Finite Group Databasebeta

← #38768133

Subgroups — #38768133

Order 25472.

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

normal subgroups only

This group has 240 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#27164672C211yesmaximal, characteristic
2#27164414C211yesmaximal, characteristic, complemented
3#27164220C211yesmaximal, characteristic, complemented
4#27163765C211yesmaximal, characteristic, complemented
5D4⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic, complemented
7#474×C199C211yesmaximal, characteristic, complemented
8C2×#15691885—21no—
9C2×#15691873—21no—
10#20263955C2211yescharacteristic
11#20263939—22no—
12#20263935—21no—
13#20263929—21no—
14C4×D796C2211yescharacteristic
15C2×C4⋊Dic199—21no—
16#20263886—22no—
17C4⋊(C199⋊C8)C2211yescharacteristic
18#20263879—21no—
19C4×C199⋊C8C2211yescharacteristic
20C4⋊Q8×C199C2211yescharacteristic
21C4×D4×C199C2211yescharacteristic
22#107×C199—21no—
23C4⋊C8×C199C2211yescharacteristic
24#102×C199—21no—
25#15691885—42no—
26#15691885—42no—
27C2×Dic398—21no—
28C2×D796D411yescharacteristic
29C2×C22⋊D199—22no—
30C22×Dic199—22no—
31C2×C4×D199—21no—
32#15691873—42no—
33C2×C199⋊C8—21no—
34C2×C199⋊C8—21no—
35C4⋊Dic199—22no—
36C4⋊Dic199—21no—
37C4⋊Dic199D412yes—
38C4⋊Dic199D411yescharacteristic
39(C2×D199)⋊C4—22no—
40(C2×D199)⋊C4—21no—
41C2×Q8×C199—21no—
42C2×D4×C199D411yescharacteristic
43C22×C4×C199—21no—
44C2×C8×C199—21no—
45C4⋊C4×C199—21no—
46C4⋊C4×C199—21no—
47C4⋊C4×C199D411yescharacteristic
48C22⋊C4×C199—21no—
49C42×C199C2311yescharacteristic
50C22×D199—21no—
51Dic398—41no—
52Dic398—41no—
53D796—21no—
54D796—21no—
55C22⋊D199—44no—
56C2×Dic199—21no—
57C2×Dic199—22no—
58C2×Dic199—44no—
59C4×D199—41no—
60C4×D199—41no—
61C199⋊C8—22no—
62C199⋊C8—41no—
63C23×C199—21no—
64Q8×C199—41no—
65Q8×C199—41no—
66D4×C199—21no—
67D4×C199SD1612yescomplemented
68C2×C4×C199—21no—
69C2×C4×C199—41no—
70C2×C4×C199—41no—
71C2×C4×C199—41no—
72C2×C4×C199C2×D411yescharacteristic
73C2×C4×C199C2×D411yescharacteristic
74C2×C4×C199C2×D411yescharacteristic
75C8×C199—41no—
76C22×C199—22no—
77C22×C199—41no—
78C22×C199—41no—
79C22×C199#12011yescharacteristic
80C2×D199—41no—
81C2×D199—41no—
82C2×D199—41no—
83C4×C199—21no—
84C4×C199—41no—
85C4×C199—81no—
86C4×C199C2×SD1611yescharacteristic
87C4×C199#13511yescharacteristic
88C4×C199AGL(1,ℤ/8)11yescharacteristic
89C4×C199AGL(1,ℤ/8)11yescharacteristic
90Dic199—41no—
91Dic199—42no—
92C2×C199—42no—
93C2×C199#44911yescharacteristic
94C2×C199#45011yescharacteristic
95C2×C199#45611yescharacteristic
96D199—81no—
97C199#165411yescharacteristic, complemented, Sylow
98#1654—1991nomaximal, Sylow
99#547—1991no—
100#498—1991no—
101#474D19911yescharacteristic, complemented
102#461—1991no—
103#332—1991no—
104D4⋊C8—1991no—
105D4⋊C8—1991no—
106C2×#41—3981no—
107C2×SD16—3981no—
108C4⋊Q8—1991no—
109C4⋊Q8C2×D19911yescharacteristic
110#123—3982no—
111C22⋊Q8—3981no—
112#121—3981no—
113C4×D4—1991no—
114C4×D4C2×D19911yescharacteristic
115C2×C4⋊C4—3981no—
116#107—21no—
117#106—3982no—
118C4⋊C8—1991no—
119C4⋊C8C2×D19911yescharacteristic
120#103—3981no—
121#102—21no—
122C4×C8—1991no—
123#41—7962no—
124#41—7962no—
125C2×Q8—21no—
126C2×Q8—3981no—
127C2×D4—3982no—
128C2×D4—1991no—
129C2×D4C22⋊D19911yescharacteristic
130C22×C4—21no—
131C22×C4—3982no—
132C22×C4—3981no—
133SD16—7962no—
134C2×C8—21no—
135C2×C8—3981no—
136C2×C8—3981no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—3982no—
140C4⋊C4—3981no—
141C4⋊C4—1991no—
142C4⋊C4—1992no—
143C4⋊C4C22⋊D19911yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—3982no—
146C22⋊C4—3981no—
147C42C22×D19911yescharacteristic
148C23—21no—
149C23—3981no—
150Q8—41no—
151Q8—41no—
152Q8—7961no—
153Q8—7961no—
154D4—21no—
155D4—7964no—
156D4—3981no—
157D4—3981no—
158D4#1569187312yescomplemented
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—7964no—
164C2×C4—7961no—
165C2×C4—7961no—
166C2×C4—3981no—
167C2×C4—3982no—
168C2×C4D4×D19911yescharacteristic
169C2×C4D4×D19911yescharacteristic
170C2×C4C2×C22⋊D19911yescharacteristic
171C8—41no—
172C8—7961no—
173C8—3982no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—7961no—
178C22—7961no—
179C22—7961no—
180C22#2026392311yescharacteristic
181C4—21no—
182C4—41no—
183C4—81no—
184C4—7961no—
185C4—7962no—
186C4C2×#1569187311yescharacteristic
187C4#2026397411yescharacteristic
188C4#2026397911yescharacteristic
189C4#2026398011yescharacteristic
190C2—42no—
191C2—15921no—
192C2#2716416011yescharacteristic
193C2#2716416811yescharacteristic
194C2#2716419911yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#474×C199C2
socleC22×C199#120
prime radical#27164672C2
hypercenter#474D199
hypocenterC199#1654
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #27164414C2semidirect
3. #27164220C2semidirect
4. #27163765C2semidirect
5. D4⋊(C199⋊C8)C2semidirect
6. #27163745C2semidirect
7. #474×C199C2semidirect
67. D4×C199SD16semidirect
97. C199#1654semidirect
101. #474D199semidirect
158. D4#15691873semidirect

Maximal subgroups

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

subgroupindex
#271646722
#271644142
#271642202
#271637652
D4⋊(C199⋊C8)2
#271637452
#474×C1992
#1654199