Finite Group Databasebeta

← #38767775

Subgroups — #38767775

Order 25472.

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

normal subgroups only

This group has 316 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#27164644C211yesmaximal, characteristic, complemented
2#27164171C212yesmaximal, complemented
3#27163927C211yesmaximal, characteristic
4#27163756C212yesmaximal, complemented
5#323×C199C211yesmaximal, characteristic, complemented
6C22×D796C2211yescharacteristic
7C22×C22⋊D199—21no—
8C2×#15691875—22no—
9#20263934—21no—
10#20263934C2212yes—
11#20263932—22no—
12#20263932—22no—
13C2×C22⋊Dic199—21no—
14C2×(C2×D199)⋊C4—21no—
15#20263877—22no—
16C199⋊#95C2211yescharacteristic
17C2×C22⋊C4×C199C2211yescharacteristic
18C22⋊C8×C199C2212yes—
19C23×D199—21no—
20C2×Dic398—21no—
21C2×Dic398D412yescomplemented
22C2×D796—22no—
23C2×D796—42no—
24C2×C22⋊D199—42no—
25C2×C22⋊D199—42no—
26C2×C22⋊D199—42no—
27C22×Dic199—21no—
28C22×Dic199—21no—
29C22×Dic199—21no—
30#15691875—44no—
31C4⋊Dic199—21no—
32C4⋊Dic199D412yescomplemented
33Dic199⋊C4—22no—
34Dic199⋊C4—22no—
35C22⋊Dic199—22no—
36C22⋊Dic199—22no—
37(C2×D199)⋊C4—42no—
38C24×C199D411yescharacteristic
39C22×C4×C199D411yescharacteristic
40C22×C4×C199C2311yescharacteristic
41C2×C8×C199—22no—
42C22⋊C4×C199—22no—
43C22×D199—22no—
44C22×D199—41no—
45C22×D199—41no—
46C22×D199—41no—
47C22×D199—41no—
48C22×D199—41no—
49C22×D199—41no—
50Dic398—42no—
51Dic398—81no—
52D796—24no—
53D796—42no—
54C22⋊D199—48no—
55C2×Dic199—22no—
56C2×Dic199—22no—
57C2×Dic199—22no—
58C2×Dic199—41no—
59C2×Dic199—41no—
60C2×Dic199—81no—
61C2×Dic199—81no—
62C23×C199—21no—
63C23×C199—21no—
64C23×C199—41no—
65C23×C199—41no—
66C23×C199SD1612yescomplemented
67C23×C199C2×D411yescharacteristic
68C2×C4×C199—21no—
69C2×C4×C199—41no—
70C2×C4×C199—41no—
71C2×C4×C199C2×D412yes—
72C8×C199—42no—
73C22×C199—21no—
74C22×C199—21no—
75C22×C199—22no—
76C22×C199—42no—
77C22×C199—41no—
78C22×C199—41no—
79C22×C199—81no—
80C22×C199#12011yescharacteristic
81C22×C199C2×SD1611yescharacteristic
82C22×C199#13711yescharacteristic
83C2×D199—42no—
84C2×D199—42no—
85C2×D199—42no—
86C2×D199—81no—
87C2×D199—81no—
88C2×D199—81no—
89C2×D199—81no—
90C4×C199—22no—
91C4×C199—41no—
92Dic199—42no—
93Dic199—42no—
94Dic199—42no—
95C2×C199—21no—
96C2×C199—21no—
97C2×C199—42no—
98C2×C199#44811yescharacteristic
99C2×C199#45311yescharacteristic
100C2×C199B(4,𝔽2)11yescharacteristic
101D199—82no—
102C199#159611yescharacteristic, complemented, Sylow
103#1596—1991nomaximal, Sylow
104#543—1991no—
105#450—1992no—
106#392—1991no—
107#328—1992no—
108#323D19911yescharacteristic, complemented
109C22×D4—3981no—
110C22×D4—1991no—
111C2×SD16—3982no—
112C22⋊Q8—3982no—
113C22⋊Q8—3982no—
114C22⋊Q8—3981no—
115C22⋊Q8—1992no—
116C2×C22⋊C4—3981no—
117C2×C22⋊C4—3981no—
118C2×C22⋊C4C2×D19911yescharacteristic
119#102—3982no—
120C22⋊C8C2×D19912yes—
121#95—1991no—
122C24—3981no—
123C24D79611yescharacteristic
124C2×Q8—3981no—
125C2×Q8—1992no—
126C2×D4—7962no—
127C2×D4—7962no—
128C2×D4—7962no—
129C2×D4—7962no—
130C2×D4—3982no—
131C22×C4—3981no—
132C22×C4—3981no—
133C22×C4—3981no—
134C22×C4D79611yescharacteristic
135C22×C4C22×D19911yescharacteristic
136SD16—7964no—
137C2×C8—22no—
138C4⋊C4—3982no—
139C4⋊C4—3981no—
140C4⋊C4—3982no—
141C4⋊C4—1992no—
142C22⋊C4—22no—
143C22⋊C4—7962no—
144C22⋊C4—3982no—
145C22⋊C4—3982no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—7961no—
151C23—7961no—
152C23—7961no—
153C23—7961no—
154C23—7961no—
155C23—7961no—
156C23—3982no—
157C23#1569187512yescomplemented
158C23C2×D79611yescharacteristic
159Q8—7962no—
160Q8—15921no—
161D4—7962no—
162D4—7968no—
163D4—3984no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—7961no—
168C2×C4—7961no—
169C2×C4—15921no—
170C2×C4—15921no—
171C2×C4—3982no—
172C2×C4—3982no—
173C2×C4—3982no—
174C2×C4D4×D19912yes—
175C8—42no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—7962no—
184C22—7962no—
185C22—7962no—
186C22—15921no—
187C22—15921no—
188C22—15921no—
189C22—15921no—
190C22#2026392511yescharacteristic
191C22C2×#1569187511yescharacteristic
192C22#2026398811yescharacteristic
193C4—22no—
194C4—41no—
195C4—7962no—
196C4—7962no—
197C4—7962no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—15922no—
202C2#2716415911yescharacteristic
203C2#2716418811yescharacteristic
204C2#2716420611yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#323×C199C2
socleC22×C199#120
prime radical#27163927C2
hypercenter#323D199
hypocenterC199#1596
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
1. #27164644C2semidirect
2. #27164171C2semidirect
4. #27163756C2semidirect
5. #323×C199C2semidirect
21. C2×Dic398D4semidirect
32. C4⋊Dic199D4semidirect
66. C23×C199SD16semidirect
102. C199#1596semidirect
108. #323D199semidirect
157. C23#15691875semidirect

Maximal subgroups

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

subgroupindex
#271646442
#271641712
#271639272
#271637562
#323×C1992
#1596199