Finite Group Databasebeta

← #38768121

Subgroups — #38768121

Order 25472.

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

normal subgroups only

This group has 200 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#27164742C211yesmaximal, characteristic
2#27164414C211yesmaximal, characteristic, complemented
3#27164299C211yesmaximal, characteristic
4#27164229C211yesmaximal, characteristic, complemented
5#27163748C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic
7#332×C199C211yesmaximal, characteristic, complemented
8C2×Dic796—21no—
9#20263955C2211yescharacteristic
10#20263950—22no—
11#20263944—21no—
12#20263935—21no—
13Q8×Dic199—21no—
14C4×Dic398C2211yescharacteristic
15C4×D796C2211yescharacteristic
16#20263913—21no—
17#20263889—21no—
18#20263886—22no—
19#20263880—21no—
20#20263877—21no—
21C4×C199⋊C8C2211yescharacteristic
22C4⋊Q8×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4⋊C8×C199C2211yescharacteristic
25C2×Dic398—21no—
26C2×Dic398D411yescharacteristic
27C2×D796D411yescharacteristic
28C2×C4×D199—21no—
29Dic796—42no—
30C2×C199⋊C8—21no—
31C4⋊Dic199—22no—
32C4⋊Dic199—21no—
33C4⋊Dic199—21no—
34C4⋊Dic199D411yescharacteristic
35C4⋊Dic199D411yescharacteristic
36Dic199⋊C4—21no—
37(C2×D199)⋊C4—22no—
38(C2×D199)⋊C4—22no—
39(C2×D199)⋊C4—21no—
40C4×Dic199—22no—
41C4×Dic199—21no—
42C2×Q8×C199—21no—
43C2×C8×C199—21no—
44C2×C8×C199—21no—
45C4⋊C4×C199—21no—
46C4⋊C4×C199D412yes—
47C42×C199C2311yescharacteristic
48C22×D199—21no—
49Dic398—21no—
50Dic398—21no—
51Dic398—41no—
52Dic398—41no—
53D796—21no—
54D796—21no—
55C2×Dic199—21no—
56C2×Dic199—21no—
57C2×Dic199—21no—
58C2×Dic199—41no—
59C4×D199—41no—
60C4×D199—41no—
61C199⋊C8—22no—
62Q8×C199—41no—
63Q8×C199—41no—
64C2×C4×C199—22no—
65C2×C4×C199C2×D411yescharacteristic
66C2×C4×C199C2×D411yescharacteristic
67C2×C4×C199C2×D411yescharacteristic
68C8×C199—41no—
69C8×C199—41no—
70C22×C199#12011yescharacteristic
71C2×D199—41no—
72C2×D199—41no—
73C2×D199—41no—
74C4×C199—21no—
75C4×C199—42no—
76C4×C199#13511yescharacteristic
77C4×C199#13511yescharacteristic
78C4×C199AGL(1,ℤ/8)11yescharacteristic
79C4×C199#13711yescharacteristic
80Dic199—41no—
81Dic199—41no—
82Dic199—41no—
83Dic199—81no—
84C2×C199#44911yescharacteristic
85C2×C199#45211yescharacteristic
86C2×C199#45611yescharacteristic
87D199—81no—
88C199#165211yescharacteristic, complemented, Sylow
89#1652—1991nomaximal, Sylow
90#556—1991no—
91#498—1991no—
92#474—1991no—
93#462—1991no—
94#332D19911yescharacteristic, complemented
95Q8⋊C8—1991no—
96D4⋊C8—1991no—
97C2×Q16—3981no—
98C4⋊Q8—1991no—
99C4⋊Q8C2×D19911yescharacteristic
100#126—3982no—
101#124—3981no—
102C22⋊Q8—3981no—
103C4×Q8—3981no—
104C4×Q8—1991no—
105C4×D4—1991no—
106#117—3981no—
107#107—3981no—
108#106—3982no—
109C4⋊C8C2×D19911yescharacteristic
110C4⋊C8C2×D19911yescharacteristic
111#103—3981no—
112#102—3981no—
113C4×C8—1991no—
114C2×Q8—21no—
115C2×Q8—3981no—
116C2×Q8—1991no—
117C2×D4—1991no—
118C22×C4—3981no—
119Q16—7962no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—3981no—
123C4⋊C4—21no—
124C4⋊C4—3982no—
125C4⋊C4—3981no—
126C4⋊C4—3981no—
127C4⋊C4—3981no—
128C4⋊C4—1991no—
129C4⋊C4—1991no—
130C4⋊C4D79612yes—
131C22⋊C4—3982no—
132C22⋊C4—3982no—
133C22⋊C4—3981no—
134C42—3982no—
135C42—3981no—
136C42C22×D19911yescharacteristic
137C23—3981no—
138Q8—41no—
139Q8—41no—
140Q8—7961no—
141Q8—7961no—
142Q8—3981no—
143Q8—3981no—
144D4—3981no—
145D4—3981no—
146C2×C4—22no—
147C2×C4—7961no—
148C2×C4—7961no—
149C2×C4—7961no—
150C2×C4—3981no—
151C2×C4—3981no—
152C2×C4—3981no—
153C2×C4D4×D19911yescharacteristic
154C2×C4D4×D19911yescharacteristic
155C2×C4C2×D79611yescharacteristic
156C8—41no—
157C8—41no—
158C8—3982no—
159C22—7961no—
160C22—7961no—
161C22—7961no—
162C22#2026392511yescharacteristic
163C4—21no—
164C4—42no—
165C4—7961no—
166C4—7961no—
167C4—7961no—
168C4—15921no—
169C4#2026397411yescharacteristic
170C4#2026397811yescharacteristic
171C4#2026398211yescharacteristic
172C4#2026398411yescharacteristic
173C2—15921no—
174C2#2716416611yescharacteristic
175C2#2716418111yescharacteristic
176C2#2716420311yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#332×C199C2
socleC22×C199#120
prime radicalC2×D796D4
hypercenter#332D199
hypocenterC199#1652
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #27164414C2semidirect
4. #27164229C2semidirect
5. #27163748C2semidirect
7. #332×C199C2semidirect
88. C199#1652semidirect
94. #332D199semidirect

Maximal subgroups

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

subgroupindex
#271647422
#271644142
#271642992
#271642292
#271637482
#271637452
#332×C1992
#1652199