Finite Group Databasebeta

← #38767774

Subgroups — #38767774

Order 25472.

316 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 51 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#27164643C211yesmaximal, characteristic, complemented
2#27164171C211yesmaximal, characteristic, complemented
3#27164168C211yesmaximal, characteristic, complemented
4#27163928C211yesmaximal, characteristic
5#27163754C211yesmaximal, characteristic, complemented
6#27163741C211yesmaximal, characteristic, complemented
7#328×C199C211yesmaximal, characteristic, complemented
8C22×D796—21no—
9C22×D796C2211yescharacteristic
10C2×#15691875—21no—
11C2×#15691873—21no—
12#20263935—21no—
13#20263934C2211yescharacteristic
14#20263933—21no—
15#20263933—21no—
16#20263932—21no—
17#20263931—21no—
18C2×(C2×D199)⋊C4—21no—
19C2×(C2×D199)⋊C4C2211yescharacteristic
20C22⋊C4×D199—21no—
21#20263877—21no—
22#20263875—21no—
23C22⋊(C199⋊C8)C2211yescharacteristic
24C22⋊Q8×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic
26#95×C199C2211yescharacteristic
27C23×D199—21no—
28C23×D199D411yescharacteristic
29C2×Dic398—21no—
30C2×Dic398D411yescharacteristic, complemented
31C2×D796—21no—
32C2×D796—21no—
33C2×D796—42no—
34C2×D796—42no—
35C2×D796—42no—
36C2×D796—42no—
37C22×Dic199D411yescharacteristic
38C2×C4×D199—21no—
39C2×C4×D199—21no—
40#15691875—42no—
41#15691873—42no—
42C2×C199⋊C8—21no—
43C4⋊Dic199—21no—
44C4⋊Dic199D411yescharacteristic, complemented
45Dic199⋊C4—21no—
46Dic199⋊C4—21no—
47Dic199⋊C4—21no—
48C22⋊Dic199—21no—
49(C2×D199)⋊C4—21no—
50(C2×D199)⋊C4—21no—
51(C2×D199)⋊C4—21no—
52(C2×D199)⋊C4—21no—
53(C2×D199)⋊C4—42no—
54C2×Q8×C199D411yescharacteristic, complemented
55C22×C4×C199—21no—
56C22×C4×C199C2311yescharacteristic
57C2×C8×C199—21no—
58C4⋊C4×C199—21no—
59C4⋊C4×C199D411yescharacteristic, complemented
60C22⋊C4×C199—21no—
61C22×D199—21no—
62C22×D199—21no—
63C22×D199—22no—
64C22×D199—41no—
65C22×D199—41no—
66C22×D199—41no—
67C22×D199—41no—
68C22×D199—41no—
69C22×D199—41no—
70C22×D199—41no—
71C22×D199—41no—
72C22×D199SD1612yescomplemented
73Dic398—41no—
74Dic398—81no—
75D796—22no—
76D796—22no—
77D796—41no—
78D796—41no—
79D796—48no—
80C2×Dic199—21no—
81C2×Dic199—21no—
82C2×Dic199—21no—
83C2×Dic199—41no—
84C4×D199—81no—
85C4×D199—81no—
86C199⋊C8—41no—
87C23×C199C2×D411yescharacteristic
88Q8×C199—41no—
89C2×C4×C199—22no—
90C2×C4×C199—21no—
91C2×C4×C199—21no—
92C2×C4×C199—41no—
93C2×C4×C199—41no—
94C2×C4×C199—41no—
95C2×C4×C199C2×D411yescharacteristic
96C2×C4×C199C2×D411yescharacteristic
97C8×C199—41no—
98C22×C199—21no—
99C22×C199—21no—
100C22×C199#12011yescharacteristic
101C22×C199C2×SD1611yescharacteristic
102C22×C199#13711yescharacteristic
103C2×D199—22no—
104C2×D199—41no—
105C2×D199—41no—
106C2×D199—42no—
107C2×D199—42no—
108C2×D199—42no—
109C2×D199—42no—
110C2×D199—81no—
111C2×D199—81no—
112C2×D199—81no—
113C2×D199—81no—
114C2×D199—81no—
115C4×C199—21no—
116C4×C199—21no—
117C4×C199—42no—
118C4×C199—41no—
119C4×C199—41no—
120Dic199—41no—
121Dic199—41no—
122Dic199—41no—
123C2×C199—21no—
124C2×C199—21no—
125C2×C199#44811yescharacteristic
126C2×C199#45311yescharacteristic
127C2×C199B(4,𝔽2)11yescharacteristic
128D199—42no—
129D199—82no—
