Finite Group Databasebeta

← #38767800

Subgroups — #38767800

Order 25472.

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

normal subgroups only

This group has 308 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#27164576C211yesmaximal, characteristic
2#27164157C211yesmaximal, characteristic, complemented
3#27163929C211yesmaximal, characteristic, complemented
4#27163753C211yesmaximal, characteristic, complemented
5#27163752C211yesmaximal, characteristic, complemented
6#27163741C211yesmaximal, characteristic, complemented
7#448×C199C211yesmaximal, characteristic, complemented
8C2×D4×D199—21no—
9C2×#15691873—21no—
10#20263940—21no—
11#20263937—21no—
12#20263929C2211yescharacteristic
13#20263927—21no—
14#20263925—21no—
15#20263923—21no—
16C2×(C2×D199)⋊C4—21no—
17C2×(C2×D199)⋊C4C2211yescharacteristic
18#20263879—21no—
19C22⋊(C199⋊C8)C2211yescharacteristic, complemented
20C199⋊#95—21no—
21C199⋊#95C2211yescharacteristic, complemented
22C22×Q8×C199C2211yescharacteristic, complemented
23C2×SD16×C199—21no—
24#121×C199C2211yescharacteristic
25#103×C199—21no—
26C22⋊C8×C199C2211yescharacteristic, complemented
27C23×D199—21no—
28C23×D199D411yescharacteristic
29C2×D796—21no—
30C2×D796D411yescharacteristic
31C2×C22⋊D199—21no—
32C2×C22⋊D199—21no—
33C2×C22⋊D199—21no—
34D4×D199—42no—
35D4×D199—42no—
36C22×Dic199—21no—
37C22×Dic199D411yescharacteristic
38C2×C4×D199—21no—
39#15691873—42no—
40C2×C199⋊C8—21no—
41C4⋊Dic199D411yescharacteristic
42Dic199⋊C4—21no—
43C22⋊Dic199—21no—
44(C2×D199)⋊C4—21no—
45(C2×D199)⋊C4—21no—
46(C2×D199)⋊C4—21no—
47(C2×D199)⋊C4—21no—
48(C2×D199)⋊C4—42no—
49C2×Q8×C199—21no—
50C2×Q8×C199—21no—
51C2×Q8×C199—42no—
52C2×D4×C199—21no—
53C2×D4×C199D411yescharacteristic
54C22×C4×C199—21no—
55C22×C4×C199C2311yescharacteristic
56SD16×C199—42no—
57C2×C8×C199—21no—
58C4⋊C4×C199D411yescharacteristic
59C22⋊C4×C199—21no—
60C22×D199—21no—
61C22×D199—21no—
62C22×D199—21no—
63C22×D199—42no—
64C22×D199—41no—
65C22×D199—41no—
66C22×D199—81no—
67C22×D199—81no—
68C22×D199SD1612yescomplemented
69D796—41no—
70D796—42no—
71C22⋊D199—42no—
72C22⋊D199—41no—
73C22⋊D199—41no—
74C22⋊D199—81no—
75C2×Dic199—21no—
76C2×Dic199—21no—
77C2×Dic199—41no—
78C2×Dic199—41no—
79C2×Dic199—41no—
80C2×Dic199—41no—
81C4×D199—42no—
82C199⋊C8—41no—
83C23×C199—21no—
84C23×C199C2×D411yescharacteristic
85Q8×C199—22no—
86Q8×C199—22no—
87Q8×C199—41no—
88Q8×C199—41no—
89D4×C199—41no—
90D4×C199—41no—
91D4×C199—41no—
92C2×C4×C199—22no—
93C2×C4×C199—21no—
94C2×C4×C199—41no—
95C2×C4×C199—41no—
96C2×C4×C199—41no—
97C2×C4×C199C2×D411yescharacteristic
98C2×C4×C199C2×D411yescharacteristic
99C8×C199—41no—
100C22×C199—21no—
101C22×C199—21no—
102C22×C199—41no—
103C22×C199—41no—
104C22×C199—41no—
105C22×C199#12011yescharacteristic
106C22×C199C2×SD1611yescharacteristic
107C22×C199AGL(1,ℤ/8)11yescharacteristic
108C2×D199—22no—
109C2×D199—41no—
110C2×D199—41no—
111C2×D199—42no—
112C2×D199—41no—
113C2×D199—41no—
114C2×D199—41no—
115C2×D199—81no—
116C2×D199—84no—
117C4×C199—21no—
118C4×C199—21no—
119C4×C199—41no—
120C4×C199—42no—
121Dic199—41no—
122Dic199—41no—
123Dic199—81no—
124C2×C199—21no—
125C2×C199—21no—
126C2×C199—81no—
127C2×C199#45011yescharacteristic
128C2×C199#45511yescharacteristic
