Finite Group Databasebeta

← #38768160

Subgroups — #38768160

Order 25472.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#27164772C211yesmaximal, characteristic
2#27164424C211yesmaximal, characteristic, complemented
3#27164410C211yesmaximal, characteristic
4#27163760C211yesmaximal, characteristic, complemented
5#27163759C211yesmaximal, characteristic, complemented
6#27163757C211yesmaximal, characteristic
7#501×C199C211yesmaximal, characteristic, complemented
8C2×#15691876—21no—
9C2×#15691873—21no—
10#20263954—21no—
11#20263950—21no—
12#20263950—21no—
13#20263947C2211yescharacteristic
14#20263943C2211yescharacteristic
15#20263940—21no—
16#20263935—21no—
17#20263933—21no—
18#20263888—21no—
19#20263886—21no—
20#20263866C2211yescharacteristic
21#20263866C2211yescharacteristic
22C4⋊Q8×C199C2211yescharacteristic
23#125×C199C2211yescharacteristic
24#107×C199—21no—
25#106×C199—21no—
26#97×C199C2211yescharacteristic
27C2×Dic398—21no—
28C2×Dic398D411yescharacteristic
29C2×D796D411yescharacteristic
30C2×C4×D199—21no—
31#15691876—42no—
32#15691873—42no—
33C2×C199⋊C8—21no—
34C2×C199⋊C8—21no—
35C4⋊Dic199—21no—
36C4⋊Dic199D411yescharacteristic
37C4⋊Dic199D411yescharacteristic
38Dic199⋊C4—21no—
39Dic199⋊C4—21no—
40Dic199⋊C4—21no—
41(C2×D199)⋊C4—21no—
42(C2×D199)⋊C4—21no—
43(C2×D199)⋊C4—21no—
44(C2×D199)⋊C4—21no—
45(C2×D199)⋊C4—21no—
46C4×Dic199—21no—
47C2×Q8×C199—21no—
48C2×C8×C199—21no—
49C4⋊C4×C199—21no—
50C4⋊C4×C199—21no—
51C4⋊C4×C199—21no—
52C4⋊C4×C199—21no—
53C4⋊C4×C199D411yescharacteristic
54C4⋊C4×C199D411yescharacteristic
55C42×C199C2311yescharacteristic
56C22×D199—21no—
57Dic398—41no—
58Dic398—81no—
59D796—41no—
60C2×Dic199—21no—
61C2×Dic199—21no—
62C2×Dic199—21no—
63C4×D199—81no—
64C199⋊C8—22no—
65C199⋊C8—22no—
66Q8×C199—41no—
67Q8×C199—41no—
68C2×C4×C199—21no—
69C2×C4×C199—21no—
70C2×C4×C199—41no—
71C2×C4×C199C2×D411yescharacteristic
72C2×C4×C199C2×D411yescharacteristic
73C2×C4×C199C2×D411yescharacteristic
74C8×C199—22no—
75C22×C199#12011yescharacteristic
76C2×D199—41no—
77C2×D199—41no—
78C2×D199—41no—
79C4×C199—21no—
80C4×C199—21no—
81C4×C199—21no—
82C4×C199—41no—
83C4×C199—41no—
84C4×C199—81no—
85Dic199—41no—
86Dic199—41no—
87Dic199—41no—
88C2×C199#45311yescharacteristic
89C2×C199#45611yescharacteristic
90C2×C199#45611yescharacteristic
91D199—81no—
92C199#166011yescharacteristic, complemented, Sylow
93#1660—1991nomaximal, Sylow
94#563—1991no—
95#501—1991no—
96#501D19911yescharacteristic, complemented
97#497—1991no—
98#330—1991no—
99#330—1991no—
100#329—1991no—
101C2×Q16—3981no—
102C2×SD16—3981no—
103C4⋊Q8—3981no—
104C4⋊Q8C2×D19911yescharacteristic
105#126—3981no—
106#126—3981no—
107#125—1991no—
108#125C2×D19911yescharacteristic
109#124—1991no—
110#123—3981no—
111C22⋊Q8—3981no—
112C22⋊Q8—3981no—
113#107—21no—
114#107—3981no—
115#106—21no—
116#106—3981no—
117#97—1991no—
118#97—1991no—
119#97C2×D19911yescharacteristic
120C2×Q8—21no—
121C2×Q8—3981no—
122C2×Q8—1991no—
123C2×D4—1991no—
124C22×C4—3981no—
125Q16—7962no—
126SD16—7962no—
127C2×C8—21no—
128C2×C8—3981no—
129C2×C8—3981no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—3981no—
135C4⋊C4—3981no—
136C4⋊C4—3981no—
137C4⋊C4—3981no—
138C4⋊C4—1991no—
139C4⋊C4—1991no—
140C4⋊C4C22⋊D19911yescharacteristic
141C4⋊C4C22⋊D19911yescharacteristic
142C22⋊C4—3981no—
143C22⋊C4—3981no—
144C22⋊C4—3981no—
145C22⋊C4—3981no—
146C22⋊C4—3981no—
147C42—3981no—
148C42C22×D19911yescharacteristic
149C23—3981no—
150Q8—41no—
151Q8—41no—
152Q8—7961no—
153Q8—15921no—
154D4—7961no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—15921no—
159C2×C4—3981no—
160C2×C4—3981no—
161C2×C4—3981no—
162C2×C4D4×D19911yescharacteristic
163C2×C4D4×D19911yescharacteristic
164C2×C4C2×C22⋊D19911yescharacteristic
165C8—22no—
166C8—3982no—
167C8—3982no—
168C22—7961no—
169C22—7961no—
170C22—7961no—
171C22#2026392311yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—81no—
178C4—7961no—
179C4—7961no—
180C4—7961no—
181C2—15921no—
182C2#2716418411yescharacteristic
183C2#2716419911yescharacteristic
184C2#2716420011yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#501×C199C2
socleC22×C199#120
prime radicalC2×D796D4
hypercenter#501D199
hypocenterC199#1660
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #27164424C2semidirect
4. #27163760C2semidirect
5. #27163759C2semidirect
7. #501×C199C2semidirect
92. C199#1660semidirect
96. #501D199semidirect

Maximal subgroups

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

subgroupindex
#271647722
#271644242
#271644102
#271637602
#271637592
#271637572
#501×C1992
#1660199