Finite Group Databasebeta

← #38767787

Subgroups — #38767787

Order 25472.

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

normal subgroups only

This group has 310 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#27164599C211yesmaximal, characteristic, complemented
2#27164154C212yesmaximal, complemented
3#27163935C211yesmaximal, characteristic, complemented
4#27163753C212yesmaximal, complemented
5#324×C199C211yesmaximal, characteristic, complemented
6C2×#15691886—21no—
7C22×D796C2211yescharacteristic
8C2×D1592—22no—
9#20263935—22no—
10#20263930—22no—
11#20263929—21no—
12#20263929C2212yes—
13C2×(C2×D199)⋊C4—21no—
14C2×(C2×D199)⋊C4—21no—
15#20263877—22no—
16C199⋊#95C2211yescharacteristic
17C2×C4⋊C4×C199C2211yescharacteristic
18C22⋊C8×C199C2212yes—
19C23×D199—21no—
20#15691886—42no—
21#15691886—42no—
22C2×Dic398—21no—
23C2×D796—21no—
24C2×D796—21no—
25C2×D796—22no—
26C2×D796—42no—
27C2×D796D412yes—
28C2×C22⋊D199—22no—
29C22×Dic199—21no—
30C2×C4×D199—22no—
31D1592—44no—
32C4⋊Dic199—21no—
33C4⋊Dic199D412yes—
34(C2×D199)⋊C4—22no—
35(C2×D199)⋊C4—22no—
36(C2×D199)⋊C4—42no—
37(C2×D199)⋊C4—42no—
38C22×C4×C199D411yescharacteristic
39C22×C4×C199D411yescharacteristic
40C22×C4×C199C2311yescharacteristic
41C2×C8×C199—22no—
42C4⋊C4×C199—22no—
43C22×D199—22no—
44C22×D199—22no—
45C22×D199—41no—
46C22×D199—41no—
47C22×D199—41no—
48C22×D199—41no—
49C22×D199—41no—
50C22×D199—41no—
51Dic398—42no—
52D796—24no—
53D796—42no—
54D796—42no—
55D796—42no—
56D796—81no—
57C22⋊D199—42no—
58C22⋊D199—42no—
59C2×Dic199—22no—
60C2×Dic199—41no—
61C2×Dic199—41no—
62C2×Dic199—41no—
63C4×D199—44no—
64C23×C199C2×D411yescharacteristic
65C2×C4×C199—21no—
66C2×C4×C199—21no—
67C2×C4×C199—21no—
68C2×C4×C199—41no—
69C2×C4×C199—41no—
70C2×C4×C199D812yescomplemented
71C2×C4×C199C2×D412yes—
72C8×C199—42no—
73C22×C199—21no—
74C22×C199—21no—
75C22×C199#12011yescharacteristic
76C22×C199C2×D811yescharacteristic
77C22×C199AGL(1,ℤ/8)11yescharacteristic
78C2×D199—42no—
79C2×D199—42no—
80C2×D199—42no—
81C2×D199—42no—
82C2×D199—42no—
83C2×D199—42no—
84C2×D199—81no—
85C2×D199—81no—
86C2×D199—81no—
87C2×D199—81no—
88C4×C199—22no—
89C4×C199—22no—
90C4×C199—41no—
91Dic199—42no—
92Dic199—81no—
93C2×C199—21no—
94C2×C199—21no—
95C2×C199#44711yescharacteristic
96C2×C199#45411yescharacteristic
97C2×C199B(4,𝔽2)11yescharacteristic
98D199—82no—
99D199—82no—
100C199#159811yescharacteristic, complemented, Sylow
101#1598—1991nomaximal, Sylow
102#537—1991no—
103#447—1992no—
104#394—1991no—
105#327—1992no—
106#324D19911yescharacteristic, complemented
107C2×#41—3981no—
108C22×D4—1991no—
109C2×D8—3982no—
110C22⋊Q8—3982no—
111#121—3982no—
112#121—3981no—
113#121—1992no—
114C2×C4⋊C4C2×D19911yescharacteristic
115C2×C22⋊C4—3981no—
116C2×C22⋊C4—3981no—
117#102—3982no—
118C22⋊C8C2×D19912yes—
119#95—1991no—
120C24—3981no—
121#41—7962no—
122#41—7962no—
123C2×Q8—3981no—
124C2×D4—7962no—
125C2×D4—3981no—
126C2×D4—3981no—
127C2×D4—3982no—
128C2×D4—3982no—
129C2×D4—1992no—
130C22×C4—3982no—
131C22×C4—3981no—
132C22×C4D79611yescharacteristic
133C22×C4D79611yescharacteristic
134C22×C4C22×D19911yescharacteristic
135D8—7964no—
136C2×C8—22no—
137C4⋊C4—22no—
138C4⋊C4—3981no—
139C4⋊C4—1992no—
140C22⋊C4—7962no—
141C22⋊C4—7962no—
142C22⋊C4—3982no—
143C22⋊C4—3982no—
144C23—7961no—
145C23—7961no—
146C23—7961no—
147C23—7961no—
148C23—7961no—
149C23—7961no—
150C23—3982no—
151C23—3982no—
152C23C2×D79611yescharacteristic
153Q8—7962no—
154D4—7962no—
155D4—7962no—
156D4—7962no—
157D4—7962no—
158D4—7962no—
159D4—15921no—
160D4—3984no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—7964no—
167C2×C4—7961no—
168C2×C4—7961no—
169C2×C4—7961no—
170C2×C4—3982no—
171C2×C4D159212yescomplemented
172C2×C4D4×D19912yes—
173C8—42no—
174C22—21no—
175C22—21no—
176C22—7962no—
177C22—7962no—
178C22—7962no—
179C22—7962no—
180C22—7962no—
181C22—7962no—
182C22—15921no—
183C22—15921no—
184C22—15921no—
185C22—15921no—
186C22#2026392511yescharacteristic
187C22C2×D159211yescharacteristic
188C22#2026398311yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—7962no—
193C4—15921no—
194C2—21no—
195C2—21no—
196C2—15922no—
197C2—15922no—
198C2#2716415411yescharacteristic
199C2#2716419311yescharacteristic
200C2#2716420611yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#324×C199C2
socleC22×C199#120
hypercenter#324D199
hypocenterC199#1598
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #27164599C2semidirect
2. #27164154C2semidirect
3. #27163935C2semidirect
4. #27163753C2semidirect
5. #324×C199C2semidirect
70. C2×C4×C199D8semidirect
100. C199#1598semidirect
106. #324D199semidirect
171. C2×C4D1592semidirect

Maximal subgroups

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

subgroupindex
#271645992
#271641542
#271639352
#271637532
#324×C1992
#1598199