129C2×C199B(4,𝔽2)11yescharacteristic
130D199—42no—
131D199—81no—
132C199#160111yescharacteristic, complemented, Sylow
133#1601—1991nomaximal, Sylow
134#534—1991no—
135#448—1991no—
136#448D19911yescharacteristic, complemented
137#393—1991no—
138#327—1991no—
139#327—1991no—
140#323—1991no—
141C22×Q8C2×D19911yescharacteristic, complemented
142C22×D4—3981no—
143C2×SD16—21no—
144C2×SD16—3981no—
145#123—3981no—
146#123—3981no—
147#121—3981no—
148#121—1991no—
149#121C2×D19911yescharacteristic
150#120—3981no—
151#120—3981no—
152C2×C22⋊C4—3981no—
153C2×C22⋊C4—1991no—
154#103—21no—
155#103—3981no—
156C22⋊C8—1991no—
157C22⋊C8C2×D19911yescharacteristic, complemented
158#95—3981no—
159#95—1991no—
160C24—3981no—
161C24—1991no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—42no—
165C2×D4—21no—
166C2×D4—7962no—
167C2×D4—7962no—
168C2×D4—3981no—
169C2×D4—3981no—
170C2×D4—3981no—
171C2×D4—3981no—
172C2×D4—1991no—
173C2×D4C22⋊D19911yescharacteristic
174C22×C4—21no—
175C22×C4—3981no—
176C22×C4—3981no—
177C22×C4—1991no—
178C22×C4C22×D19911yescharacteristic
179SD16—42no—
180SD16—7962no—
181C2×C8—21no—
182C2×C8—3981no—
183C4⋊C4—3981no—
184C4⋊C4—1991no—
185C4⋊C4C22⋊D19911yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—7962no—
188C22⋊C4—3981no—
189C22⋊C4—3981no—
190C22⋊C4—3981no—
191C22⋊C4—3981no—
192C22⋊C4—3981no—
193C23—21no—
194C23—7961no—
195C23—7961no—
196C23—7962no—
197C23—15921no—
198C23—15921no—
199C23—3981no—
200C23—3981no—
201C23—3981no—
202C23—1992no—
203C23D4×D19911yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—41no—
207Q8—41no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—7961no—
212D4—7962no—
213D4—7962no—
214D4—7961no—
215D4—7961no—
216D4—15921no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—7962no—
223C2×C4—7961no—
224C2×C4—7961no—
225C2×C4—7961no—
226C2×C4—7961no—
227C2×C4—3981no—
228C2×C4—3981no—
229C2×C4D4×D19911yescharacteristic
230C2×C4C2×C22⋊D19911yescharacteristic
231C8—41no—
232C8—7961no—
233C22—21no—
234C22—21no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—7962no—
239C22—7961no—
240C22—7961no—
241C22—7961no—
242C22—7961no—
243C22—7961no—
244C22—15921no—
245C22—15924no—
246C22—3982no—
247C22#2026392311yescharacteristic
248C22SD16×D19911yescharacteristic
249C22#2026398211yescharacteristic
250C4—21no—
251C4—21no—
252C4—41no—
253C4—42no—
254C4—7961no—
255C4—7961no—
256C4—15921no—
257C2—21no—
258C2—21no—
259C2—81no—
260C2—7962no—
261C2—15921no—
262C2#2716416711yescharacteristic
263C2#2716419411yescharacteristic
264C2#2716420411yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#448×C199C2
socleC22×C199#120
prime radical#27164576C2
hypercenter#448D199
hypocenterC199#1601
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #27164157C2semidirect
3. #27163929C2semidirect
4. #27163753C2semidirect
5. #27163752C2semidirect
6. #27163741C2semidirect
7. #448×C199C2semidirect
19. C22⋊(C199⋊C8)C22semidirect
21. C199⋊#95C22semidirect
22. C22×Q8×C199C22semidirect
26. C22⋊C8×C199C22semidirect
68. C22×D199SD16semidirect
132. C199#1601semidirect
136. #448D199semidirect
141. C22×Q8C2×D199semidirect
157. C22⋊C8C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271645762
#271641572
#271639292
#271637532
#271637522
#271637412
#448×C1992
#1601199