Finite Group Databasebeta

← #38767428

Subgroups — #38767428

Order 25472.

218 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 69 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#27164552C211yesmaximal, characteristic
2#27163886C211yesmaximal, characteristic, complemented
3#27163756C212yesmaximal, complemented
4#27163753C212yesmaximal
5#431×C199C211yesmaximal, characteristic, complemented
6#20263949—21no—
7#20263947C411yescharacteristic, complemented
8#20263943C411yescharacteristic, complemented
9#20263938—21no—
10#20263934C2211yescharacteristic
11#20263929C2211yescharacteristic
12C4×Dic398C411yescharacteristic, complemented
13C4×D796C411yescharacteristic, complemented
14C4×C22⋊D199—21no—
15#20263912—21no—
16C199⋊#95—21no—
17C199⋊#95—21no—
18C199⋊#95C2212yes—
19C2×C42×C199C2211yescharacteristic
20C4⋊C8×C199—21no—
21C22⋊C8×C199C2212yes—
22C4×C8×C199—21no—
23C2×Dic398C2×C411yescharacteristic
24C2×D796C2×C411yescharacteristic
25C2×C22⋊D199—21no—
26C22×Dic199—22no—
27C2×C4×D199—21no—
28C4⋊Dic199C2×C411yescharacteristic
29C4⋊Dic199C2×C411yescharacteristic
30Dic199⋊C4—21no—
31Dic199⋊C4—21no—
32Dic199⋊C4—21no—
33Dic199⋊C4—21no—
34C22⋊Dic199—21no—
35(C2×D199)⋊C4—21no—
36(C2×D199)⋊C4—21no—
37(C2×D199)⋊C4—21no—
38(C2×D199)⋊C4—21no—
39C4×Dic199—21no—
40C22×C4×C199D411yescharacteristic
41C22×C4×C199D411yescharacteristic
42C22×C4×C199C2311yescharacteristic
43C2×C8×C199—22no—
44C42×C199C2×C411yescharacteristic
45C42×C199C2×C411yescharacteristic
46C42×C199D411yescharacteristic
47C42×C199D411yescharacteristic
48C22×D199—21no—
49Dic398—41no—
50D796—41no—
51C22⋊D199—42no—
52C2×Dic199—21no—
53C2×Dic199—21no—
54C2×Dic199—21no—
55C2×Dic199—42no—
56C2×Dic199—42no—
57C2×Dic199—42no—
58C4×D199—42no—
59C23×C199C2×D411yescharacteristic
60C2×C4×C199—22no—
61C2×C4×C199—41no—
62C2×C4×C199—41no—
63C2×C4×C199C22⋊C411yescharacteristic
64C2×C4×C199C22⋊C411yescharacteristic
65C2×C4×C199C22⋊C411yescharacteristic
66C2×C4×C199C22⋊C411yescharacteristic
67C2×C4×C199C22×C411yescharacteristic
68C2×C4×C199C2×D411yescharacteristic
69C8×C199—42no—
70C22×C199—21no—
71C22×C199—21no—
72C22×C199C2×C22⋊C411yescharacteristic
73C22×C199#13512yes—
74C2×D199—41no—
75C2×D199—41no—
76C2×D199—41no—
77C4×C199—21no—
78C4×C199—21no—
79C4×C199—21no—
80C4×C199—21no—
81C4×C199—21no—
82C4×C199—21no—
83Dic199—41no—
84Dic199—41no—
85Dic199—41no—
86Dic199—82no—
87C2×C199—22no—
88C2×C199#41011yescharacteristic
89C2×C199#41611yescharacteristic
90C2×C199#42111yescharacteristic
91D199—81no—
92C199#152111yescharacteristic, complemented, Sylow
93#1521—1991nomaximal, Sylow
94#529—1991no—
95#431D19911yescharacteristic, complemented
96#383—1991no—
97#328—1992no—
98#327—1992no—
99#126—3981no—
100#125—1991no—
101#124—1991no—
102#123—3981no—
103C22⋊Q8—1991no—
104#121—1991no—
105C4×Q8—1991no—
106C4×D4—3981no—
107C4×D4—1991no—
108#117—3981no—
109C2×C42C2×D19911yescharacteristic
110C4⋊C8—21no—
111C22⋊C8C2×D19912yes—
112C4×C8—21no—
113#95—3981no—
114#95—3981no—
115#95—1992no—
116C2×Q8—1991no—
117C2×D4—3981no—
118C2×D4—1991no—
119C22×C4—3982no—
120C22×C4—3981no—
121C22×C4D79611yescharacteristic
122C22×C4D79611yescharacteristic
123C22×C4C22×D19911yescharacteristic
124C2×C8—22no—
125C4⋊C4—3981no—
126C4⋊C4—3981no—
127C4⋊C4—3981no—
128C4⋊C4—3981no—
129C4⋊C4—1991no—
130C4⋊C4—1991no—
131C22⋊C4—3981no—
132C22⋊C4—3981no—
133C22⋊C4—3981no—
134C22⋊C4—3981no—
135C22⋊C4—3981no—
136C42—3981no—
137C42C4×D19911yescharacteristic
138C42C4×D19911yescharacteristic
139C42C22⋊D19911yescharacteristic
140C42C22⋊D19911yescharacteristic
141C23—3981no—
142C23C2×D79611yescharacteristic
143Q8—7961no—
144D4—7961no—
145D4—7962no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—7962no—
150C2×C4—7962no—
151C2×C4—7962no—
152C2×C4—7962no—
153C2×C4—3981no—
154C2×C4—3981no—
155C2×C4—3981no—
156C2×C4(C2×D199)⋊C411yescharacteristic
157C2×C4(C2×D199)⋊C411yescharacteristic
158C2×C4(C2×D199)⋊C411yescharacteristic
159C2×C4(C2×D199)⋊C411yescharacteristic
160C2×C4C2×C4×D19911yescharacteristic
161C2×C4C2×C22⋊D19911yescharacteristic
162C8—42no—
163C22—21no—
164C22—21no—
165C22—7961no—
166C22—7961no—
167C22—7961no—
168C22C2×(C2×D199)⋊C411yescharacteristic
169C22#2026397712yes—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—7961no—
177C4—7961no—
178C4—7961no—
179C4—15922no—
180C2—22no—
181C2—15921no—
182C2#2716399111yescharacteristic
183C2#2716401511yescharacteristic
184C2#2716403411yescharacteristic
subgroupG/H
centerC22C2×(C2×D199)⋊C4
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#431×C199C2
socleC22×C199C2×C22⋊C4
prime radical#20263929C22
cosocleC22×C4×C199C23
hypercenter#431D199
hypocenterC199#1521
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. #27163886C2semidirect
3. #27163756C2semidirect
5. #431×C199C2semidirect
7. #20263947C4semidirect
8. #20263943C4semidirect
12. C4×Dic398C4semidirect
13. C4×D796C4semidirect
92. C199#1521semidirect
95. #431D199semidirect

Maximal subgroups

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

subgroupindex
#271645522
#271638862
#271637562
#271637532
#431×C1992
#1521199