Finite Group Databasebeta

← #38767392

Subgroups — #38767392

Order 25472.

286 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 63 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#27164531C211yesmaximal, characteristic
2#27163980C211yesmaximal, characteristic, complemented
3#27163877C211yesmaximal, characteristic, complemented
4#27163755C211yesmaximal, characteristic, complemented
5#27163753C211yesmaximal, characteristic, complemented
6#27163751C211yesmaximal, characteristic, complemented
7#328×C199C211yesmaximal, characteristic, complemented
8C2×#15691887C411yescharacteristic, complemented
9C22×C4×D199C2211yescharacteristic
10C2×#15691869—21no—
11#20263940—21no—
12#20263931C411yescharacteristic, complemented
13#20263930C411yescharacteristic, complemented
14#20263929C2211yescharacteristic
15#20263925—21no—
16#20263917—21no—
17(C22⋊D199)⋊C4—21no—
18#20263913C411yescharacteristic, complemented
19C2×Dic199⋊C4—21no—
20C2×(C2×D199)⋊C4—21no—
21C22⋊(C199⋊C8)C2211yescharacteristic
22(C2×D199)⋊C8—21no—
23C199⋊#95C2211yescharacteristic
24C22⋊Q8×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic
26#95×C199C2211yescharacteristic
27C23×D199D411yescharacteristic
28#15691887—42no—
29C2×D796—21no—
30C2×D796—21no—
31C2×D796C2×C411yescharacteristic
32C2×C22⋊D199—21no—
33C22×Dic199—21no—
34C22×Dic199D411yescharacteristic
35C2×C4×D199—21no—
36C2×C4×D199—42no—
37C2×C4×D199C2×C411yescharacteristic
38C2×C4×D199C2×C411yescharacteristic
39C2×C4×D199D411yescharacteristic
40C2×C4×D199D411yescharacteristic
41#15691869—42no—
42C2×C199⋊C8—21no—
43C4⋊Dic199C2×C411yescharacteristic
44Dic199⋊C4—21no—
45Dic199⋊C4—42no—
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—
51C4×Dic199—21no—
52C2×Q8×C199C2×C411yescharacteristic
53C22×C4×C199—21no—
54C22×C4×C199C2311yescharacteristic
55C2×C8×C199—21no—
56C4⋊C4×C199—21no—
57C4⋊C4×C199C2×C411yescharacteristic
58C22⋊C4×C199—21no—
59C22×D199—21no—
60C22×D199—21no—
61C22×D199—21no—
62C22×D199—41no—
63C22×D199—41no—
64C22×D199C22⋊C411yescharacteristic
65C22×D199C22⋊C411yescharacteristic
66D796—41no—
67D796—42no—
68D796—42no—
69C22⋊D199—41no—
70C22⋊D199—41no—
71C2×Dic199—22no—
72C2×Dic199—21no—
73C2×Dic199—41no—
74C2×Dic199—41no—
75C2×Dic199—41no—
76C2×Dic199C22⋊C411yescharacteristic
77C2×Dic199C22⋊C411yescharacteristic
78C4×D199—22no—
79C4×D199—22no—
80C4×D199—41no—
81C4×D199—41no—
82C4×D199—42no—
83C199⋊C8—41no—
84C23×C199C2×D411yescharacteristic
85Q8×C199—41no—
86C2×C4×C199—21no—
87C2×C4×C199—21no—
88C2×C4×C199—41no—
89C2×C4×C199—41no—
90C2×C4×C199—41no—
91C2×C4×C199—41no—
92C2×C4×C199C22×C411yescharacteristic
93C2×C4×C199C2×D411yescharacteristic
94C8×C199—41no—
95C22×C199—21no—
96C22×C199—21no—
97C22×C199C2×C22⋊C411yescharacteristic
98C22×C199AGL(1,ℤ/8)11yescharacteristic
99C22×C199#13711yescharacteristic
100C2×D199—21no—
101C2×D199—21no—
102C2×D199—21no—
103C2×D199—21no—
104C2×D199—21no—
105C2×D199—21no—
106C2×D199—42no—
107C2×D199—42no—
108C2×D199—41no—
109C2×D199—41no—
110C2×D199—41no—
111C4×C199—21no—
112C4×C199—21no—
113C4×C199—41no—
114C4×C199—41no—
115C4×C199—81no—
116Dic199—21no—
117Dic199—41no—
118Dic199—81no—
119Dic199#9912yes—
120C2×C199—21no—
121C2×C199—21no—
122C2×C199C2×#9911yescharacteristic
123C2×C199#41711yescharacteristic
