Finite Group Databasebeta

← #38769370

Subgroups — #38769370

Order 25472.

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

normal subgroups only

This group has 312 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
2#27164589C211yesmaximal, characteristic
3#27164026C212yesmaximal, complemented
4#27163774C212yesmaximal, complemented
5#410×C199C211yesmaximal, characteristic
6C22×Dic398C2211yescharacteristic
7#20263962—24no—
8C2×#15691870—22no—
9C2×#15691870C2212yes—
10#20263953C412yescomplemented
11#20263942C412yescomplemented
12#20263936—22no—
13#20263934—22no—
14#20263914C2211yescharacteristic
15#20263878—24no—
16C2×#41×C199C2211yescharacteristic
17#117×C199C2212yes—
18#99×C199—22no—
19C2×Dic398—21no—
20C2×Dic398—22no—
21C2×Dic398—22no—
22C2×Dic398C2×C412yes—
23C22×Dic199—21no—
24#15691870—24no—
25#15691870—22no—
26#15691870—22no—
27C2×C199⋊C8—24no—
28C2×C199⋊C8D414yes—
29C4⋊Dic199—21no—
30Dic199⋊C4—24no—
31C22⋊Dic199—24no—
32C22⋊Dic199—21no—
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—
44Dic398—24no—
45Dic398—41no—
46Dic398—41no—
47C2×Dic199—22no—
48C2×Dic199—22no—
49C2×Dic199—42no—
50C199⋊C8—24no—
51C23×C199C22⋊C412yes—
52C23×C199C2×D411yescharacteristic
53Q8×C199—22no—
54D4×C199—24no—
55D4×C199—22no—
56C2×C4×C199—22no—
57C2×C4×C199—22no—
58C2×C4×C199—42no—
59C2×C4×C199C22⋊C412yes—
60C2×C4×C199C22×C411yescharacteristic
61C2×C4×C199C2×D412yes—
62C2×C4×C199C2×D411yescharacteristic
63C2×C4×C199#4112yes—
64C22×C199—22no—
65C22×C199—41no—
66C22×C199—42no—
67C22×C199C2×C22⋊C411yescharacteristic
68C22×C199#12112yes—
69C4×C199—22no—
70C4×C199—82no—
71C4×C199C4×D412yes—
72C4×C199#12011yescharacteristic
73C4×C199#12311yescharacteristic
74Dic199—42no—
75Dic199—42no—
76C2×C199—21no—
77C2×C199—22no—
78C2×C199—42no—
79C2×C199#38611yescharacteristic
80C199#190911yescharacteristic, complemented, Sylow
81#1909—1991nomaximal, Sylow
82#568—1991no—
83#536—1991no—
84#419—1992no—
85#410D19911yescharacteristic
86H(3,ℤ/4)—1992no—
87C2×#41C2×D19911yescharacteristic
88C22×Q8—1991no—
89#131—3984no—
90C2×M16—3982no—
91C2×M16—1992no—
92C4⋊Q8—1992no—
93#124—1992no—
94#123—3982no—
95C22⋊Q8—3982no—
96#117—1991no—
97#117C2×D19912yes—
98#103—3984no—
99#99—22no—
100#41—22no—
101#41—22no—
102C2×Q8—3982no—
103C2×Q8—3982no—
104C2×Q8—3981no—
105C2×Q8—1992no—
106C2×Q8C4×D19911yescharacteristic
107C2×D4C4×D19911yescharacteristic
108C2×D4C22⋊D19912yes—
109C22×C4—3981no—
110C22×C4D79612yes—
111C22×C4C22×D19911yescharacteristic
112M16—3984no—
113M16—3982no—
114M16—3982no—
115C2×C8—3984no—
116C2×C8—1994no—
117C4⋊C4—22no—
118C4⋊C4—3984no—
119C4⋊C4—3981no—
120C22⋊C4—22no—
121C22⋊C4—3984no—
122C22⋊C4—3981no—
123C42—22no—
124C42—1992no—
125C23(C2×D199)⋊C412yes—
126C23C2×C22⋊D19911yescharacteristic
127Q8—22no—
128Q8—7961no—
129Q8—7961no—
130Q8—3984no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—7962no—
137C2×C4—3982no—
138C2×C4—3982no—
139C2×C4(C2×D199)⋊C412yes—
140C2×C4C2×C4×D19911yescharacteristic
141C2×C4D4×D19912yes—
142C2×C4C2×D79611yescharacteristic
143C2×C4#1569188512yes—
144C8—3984no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×(C2×D199)⋊C411yescharacteristic
149C22#2026392712yes—
150C4—22no—
151C4—82no—
152C4—7962no—
153C4—7962no—
154C4(C22⋊D199)⋊C412yes—
155C4#2026392511yescharacteristic
156C4#2026393911yescharacteristic
157C2—22no—
158C2—21no—
159C2—42no—
160C2#2716389511yescharacteristic
subgroupG/H
centerC2#27163895
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#410×C199C2
socleC2×C199#386
prime radicalC2×#41×C199C22
cosocleC22×C4×C199C23
hypercenter#410D199
hypocenterC199#1909
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
3. #27164026C2semidirect
4. #27163774C2semidirect
10. #20263953C4semidirect
11. #20263942C4semidirect
80. C199#1909semidirect

Maximal subgroups

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

subgroupindex
#271647852
#271645892
#271640262
#271637742
#410×C1992
#1909199