Finite Group Databasebeta

← #38767804

Subgroups — #38767804

Order 25472.

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

normal subgroups only

This group has 308 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#27164580C211yesmaximal, characteristic
2#27164159C212yesmaximal, complemented
3#27163930C211yesmaximal, characteristic, complemented
4#27163753C212yesmaximal, complemented
5#323×C199C211yesmaximal, characteristic, complemented
6C22×Dic398C2211yescharacteristic, complemented
7C22×C22⋊D199—21no—
8C2×#15691875—22no—
9#20263939—22no—
10#20263929—21no—
11#20263929C2212yes—
12#20263925—22no—
13C2×C22⋊Dic199—21no—
14#20263880—22no—
15C199⋊#95—21no—
16C199⋊#95C2211yescharacteristic, complemented
17C2×C22⋊C4×C199C2211yescharacteristic
18C22⋊C8×C199C2212yescomplemented
19C23×D199—21no—
20C2×Dic398—22no—
21C2×Dic398—42no—
22C2×D796—21no—
23C2×D796D412yes—
24C2×C22⋊D199—22no—
25C2×C22⋊D199—22no—
26C2×C22⋊D199—42no—
27C2×C22⋊D199—42no—
28C22×Dic199—21no—
29C22×Dic199—21no—
30C22×Dic199—21no—
31#15691875—44no—
32C4⋊Dic199—21no—
33C4⋊Dic199D412yes—
34C22⋊Dic199—42no—
35(C2×D199)⋊C4—22no—
36(C2×D199)⋊C4—22no—
37C24×C199D411yescharacteristic
38C22×C4×C199D411yescharacteristic
39C22×C4×C199C2311yescharacteristic
40C2×C8×C199—22no—
41C22⋊C4×C199—22no—
42C22×D199—22no—
43C22×D199—42no—
44C22×D199—81no—
45C22×D199—81no—
46Dic398—24no—
47Dic398—42no—
48D796—42no—
49D796—81no—
50C22⋊D199—44no—
51C22⋊D199—42no—
52C22⋊D199—42no—
53C2×Dic199—22no—
54C2×Dic199—22no—
55C2×Dic199—42no—
56C2×Dic199—41no—
57C2×Dic199—41no—
58C2×Dic199—41no—
59C2×Dic199—41no—
60C2×Dic199—41no—
61C23×C199—21no—
62C23×C199—21no—
63C23×C199—41no—
64C23×C199—41no—
65C23×C199SD1612yescomplemented
66C23×C199C2×D411yescharacteristic
67C2×C4×C199—21no—
68C2×C4×C199—41no—
69C2×C4×C199—41no—
70C2×C4×C199C2×D412yes—
71C8×C199—42no—
72C22×C199—21no—
73C22×C199—21no—
74C22×C199—22no—
75C22×C199—42no—
76C22×C199—41no—
77C22×C199—41no—
78C22×C199—81no—
79C22×C199#12011yescharacteristic
80C22×C199C2×SD1611yescharacteristic
81C22×C199AGL(1,ℤ/8)11yescharacteristic
82C2×D199—42no—
83C2×D199—42no—
84C2×D199—42no—
85C2×D199—84no—
86C4×C199—22no—
87C4×C199—41no—
88Dic199—42no—
89Dic199—42no—
90Dic199—81no—
91C2×C199—21no—
92C2×C199—21no—
93C2×C199—42no—
94C2×C199#45011yescharacteristic
95C2×C199#45511yescharacteristic
96C2×C199B(4,𝔽2)11yescharacteristic
97D199—82no—
98C199#160111yescharacteristic, complemented, Sylow
99#1601—1991nomaximal, Sylow
100#534—1991no—
101#448—1992no—
102#393—1991no—
103#327—1992no—
104#323D19911yescharacteristic, complemented
105C22×Q8—1991no—
106C22×D4—3981no—
107C2×SD16—3982no—
108#123—3982no—
109#121—3981no—
110#121—1992no—
111#120—3982no—
112C2×C22⋊C4—3981no—
113C2×C22⋊C4C2×D19911yescharacteristic
114#103—3982no—
115C22⋊C8C2×D19912yescomplemented
116#95—3981no—
117#95—1991no—
118C24—3981no—
119C24D79611yescharacteristic
120C2×Q8—7962no—
121C2×Q8—3982no—
122C2×D4—7962no—
123C2×D4—7962no—
124C2×D4—3981no—
125C2×D4—3982no—
126C2×D4—3982no—
127C2×D4—1992no—
128C22×C4—3981no—
129C22×C4—3981no—
130C22×C4—3981no—
131C22×C4D79611yescharacteristic
132C22×C4C22×D19911yescharacteristic
133SD16—7964no—
134C2×C8—22no—
135C4⋊C4—3981no—
136C4⋊C4—1992no—
137C22⋊C4—22no—
138C22⋊C4—7962no—
139C22⋊C4—3982no—
140C22⋊C4—3982no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—7962no—
146C23—15921no—
147C23—15921no—
148C23—3982no—
149C23#1569187512yescomplemented
150C23C2×D79611yescharacteristic
151Q8—7962no—
152Q8—3984no—
153D4—7962no—
154D4—7964no—
155D4—7962no—
156D4—7962no—
157D4—15921no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—7962no—
162C2×C4—7961no—
163C2×C4—7961no—
164C2×C4—7961no—
165C2×C4—7961no—
166C2×C4—7961no—
167C2×C4—3982no—
168C2×C4—3982no—
169C2×C4D4×D19912yes—
170C8—42no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—81no—
178C22—7962no—
179C22—7962no—
180C22—7962no—
181C22—15924no—
182C22#2026392511yescharacteristic
183C22C2×#1569187511yescharacteristic
184C22#2026398311yescharacteristic
185C4—22no—
186C4—41no—
187C4—7962no—
188C4—7962no—
189C4—15921no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—15922no—
194C2#2716417111yescharacteristic
195C2#2716419811yescharacteristic
196C2#2716420611yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#323×C199C2
socleC22×C199#120
prime radical#27164580C2
hypercenter#323D199
hypocenterC199#1601
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #27164159C2semidirect
3. #27163930C2semidirect
4. #27163753C2semidirect
5. #323×C199C2semidirect
6. C22×Dic398C22semidirect
16. C199⋊#95C22semidirect
18. C22⋊C8×C199C22semidirect
65. C23×C199SD16semidirect
98. C199#1601semidirect
104. #323D199semidirect
115. C22⋊C8C2×D199semidirect
149. C23#15691875semidirect

Maximal subgroups

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

subgroupindex
#271645802
#271641592
#271639302
#271637532
#323×C1992
#1601199