130C199#159611yescharacteristic, complemented, Sylow
131#1596—1991nomaximal, Sylow
132#543—1991no—
133#450—1991no—
134#450—1991no—
135#392—1991no—
136#328—1991no—
137#328D19911yescharacteristic, complemented
138#323—1991no—
139C22×D4—3981no—
140C22×D4—1991no—
141C2×SD16—3981no—
142C2×SD16—3981no—
143C22⋊Q8—3981no—
144C22⋊Q8—3981no—
145C22⋊Q8—3981no—
146C22⋊Q8—3981no—
147C22⋊Q8—3981no—
148C22⋊Q8—1991no—
149C22⋊Q8C2×D19911yescharacteristic
150C2×C22⋊C4—3981no—
151C2×C22⋊C4—3981no—
152C2×C22⋊C4—1991no—
153#102—3981no—
154#102—3981no—
155C22⋊C8—1991no—
156C22⋊C8C2×D19911yescharacteristic
157#95C2×D19911yescharacteristic
158C24—3981no—
159C24—1991no—
160C2×Q8—3981no—
161C2×Q8—1991no—
162C2×Q8D79611yescharacteristic, complemented
163C2×D4—7962no—
164C2×D4—7962no—
165C2×D4—7962no—
166C2×D4—7962no—
167C2×D4—3981no—
168C2×D4—3981no—
169C22×C4—21no—
170C22×C4—3981no—
171C22×C4—3981no—
172C22×C4—1991no—
173C22×C4C22×D19911yescharacteristic
174SD16—7962no—
175SD16—7962no—
176C2×C8—21no—
177C2×C8—3981no—
178C4⋊C4—21no—
179C4⋊C4—3981no—
180C4⋊C4—3981no—
181C4⋊C4—3981no—
182C4⋊C4—3981no—
183C4⋊C4—1991no—
184C4⋊C4D79611yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—7962no—
187C22⋊C4—3981no—
188C22⋊C4—3981no—
189C22⋊C4—3981no—
190C22⋊C4—3981no—
191C22⋊C4—3981no—
192C23—7961no—
193C23—7961no—
194C23—7961no—
195C23—7961no—
196C23—7961no—
197C23—7961no—
198C23—7961no—
199C23—7961no—
200C23—3982no—
201C23—3981no—
202C23—3981no—
203C23—1992no—
204C23D4×D19911yescharacteristic
205Q8—41no—
206Q8—7961no—
207Q8—15921no—
208D4—7961no—
209D4—7961no—
210D4—7968no—
211D4—3982no—
212D4—3982no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—7961no—
220C2×C4—15921no—
221C2×C4—15921no—
222C2×C4—3981no—
223C2×C4—3981no—
224C2×C4—3981no—
225C2×C4D4×D19911yescharacteristic
226C2×C4C2×D79611yescharacteristic
227C8—41no—
228C8—7961no—
229C22—21no—
230C22—21no—
231C22—7962no—
232C22—7962no—
233C22—7962no—
234C22—7961no—
235C22—7961no—
236C22—7962no—
237C22—15921no—
238C22—15921no—
239C22—15921no—
240C22—15921no—
241C22—15921no—
242C22—3982no—
243C22#2026392511yescharacteristic
244C22SD16×D19911yescharacteristic
245C22#2026398711yescharacteristic
246C4—21no—
247C4—21no—
248C4—42no—
249C4—41no—
250C4—41no—
251C4—7961no—
252C4—7961no—
253C4—7961no—
254C2—21no—
255C2—21no—
256C2—7962no—
257C2—15922no—
258C2#2716415811yescharacteristic
259C2#2716418711yescharacteristic
260C2#2716420611yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#328×C199C2
socleC22×C199#120
prime radical#27163928C2
hypercenter#328D199
hypocenterC199#1596
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #27164643C2semidirect
2. #27164171C2semidirect
3. #27164168C2semidirect
5. #27163754C2semidirect
6. #27163741C2semidirect
7. #328×C199C2semidirect
30. C2×Dic398D4semidirect
44. C4⋊Dic199D4semidirect
54. C2×Q8×C199D4semidirect
59. C4⋊C4×C199D4semidirect
72. C22×D199SD16semidirect
130. C199#1596semidirect
137. #328D199semidirect
162. C2×Q8D796semidirect
184. C4⋊C4D796semidirect

Maximal subgroups

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

subgroupindex
#271646432
#271641712
#271641682
#271639282
#271637542
#271637412
#328×C1992
#1596199