Finite Group Databasebeta

← #38767959

Subgroups — #38767959

Order 25472.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#27164741C211yesmaximal, characteristic, complemented
2#27164364C211yesmaximal, characteristic, complemented
3#27164210C211yesmaximal, characteristic
4#27163763C211yesmaximal, characteristic, complemented
5Q8⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic, complemented
7#460×C199C211yesmaximal, characteristic, complemented
8C2×#15691871—21no—
9#20263955C2211yescharacteristic
10#20263950—22no—
11#20263943—21no—
12#20263931—21no—
13C4×Dic398—21no—
14C4×D796C2211yescharacteristic
15#20263913—21no—
16C4⋊(C199⋊C8)C2211yescharacteristic
17#20263876—21no—
18#20263876—22no—
19C4×C199⋊C8C2211yescharacteristic
20C2×SD16×C199—21no—
21#127×C199C2211yescharacteristic
22C4×Q8×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24#102×C199—21no—
25C2×Dic398—21no—
26C2×D796D411yescharacteristic
27C2×C4×D199—21no—
28#15691871—42no—
29C2×C199⋊C8—21no—
30C2×C199⋊C8—21no—
31C4⋊Dic199—21no—
32C4⋊Dic199D412yes—
33C4⋊Dic199D411yescharacteristic
34Dic199⋊C4—22no—
35(C2×D199)⋊C4—22no—
36(C2×D199)⋊C4—22no—
37(C2×D199)⋊C4—21no—
38C4×Dic199—22no—
39C2×Q8×C199D411yescharacteristic
40C2×D4×C199—21no—
41C2×D4×C199—21no—
42C2×D4×C199—21no—
43SD16×C199—42no—
44C2×C8×C199—21no—
45C4⋊C4×C199—21no—
46C4⋊C4×C199D411yescharacteristic
47C42×C199—21no—
48C42×C199C2311yescharacteristic
49C22×D199—21no—
50Dic398—41no—
51Dic398—41no—
52D796—21no—
53D796—21no—
54C2×Dic199—21no—
55C2×Dic199—22no—
56C4×D199—41no—
57C4×D199—41no—
58C199⋊C8—41no—
59C199⋊C8—41no—
60C23×C199—41no—
61Q8×C199—21no—
62Q8×C199—21no—
63D4×C199—42no—
64D4×C199—41no—
65D4×C199—41no—
66D4×C199—41no—
67D4×C199—41no—
68C2×C4×C199—21no—
69C2×C4×C199—21no—
70C2×C4×C199C2×D411yescharacteristic
71C2×C4×C199C2×D411yescharacteristic
72C2×C4×C199C2×D411yescharacteristic
73C8×C199—41no—
74C22×C199—81no—
75C22×C199—81no—
76C22×C199—81no—
77C22×C199#12011yescharacteristic
78C2×D199—41no—
79C2×D199—41no—
80C2×D199—41no—
81C4×C199—21no—
82C4×C199—41no—
83C4×C199—41no—
84C4×C199#13511yescharacteristic
85C4×C199#13511yescharacteristic
86C4×C199AGL(1,ℤ/8)11yescharacteristic
87C4×C199AGL(1,ℤ/8)11yescharacteristic
88Dic199—41no—
89Dic199—42no—
90C2×C199—161no—
91C2×C199#44911yescharacteristic
92C2×C199#44911yescharacteristic
93C2×C199#45411yescharacteristic
94D199—81no—
95C199#162811yescharacteristic, complemented, Sylow
96#1628—1991nomaximal, Sylow
97#556—1991no—
98#486—1991no—
99#460D19911yescharacteristic, complemented
100#459—1991no—
101#332—1991no—
102Q8⋊C8—1991no—
103D4⋊C8—1991no—
104C2×SD16—21no—
105C2×D8—3981no—
106C4⋊Q8—1991no—
107#127C2×D19911yescharacteristic
108#126—3982no—
109#124—3981no—
110C22⋊Q8—3981no—
111C4×Q8—3981no—
112C4×Q8C2×D19911yescharacteristic
113C4×D4—1991no—
114#117—3981no—
115C4⋊C8—1991no—
116C4⋊C8C2×D19911yescharacteristic
117#102—21no—
118#102—3981no—
119#102—3982no—
120C4×C8—1991no—
121C2×Q8—3981no—
122C2×Q8C22⋊D19911yescharacteristic
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—1991no—
127C22×C4—3981no—
128SD16—42no—
129D8—7962no—
130C2×C8—21no—
131C2×C8—3981no—
132C2×C8—3981no—
133C4⋊C4—21no—
134C4⋊C4—3982no—
135C4⋊C4—3981no—
136C4⋊C4—1991no—
137C4⋊C4—1992no—
138C4⋊C4C22⋊D19911yescharacteristic
139C22⋊C4—3982no—
140C22⋊C4—3982no—
141C22⋊C4—3981no—
142C42—21no—
143C42—3982no—
144C42C22×D19911yescharacteristic
145C23—41no—
146C23—3981no—
147Q8—21no—
148Q8—21no—
149Q8—7961no—
150Q8—7961no—
151D4—42no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—3981no—
157D4—3981no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—7961no—
161C2×C4—7961no—
162C2×C4—3981no—
163C2×C4—3982no—
164C2×C4D4×D19911yescharacteristic
165C2×C4D4×D19911yescharacteristic
166C2×C4C2×C22⋊D19911yescharacteristic
167C8—41no—
168C8—7961no—
169C8—7961no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—7961no—
174C22—7961no—
175C22—7961no—
176C22#2026392311yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—7961no—
181C4—7962no—
182C4#2026397511yescharacteristic
183C4#2026397611yescharacteristic
184C4#2026398111yescharacteristic
185C4#2026398211yescharacteristic
186C2—161no—
187C2—15921no—
188C2#2716416111yescharacteristic
189C2#2716416211yescharacteristic
190C2#2716418911yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#460×C199C2
socleC22×C199#120
prime radical#27164210C2
hypercenter#460D199
hypocenterC199#1628
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #27164741C2semidirect
2. #27164364C2semidirect
4. #27163763C2semidirect
5. Q8⋊(C199⋊C8)C2semidirect
6. #27163745C2semidirect
7. #460×C199C2semidirect
95. C199#1628semidirect
99. #460D199semidirect

Maximal subgroups

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

subgroupindex
#271647412
#271643642
#271642102
#271637632
Q8⋊(C199⋊C8)2
#271637452
#460×C1992
#1628199