Finite Group Databasebeta

← #38768131

Subgroups — #38768131

Order 25472.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#27164671C211yesmaximal, characteristic
2#27164295C211yesmaximal, characteristic, complemented
3#27164220C211yesmaximal, characteristic, complemented
4#27163764C211yesmaximal, characteristic, complemented
5#27163747C211yesmaximal, characteristic, complemented
6#27163747C211yesmaximal, characteristic, complemented
7#498×C199C211yesmaximal, characteristic, complemented
8C2×#15691887—21no—
9C2×#15691873—21no—
10#20263940—22no—
11#20263931—21no—
12#20263930—21no—
13C4×D796C2211yescharacteristic
14C4×D796C2211yescharacteristic
15C4⋊C4×D199—21no—
16#20263888—21no—
17C4⋊(C199⋊C8)C2211yescharacteristic
18C4⋊(C199⋊C8)C2211yescharacteristic
19#20263879—21no—
20#20263875—21no—
21C4⋊Q8×C199C2211yescharacteristic
22C4⋊Q8×C199C2211yescharacteristic
23#106×C199—22no—
24C4×C8×C199C2211yescharacteristic
25#15691887—42no—
26#15691887—42no—
27C2×D796—22no—
28C2×D796D411yescharacteristic
29C2×D796D411yescharacteristic
30C2×C4×D199—22no—
31C2×C4×D199—21no—
32C2×C4×D199—21no—
33#15691873—42no—
34C2×C199⋊C8—21no—
35C2×C199⋊C8—21no—
36C4⋊Dic199D411yescharacteristic
37C4⋊Dic199D411yescharacteristic
38Dic199⋊C4—22no—
39(C2×D199)⋊C4—22no—
40(C2×D199)⋊C4—21no—
41(C2×D199)⋊C4—21no—
42C2×Q8×C199—21no—
43C2×Q8×C199—21no—
44C2×C8×C199—21no—
45C4⋊C4×C199—21no—
46C4⋊C4×C199—21no—
47C4⋊C4×C199—21no—
48C4⋊C4×C199D412yes—
49C42×C199C2311yescharacteristic
50C22×D199—21no—
51C22×D199—21no—
52D796—21no—
53D796—21no—
54D796—21no—
55D796—44no—
56D796SD1612yescomplemented
57C2×Dic199—21no—
58C2×Dic199—21no—
59C4×D199—44no—
60C4×D199—41no—
61C4×D199—41no—
62C4×D199—41no—
63C4×D199—41no—
64C199⋊C8—41no—
65C199⋊C8—41no—
66Q8×C199—41no—
67Q8×C199—41no—
68Q8×C199—41no—
69Q8×C199—41no—
70C2×C4×C199—22no—
71C2×C4×C199—41no—
72C2×C4×C199C2×D411yescharacteristic
73C2×C4×C199C2×D411yescharacteristic
74C2×C4×C199C2×D411yescharacteristic
75C8×C199—22no—
76C22×C199#12011yescharacteristic
77C2×D199—22no—
78C2×D199—41no—
79C2×D199—41no—
80C2×D199—41no—
81C2×D199—41no—
82C2×D199—41no—
83C4×C199—21no—
84C4×C199—42no—
85C4×C199—81no—
86C4×C199C2×SD1611yescharacteristic
87C4×C199#13511yescharacteristic
88C4×C199AGL(1,ℤ/8)11yescharacteristic
89C4×C199AGL(1,ℤ/8)11yescharacteristic
90Dic199—41no—
91Dic199—41no—
92C2×C199#44911yescharacteristic
93C2×C199#45011yescharacteristic
94C2×C199#45611yescharacteristic
95D199—42no—
96D199—81no—
97C199#165411yescharacteristic, complemented, Sylow
98#1654—1991nomaximal, Sylow
99#547—1991no—
100#498D19911yescharacteristic, complemented
101#474—1991no—
102#461—1991no—
103#332—1991no—
104D4⋊C8—1991no—
105D4⋊C8—1991no—
106C2×#41—3981no—
107C2×SD16—3981no—
108C4⋊Q8C2×D19911yescharacteristic
109C4⋊Q8C2×D19911yescharacteristic
110#123—3982no—
111C22⋊Q8—3981no—
112#121—3981no—
113C4×D4—1991no—
114C4×D4—1991no—
115C2×C4⋊C4—3981no—
116#107—3981no—
117#106—22no—
118C4⋊C8—1991no—
119C4⋊C8—1991no—
120#103—3981no—
121#102—3981no—
122C4×C8C2×D19911yescharacteristic
123#41—7962no—
124#41—7962no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×D4—3982no—
128C2×D4—1991no—
129C2×D4—1991no—
130C22×C4—3982no—
131C22×C4—3981no—
132C22×C4—3981no—
133SD16—7962no—
134C2×C8—21no—
135C2×C8—3981no—
136C2×C8—3981no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—3982no—
141C4⋊C4—1991no—
142C4⋊C4—1991no—
143C4⋊C4C22⋊D19912yes—
144C22⋊C4—3982no—
145C22⋊C4—3981no—
146C22⋊C4—3981no—
147C42C22×D19911yescharacteristic
148C23—3981no—
149C23—3981no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153Q8—41no—
154D4—7964no—
155D4—3981no—
156D4—3981no—
157D4—3981no—
158D4—1992no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—7964no—
162C2×C4—7961no—
163C2×C4—7961no—
164C2×C4—7961no—
165C2×C4—7961no—
166C2×C4—3981no—
167C2×C4—3981no—
168C2×C4D4×D19911yescharacteristic
169C2×C4D4×D19911yescharacteristic
170C2×C4C2×C22⋊D19911yescharacteristic
171C8—22no—
172C8—7961no—
173C8—7961no—
174C22—7961no—
175C22—7961no—
176C22—7961no—
177C22—7961no—
178C22—7961no—
179C22—3982no—
180C22#2026392311yescharacteristic
181C4—21no—
182C4—42no—
183C4—81no—
184C4—7961no—
185C4—7961no—
186C4SD16×D19911yescharacteristic
187C4#2026397611yescharacteristic
188C4#2026398211yescharacteristic
189C4#2026398211yescharacteristic
190C2—7962no—
191C2—15921no—
192C2#2716416211yescharacteristic
193C2#2716416711yescharacteristic
194C2#2716420011yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#498×C199C2
socleC22×C199#120
prime radical#27164671C2
hypercenter#498D199
hypocenterC199#1654
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. #27164295C2semidirect
3. #27164220C2semidirect
4. #27163764C2semidirect
5. #27163747C2semidirect
6. #27163747C2semidirect
7. #498×C199C2semidirect
56. D796SD16semidirect
97. C199#1654semidirect
100. #498D199semidirect

Maximal subgroups

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

subgroupindex
#271646712
#271642952
#271642202
#271637642
#271637472
#498×C1992
#1654199