Finite Group Databasebeta

← #38767423

Subgroups — #38767423

Order 25472.

288 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×#20263934C211yesmaximal, characteristic
2#27163878C211yesmaximal, characteristic
3#27163756C214yesmaximal, complemented
4C2×C22⋊C8×C199C211yesmaximal, characteristic
5C22×Dic398C411yescharacteristic, complemented
6#20263934—21no—
7#20263934—21no—
8#20263934C412yescomplemented
9#20263934C2212yes—
10C2×C4⋊Dic199—21no—
11C2×C4⋊Dic199C411yescharacteristic, complemented
12C2×Dic199⋊C4—21no—
13C2×C22⋊Dic199—21no—
14C2×C22⋊Dic199—21no—
15C199⋊#95C2212yes—
16C23×C4×C199C2211yescharacteristic
17C22×C8×C199—21no—
18C22⋊C8×C199—21no—
19C22⋊C8×C199C2212yes—
20C2×Dic398—21no—
21C2×Dic398—42no—
22C2×Dic398C2×C412yes—
23C22×Dic199—21no—
24C22×Dic199—21no—
25C22×Dic199—22no—
26C4⋊Dic199—21no—
27C4⋊Dic199—42no—
28C4⋊Dic199C2×C412yes—
29Dic199⋊C4—22no—
30Dic199⋊C4—22no—
31C22⋊Dic199—22no—
32C22⋊Dic199—22no—
33C22⋊Dic199—42no—
34C24×C199D411yescharacteristic
35C22×C4×C199—42no—
36C22×C4×C199C2×C411yescharacteristic
37C22×C4×C199C2×C411yescharacteristic
38C22×C4×C199D411yescharacteristic
39C22×C4×C199D411yescharacteristic
40C22×C4×C199D411yescharacteristic
41C22×C4×C199C2311yescharacteristic
42C2×C8×C199—22no—
43C2×C8×C199—42no—
44Dic398—42no—
45Dic398—42no—
46C2×Dic199—22no—
47C2×Dic199—22no—
48C2×Dic199—42no—
49C2×Dic199—42no—
50C2×Dic199—42no—
51C2×Dic199—42no—
52C2×Dic199—42no—
53C23×C199—41no—
54C23×C199—41no—
55C23×C199C22⋊C411yescharacteristic
56C23×C199C22⋊C411yescharacteristic
57C23×C199SD1612yes—
58C23×C199Q1612yes—
59C23×C199C2×D411yescharacteristic
60C2×C4×C199—22no—
61C2×C4×C199—22no—
62C2×C4×C199—22no—
63C2×C4×C199—41no—
64C2×C4×C199—41no—
65C2×C4×C199—41no—
66C2×C4×C199C22⋊C411yescharacteristic
67C2×C4×C199C22⋊C411yescharacteristic
68C2×C4×C199C22×C411yescharacteristic
69C2×C4×C199C2×D411yescharacteristic
70C8×C199—42no—
71C22×C199—21no—
72C22×C199—21no—
73C22×C199—21no—
74C22×C199—21no—
75C22×C199—21no—
76C22×C199—21no—
77C22×C199—42no—
78C22×C199—42no—
79C22×C199#10312yes—
80C22×C199#10312yes—
81C22×C199C2×C22⋊C411yescharacteristic
82C22×C199C2×SD1611yescharacteristic
83C22×C199C2×Q1611yescharacteristic
84C4×C199—21no—
85C4×C199—21no—
86C4×C199—21no—
87C4×C199#9912yes—
88Dic199—42no—
89Dic199—42no—
90Dic199—82no—
91C2×C199—21no—
92C2×C199—22no—
93C2×C199—22no—
94C2×C199—21no—
95C2×C199C2×#9911yescharacteristic
96C2×C199C2×#10311yescharacteristic
97C2×C199#42111yescharacteristic
98C199#152011yescharacteristic, complemented, Sylow
99#1520—1991nomaximal, Sylow
100C2×C22⋊Q8—1991no—
101C2×C22⋊C8D19911yescharacteristic
102#380—1991no—
103#328—1994no—
104C22×Q8—1991no—
105C23×C4C2×D19911yescharacteristic
106C22×C8—21no—
107C22⋊Q8—3981no—
108C22⋊Q8—3981no—
109C22⋊Q8—1992no—
110C22⋊Q8—1992no—
111C2×C4⋊C4—3981no—
112C2×C4⋊C4—3981no—
113C2×C4⋊C4—1991no—
114C2×C22⋊C4—3981no—
115C2×C22⋊C4—3981no—
116C22⋊C8—21no—
117C22⋊C8C2×D19912yes—
118#95—1992no—
119C24D79611yescharacteristic
120C2×Q8—7962no—
121C2×Q8—3981no—
122C2×Q8—1992no—
123C22×C4—42no—
124C22×C4—3982no—
125C22×C4—3981no—
126C22×C4—3981no—
127C22×C4C4×D19911yescharacteristic
128C22×C4C4×D19911yescharacteristic
129C22×C4C22⋊D19911yescharacteristic
130C22×C4C22⋊D19911yescharacteristic
131C22×C4D79611yescharacteristic
132C22×C4C22×D19911yescharacteristic
133C2×C8—22no—
134C2×C8—42no—
135C4⋊C4—7962no—
136C4⋊C4—3982no—
137C4⋊C4—3981no—
138C4⋊C4—3982no—
139C4⋊C4—1992no—
140C22⋊C4—7962no—
141C22⋊C4—3982no—
142C22⋊C4—3982no—
143C23—41no—
144C23—41no—
145C23(C2×D199)⋊C411yescharacteristic
146C23(C2×D199)⋊C411yescharacteristic
147C23#1569187512yes—
148C23Dic79612yes—
149C23C2×D79611yescharacteristic
150Q8—7962no—
151Q8—7962no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—7962no—
159C2×C4—7962no—
160C2×C4—7962no—
161C2×C4—7962no—
162C2×C4—7962no—
163C2×C4—3982no—
164C2×C4—3982no—
165C2×C4(C2×D199)⋊C411yescharacteristic
166C2×C4(C2×D199)⋊C411yescharacteristic
167C2×C4C2×C4×D19911yescharacteristic
168C2×C4C2×C22⋊D19911yescharacteristic
169C8—42no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22#2026388012yes—
179C22#2026388012yes—
180C22C2×(C2×D199)⋊C411yescharacteristic
181C22C2×#1569187511yescharacteristic
182C22C2×Dic79611yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—7962no—
187C4—7962no—
188C4—15922no—
189C4#2026386912yes—
190C2—21no—
191C2—21no—
192C2—22no—
193C2—22no—
194C2C2×#2026386911yescharacteristic
195C2C2×#2026388011yescharacteristic
196C2#2716403411yescharacteristic
subgroupG/H
centerC22C2×(C2×D199)⋊C4
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
FittingC2×C22⋊C8×C199C2
socleC22×C199C2×C22⋊C4
prime radicalC24×C199D4
cosocleC22×C4×C199C23
hypercenterC2×C22⋊C8D199
hypocenterC199#1520
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
3. #27163756C2semidirect
5. C22×Dic398C4semidirect
8. #20263934C4semidirect
11. C2×C4⋊Dic199C4semidirect
98. C199#1520semidirect

Maximal subgroups

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

subgroupindex
C2×#202639342
#271638782
#271637562
C2×C22⋊C8×C1992
#1520199