Finite Group Databasebeta

← #38769387

Subgroups — #38769387

Order 25472.

314 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 89 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#27164592C211yesmaximal, characteristic
2#27164036C212yesmaximal
3C2×#20263874C211yesmaximal, characteristic, complemented
4#27163774C212yesmaximal, complemented
5#519×C199C211yesmaximal, characteristic, complemented
6C2×#15691886C2211yescharacteristic
7C2×#15691870C2212yes—
8#20263953C412yescomplemented
9#20263944C412yescomplemented
10#20263938—22no—
11#20263931—22no—
12#20263914C2211yescharacteristic
13#20263883—24no—
14#20263874—22no—
15C22×Q8×C199C2211yescharacteristic
16C4×Q8×C199—24no—
17#117×C199—22no—
18#117×C199C2212yes—
19#15691886—22no—
20#15691886—22no—
21C2×Dic398C2×C411yescharacteristic
22C2×D796C2×C411yescharacteristic
23C2×C22⋊D199—21no—
24C2×C4×D199—21no—
25#15691870—22no—
26#15691870—22no—
27C2×C199⋊C8—22no—
28C4⋊Dic199—21no—
29Dic199⋊C4—24no—
30C22⋊Dic199—21no—
31(C2×D199)⋊C4—24no—
32C4×Dic199C2×C412yes—
33C2×Q8×C199—22no—
34C2×Q8×C199—22no—
35C2×Q8×C199C2×C412yes—
36C2×Q8×C199D412yes—
37C22×C4×C199D412yes—
38C22×C4×C199C2311yescharacteristic
39C4⋊C4×C199—24no—
40C4⋊C4×C199—22no—
41C22⋊C4×C199—22no—
42C42×C199—24no—
43C42×C199D414yes—
44C22×D199—21no—
45Dic398—22no—
46D796—22no—
47C22⋊D199—41no—
48C22⋊D199—41no—
49C2×Dic199—22no—
50C2×Dic199—21no—
51C4×D199—42no—
52C199⋊C8—42no—
53C23×C199C2×D411yescharacteristic
54Q8×C199—24no—
55Q8×C199—24no—
56C2×C4×C199—22no—
57C2×C4×C199—22no—
58C2×C4×C199—24no—
59C2×C4×C199C22⋊C414yes—
60C2×C4×C199C22×C411yescharacteristic
61C2×C4×C199C2×D412yes—
62C2×C4×C199C2×D411yescharacteristic
63C2×C4×C199#4112yes—
64C22×C199—41no—
65C22×C199C2×C22⋊C411yescharacteristic
66C22×C199#12011yescharacteristic
67C22×C199#12311yescharacteristic
68C2×D199—42no—
69C2×D199—41no—
70C4×C199—24no—
71C4×C199—44no—
72C4×C199C4×D412yes—
73C4×C199#12112yes—
74Dic199—42no—
75Dic199—41no—
76C2×C199—21no—
77C2×C199—21no—
78C2×C199—21no—
79C2×C199#38611yescharacteristic
80D199—81no—
81C199#191211yescharacteristic, complemented, Sylow
82#1912—1991nomaximal, Sylow
83#536—1991no—
84#519D19911yescharacteristic, complemented
85#421—1992no—
86C2×#101—1991no—
87H(3,ℤ/4)—1992no—
88C2×#41—1991no—
89C22×Q8C2×D19911yescharacteristic
90C2×M16—1992no—
91C4⋊Q8—1992no—
92#124—1992no—
93#123—3982no—
94C22⋊Q8—3982no—
95C4×Q8—24no—
96#117—22no—
97#117—1991no—
98#117C2×D19912yes—
99#104—3984no—
100#101—3982no—
101#41—3982no—
102#41—3982no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—1991no—
106C2×Q8C4×D19912yes—
107C2×Q8D79612yes—
108C2×D4—3981no—
109C2×D4—1991no—
110C22×C4—3981no—
111C22×C4C22⋊D19912yes—
112C22×C4C22×D19911yescharacteristic
113M16—3982no—
114M16—3982no—
115C2×C8—3982no—
116C4⋊C4—24no—
117C4⋊C4—22no—
118C4⋊C4—3984no—
119C4⋊C4—3981no—
120C22⋊C4—22no—
121C22⋊C4—3981no—
122C22⋊C4—3984no—
123C42—24no—
124C42—1992no—
125C42C22⋊D19914yes—
126C23—3981no—
127C23C2×C22⋊D19911yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—3982no—
131D4—7961no—
132D4—7961no—
133D4—3982no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—24no—
137C2×C4—7962no—
138C2×C4—3982no—
139C2×C4—3981no—
140C2×C4(C2×D199)⋊C414yes—
141C2×C4C2×C4×D19911yescharacteristic
142C2×C4C2×C22⋊D19912yes—
143C2×C4C2×D79611yescharacteristic
144C2×C4#1569188612yes—
145C8—7962no—
146C22—41no—
147C22—7962no—
148C22—7961no—
149C22C2×(C2×D199)⋊C411yescharacteristic
150C22#2026392411yescharacteristic
151C22#2026393811yescharacteristic
152C4—24no—
153C4—44no—
154C4—7962no—
155C4—7961no—
156C4C4×C22⋊D19912yes—
157C4#2026392912yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—15921no—
162C2#2716389711yescharacteristic
subgroupG/H
centerC2#27163897
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#519×C199C2
socleC2×C199#386
prime radicalC2×#15691886C22
cosocleC22×C4×C199C23
hypercenter#519D199
hypocenterC199#1912
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22×C4 ◁ #519

Complements

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

NH ≅ G/Ntype
3. C2×#20263874C2semidirect
4. #27163774C2semidirect
5. #519×C199C2semidirect
8. #20263953C4semidirect
9. #20263944C4semidirect
81. C199#1912semidirect
84. #519D199semidirect

Maximal subgroups

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

subgroupindex
#271645922
#271640362
C2×#202638742
#271637742
#519×C1992
#1912199