Finite Group Databasebeta

← #38767871

Subgroups — #38767871

Order 25472.

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

normal subgroups only

This group has 288 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#27164337C211yesmaximal, characteristic, complemented
3#27164334C211yesmaximal, characteristic, complemented
4#27163955C211yesmaximal, characteristic, complemented
5#27163751C211yesmaximal, characteristic, complemented
6#27163741C211yesmaximal, characteristic, complemented
7#327×C199C211yesmaximal, characteristic, complemented
8C2×D4×D199—21no—
9#20263940—21no—
10#20263937—21no—
11#20263929C2211yescharacteristic
12#20263927—21no—
13#20263925—21no—
14#20263923—21no—
15C2×C4⋊Dic199—21no—
16C2×C4⋊Dic199C2211yescharacteristic, complemented
17C2×(C2×D199)⋊C4—21no—
18C2×(C2×D199)⋊C4C2211yescharacteristic
19#20263887—21no—
20#20263886—21no—
21#20263877—21no—
22#20263876—21no—
23C22⋊(C199⋊C8)C2211yescharacteristic, complemented
24#121×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic, complemented
26#95×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—
39C2×C199⋊C8—21no—
40C4⋊Dic199—21no—
41C4⋊Dic199—21no—
42C4⋊Dic199—42no—
43C4⋊Dic199D411yescharacteristic
44Dic199⋊C4—21no—
45C22⋊Dic199—21no—
46(C2×D199)⋊C4—21no—
47(C2×D199)⋊C4—21no—
48(C2×D199)⋊C4—21no—
49(C2×D199)⋊C4—21no—
50(C2×D199)⋊C4—42no—
51C2×D4×C199—21no—
52C2×D4×C199D411yescharacteristic
53C22×C4×C199—21no—
54C22×C4×C199C2311yescharacteristic
55C2×C8×C199—21no—
56C4⋊C4×C199D411yescharacteristic
57C22⋊C4×C199—21no—
58C22×D199—21no—
59C22×D199—21no—
60C22×D199—21no—
61C22×D199—42no—
62C22×D199—41no—
63C22×D199—41no—
64C22×D199—81no—
65C22×D199—81no—
66C22×D199SD1612yes—
67D796—41no—
68D796—42no—
69C22⋊D199—42no—
70C22⋊D199—41no—
71C22⋊D199—41no—
72C22⋊D199—81no—
73C2×Dic199—21no—
74C2×Dic199—21no—
75C2×Dic199—41no—
76C2×Dic199—41no—
77C2×Dic199—41no—
78C2×Dic199—41no—
79C4×D199—42no—
80C199⋊C8—41no—
81C23×C199—21no—
82C23×C199C2×D411yescharacteristic
83D4×C199—41no—
84D4×C199—41no—
85D4×C199—41no—
86C2×C4×C199—22no—
87C2×C4×C199—21no—
88C2×C4×C199—41no—
89C2×C4×C199—41no—
90C2×C4×C199—41no—
91C2×C4×C199C2×D411yescharacteristic
92C2×C4×C199C2×D411yescharacteristic
93C8×C199—41no—
94C22×C199—21no—
95C22×C199—21no—
96C22×C199—41no—
97C22×C199—41no—
98C22×C199—41no—
99C22×C199#12011yescharacteristic
100C22×C199C2×SD1611yescharacteristic
101C22×C199AGL(1,ℤ/8)11yescharacteristic
102C2×D199—22no—
103C2×D199—41no—
104C2×D199—41no—
105C2×D199—42no—
106C2×D199—41no—
107C2×D199—41no—
108C2×D199—41no—
109C2×D199—81no—
110C2×D199—84no—
111C4×C199—21no—
112C4×C199—21no—
113C4×C199—42no—
114C4×C199—41no—
115Dic199—41no—
116Dic199—41no—
117Dic199—81no—
118C2×C199—21no—
119C2×C199—21no—
120C2×C199—81no—
121C2×C199#45011yescharacteristic
122C2×C199#45311yescharacteristic
123C2×C199#45811yescharacteristic
124D199—42no—
125D199—81no—
126C199#161311yescharacteristic, complemented, Sylow
127#1613—1991nomaximal, Sylow
