Finite Group Databasebeta

← #38768181

Subgroups — #38768181

Order 25472.

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

normal subgroups only

This group has 184 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#27164380C211yesmaximal, characteristic
4#27163760C211yesmaximal, characteristic, complemented
5#27163759C211yesmaximal, characteristic, complemented
6#27163757C211yesmaximal, characteristic
7#491×C199C211yesmaximal, characteristic, complemented
8#20263954—21no—
9#20263950—21no—
10#20263950—21no—
11#20263947C2211yescharacteristic
12#20263943C2211yescharacteristic
13#20263940—21no—
14#20263935—21no—
15#20263933—21no—
16#20263888—21no—
17#20263886—21no—
18#20263878—21no—
19#20263875—21no—
20#20263866C2211yescharacteristic
21#20263866C2211yescharacteristic
22C4⋊Q8×C199C2211yescharacteristic
23#125×C199C2211yescharacteristic
24#103×C199—21no—
25#103×C199—21no—
26#97×C199C2211yescharacteristic
27C2×Dic398—21no—
28C2×Dic398D411yescharacteristic
29C2×D796D411yescharacteristic
30C2×C4×D199—21no—
31C2×C199⋊C8—21no—
32C2×C199⋊C8—21no—
33C4⋊Dic199—21no—
34C4⋊Dic199D411yescharacteristic
35C4⋊Dic199D411yescharacteristic
36Dic199⋊C4—21no—
37Dic199⋊C4—21no—
38Dic199⋊C4—21no—
39(C2×D199)⋊C4—21no—
40(C2×D199)⋊C4—21no—
41(C2×D199)⋊C4—21no—
42(C2×D199)⋊C4—21no—
43(C2×D199)⋊C4—21no—
44C4×Dic199—21no—
45C2×Q8×C199—21no—
46C2×C8×C199—21no—
47C4⋊C4×C199—21no—
48C4⋊C4×C199—21no—
49C4⋊C4×C199—21no—
50C4⋊C4×C199—21no—
51C4⋊C4×C199D411yescharacteristic
52C4⋊C4×C199D411yescharacteristic
53C42×C199C2311yescharacteristic
54C22×D199—21no—
55Dic398—41no—
56Dic398—81no—
57D796—41no—
58C2×Dic199—21no—
59C2×Dic199—21no—
60C2×Dic199—21no—
61C4×D199—81no—
62C199⋊C8—22no—
63C199⋊C8—41no—
64Q8×C199—42no—
65C2×C4×C199—21no—
66C2×C4×C199—21no—
67C2×C4×C199—41no—
68C2×C4×C199C2×D411yescharacteristic
69C2×C4×C199C2×D411yescharacteristic
70C2×C4×C199C2×D411yescharacteristic
71C8×C199—41no—
72C22×C199#12011yescharacteristic
73C2×D199—41no—
74C2×D199—41no—
75C2×D199—41no—
76C4×C199—21no—
77C4×C199—21no—
78C4×C199—21no—
79C4×C199—41no—
80C4×C199—41no—
81C4×C199—81no—
82Dic199—41no—
83Dic199—41no—
84Dic199—41no—
85C2×C199#45411yescharacteristic
86C2×C199#45511yescharacteristic
87C2×C199#45611yescharacteristic
88D199—81no—
89C199#166411yescharacteristic, complemented, Sylow
90#1664—1991nomaximal, Sylow
91#563—1991no—
92#501—1991no—
93#491D19911yescharacteristic, complemented
94#489—1991no—
95#330—1991no—
96#330—1991no—
97#329—1991no—
98C4⋊Q8—3981no—
99C4⋊Q8C2×D19911yescharacteristic
100#126—3981no—
101#126—3981no—
102#125—1991no—
103#125C2×D19911yescharacteristic
104#124—1991no—
105#123—3981no—
106C22⋊Q8—3981no—
107C22⋊Q8—3981no—
108#107—3981no—
109#106—3981no—
110#103—21no—
111#103—21no—
112#103—3981no—
113#102—3981no—
114#97—1991no—
115#97—1991no—
116#97C2×D19911yescharacteristic
117C2×Q8—21no—
118C2×Q8—3981no—
119C2×Q8—1991no—
120C2×D4—1991no—
121C22×C4—3981no—
122C2×C8—21no—
123C2×C8—3981no—
124C2×C8—3981no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—3981no—
130C4⋊C4—3981no—
131C4⋊C4—3981no—
132C4⋊C4—3981no—
133C4⋊C4—1991no—
134C4⋊C4—1991no—
135C4⋊C4C22⋊D19911yescharacteristic
136C4⋊C4C22⋊D19911yescharacteristic
137C22⋊C4—3981no—
138C22⋊C4—3981no—
139C22⋊C4—3981no—
140C22⋊C4—3981no—
141C22⋊C4—3981no—
142C42—3981no—
143C42C22×D19911yescharacteristic
144C23—3981no—
145Q8—42no—
146Q8—7961no—
147Q8—15921no—
148D4—7961no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—15921no—
153C2×C4—3981no—
154C2×C4—3981no—
155C2×C4—3981no—
156C2×C4D4×D19911yescharacteristic
157C2×C4D4×D19911yescharacteristic
158C2×C4C2×C22⋊D19911yescharacteristic
159C8—41no—
160C8—7961no—
161C8—3982no—
162C22—7961no—
163C22—7961no—
164C22—7961no—
165C22#2026392311yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—81no—
172C4—7961no—
173C4—7961no—
174C4—7961no—
175C2—15921no—
176C2#2716419011yescharacteristic
177C2#2716419411yescharacteristic
178C2#2716419911yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#491×C199C2
socleC22×C199#120
prime radicalC2×D796D4
hypercenter#491D199
hypocenterC199#1664
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #27164424C2semidirect
4. #27163760C2semidirect
5. #27163759C2semidirect
7. #491×C199C2semidirect
89. C199#1664semidirect
93. #491D199semidirect

Maximal subgroups

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

subgroupindex
#271647722
#271644242
#271643802
#271637602
#271637592
#271637572
#491×C1992
#1664199