Finite Group Databasebeta

← #38768049

Subgroups — #38768049

Order 25472.

250 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 57 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#27164690C211yesmaximal, characteristic
2#27164366C211yesmaximal, characteristic, complemented
3#27164220C211yesmaximal, characteristic, complemented
4#27163761C211yesmaximal, characteristic, complemented
5Q8⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic, complemented
7#475×C199C211yesmaximal, characteristic, complemented
8C2×#15691887—21no—
9C2×#15691873—21no—
10#20263952—21no—
11#20263952C2211yescharacteristic
12#20263930—22no—
13#20263930—21no—
14C4×D796—21no—
15C4×D796C2211yescharacteristic
16C4⋊(C199⋊C8)C2211yescharacteristic
17#20263879—21no—
18#20263875—22no—
19C4×C199⋊C8C2211yescharacteristic
20C4⋊Q8×C199C2211yescharacteristic
21C4×Q8×C199C2211yescharacteristic
22#106×C199—21no—
23C4⋊C8×C199C2211yescharacteristic
24#103×C199—21no—
25#15691887—42no—
26#15691887—42no—
27C2×D796—22no—
28C2×D796—22no—
29C2×D796—21no—
30C2×D796—21no—
31C2×D796D411yescharacteristic
32C2×D796D412yes—
33C2×C4×D199—22no—
34C2×C4×D199—21no—
35#15691873—42no—
36C2×C199⋊C8—21no—
37C2×C199⋊C8—21no—
38C4⋊Dic199D411yescharacteristic
39(C2×D199)⋊C4—22no—
40(C2×D199)⋊C4—21no—
41C2×Q8×C199—21no—
42C2×Q8×C199D411yescharacteristic
43C2×C8×C199—21no—
44C4⋊C4×C199—21no—
45C4⋊C4×C199—21no—
46C4⋊C4×C199—21no—
47C4⋊C4×C199D411yescharacteristic
48C42×C199—21no—
49C42×C199C2311yescharacteristic
50C22×D199—21no—
51C22×D199—22no—
52D796—21no—
53D796—21no—
54D796—42no—
55D796—44no—
56D796—41no—
57D796—41no—
58D796—41no—
59D796—41no—
60D796—82no—
61C2×Dic199—21no—
62C4×D199—44no—
63C4×D199—41no—
64C4×D199—41no—
65C199⋊C8—41no—
66C199⋊C8—41no—
67Q8×C199—21no—
68Q8×C199—41no—
69Q8×C199—41no—
70Q8×C199SD1612yescomplemented
71C2×C4×C199—21no—
72C2×C4×C199—21no—
73C2×C4×C199—41no—
74C2×C4×C199C2×D411yescharacteristic
75C2×C4×C199C2×D411yescharacteristic
76C2×C4×C199C2×D411yescharacteristic
77C8×C199—41no—
78C22×C199#12011yescharacteristic
79C2×D199—41no—
80C2×D199—42no—
81C2×D199—42no—
82C2×D199—41no—
83C2×D199—42no—
84C2×D199—41no—
85C4×C199—22no—
86C4×C199—21no—
87C4×C199—41no—
88C4×C199—81no—
89C4×C199C2×SD1611yescharacteristic
90C4×C199#13511yescharacteristic
91C4×C199AGL(1,ℤ/8)11yescharacteristic
92C4×C199AGL(1,ℤ/8)11yescharacteristic
93Dic199—41no—
94C2×C199#44911yescharacteristic
95C2×C199#45011yescharacteristic
96C2×C199#45511yescharacteristic
97D199—81no—
98D199—82no—
99C199#164211yescharacteristic, complemented, Sylow
100#1642—1991nomaximal, Sylow
101#550—1991no—
102#486—1991no—
103#475D19911yescharacteristic, complemented
104#461—1991no—
105#331—1991no—
106Q8⋊C8—1991no—
107D4⋊C8—1991no—
108C2×#41—3981no—
109C2×SD16—3981no—
110C4⋊Q8C2×D19911yescharacteristic
111#127—3981no—
112#127—1991no—
113#121—3982no—
114#121—3981no—
115C4×Q8C2×D19911yescharacteristic
116C4×D4—3981no—
117C4×D4—1991no—
118#106—21no—
119C4⋊C8—1991no—
120C4⋊C8C2×D19911yescharacteristic
121#103—21no—
122#103—3981no—
123#102—3982no—
124C4×C8—1991no—
125#41—7962no—
126#41—7962no—
127C2×Q8—21no—
128C2×Q8C22⋊D19911yescharacteristic
129C2×D4—3982no—
130C2×D4—3982no—
131C2×D4—3981no—
132C2×D4—3981no—
133C2×D4—1992no—
134C2×D4—1991no—
135C22×C4—3982no—
136C22×C4—3981no—
137SD16—7962no—
138C2×C8—21no—
139C2×C8—3981no—
140C2×C8—3981no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—1991no—
145C4⋊C4C22⋊D19911yescharacteristic
146C22⋊C4—3982no—
147C22⋊C4—3981no—
148C42—21no—
149C42C22×D19911yescharacteristic
150C23—3982no—
151C23—3981no—
152Q8—21no—
153Q8—41no—
154Q8—41no—
155Q8#1569187312yescomplemented
156D4—7962no—
157D4—7964no—
158D4—7961no—
159D4—7961no—
160D4—7961no—
161D4—7961no—
162D4—15922no—
163D4—3981no—
164D4—3981no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—7964no—
169C2×C4—7961no—
170C2×C4—7961no—
171C2×C4—3981no—
172C2×C4D4×D19911yescharacteristic
173C2×C4D4×D19911yescharacteristic
174C2×C4C2×C22⋊D19911yescharacteristic
175C8—41no—
176C8—7961no—
177C8—7961no—
178C22—7962no—
179C22—7961no—
180C22—7962no—
181C22—7961no—
182C22—7962no—
183C22—7961no—
184C22#2026392311yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—81no—
189C4—7961no—
190C4C2×#1569187311yescharacteristic
191C4#2026397611yescharacteristic
192C4#2026398011yescharacteristic
193C4#2026398211yescharacteristic
194C2—15922no—
195C2—15921no—
196C2#2716416211yescharacteristic
197C2#2716416811yescharacteristic
198C2#2716419411yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#475×C199C2
socleC22×C199#120
prime radical#27164690C2
hypercenter#475D199
hypocenterC199#1642
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #27164366C2semidirect
3. #27164220C2semidirect
4. #27163761C2semidirect
5. Q8⋊(C199⋊C8)C2semidirect
6. #27163745C2semidirect
7. #475×C199C2semidirect
70. Q8×C199SD16semidirect
99. C199#1642semidirect
103. #475D199semidirect
155. Q8#15691873semidirect

Maximal subgroups

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

subgroupindex
#271646902
#271643662
#271642202
#271637612
Q8⋊(C199⋊C8)2
#271637452
#475×C1992
#1642199