Finite Group Databasebeta

← #38770521

Subgroups — #38770521

Order 25472.

270 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 65 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#27164582C211yesmaximal, characteristic
2#27163992C211yesmaximal, characteristic, complemented
3#27163810C214yesmaximal, complemented
4#410×C199C211yesmaximal, characteristic, complemented
5C22×D796C411yescharacteristic, complemented
6#20263951C2212yes—
7#20263944C412yescomplemented
8#20263929—21no—
9#20263929—21no—
10#20263923—21no—
11#20263923—21no—
12#20263914—21no—
13#20263914C411yescharacteristic, complemented
14#20263870—21no—
15#20263870C2212yescomplemented
16C2×#41×C199C2211yescharacteristic
17#117×C199—21no—
18#99×C199—21no—
19#99×C199C2212yescomplemented
20C23×D199—21no—
21C2×D796—21no—
22C2×D796—41no—
23C2×D796—41no—
24C2×D796C2×C412yes—
25C2×C22⋊D199—22no—
26C2×C22⋊D199—22no—
27C4⋊Dic199—21no—
28C4⋊Dic199—41no—
29C22⋊Dic199—21no—
30C22⋊Dic199—22no—
31(C2×D199)⋊C4—22no—
32(C2×D199)⋊C4—22no—
33C4×Dic199—41no—
34C4×Dic199C2×C412yes—
35#41×C199—41no—
36#41×C199—41no—
37C2×Q8×C199C2×C411yescharacteristic
38C2×D4×C199D411yescharacteristic
39C2×D4×C199D411yescharacteristic
40C2×D4×C199C2311yescharacteristic
41C22×C4×C199C2×C411yescharacteristic
42C22×C4×C199D411yescharacteristic
43C22×C4×C199D411yescharacteristic
44C4⋊C4×C199—41no—
45C22⋊C4×C199—22no—
46C42×C199—41no—
47C22×D199—22no—
48C22×D199—42no—
49C22×D199—82no—
50D796—42no—
51D796—41no—
52D796—41no—
53C22⋊D199—82no—
54C22⋊D199—82no—
55C2×Dic199—22no—
56C2×Dic199—42no—
57C23×C199C22⋊C411yescharacteristic
58C23×C199C2×D411yescharacteristic
59C23×C199C2×D411yescharacteristic
60Q8×C199—41no—
61D4×C199—41no—
62D4×C199—41no—
63D4×C199—41no—
64C2×C4×C199—21no—
65C2×C4×C199—21no—
66C2×C4×C199—21no—
67C2×C4×C199—21no—
68C2×C4×C199—21no—
69C2×C4×C199—21no—
70C2×C4×C199—42no—
71C2×C4×C199C22⋊C411yescharacteristic
72C2×C4×C199C22⋊C411yescharacteristic
73C2×C4×C199C22⋊C411yescharacteristic
74C2×C4×C199C22×C411yescharacteristic
75C22×C199—21no—
76C22×C199—21no—
77C22×C199—21no—
78C22×C199—41no—
79C22×C199—41no—
80C22×C199—41no—
81C22×C199C2×C22⋊C411yescharacteristic
82C2×D199—42no—
83C2×D199—42no—
84C2×D199—42no—
85C2×D199—84no—
86C4×C199—21no—
87C4×C199—21no—
88C4×C199—21no—
89C4×C199—82no—
90C4×C199#9911yescharacteristic
91C4×C199#9911yescharacteristic
92Dic199—42no—
93Dic199—82no—
94C2×C199—21no—
95C2×C199—41no—
96C2×C199—41no—
97C2×C199—41no—
98C2×C199C2×#9911yescharacteristic
99D199—82no—
100C199#212811yescharacteristic, complemented, Sylow
101#2128—1991nomaximal, Sylow
102#535—1991no—
103#410—1991no—
104#410D19911yescharacteristic, complemented
105#353—1994no—
106C2×#41C2×D19911yescharacteristic
107C22×D4—1991no—
108#127—1992no—
109#124—1992no—
110#121—3981no—
111#121—3981no—
112#120—3981no—
113#120—3981no—
114#117—21no—
115#117—3981no—
116#117—1991no—
117#99—21no—
118#99—3981no—
119#99—1992no—
120#99C2×D19912yescomplemented
121C24—3981no—
122#41—41no—
123#41—41no—
124C2×Q8C4×D19911yescharacteristic
125C2×D4—7961no—
126C2×D4—7961no—
127C2×D4—3982no—
128C2×D4—3982no—
129C2×D4—3981no—
130C2×D4—1992no—
131C2×D4C22⋊D19911yescharacteristic
132C2×D4D79611yescharacteristic
133C2×D4C22×D19911yescharacteristic
134C22×C4C4×D19911yescharacteristic
135C22×C4C22⋊D19911yescharacteristic
136C22×C4D79611yescharacteristic
137C4⋊C4—41no—
138C4⋊C4—7961no—
139C4⋊C4—3981no—
140C22⋊C4—22no—
141C22⋊C4—3982no—
142C22⋊C4—3981no—
143C22⋊C4—3982no—
144C22⋊C4—3982no—
145C42—41no—
146C42—7961no—
147C42—1992no—
148C23—7962no—
149C23—15922no—
150C23—3982no—
151C23(C2×D199)⋊C411yescharacteristic
152C23C2×C22⋊D19911yescharacteristic
153C23C2×D79611yescharacteristic
154Q8—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—7962no—
159D4—7961no—
160D4—7961no—
161D4—15922no—
162D4—15922no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—42no—
170C2×C4—7962no—
171C2×C4—3982no—
172C2×C4(C2×D199)⋊C411yescharacteristic
173C2×C4(C2×D199)⋊C411yescharacteristic
174C2×C4(C2×D199)⋊C411yescharacteristic
175C2×C4C2×C4×D19911yescharacteristic
176C22—21no—
177C22—21no—
178C22—21no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—7962no—
183C22—7962no—
184C22—7962no—
185C22—15924no—
186C22C2×(C2×D199)⋊C411yescharacteristic
187C4—21no—
188C4—21no—
189C4—21no—
190C4—82no—
191C4—7962no—
192C4—15922no—
193C4#2026386911yescharacteristic
194C4#2026386911yescharacteristic
195C2—21no—
196C2—41no—
197C2—41no—
198C2—41no—
199C2—15922no—
200C2C2×#2026386911yescharacteristic
subgroupG/H
centerC2C2×#20263869
commutatorC2×C4×C199C22×C4
FrattiniC2×D4C22×D199
Fitting#410×C199C2
socleC2×C199C2×#99
prime radical#27164582C2
cosocleC2×D4×C199C23
hypercenter#410D199
hypocenterC199#2128
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #27163992C2semidirect
3. #27163810C2semidirect
4. #410×C199C2semidirect
5. C22×D796C4semidirect
7. #20263944C4semidirect
13. #20263914C4semidirect
15. #20263870C22semidirect
19. #99×C199C22semidirect
100. C199#2128semidirect
104. #410D199semidirect
120. #99C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271645822
#271639922
#271638102
#410×C1992
#2128199