
16. #127×C199C4semidirect
22. C4⋊Dic199D4semidirect
36. C2×D4×C199C2×C4semidirect
41. C42×C199C2×C4semidirect
87. C199#1887semidirect
99. #127Dic199semidirect
116. C2×D4C2×Dic199semidirect
129. C42C2×Dic199semidirect

Maximal subgroups

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

subgroupindex
#271644932
#271640232
C2×#202638722
#271637782
C2×#127×C1992
#1887199