Finite Group Databasebeta

← #38770489

Subgroups — #38770489

Order 25472.

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

normal subgroups only

This group has 264 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#27164525C211yesmaximal, characteristic
2#27164002C211yesmaximal, characteristic, complemented
3#27163807C212yesmaximal, complemented
4#27163804C212yesmaximal, complemented
5#410×C199C211yesmaximal, characteristic, complemented
6C22×C4×D199C411yescharacteristic, complemented
7C2×#15691870—21no—
8#20263936C411yescharacteristic, complemented
9#20263936C2211yescharacteristic
10#20263931—21no—
11#20263926—21no—
12#20263923C411yescharacteristic, complemented
13#20263923C2211yescharacteristic
14C4×C22⋊D199—21no—
15C4×C22⋊D199—21no—
16#20263914C411yescharacteristic, complemented
17#20263874—21no—
18#20263872C2212yes—
19C2×#41×C199C2211yescharacteristic
20#117×C199—21no—
21#99×C199—21no—
22#99×C199C2212yes—
23C23×D199C2×C411yescharacteristic
24C2×C22⋊D199—21no—
25C2×C22⋊D199—21no—
26C22×Dic199C2×C411yescharacteristic
27C2×C4×D199—21no—
28C2×C4×D199—21no—
29C2×C4×D199—41no—
30C2×C4×D199—41no—
31#15691870—22no—
32#15691870—41no—
33C2×C199⋊C8—41no—
34C4⋊Dic199—21no—
35Dic199⋊C4—21no—
36Dic199⋊C4—21no—
37C22⋊Dic199—21no—
38C22⋊Dic199—21no—
39C22⋊Dic199C2×C411yescharacteristic
40C22⋊Dic199C2×C411yescharacteristic
41(C2×D199)⋊C4—21no—
42(C2×D199)⋊C4—21no—
43C4×Dic199—21no—
44#41×C199—41no—
45#41×C199—41no—
46C2×Q8×C199D411yescharacteristic
47C2×D4×C199C2×C411yescharacteristic
48C2×D4×C199D411yescharacteristic
49C2×D4×C199C2311yescharacteristic
50C22×C4×C199C2×C411yescharacteristic
51C22×C4×C199D411yescharacteristic
52C22×C4×C199D411yescharacteristic
53C4⋊C4×C199—41no—
54C22⋊C4×C199—22no—
55C42×C199—41no—
56C22×D199—21no—
57C22×D199—42no—
58C22×D199—41no—
59C22⋊D199—42no—
60C22⋊D199—42no—
61C2×Dic199—21no—
62C2×Dic199—21no—
63C2×Dic199—21no—
64C2×Dic199—41no—
65C4×D199—42no—
66C4×D199—41no—
67C4×D199—41no—
68C199⋊C8—42no—
69C23×C199C22⋊C411yescharacteristic
70C23×C199C22×C411yescharacteristic
71C23×C199C2×D411yescharacteristic
72Q8×C199—41no—
73D4×C199—41no—
74D4×C199—41no—
75D4×C199—41no—
76C2×C4×C199—21no—
77C2×C4×C199—21no—
78C2×C4×C199—21no—
79C2×C4×C199—21no—
80C2×C4×C199—21no—
81C2×C4×C199—21no—
82C2×C4×C199—42no—
83C2×C4×C199C22⋊C411yescharacteristic
84C2×C4×C199C22⋊C411yescharacteristic
85C2×C4×C199C22⋊C411yescharacteristic
86C2×C4×C199C2×D411yescharacteristic
87C22×C199—21no—
88C22×C199—21no—
89C22×C199—21no—
90C22×C199—41no—
91C22×C199—41no—
92C22×C199—41no—
93C22×C199C2×C22⋊C411yescharacteristic
94C2×D199—22no—
95C2×D199—41no—
96C2×D199—42no—
97C2×D199—41no—
98C2×D199—42no—
99C4×C199—21no—
100C4×C199—21no—
101C4×C199—21no—
102C4×C199—82no—
103C4×C199#9911yescharacteristic
104C4×C199#9911yescharacteristic
105Dic199—41no—
106Dic199—41no—
107Dic199—41no—
108C2×C199—21no—
109C2×C199—41no—
110C2×C199—41no—
