Finite Group Databasebeta

← #38768776

Subgroups — #38768776

Order 25472.

584 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#15691886C211yesmaximal, characteristic
2C2×#20263873C212yesmaximal, complemented
3C2×#20263871C212yesmaximal, complemented
4#27163982C211yesmaximal, characteristic, complemented
5#407×C199C211yesmaximal, characteristic, complemented
6C2×#15691886—24no—
7C2×#15691886—24no—
8C2×#15691886—22no—
9C2×#15691886—22no—
10C22×Dic398C2211yescharacteristic
11C22×D796C2211yescharacteristic
12C22×C22⋊D199C412yes—
13C22×C4×D199C412yes—
14C2×#15691870C2212yes—
15#20263873—24no—
16#20263871—24no—
17C22⋊(C199⋊C8)—22no—
18C23×C4×C199C2211yescharacteristic
19C2×M16×C199C2212yes—
20C22⋊C8×C199—22no—
21C23×D199C2×C412yes—
22#15691886—216no—
23#15691886—216no—
24C2×Dic398—22no—
25C2×Dic398—21no—
26C2×Dic398—21no—
27C2×Dic398D414yes—
28C2×D796—22no—
29C2×D796—21no—
30C2×D796—21no—
31C2×D796D414yes—
32C2×C22⋊D199—24no—
33C2×C22⋊D199—24no—
34C2×C22⋊D199—24no—
35C22×Dic199C2×C412yes—
36C2×C4×D199—24no—
37C2×C4×D199—24no—
38C2×C4×D199—24no—
39#15691870—22no—
40#15691870—22no—
41C2×C199⋊C8—22no—
42C24×C199C2×C411yescharacteristic
43C22×C4×C199—22no—
44C22×C4×C199—22no—
45C22×C4×C199C2×C411yescharacteristic
46C22×C4×C199D412yes—
47C22×C4×C199D412yes—
48C22×C4×C199C2311yescharacteristic
49M16×C199—22no—
50M16×C199—22no—
51C2×C8×C199—22no—
52C22×D199—22no—
53C22×D199—22no—
54C22×D199—44no—
55C22×D199C22⋊C414yes—
56Dic398—24no—
57Dic398—24no—
58D796—24no—
59D796—24no—
60C22⋊D199—216no—
61C2×Dic199—22no—
62C2×Dic199—22no—
63C2×Dic199C22⋊C414yes—
64C4×D199—216no—
65C199⋊C8—42no—
66C23×C199—21no—
67C23×C199—21no—
68C23×C199—42no—
69C23×C199C22⋊C412yes—
70C23×C199C22×C411yescharacteristic
71C2×C4×C199—21no—
72C2×C4×C199—21no—
73C2×C4×C199—24no—
74C2×C4×C199—24no—
75C2×C4×C199C22⋊C412yes—
76C2×C4×C199C2×D412yes—
77C2×C4×C199C2×D412yes—
78C2×C4×C199C2×D411yescharacteristic
79C2×C4×C199C2×D411yescharacteristic
80C8×C199—42no—
81C22×C199—22no—
82C22×C199—22no—
83C22×C199—24no—
84C22×C199—42no—
85C22×C199—42no—
86C22×C199C2×C22⋊C411yescharacteristic
87C22×C199C2×C22⋊C411yescharacteristic
88C22×C199C2×C22⋊C411yescharacteristic
89C2×D199—24no—
90C2×D199—44no—
91C2×D199—44no—
92C2×D199—44no—
93C4×C199—22no—
94C4×C199#12012yes—
95C4×C199#12012yes—
96Dic199—24no—
97C2×C199—21no—
98C2×C199—21no—
99C2×C199—42no—
100C2×C199#37911yescharacteristic
101C2×C199#41312yes—
102D199—44no—
103C199#178511yescharacteristic, complemented, Sylow
104#1785—1991nomaximal, Sylow
105C22×#41—1991no—
106C2×#101—1992no—
107C2×#100—1992no—
108#407—1991no—
109#407D19911yescharacteristic, complemented
110C2×#41—3984no—
111C2×#41—3984no—
112C2×#41—3982no—
113C2×#41—3982no—
114C22×Q8—1991no—
115C22×D4—1992no—
116C22×D4—1991no—
117C23×C4—1992no—
118C23×C4C2×D19911yescharacteristic
119C2×M16—1992no—
120C2×M16C2×D19912yes—
121#101—3984no—
122#100—3984no—
123C22⋊C8—22no—
124C22⋊C8—3982no—
125C24—1992no—
126C24C4×D19911yescharacteristic
127#41—39816no—
128#41—39816no—
129C2×Q8—3982no—
130C2×Q8—3981no—
131C2×Q8—3981no—
132C2×Q8—1994no—
133C2×D4—3984no—
134C2×D4—3984no—
135C2×D4—3984no—
136C2×D4—3982no—
137C2×D4—3981no—
138C2×D4—3981no—
139C2×D4—1994no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—3984no—
143C22×C4—3984no—
144C22×C4—3984no—
145C22×C4—1992no—
146C22×C4C4×D19911yescharacteristic
147C22×C4C22⋊D19912yes—
148C22×C4D79612yes—
149C22×C4C22×D19911yescharacteristic
150M16—22no—
151M16—22no—
152M16—3982no—
153M16—3982no—
154C2×C8—22no—
155C2×C8—3982no—
156C23—21no—
157C23—21no—
158C23—42no—
159C23—7964no—
160C23—3982no—
161C23—3982no—
162C23—1994no—
163C23(C2×D199)⋊C412yes—
164C23C2×C4×D19911yescharacteristic
165Q8—3984no—
166Q8—3984no—
167D4—39816no—
168D4—3984no—
169D4—3984no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—39816no—
175C2×C4—3982no—
176C2×C4—3982no—
177C2×C4—1994no—
178C2×C4(C2×D199)⋊C412yes—
179C2×C4D4×D19912yes—
180C2×C4D4×D19912yes—
181C2×C4C2×C22⋊D19911yescharacteristic
182C2×C4C2×D79611yescharacteristic
183C8—42no—
184C8—7962no—
185C22—22no—
186C22—24no—
187C22—22no—
188C22—42no—
189C22—42no—
190C22—7964no—
191C22—7964no—
192C22—7964no—
193C22—3984no—
194C22C22⋊C4×D19911yescharacteristic
195C22C22⋊C4×D19911yescharacteristic
196C22C2×(C2×D199)⋊C411yescharacteristic
197C4—22no—
198C4—3984no—
199C4#2026392312yes—
200C4#2026392512yes—
201C2—21no—
202C2—21no—
203C2—42no—
204C2—7964no—
205C2#2716387311yescharacteristic
206C2#2716400012yes—
subgroupG/H
centerC22C2×(C2×D199)⋊C4
commutatorC23×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#407×C199C2
socleC22×C199C2×C22⋊C4
prime radicalC22×#15691886C2
cosocleC22×C4×C199C23
hypercenter#407D199
hypocenterC199#1785
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#20263873C2semidirect
3. C2×#20263871C2semidirect
4. #27163982C2semidirect
5. #407×C199C2semidirect
103. C199#1785semidirect
109. #407D199semidirect

Maximal subgroups

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

subgroupindex
C22×#156918862
C2×#202638732
C2×#202638712
#271639822
#407×C1992
#1785199