Finite Group Databasebeta

← #38768020

Subgroups — #38768020

Order 25472.

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

normal subgroups only

This group has 194 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#27164738C211yesmaximal, characteristic
2#27164368C211yesmaximal, characteristic, complemented
3#27164310C211yesmaximal, characteristic, complemented
4#27163765C211yesmaximal, characteristic, complemented
5#27163748C211yesmaximal, characteristic, complemented
6D4⋊(C199⋊C8)C211yesmaximal, characteristic
7#476×C199C211yesmaximal, characteristic
8#20263955C2211yescharacteristic
9#20263948—22no—
10#20263942—21no—
11#20263934—21no—
12C4×Dic398C2211yescharacteristic
13#20263920—21no—
14#20263914—21no—
15#20263888—21no—
16C4⋊(C199⋊C8)C2211yescharacteristic
17#20263879—22no—
18#20263878—21no—
19C4×C199⋊C8C2211yescharacteristic
20C4⋊Q8×C199C2211yescharacteristic
21C4×D4×C199C2211yescharacteristic
22#106×C199—21no—
23C4⋊C8×C199C2211yescharacteristic
24#102×C199—21no—
25C2×Dic398—21no—
26C2×Dic398D411yescharacteristic
27C2×C199⋊C8—21no—
28C2×C199⋊C8—21no—
29C4⋊Dic199—21no—
30C4⋊Dic199D412yes—
31C4⋊Dic199D411yescharacteristic
32Dic199⋊C4—22no—
33Dic199⋊C4—21no—
34C22⋊Dic199—22no—
35C22⋊Dic199—22no—
36C4×Dic199—22no—
37C4×Dic199—21no—
38C2×Q8×C199—21no—
39C2×D4×C199D411yescharacteristic
40C22×C4×C199—21no—
41C2×C8×C199—21no—
42C4⋊C4×C199—21no—
43C4⋊C4×C199—21no—
44C4⋊C4×C199D411yescharacteristic
45C22⋊C4×C199—21no—
46C42×C199C2311yescharacteristic
47Dic398—21no—
48Dic398—21no—
49Dic398—41no—
50Dic398—41no—
51C2×Dic199—21no—
52C2×Dic199—21no—
53C2×Dic199—22no—
54C199⋊C8—41no—
55C199⋊C8—41no—
56C23×C199—21no—
57Q8×C199—42no—
58D4×C199—21no—
59D4×C199—21no—
60C2×C4×C199—21no—
61C2×C4×C199—41no—
62C2×C4×C199—41no—
63C2×C4×C199—41no—
64C2×C4×C199C2×D411yescharacteristic
65C2×C4×C199C2×D411yescharacteristic
66C2×C4×C199C2×D411yescharacteristic
67C8×C199—41no—
68C22×C199—41no—
69C22×C199—41no—
70C22×C199—41no—
71C22×C199#12011yescharacteristic
72C4×C199—21no—
73C4×C199—41no—
74C4×C199—81no—
75C4×C199#13511yescharacteristic
76C4×C199#13511yescharacteristic
77C4×C199#13711yescharacteristic
78C4×C199#13711yescharacteristic
79Dic199—41no—
80Dic199—42no—
81Dic199—41no—
82C2×C199—81no—
83C2×C199#45211yescharacteristic
84C2×C199#45211yescharacteristic
85C2×C199#45411yescharacteristic
86C199#163811yescharacteristic, complemented, Sylow
87#1638—1991nomaximal, Sylow
88#556—1991no—
89#487—1991no—
90#477—1991no—
91#476D19911yescharacteristic
92#332—1991no—
93Q8⋊C8—1991no—
94D4⋊C8—1991no—
95C4⋊Q8—1991no—
96C4⋊Q8C2×D19911yescharacteristic
97#126—3982no—
98#124—3981no—
99C22⋊Q8—3981no—
100C4×Q8—3981no—
101C4×Q8—1991no—
102C4×D4C2×D19911yescharacteristic
103#117—3981no—
104#107—3981no—
105#106—21no—
106C4⋊C8—1991no—
107C4⋊C8C2×D19911yescharacteristic
108#103—3981no—
109#103—3982no—
110#102—21no—
111C4×C8—1991no—
112C2×Q8—21no—
113C2×Q8—3981no—
114C2×Q8—1991no—
115C2×D4C22⋊D19911yescharacteristic
116C22×C4—21no—
117C2×C8—21no—
118C2×C8—3981no—
119C2×C8—3981no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—3982no—
123C4⋊C4—3981no—
124C4⋊C4—3981no—
125C4⋊C4—1992no—
126C4⋊C4—1991no—
127C4⋊C4C22⋊D19911yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—3982no—
130C22⋊C4—3982no—
131C42—3982no—
132C42—3981no—
133C42C22×D19911yescharacteristic
134C23—21no—
135Q8—42no—
136Q8—7961no—
137Q8—7961no—
138Q8—3981no—
139Q8—3981no—
140D4—21no—
141D4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—3981no—
147C2×C4—3981no—
148C2×C4—3982no—
149C2×C4D4×D19911yescharacteristic
150C2×C4D4×D19911yescharacteristic
151C2×C4C2×C22⋊D19911yescharacteristic
152C8—41no—
153C8—7961no—
154C8—7961no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22#2026392311yescharacteristic
159C4—21no—
160C4—41no—
161C4—81no—
162C4—7961no—
163C4—7962no—
164C4—7961no—
165C4#2026397511yescharacteristic
166C4#2026397611yescharacteristic
167C4#2026398411yescharacteristic
168C4#2026398611yescharacteristic
169C2—81no—
170C2#2716417711yescharacteristic
171C2#2716417811yescharacteristic
172C2#2716418911yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#476×C199C2
socleC22×C199#120
prime radicalC2×D4×C199D4
hypercenter#476D199
hypocenterC199#1638
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #27164368C2semidirect
3. #27164310C2semidirect
4. #27163765C2semidirect
5. #27163748C2semidirect
86. C199#1638semidirect

Maximal subgroups

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

subgroupindex
#271647382
#271643682
#271643102
#271637652
#271637482
D4⋊(C199⋊C8)2
#476×C1992
#1638199