111C2×C199—41no—
112C2×C199C2×#9911yescharacteristic
113D199—42no—
114C199#212011yescharacteristic, complemented, Sylow
115#2120—1991nomaximal, Sylow
116#525—1991no—
117#413—1991no—
118#410D19911yescharacteristic, complemented
119#352—1992no—
120AΣL(1,𝔽16)—1992no—
121C2×#41C2×D19911yescharacteristic
122C23×C4—1991no—
123C2×M16—3981no—
124#123—1991no—
125#123—1991no—
126C22⋊Q8—3981no—
127#121—3981no—
128#120—1991no—
129#120—1991no—
130C4×D4—3981no—
131C4×D4—3981no—
132#117—21no—
133#117—1991no—
134#101—3981no—
135#100—1992no—
136#99—21no—
137#99C2×D19912yes—
138C24—1991no—
139#41—41no—
140#41—41no—
141C2×Q8D79611yescharacteristic
142C2×D4—3981no—
143C2×D4—3981no—
144C2×D4C4×D19911yescharacteristic
145C2×D4C22⋊D19911yescharacteristic
146C2×D4C22×D19911yescharacteristic
147C22×C4—7961no—
148C22×C4—7961no—
149C22×C4—3981no—
150C22×C4—3981no—
151C22×C4—1991no—
152C22×C4C4×D19911yescharacteristic
153C22×C4C22⋊D19911yescharacteristic
154C22×C4D79611yescharacteristic
155M16—7961no—
156M16—3982no—
157C2×C8—7961no—
158C4⋊C4—41no—
159C4⋊C4—3981no—
160C4⋊C4—3981no—
161C4⋊C4—3981no—
162C22⋊C4—22no—
163C22⋊C4—3981no—
164C22⋊C4—3981no—
165C22⋊C4—3981no—
166C22⋊C4—3981no—
167C22⋊C4—1991no—
168C22⋊C4—1991no—
169C42—41no—
170C42—3981no—
171C23—7961no—
172C23—7962no—
173C23—3981no—
174C23(C2×D199)⋊C411yescharacteristic
175C23C2×C4×D19911yescharacteristic
176C23C2×C22⋊D19911yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—7962no—
182D4—7962no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—7962no—
191C2×C4—7961no—
192C2×C4—7961no—
193C2×C4—7961no—
194C2×C4—3981no—
195C2×C4—3981no—
196C2×C4—3981no—
197C2×C4(C2×D199)⋊C411yescharacteristic
198C2×C4(C2×D199)⋊C411yescharacteristic
199C2×C4(C2×D199)⋊C411yescharacteristic
200C2×C4C2×D79611yescharacteristic
201C8—7962no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—7961no—
209C22—7962no—
210C22—7962no—
211C22—7961no—
212C22—3982no—
213C22C2×(C2×D199)⋊C411yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—82no—
218C4—7961no—
219C4—7961no—
220C4—7961no—
221C4#2026386911yescharacteristic
222C4#2026386911yescharacteristic
223C2—21no—
224C2—41no—
225C2—41no—
226C2—41no—
227C2—7962no—
228C2C2×#2026386911yescharacteristic
subgroupG/H
centerC4#20263869
commutatorC23×C199C22×C4
FrattiniC2×D4C22×D199
Fitting#410×C199C2
socleC2×C199C2×#99
prime radical#27164525C2
cosocleC2×D4×C199C23
hypercenter#410D199
hypocenterC199#2120
LCSG ▷ C23×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #27164002C2semidirect
3. #27163807C2semidirect
4. #27163804C2semidirect
5. #410×C199C2semidirect
6. C22×C4×D199C4semidirect
8. #20263936C4semidirect
12. #20263923C4semidirect
16. #20263914C4semidirect
114. C199#2120semidirect
118. #410D199semidirect

Maximal subgroups

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

subgroupindex
#271645252
#271640022
#271638072
#271638042
#410×C1992
#2120199