Finite Group Databasebeta

← #38767405

Subgroups — #38767405

Order 25472.

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

normal subgroups only

This group has 284 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
2C22⋊C8×D199C211yesmaximal, characteristic, complemented
3#27163880C211yesmaximal, 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×C8×D199—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×(C2×D199)⋊C4—21no—
20C22⋊(C199⋊C8)C2211yescharacteristic
21(C2×D199)⋊C8—21no—
22C199⋊#95—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
41C2×C199⋊C8—21no—
42C8×D199—42no—
43C4⋊Dic199C2×C411yescharacteristic
44Dic199⋊C4—21no—
45(C2×D199)⋊C4—21no—
46(C2×D199)⋊C4—21no—
47(C2×D199)⋊C4—21no—
48(C2×D199)⋊C4—21no—
49(C2×D199)⋊C4—42no—
50C4×Dic199—21no—
51C2×Q8×C199C2×C411yescharacteristic
52C22×C4×C199—21no—
53C22×C4×C199C2311yescharacteristic
54C2×C8×C199—21no—
55C4⋊C4×C199—21no—
56C4⋊C4×C199C2×C411yescharacteristic
57C22⋊C4×C199—21no—
58C22×D199—21no—
59C22×D199—21no—
60C22×D199—22no—
61C22×D199—21no—
62C22×D199—41no—
63C22×D199C22⋊C411yescharacteristic
64C22×D199C22⋊C411yescharacteristic
65D796—41no—
66D796—42no—
67D796—42no—
68C22⋊D199—41no—
69C22⋊D199—41no—
70C2×Dic199—21no—
71C2×Dic199—41no—
72C2×Dic199—41no—
73C2×Dic199—41no—
74C2×Dic199—41no—
75C2×Dic199C22⋊C411yescharacteristic
76C2×Dic199C22⋊C411yescharacteristic
77C4×D199—22no—
78C4×D199—22no—
79C4×D199—41no—
80C4×D199—41no—
81C4×D199—42no—
82C199⋊C8—41no—
83C23×C199C2×D411yescharacteristic
84Q8×C199—41no—
85C2×C4×C199—21no—
86C2×C4×C199—21no—
87C2×C4×C199—41no—
88C2×C4×C199—41no—
89C2×C4×C199—41no—
90C2×C4×C199—41no—
91C2×C4×C199C22×C411yescharacteristic
92C2×C4×C199C2×D411yescharacteristic
93C8×C199—41no—
94C22×C199—21no—
95C22×C199—21no—
96C22×C199C2×C22⋊C411yescharacteristic
97C22×C199#13511yescharacteristic
98C22×C199#13511yescharacteristic
99C2×D199—21no—
100C2×D199—21no—
101C2×D199—21no—
102C2×D199—21no—
103C2×D199—21no—
104C2×D199—42no—
105C2×D199—42no—
106C2×D199—41no—
107C2×D199—41no—
108C2×D199—41no—
109C2×D199#9912yes—
110C4×C199—21no—
111C4×C199—21no—
112C4×C199—41no—
113C4×C199—41no—
114C4×C199—81no—
115Dic199—21no—
116Dic199—21no—
117Dic199—41no—
118Dic199—81no—
119C2×C199—21no—
120C2×C199—21no—
121C2×C199C2×#9911yescharacteristic
122C2×C199#41611yescharacteristic
123C2×C199#42111yescharacteristic
124D199—22no—
125D199—41no—
126D199—81no—
127C199#151611yescharacteristic, complemented, Sylow
128#1516—1991nomaximal, Sylow
129#525—1991no—
130C2×C22⋊C8—1991no—
131#381—1991no—
132#328—1991no—
133#328D19911yescharacteristic, complemented
134#327—1991no—
135#327—1991no—
136C2×#41—1991no—
137C23×C4—1991no—
138C22×C8—3981no—
139#123—3981no—
140C22⋊Q8—1991no—
141C22⋊Q8C2×D19911yescharacteristic
142#121—1991no—
143#121—1991no—
144#120—3981no—
145C4×D4—3981no—
146C4×D4—3981no—
147#117—1991no—
148C2×C22⋊C4—3981no—
149C22⋊C8—3981no—
150C22⋊C8—1991no—
151C22⋊C8C2×D19911yescharacteristic
152#95—3981no—
153#95—1991no—
154#95C2×D19911yescharacteristic
155C24—1991no—
156#41—7962no—
157C2×Q8C4×D19911yescharacteristic
158C2×D4—3981no—
159C2×D4—3981no—
160C2×D4—3981no—
161C2×D4—1991no—
162C22×C4—21no—
163C22×C4—7962no—
164C22×C4—3981no—
165C22×C4—3981no—
166C22×C4—1991no—
167C22×C4—1991no—
168C22×C4—1991no—
169C22×C4—1991no—
170C22×C4—1991no—
171C22×C4C22×D19911yescharacteristic
172C2×C8—21no—
173C2×C8—7962no—
174C2×C8—3981no—
175C4⋊C4—21no—
176C4⋊C4—3981no—
177C4⋊C4—1991no—
178C4⋊C4C4×D19911yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—7962no—
181C22⋊C4—3981no—
182C22⋊C4—3981no—
183C22⋊C4—3981no—
184C22⋊C4—3981no—
185C42—3981no—
186C23—7961no—
187C23—3982no—
188C23—3981no—
189C23—3981no—
190C23—3981no—
191C23—1991no—
192C23—1991no—
193C23D4×D19911yescharacteristic
194Q8—41no—
195D4—7961no—
196D4—7962no—
197D4—7962no—
198D4—7961no—
199D4—7961no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—7961no—
207C2×C4—7961no—
208C2×C4—7962no—
209C2×C4—7961no—
210C2×C4—7961no—
211C2×C4—7961no—
212C2×C4—7961no—
213C2×C4—3982no—
214C2×C4—3982no—
215C2×C4—3981no—
216C2×C4—1991no—
217C2×C4—1991no—
218C2×C4C2×C4×D19911yescharacteristic
219C2×C4D4×D19911yescharacteristic
220C8—41no—
221C8—7961no—
222C22—21no—
223C22—21no—
224C22—7962no—
225C22—7962no—
226C22—7961no—
227C22—7961no—
228C22—7961no—
229C22—3981no—
230C22—3981no—
231C22—3981no—
232C22—3981no—
233C22—3981no—
234C22—1992no—
235C22C22⋊C4×D19911yescharacteristic
236C22#2026397611yescharacteristic
237C22#2026397811yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—81no—
243C4—7961no—
244C4—15921no—
245C4—3981no—
246C4—3981no—
247C2—21no—
248C2—21no—
249C2—7961no—
250C2—15921no—
251C2—3982no—
252C2#99×D19911yescharacteristic
253C2#2716401311yescharacteristic
254C2#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#1516
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. C22⋊C8×D199C2semidirect
3. #27163880C2semidirect
4. #27163755C2semidirect
5. #27163753C2semidirect
6. #27163751C2semidirect
7. #328×C199C2semidirect
8. C2×#15691887C4semidirect
12. #20263931C4semidirect
13. #20263930C4semidirect
18. #20263913C4semidirect
127. C199#1516semidirect
133. #328D199semidirect

Maximal subgroups

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

subgroupindex
#271645312
C22⋊C8×D1992
#271638802
#271637552
#271637532
#271637512
#328×C1992
#1516199