Finite Group Databasebeta

← #38767792

Subgroups — #38767792

Order 25472.

266 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 53 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#27164649C211yesmaximal, characteristic, complemented
2#27164171C212yesmaximal
3#27163935C211yesmaximal, characteristic
4#27163756C212yesmaximal, complemented
5#324×C199C211yesmaximal, characteristic, complemented
6C22×D796C2211yescharacteristic
7C2×#15691875—22no—
8#20263954—22no—
9#20263946—22no—
10#20263934—21no—
11#20263934C2212yes—
12#20263914—21no—
13C2×(C2×D199)⋊C4—21no—
14C2×(C2×D199)⋊C4—21no—
15#20263877—22no—
16C199⋊#95C2211yescharacteristic
17C2×C4⋊C4×C199C2211yescharacteristic
18C22⋊C8×C199C2212yes—
19C23×D199—21no—
20C2×Dic398—21no—
21C2×Dic398D412yes—
22C2×D796—22no—
23C2×D796—42no—
24C22×Dic199—21no—
25#15691875—44no—
26C4⋊Dic199—21no—
27C4⋊Dic199—21no—
28C4⋊Dic199—21no—
29C4⋊Dic199D412yes—
30Dic199⋊C4—22no—
31Dic199⋊C4—22no—
32C22⋊Dic199—22no—
33(C2×D199)⋊C4—42no—
34(C2×D199)⋊C4—42no—
35C4×Dic199—22no—
36C22×C4×C199D411yescharacteristic
37C22×C4×C199D411yescharacteristic
38C22×C4×C199C2311yescharacteristic
39C2×C8×C199—22no—
40C4⋊C4×C199—22no—
41C22×D199—22no—
42C22×D199—41no—
43C22×D199—41no—
44C22×D199—41no—
45C22×D199—41no—
46C22×D199—41no—
47C22×D199—41no—
48Dic398—42no—
49Dic398—81no—
50D796—24no—
51D796—42no—
52C2×Dic199—22no—
53C2×Dic199—22no—
54C2×Dic199—41no—
55C2×Dic199—41no—
56C2×Dic199—41no—
57C23×C199C2×D411yescharacteristic
58C2×C4×C199—21no—
59C2×C4×C199—21no—
60C2×C4×C199—21no—
61C2×C4×C199—41no—
62C2×C4×C199—41no—
63C2×C4×C199SD1612yescomplemented
64C2×C4×C199C2×D412yes—
65C8×C199—42no—
66C22×C199—21no—
67C22×C199—21no—
68C22×C199#12011yescharacteristic
69C22×C199C2×SD1611yescharacteristic
70C22×C199#13711yescharacteristic
71C2×D199—42no—
72C2×D199—42no—
73C2×D199—42no—
74C2×D199—81no—
75C2×D199—81no—
76C2×D199—81no—
77C2×D199—81no—
78C4×C199—22no—
79C4×C199—22no—
80C4×C199—41no—
81Dic199—42no—
82Dic199—42no—
83Dic199—81no—
84C2×C199—21no—
85C2×C199—21no—
86C2×C199#44811yescharacteristic
87C2×C199#45411yescharacteristic
88C2×C199B(4,𝔽2)11yescharacteristic
89D199—82no—
90C199#159911yescharacteristic, complemented, Sylow
91#1599—1991nomaximal, Sylow
92#544—1991no—
93#450—1992no—
94#394—1991no—
95#328—1992no—
96#324D19911yescharacteristic, complemented
97C22×D4—1991no—
98C2×SD16—3982no—
99C4⋊Q8—3982no—
100#125—3982no—
101C22⋊Q8—3981no—
102C22⋊Q8—1992no—
103#117—3981no—
104C2×C4⋊C4C2×D19911yescharacteristic
105C2×C22⋊C4—3981no—
106C2×C22⋊C4—3981no—
107#102—3982no—
108C22⋊C8C2×D19912yes—
109#95—1991no—
110C24—3981no—
111C2×Q8—3981no—
112C2×Q8—1992no—
113C2×D4—7962no—
114C2×D4—3982no—
115C22×C4—3981no—
116C22×C4D79611yescharacteristic
117C22×C4D79611yescharacteristic
118C22×C4C22×D19911yescharacteristic
119SD16—7964no—
120C2×C8—22no—
121C4⋊C4—22no—
122C4⋊C4—3982no—
123C4⋊C4—3981no—
124C4⋊C4—3981no—
125C4⋊C4—3981no—
126C4⋊C4—3982no—
127C4⋊C4—1992no—
128C22⋊C4—7962no—
129C22⋊C4—7962no—
130C22⋊C4—3982no—
131C42—3982no—
132C23—7961no—
133C23—7961no—
134C23—7961no—
135C23—7961no—
136C23—7961no—
137C23—7961no—
138C23—3982no—
139C23C2×D79611yescharacteristic
140Q8—7962no—
141Q8—15921no—
142D4—7962no—
143D4—3984no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—7961no—
150C2×C4—7961no—
151C2×C4—7961no—
152C2×C4—3982no—
153C2×C4—3982no—
154C2×C4#1569187512yescomplemented
155C2×C4D4×D19912yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—7962no—
160C22—7962no—
161C22—7962no—
162C22—15921no—
163C22—15921no—
164C22—15921no—
165C22—15921no—
166C22#2026392511yescharacteristic
167C22C2×#1569187511yescharacteristic
168C22#2026398811yescharacteristic
169C4—22no—
170C4—22no—
171C4—41no—
172C4—7962no—
173C4—7962no—
174C4—15921no—
175C2—21no—
176C2—21no—
177C2—15922no—
178C2#2716415911yescharacteristic
179C2#2716419311yescharacteristic
180C2#2716420611yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#324×C199C2
socleC22×C199#120
prime radicalC22×D796C22
hypercenter#324D199
hypocenterC199#1599
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #27164649C2semidirect
4. #27163756C2semidirect
5. #324×C199C2semidirect
63. C2×C4×C199SD16semidirect
90. C199#1599semidirect
96. #324D199semidirect
154. C2×C4#15691875semidirect

Maximal subgroups

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

subgroupindex
#271646492
#271641712
#271639352
#271637562
#324×C1992
#1599199