Finite Group Databasebeta

← #38769375

Subgroups — #38769375

Order 25472.

344 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 85 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#27164785C211yesmaximal, characteristic, complemented
2#27164582C211yesmaximal, characteristic
3#27164022C212yesmaximal, complemented
4#27163774C212yesmaximal, complemented
5#410×C199C211yesmaximal, characteristic, complemented
6C22×D796C2211yescharacteristic
7#20263962—24no—
8C2×#15691870—22no—
9C2×#15691870C2212yes—
10#20263951C412yescomplemented
11#20263944C412yescomplemented
12#20263929—22no—
13#20263923—22no—
14#20263914C2211yescharacteristic
15#20263875—24no—
16C2×#41×C199C2211yescharacteristic
17#117×C199C2212yes—
18#99×C199—22no—
19C23×D199—21no—
20C2×D796—22no—
21C2×D796—21no—
22C2×D796—22no—
23C2×D796C2×C412yes—
24C2×C22⋊D199—24no—
25#15691870—24no—
26#15691870—22no—
27#15691870—22no—
28C2×C199⋊C8—24no—
29C2×C199⋊C8D414yescomplemented
30C4⋊Dic199—21no—
31C22⋊Dic199—21no—
32(C2×D199)⋊C4—24no—
33C4×Dic199C2×C412yes—
34#41×C199—22no—
35#41×C199—22no—
36C2×Q8×C199C2×C411yescharacteristic
37C2×D4×C199C2×C411yescharacteristic
38C2×D4×C199D412yes—
39C22×C4×C199D412yes—
40C22×C4×C199C2311yescharacteristic
41C4⋊C4×C199—22no—
42C22⋊C4×C199—22no—
43C42×C199—22no—
44C22×D199—22no—
45C22×D199—42no—
46C22×D199—82no—
47D796—24no—
48D796—41no—
49D796—41no—
50C22⋊D199—84no—
51C2×Dic199—22no—
52C199⋊C8—24no—
53C23×C199C22⋊C412yes—
54C23×C199C2×D411yescharacteristic
55Q8×C199—22no—
56D4×C199—24no—
57D4×C199—22no—
58C2×C4×C199—22no—
59C2×C4×C199—22no—
60C2×C4×C199—42no—
61C2×C4×C199C22⋊C412yes—
62C2×C4×C199C22×C411yescharacteristic
63C2×C4×C199C2×D412yes—
64C2×C4×C199C2×D411yescharacteristic
65C2×C4×C199#4112yes—
66C22×C199—22no—
67C22×C199—41no—
68C22×C199—42no—
69C22×C199C2×C22⋊C411yescharacteristic
70C22×C199#12112yes—
71C2×D199—44no—
72C2×D199—42no—
73C2×D199—84no—
74C4×C199—22no—
75C4×C199—82no—
76C4×C199C4×D412yes—
77C4×C199#12011yescharacteristic
78C4×C199#12311yescharacteristic
79Dic199—42no—
80C2×C199—21no—
81C2×C199—22no—
82C2×C199—42no—
83C2×C199#38611yescharacteristic
84D199—82no—
85C199#191011yescharacteristic, complemented, Sylow
86#1910—1991nomaximal, Sylow
87#568—1991no—
88#535—1991no—
89#418—1992no—
90#410D19911yescharacteristic, complemented
91H(3,ℤ/4)—1992no—
92C2×#41C2×D19911yescharacteristic
93C22×D4—1991no—
94#131—3984no—
95C2×M16—3982no—
96C2×M16—1992no—
97#127—1992no—
98#124—1992no—
99#121—3982no—
100#120—3982no—
101#117—1991no—
102#117C2×D19912yes—
103#102—3984no—
104#99—22no—
105C24—3981no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D19911yescharacteristic
109C2×D4—3984no—
110C2×D4—3982no—
111C2×D4—3981no—
112C2×D4—3982no—
113C2×D4—1992no—
114C2×D4C4×D19911yescharacteristic
115C2×D4C22⋊D19912yes—
116C22×C4D79612yes—
117C22×C4C22×D19911yescharacteristic
118M16—3984no—
119M16—3982no—
120M16—3982no—
121C2×C8—3984no—
122C2×C8—1994no—
123C4⋊C4—22no—
124C4⋊C4—3981no—
125C22⋊C4—22no—
126C22⋊C4—3981no—
127C22⋊C4—3984no—
128C42—22no—
129C42—1992no—
130C23—7962no—
131C23—15922no—
132C23—3982no—
133C23(C2×D199)⋊C412yes—
134C23C2×C22⋊D19911yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—7961no—
139D4—7961no—
140D4—15924no—
141D4—3984no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—3982no—
146C2×C4(C2×D199)⋊C412yes—
147C2×C4C2×C4×D19911yescharacteristic
148C2×C4D4×D19912yes—
149C2×C4C2×D79611yescharacteristic
150C2×C4#1569188512yes—
151C8—3984no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—7964no—
156C22—7962no—
157C22—15924no—
158C22C2×(C2×D199)⋊C411yescharacteristic
159C22#2026392712yes—
160C4—22no—
161C4—82no—
162C4—7962no—
163C4(C22⋊D199)⋊C412yes—
164C4#2026392511yescharacteristic
165C4#2026393911yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—15922no—
170C2#2716389511yescharacteristic
subgroupG/H
centerC2#27163895
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#410×C199C2
socleC2×C199#386
prime radical#27164582C2
cosocleC22×C4×C199C23
hypercenter#410D199
hypocenterC199#1910
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #27164785C2semidirect
3. #27164022C2semidirect
4. #27163774C2semidirect
5. #410×C199C2semidirect
10. #20263951C4semidirect
11. #20263944C4semidirect
29. C2×C199⋊C8D4semidirect
85. C199#1910semidirect
90. #410D199semidirect

Maximal subgroups

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

subgroupindex
#271647852
#271645822
#271640222
#271637742
#410×C1992
#1910199