128#534—1991no—
129#481—1991no—
130#481—1991no—
131#399—1991no—
132#327—1991no—
133#327D19911yescharacteristic, complemented
134#323—1991no—
135C22×D4—3981no—
136#123—3981no—
137#123—3981no—
138#121—3981no—
139#121—1991no—
140#121C2×D19911yescharacteristic
141#120—3981no—
142#120—3981no—
143C2×C4⋊C4—3981no—
144C2×C4⋊C4—1991no—
145C2×C22⋊C4—3981no—
146C2×C22⋊C4—1991no—
147#106—3981no—
148#106—3981no—
149#102—3981no—
150#102—3981no—
151C22⋊C8—1991no—
152C22⋊C8C2×D19911yescharacteristic, complemented
153#95C2×D19911yescharacteristic, complemented
154C24—3981no—
155C24—1991no—
156C2×D4—21no—
157C2×D4—7962no—
158C2×D4—7962no—
159C2×D4—3981no—
160C2×D4—3981no—
161C2×D4—3981no—
162C2×D4—3981no—
163C2×D4—1991no—
164C2×D4D79611yescharacteristic
165C22×C4—21no—
166C22×C4—3981no—
167C22×C4—3981no—
168C22×C4—1991no—
169C22×C4C22×D19911yescharacteristic
170C2×C8—21no—
171C2×C8—3981no—
172C4⋊C4—7962no—
173C4⋊C4—3981no—
174C4⋊C4—3981no—
175C4⋊C4—3981no—
176C4⋊C4—1991no—
177C4⋊C4D79611yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—7962no—
180C22⋊C4—3981no—
181C22⋊C4—3981no—
182C22⋊C4—3981no—
183C22⋊C4—3981no—
184C22⋊C4—3981no—
185C23—21no—
186C23—7961no—
187C23—7961no—
188C23—7962no—
189C23—15921no—
190C23—15921no—
191C23—3981no—
192C23—3981no—
193C23—3981no—
194C23—1992no—
195C23D4×D19911yescharacteristic
196D4—41no—
197D4—41no—
198D4—41no—
199D4—7961no—
200D4—7962no—
201D4—7962no—
202D4—7961no—
203D4—7961no—
204D4—15921no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—7962no—
211C2×C4—7961no—
212C2×C4—7961no—
213C2×C4—7961no—
214C2×C4—7961no—
215C2×C4—3981no—
216C2×C4—3981no—
217C2×C4D4×D19911yescharacteristic
218C2×C4C2×D79611yescharacteristic
219C8—41no—
220C8—7961no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—7962no—
227C22—7961no—
228C22—7961no—
229C22—7961no—
230C22—7961no—
231C22—7961no—
232C22—15921no—
233C22—15924no—
234C22—3982no—
235C22#2026392511yescharacteristic
236C22SD16×D19911yescharacteristic
237C22#2026397911yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—41no—
242C4—7961no—
243C4—7961no—
244C4—15921no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—7962no—
249C2—15921no—
250C2#2716417011yescharacteristic
251C2#2716418711yescharacteristic
252C2#2716420911yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#327×C199C2
socleC22×C199#120
prime radical#27164576C2
hypercenter#327D199
hypocenterC199#1613
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #27164337C2semidirect
3. #27164334C2semidirect
4. #27163955C2semidirect
5. #27163751C2semidirect
6. #27163741C2semidirect
7. #327×C199C2semidirect
16. C2×C4⋊Dic199C22semidirect
23. C22⋊(C199⋊C8)C22semidirect
25. C22⋊C8×C199C22semidirect
26. #95×C199C22semidirect
126. C199#1613semidirect
133. #327D199semidirect
152. C22⋊C8C2×D199semidirect
153. #95C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271645762
#271643372
#271643342
#271639552
#271637512
#271637412
#327×C1992
#1613199