Finite Group Databasebeta

← #38767388

Subgroups — #38767388

Order 25472.

222 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 77 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#27164075C211yesmaximal, characteristic, complemented
2#27163884C211yesmaximal, characteristic, complemented
3#27163755C212yesmaximal, complemented
4#27163752C212yesmaximal
5#529×C199C211yesmaximal, characteristic
6C2×C4⋊Dic199—21no—
7C4⋊(C199⋊C8)—21no—
8C22⋊(C199⋊C8)C2212yes—
9#20263866—21no—
10C199⋊#95—21no—
11C199⋊#95C2212yes—
12#126×C199—21no—
13#125×C199C411yescharacteristic, complemented
14#124×C199C411yescharacteristic, complemented
15#123×C199—21no—
16C22⋊Q8×C199C2211yescharacteristic
17#121×C199C2211yescharacteristic
18C4×Q8×C199C411yescharacteristic, complemented
19C4×D4×C199—21no—
20C4×D4×C199C411yescharacteristic, complemented
21#117×C199—21no—
22C2×C42×C199C2211yescharacteristic
23C22×Dic199—22no—
24C2×C199⋊C8—22no—
25C4⋊Dic199—42no—
26C2×Q8×C199C2×C411yescharacteristic
27C2×D4×C199—21no—
28C2×D4×C199C2×C411yescharacteristic
29C22×C4×C199—21no—
30C22×C4×C199D411yescharacteristic
31C22×C4×C199D411yescharacteristic
32C22×C4×C199C2311yescharacteristic
33C4⋊C4×C199—21no—
34C4⋊C4×C199—21no—
35C4⋊C4×C199—21no—
36C4⋊C4×C199—21no—
37C4⋊C4×C199C2×C411yescharacteristic
38C4⋊C4×C199C2×C411yescharacteristic
39C22⋊C4×C199—21no—
40C22⋊C4×C199—21no—
41C22⋊C4×C199—21no—
42C22⋊C4×C199—21no—
43C22⋊C4×C199—21no—
44C42×C199—21no—
45C42×C199C2×C411yescharacteristic
46C42×C199C2×C411yescharacteristic
47C42×C199D411yescharacteristic
48C42×C199D411yescharacteristic
49C2×Dic199—42no—
50C2×Dic199—42no—
51C2×Dic199—42no—
52C199⋊C8—42no—
53C23×C199—21no—
54C23×C199C2×D411yescharacteristic
55Q8×C199—41no—
56D4×C199—41no—
57D4×C199—41no—
58D4×C199—41no—
59C2×C4×C199—21no—
60C2×C4×C199—21no—
61C2×C4×C199—21no—
62C2×C4×C199—21no—
63C2×C4×C199—21no—
64C2×C4×C199—41no—
65C2×C4×C199—41no—
66C2×C4×C199—42no—
67C2×C4×C199C22⋊C411yescharacteristic
68C2×C4×C199C22⋊C411yescharacteristic
69C2×C4×C199C22⋊C411yescharacteristic
70C2×C4×C199C22⋊C411yescharacteristic
71C2×C4×C199C22×C411yescharacteristic
72C2×C4×C199C2×D411yescharacteristic
73C22×C199—21no—
74C22×C199—21no—
75C22×C199—41no—
76C22×C199—41no—
77C22×C199—41no—
78C22×C199C2×C22⋊C411yescharacteristic
79C22×C199AGL(1,ℤ/8)11yescharacteristic
80C22×C199#13711yescharacteristic
81C4×C199—21no—
82C4×C199—21no—
83C4×C199—21no—
84C4×C199—21no—
85C4×C199—21no—
86C4×C199—21no—
87C4×C199—41no—
88C4×C199—41no—
89C4×C199—41no—
90Dic199—82no—
91C2×C199—21no—
92C2×C199—21no—
93C2×C199—81no—
94C2×C199#41011yescharacteristic
95C2×C199#41711yescharacteristic
96C2×C199#42111yescharacteristic
97C199#151211yescharacteristic, complemented, Sylow
98#1512—1991nomaximal, Sylow
99#529D19911yescharacteristic
100#432—1991no—
101#382—1991no—
102#328—1992no—
103#327—1992no—
104#126—21no—
105#125Dic19911yescharacteristic, complemented
106#124Dic19911yescharacteristic, complemented
107#123—21no—
108C22⋊Q8C2×D19911yescharacteristic
109#121C2×D19911yescharacteristic
110C4×Q8Dic19911yescharacteristic, complemented
111C4×D4—21no—
112C4×D4Dic19911yescharacteristic, complemented
113#117—21no—
114C2×C4⋊C4—3981no—
115C2×C42C2×D19911yescharacteristic
116C4⋊C8—3981no—
117C22⋊C8—1992no—
118#97—3981no—
119#95—3981no—
120#95—1992no—
121C2×Q8C2×Dic19911yescharacteristic
122C2×D4—21no—
123C2×D4C2×Dic19911yescharacteristic
124C22×C4—21no—
125C22×C4—3982no—
126C22×C4C22⋊D19911yescharacteristic
127C22×C4C22⋊D19911yescharacteristic
128C22×C4C22×D19911yescharacteristic
129C2×C8—3982no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—7962no—
135C4⋊C4C2×Dic19911yescharacteristic
136C4⋊C4C2×Dic19911yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C2×Dic19911yescharacteristic
144C42C2×Dic19911yescharacteristic
145C42C22⋊D19911yescharacteristic
146C42C22⋊D19911yescharacteristic
147C23—21no—
148C23C2×C22⋊D19911yescharacteristic
149Q8—41no—
150D4—41no—
151D4—41no—
152D4—41no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—7962no—
162C2×C4—7962no—
163C2×C4—7962no—
164C2×C4C22⋊Dic19911yescharacteristic
165C2×C4C22⋊Dic19911yescharacteristic
166C2×C4C22⋊Dic19911yescharacteristic
167C2×C4C22⋊Dic19911yescharacteristic
168C2×C4C22×Dic19911yescharacteristic
169C2×C4C2×C22⋊D19911yescharacteristic
170C8—7962no—
171C22—21no—
172C22—21no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22C2×C22⋊Dic19911yescharacteristic
177C22#2026398011yescharacteristic
178C22#2026398511yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—15922no—
189C2—21no—
190C2—21no—
191C2—81no—
192C2#2716399211yescharacteristic
193C2#2716401911yescharacteristic
194C2#2716403511yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic199
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#529×C199C2
socleC22×C199C2×C22⋊C4
prime radical#121×C199C22
cosocleC22×C4×C199C23
hypercenter#529D199
hypocenterC199#1512
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #27164075C2semidirect
2. #27163884C2semidirect
3. #27163755C2semidirect
13. #125×C199C4semidirect
14. #124×C199C4semidirect
18. C4×Q8×C199C4semidirect
20. C4×D4×C199C4semidirect
97. C199#1512semidirect
105. #125Dic199semidirect
106. #124Dic199semidirect
110. C4×Q8Dic199semidirect
112. C4×D4Dic199semidirect

Maximal subgroups

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

subgroupindex
#271640752
#271638842
#271637552
#271637522
#529×C1992
#1512199