Finite Group Databasebeta

← #38767198

Subgroups — #38767198

Order 25472.

280 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 81 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#27163982C211yesmaximal, characteristic, complemented
2C2×C22⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
3#27163744C212yesmaximal, complemented
4#27163742C212yesmaximal, complemented
5C2×#123×C199C211yesmaximal, characteristic
6C2×#15691870—21no—
7C22×C199⋊C8—21no—
8C22⋊(C199⋊C8)—21no—
9C22⋊(C199⋊C8)—21no—
10C22⋊(C199⋊C8)C2212yes—
11C22⋊(C199⋊C8)C2212yes—
12C22×D4×C199C411yescharacteristic
13C23×C4×C199C2211yescharacteristic
14#123×C199—21no—
15#123×C199—21no—
16#123×C199—21no—
17#123×C199—21no—
18C2×C4⋊C4×C199C411yescharacteristic
19C2×C4⋊C4×C199C2211yescharacteristic
20C2×C22⋊C4×C199C411yescharacteristic
21C2×C22⋊C4×C199C411yescharacteristic
22C2×C22⋊C4×C199C2211yescharacteristic
23#15691870—42no—
24C2×C199⋊C8—22no—
25C2×C199⋊C8—22no—
26C2×C199⋊C8—42no—
27C24×C199C2×C411yescharacteristic
28C24×C199C2×C411yescharacteristic
29C2×D4×C199—21no—
30C2×D4×C199—21no—
31C2×D4×C199—42no—
32C22×C4×C199—42no—
33C22×C4×C199C2×C411yescharacteristic
34C22×C4×C199C2×C411yescharacteristic
35C22×C4×C199C2×C411yescharacteristic
36C22×C4×C199C2×C411yescharacteristic
37C22×C4×C199D411yescharacteristic
38C22×C4×C199D411yescharacteristic
39C22×C4×C199D411yescharacteristic
40C22×C4×C199D411yescharacteristic
41C22×C4×C199C2311yescharacteristic
42C4⋊C4×C199—21no—
43C4⋊C4×C199—21no—
44C4⋊C4×C199—21no—
45C4⋊C4×C199—21no—
46C22⋊C4×C199—21no—
47C22⋊C4×C199—21no—
48C22⋊C4×C199—21no—
49C22⋊C4×C199—21no—
50C22⋊C4×C199—21no—
51C22⋊C4×C199—21no—
52C199⋊C8—42no—
53C199⋊C8—42no—
54C23×C199—22no—
55C23×C199—22no—
56C23×C199—21no—
57C23×C199—22no—
58C23×C199—41no—
59C23×C199—41no—
60C23×C199—41no—
61C23×C199C22⋊C411yescharacteristic
62C23×C199C22⋊C411yescharacteristic
63C23×C199C22×C411yescharacteristic
64D4×C199—42no—
65D4×C199—42no—
66C2×C4×C199—22no—
67C2×C4×C199—22no—
68C2×C4×C199—21no—
69C2×C4×C199—21no—
70C2×C4×C199—21no—
71C2×C4×C199—22no—
72C2×C4×C199—41no—
73C2×C4×C199—41no—
74C2×C4×C199—41no—
75C2×C4×C199—41no—
76C2×C4×C199—41no—
77C2×C4×C199—41no—
78C2×C4×C199C22⋊C411yescharacteristic
79C2×C4×C199C22⋊C411yescharacteristic
80C2×C4×C199C2×D411yescharacteristic
81C2×C4×C199C2×D411yescharacteristic
82C22×C199—21no—
83C22×C199—21no—
84C22×C199—21no—
85C22×C199—21no—
86C22×C199—21no—
87C22×C199—21no—
88C22×C199—21no—
89C22×C199—44no—
90C22×C199—41no—
91C22×C199—41no—
92C22×C199—41no—
93C22×C199—82no—
94C22×C199#9912yes—
95C22×C199C2×C22⋊C411yescharacteristic
96C22×C199#13112yes—
97C4×C199—21no—
98C4×C199—21no—
99C4×C199—21no—
100C4×C199—21no—
101C4×C199—41no—
102C4×C199—41no—
103C4×C199—41no—
104C2×C199—22no—
105C2×C199—22no—
106C2×C199—41no—
107C2×C199—81no—
108C2×C199#40811yescharacteristic
109C2×C199C2×#9911yescharacteristic
110C2×C199#41311yescharacteristic
111C199#146811yescharacteristic, complemented, Sylow
112#1468—1991nomaximal, Sylow
113C2×#123D19911yescharacteristic
114#407—1991no—
115C2×C22⋊C8—1991no—
116#324—1992no—
117#323—1992no—
118C22×D4Dic19911yescharacteristic
119C23×C4C2×D19911yescharacteristic
120C2×M16—3981no—
121C22×C8—3981no—
122#123—21no—
123#123—21no—
124#123—21no—
125#123—21no—
126C2×C4⋊C4Dic19911yescharacteristic
127C2×C4⋊C4C2×D19911yescharacteristic
128C2×C22⋊C4Dic19911yescharacteristic
129C2×C22⋊C4Dic19911yescharacteristic
130C2×C22⋊C4C2×D19911yescharacteristic
131C22⋊C8—3981no—
132C22⋊C8—3981no—
133C22⋊C8—1992no—
134C22⋊C8—1992no—
135C24C2×Dic19911yescharacteristic
136C24C2×Dic19911yescharacteristic
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—42no—
140C22×C4—42no—
141C22×C4C2×Dic19911yescharacteristic
142C22×C4C2×Dic19911yescharacteristic
143C22×C4C2×Dic19911yescharacteristic
144C22×C4C2×Dic19911yescharacteristic
145C22×C4C22⋊D19911yescharacteristic
146C22×C4C22⋊D19911yescharacteristic
147C22×C4C22⋊D19911yescharacteristic
148C22×C4C22⋊D19911yescharacteristic
149C22×C4C22×D19911yescharacteristic
150M16—7962no—
151C2×C8—7962no—
152C2×C8—3982no—
153C2×C8—3982no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C23—22no—
165C23—22no—
166C23—21no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23C22⋊Dic19911yescharacteristic
172C23C22⋊Dic19911yescharacteristic
173C23C22×Dic19911yescharacteristic
174D4—42no—
175D4—42no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4C22⋊Dic19911yescharacteristic
189C2×C4C22⋊Dic19911yescharacteristic
190C2×C4C2×C22⋊D19911yescharacteristic
191C2×C4C2×C22⋊D19911yescharacteristic
192C8—7962no—
193C8—7962no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—82no—
206C22#2026387012yes—
207C22C2×C22⋊Dic19911yescharacteristic
208C22#2026396212yes—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—41no—
214C4—41no—
215C4—41no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—81no—
220C2#2716398511yescharacteristic
221C2C2×#2026387011yescharacteristic
222C2#2716400211yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic199
commutatorC23×C199C22×C4
FrattiniC22×C4C22×D199
FittingC2×#123×C199C2
socleC22×C199C2×C22⋊C4
prime radicalC22×D4×C199C4
cosocleC22×C4×C199C23
hypercenterC2×#123D199
hypocenterC199#1468
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #27163982C2semidirect
2. C2×C22⋊(C199⋊C8)C2semidirect
3. #27163744C2semidirect
4. #27163742C2semidirect
111. C199#1468semidirect

Maximal subgroups

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

subgroupindex
#271639822
C2×C22⋊(C199⋊C8)2
#271637442
#271637422
C2×#123×C1992
#1468199