124C2×C199#42111yescharacteristic
125D199—41no—
126D199—41no—
127D199—81no—
128C199#151311yescharacteristic, complemented, Sylow
129#1513—1991nomaximal, Sylow
130#525—1991no—
131#407—1991no—
132#380—1991no—
133#328—1991no—
134#328D19911yescharacteristic, complemented
135#327—1991no—
136#327—1991no—
137C2×#41—1991no—
138C23×C4—1991no—
139C2×M16—3981no—
140#123—3981no—
141C22⋊Q8—1991no—
142C22⋊Q8C2×D19911yescharacteristic
143#121—1991no—
144#121—1991no—
145#120—3981no—
146C4×D4—3981no—
147C4×D4—3981no—
148#117—1991no—
149C2×C4⋊C4—3981no—
150C2×C22⋊C4—3981no—
151C22⋊C8—3981no—
152C22⋊C8—1991no—
153C22⋊C8C2×D19911yescharacteristic
154#95—1991no—
155#95C2×D19911yescharacteristic
156C24—1991no—
157#41—7962no—
158C2×Q8C4×D19911yescharacteristic
159C2×D4—3981no—
160C2×D4—3981no—
161C2×D4—3981no—
162C2×D4—1991no—
163C22×C4—21no—
164C22×C4—7962no—
165C22×C4—3981no—
166C22×C4—3981no—
167C22×C4—1991no—
168C22×C4—1991no—
169C22×C4—1991no—
170C22×C4—1991no—
171C22×C4—1991no—
172C22×C4C22×D19911yescharacteristic
173M16—7962no—
174C2×C8—21no—
175C2×C8—3981no—
176C4⋊C4—21no—
177C4⋊C4—7962no—
178C4⋊C4—3981no—
179C4⋊C4—1991no—
180C4⋊C4C4×D19911yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—7962no—
183C22⋊C4—3981no—
184C22⋊C4—3981no—
185C22⋊C4—3981no—
186C22⋊C4—3981no—
187C42—3981no—
188C23—7961no—
189C23—7961no—
190C23—3981no—
191C23—3981no—
192C23—3981no—
193C23—1991no—
194C23—1991no—
195C23D4×D19911yescharacteristic
196Q8—41no—
197D4—7961no—
198D4—7962no—
199D4—7962no—
200D4—7961no—
201D4—7961no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—7961no—
209C2×C4—7961no—
210C2×C4—7962no—
211C2×C4—7961no—
212C2×C4—7961no—
213C2×C4—7961no—
214C2×C4—3982no—
215C2×C4—3982no—
216C2×C4—3981no—
217C2×C4—3982no—
218C2×C4—1991no—
219C2×C4—1991no—
220C2×C4C2×C4×D19911yescharacteristic
221C2×C4D4×D19911yescharacteristic
222C8—41no—
223C8—7961no—
224C22—21no—
225C22—21no—
226C22—7962no—
227C22—7962no—
228C22—7961no—
229C22—7961no—
230C22—7961no—
231C22—3981no—
232C22—3981no—
233C22—3981no—
234C22—3981no—
235C22—3981no—
236C22—3981no—
237C22C22⋊C4×D19911yescharacteristic
238C22#2026398211yescharacteristic
239C22#2026398711yescharacteristic
240C4—21no—
241C4—21no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—7961no—
246C4—15921no—
247C4—3981no—
248C4—1992no—
249C2—21no—
250C2—21no—
251C2—7961no—
252C2—7961no—
253C2—15921no—
254C2#99×D19911yescharacteristic
255C2#2716401811yescharacteristic
256C2#2716403311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D199
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#328×C199C2
socleC22×C199C2×C22⋊C4
prime radical#27164531C2
cosocleC22×C4×C199C23
hypercenter#328D199
hypocenterC199#1513
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #27163980C2semidirect
3. #27163877C2semidirect
4. #27163755C2semidirect
5. #27163753C2semidirect
6. #27163751C2semidirect
7. #328×C199C2semidirect
8. C2×#15691887C4semidirect
12. #20263931C4semidirect
13. #20263930C4semidirect
18. #20263913C4semidirect
128. C199#1513semidirect
134. #328D199semidirect

Maximal subgroups

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

subgroupindex
#271645312
#271639802
#271638772
#271637552
#271637532
#271637512
#328×C1992
#1513199