Finite Group Databasebeta

← #38768165

Subgroups — #38768165

Order 25472.

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

normal subgroups only

This group has 208 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#27164766C211yesmaximal, characteristic
2#27164410C211yesmaximal, characteristic, complemented
3#27164380C211yesmaximal, characteristic, complemented
4#27163760C211yesmaximal, characteristic, complemented
5#27163757C211yesmaximal, characteristic, complemented
6#27163757C211yesmaximal, characteristic, complemented
7#501×C199C211yesmaximal, characteristic, complemented
8C2×#15691876—21no—
9C2×#15691873—21no—
10#20263943C2211yescharacteristic
11#20263943C2211yescharacteristic
12#20263940—21no—
13#20263940—21no—
14#20263933—21no—
15#20263933—21no—
16#20263930—21no—
17#20263930—21no—
18#20263878—21no—
19#20263875—21no—
20#20263866C2211yescharacteristic
21#20263866C2211yescharacteristic
22C4⋊Q8×C199C2211yescharacteristic
23#125×C199C2211yescharacteristic
24#107×C199—21no—
25#106×C199—21no—
26#97×C199C2211yescharacteristic
27C2×Dic398D411yescharacteristic
28C2×Dic398D411yescharacteristic
29C2×D796—21no—
30C2×D796—21no—
31C2×D796D411yescharacteristic
32C2×D796D411yescharacteristic
33C2×C4×D199—21no—
34C2×C4×D199—21no—
35#15691876—42no—
36#15691873—42no—
37C2×C199⋊C8—21no—
38C2×C199⋊C8—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—
46(C2×D199)⋊C4—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—
57C22×D199—21no—
58Dic398—41no—
59Dic398—41no—
60D796—41no—
61D796—41no—
62D796—81no—
63D796—81no—
64C2×Dic199—21no—
65C2×Dic199—21no—
66C4×D199—81no—
67C4×D199—81no—
68C199⋊C8—22no—
69C199⋊C8—41no—
70Q8×C199—41no—
71Q8×C199—41no—
72C2×C4×C199—21no—
73C2×C4×C199—21no—
74C2×C4×C199—41no—
75C2×C4×C199C2×D411yescharacteristic
76C2×C4×C199C2×D411yescharacteristic
77C2×C4×C199C2×D411yescharacteristic
78C8×C199—22no—
79C22×C199#12011yescharacteristic
80C2×D199—41no—
81C2×D199—41no—
82C2×D199—41no—
83C2×D199—41no—
84C2×D199—41no—
85C2×D199—41no—
86C4×C199—21no—
87C4×C199—21no—
88C4×C199—21no—
89C4×C199—41no—
90C4×C199—41no—
91C4×C199—81no—
92Dic199—41no—
93Dic199—41no—
94C2×C199#45311yescharacteristic
95C2×C199#45511yescharacteristic
96C2×C199#45611yescharacteristic
97D199—81no—
98D199—81no—
99C199#166111yescharacteristic, complemented, Sylow
100#1661—1991nomaximal, Sylow
101#562—1991no—
102#501D19911yescharacteristic, complemented
103#497—1991no—
104#489—1991no—
105#330—1991no—
106#329—1991no—
107#329—1991no—
108C2×Q16—3981no—
109C2×SD16—3981no—
110C4⋊Q8C2×D19911yescharacteristic
111#125C2×D19911yescharacteristic
112#124—1991no—
113#124—1991no—
114#123—3981no—
115#123—3981no—
116C22⋊Q8—3981no—
117C22⋊Q8—3981no—
118#121—3981no—
119#121—3981no—
120#107—21no—
121#106—21no—
122#103—3981no—
123#102—3981no—
124#97—1991no—
125#97—1991no—
126#97C2×D19911yescharacteristic
127C2×Q8—21no—
128C2×Q8—1991no—
129C2×Q8—1991no—
130C2×D4—3981no—
131C2×D4—3981no—
132C2×D4—1991no—
133C2×D4—1991no—
134C22×C4—3981no—
135C22×C4—3981no—
136Q16—7962no—
137SD16—7962no—
138C2×C8—21no—
139C2×C8—3981no—
140C2×C8—3981no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—3981no—
146C4⋊C4—3981no—
147C4⋊C4C22⋊D19911yescharacteristic
148C4⋊C4C22⋊D19911yescharacteristic
149C22⋊C4—3981no—
150C22⋊C4—3981no—
151C22⋊C4—3981no—
152C22⋊C4—3981no—
153C22⋊C4—3981no—
154C22⋊C4—3981no—
155C42C22×D19911yescharacteristic
156C23—3981no—
157C23—3981no—
158Q8—41no—
159Q8—41no—
160Q8—7961no—
161Q8—7961no—
162D4—7961no—
163D4—7961no—
164D4—15921no—
165D4—15921no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—15921no—
170C2×C4—15921no—
171C2×C4—3981no—
172C2×C4—3981no—
173C2×C4D4×D19911yescharacteristic
174C2×C4D4×D19911yescharacteristic
175C2×C4C2×C22⋊D19911yescharacteristic
176C8—22no—
177C8—7961no—
178C8—3982no—
179C22—7961no—
180C22—7961no—
181C22—7961no—
182C22—7961no—
183C22—7961no—
184C22—7961no—
185C22#2026392311yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—41no—
191C4—81no—
192C4—7961no—
193C4—7961no—
194C2—15921no—
195C2—15921no—
196C2#2716418411yescharacteristic
197C2#2716419411yescharacteristic
198C2#2716420011yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#501×C199C2
socleC22×C199#120
prime radical#27164766C2
hypercenter#501D199
hypocenterC199#1661
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. #27164410C2semidirect
3. #27164380C2semidirect
4. #27163760C2semidirect
5. #27163757C2semidirect
6. #27163757C2semidirect
7. #501×C199C2semidirect
99. C199#1661semidirect
102. #501D199semidirect

Maximal subgroups

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

subgroupindex
#271647662
#271644102
#271643802
#271637602
#271637572
#501×C1992
#1661199