Finite Group Databasebeta

← #38767338

Subgroups — #38767338

Order 25472.

304 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 65 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#27164528C211yesmaximal, characteristic
2C22⋊C8×D199C211yesmaximal, characteristic, complemented
3#27163865C211yesmaximal, characteristic, complemented
4#27163756C211yesmaximal, characteristic, complemented
5#27163754C211yesmaximal, characteristic, complemented
6#27163752C211yesmaximal, characteristic, complemented
7#327×C199C211yesmaximal, characteristic, complemented
8C2×#15691885C411yescharacteristic, complemented
9C22×C4×D199C2211yescharacteristic
10C2×C8×D199—21no—
11#20263937—21no—
12#20263935C411yescharacteristic, complemented
13#20263934C2211yescharacteristic
14#20263926C411yescharacteristic, complemented
15#20263923—21no—
16D4×Dic199—21no—
17(C22⋊D199)⋊C4—21no—
18#20263913C411yescharacteristic, complemented
19C2×(C2×D199)⋊C4—21no—
20C2×C4×Dic199—21no—
21C22⋊(C199⋊C8)C2211yescharacteristic
22(C2×D199)⋊C8—21no—
23C199⋊#95C2211yescharacteristic
24#121×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic
26#95×C199C2211yescharacteristic
27C23×D199D411yescharacteristic
28#15691885—42no—
29C2×Dic398C2×C411yescharacteristic
30C2×C22⋊D199—21no—
31C2×C22⋊D199—21no—
32C22×Dic199—21no—
33C22×Dic199—21no—
34C22×Dic199D411yescharacteristic
35C2×C4×D199—42no—
36C2×C4×D199C2×C411yescharacteristic
37C2×C4×D199C2×C411yescharacteristic
38C2×C4×D199D411yescharacteristic
39C2×C4×D199D411yescharacteristic
40C2×C199⋊C8—21no—
41C8×D199—42no—
42C4⋊Dic199—21no—
43C4⋊Dic199C2×C411yescharacteristic
44Dic199⋊C4—21no—
45C22⋊Dic199—21no—
46C22⋊Dic199—21no—
47(C2×D199)⋊C4—24no—
48(C2×D199)⋊C4—21no—
49(C2×D199)⋊C4—21no—
50C4×Dic199—21no—
51C4×Dic199—42no—
52C2×D4×C199—21no—
53C2×D4×C199C2×C411yescharacteristic
54C22×C4×C199—21no—
55C22×C4×C199C2311yescharacteristic
56C2×C8×C199—21no—
57C4⋊C4×C199C2×C411yescharacteristic
58C22⋊C4×C199—21no—
59C22×D199—21no—
60C22×D199—21no—
61C22×D199—22no—
62C22×D199—41no—
63C22×D199C22⋊C411yescharacteristic
64C22×D199C22⋊C411yescharacteristic
65Dic398—41no—
66C22⋊D199—42no—
67C22⋊D199—42no—
68C2×Dic199—22no—
69C2×Dic199—22no—
70C2×Dic199—21no—
71C2×Dic199—21no—
72C2×Dic199—41no—
73C2×Dic199—41no—
74C2×Dic199—42no—
75C2×Dic199C22⋊C411yescharacteristic
76C2×Dic199C22⋊C411yescharacteristic
77C4×D199—22no—
78C4×D199—22no—
79C4×D199—41no—
80C4×D199—41no—
81C199⋊C8—41no—
82C23×C199—21no—
83C23×C199C2×D411yescharacteristic
84D4×C199—41no—
85D4×C199—41no—
86D4×C199—41no—
87C2×C4×C199—22no—
88C2×C4×C199—21no—
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×C199—41no—
98C22×C199—41no—
99C22×C199—41no—
100C22×C199C2×C22⋊C411yescharacteristic
101C22×C199#13511yescharacteristic
102C22×C199#13511yescharacteristic
103C2×D199—21no—
104C2×D199—21no—
105C2×D199—21no—
106C2×D199—21no—
107C2×D199—21no—
108C2×D199—42no—
109C2×D199—42no—
110C2×D199#10412yes—
111C4×C199—21no—
112C4×C199—21no—
113C4×C199—42no—
114C4×C199—41no—
115Dic199—21no—
116Dic199—42no—
117Dic199—41no—
118Dic199—41no—
119Dic199#9912yes—
120C2×C199—21no—
121C2×C199—21no—
122C2×C199—81no—
123C2×C199C2×#9911yescharacteristic
124C2×C199#41611yescharacteristic
125C2×C199C2×#10411yescharacteristic
126D199—22no—
127D199—41no—
128C199#150111yescharacteristic, complemented, Sylow
129#1501—1991nomaximal, Sylow
130#525—1991no—
131C2×C22⋊C8—1991no—
132C4×C22⋊C4—1991no—
133#328—1991no—
134#328—1991no—
135#327—1991no—
136#327D19911yescharacteristic, complemented
137C2×#41—1991no—
138C23×C4—1991no—
139C22×C8—3981no—
140#123—3981no—
141C22⋊Q8—1991no—
142C22⋊Q8—1991no—
143#121—1991no—
144#121C2×D19911yescharacteristic
145#120—3981no—
146C4×D4—3981no—
147C4×D4—3981no—
148#117—1991no—
149C2×C22⋊C4—3981no—
150C2×C42—3981no—
151C22⋊C8—3981no—
152C22⋊C8—1991no—
153C22⋊C8C2×D19911yescharacteristic
154#95—1991no—
155#95C2×D19911yescharacteristic
156C24—1991no—
157#41—7962no—
158C2×Q8—1991no—
159C2×D4—21no—
160C2×D4—3981no—
161C2×D4—3981no—
162C2×D4C4×D19911yescharacteristic
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
173C2×C8—21no—
174C2×C8—7962no—
175C2×C8—3981no—
176C4⋊C4—3981no—
177C4⋊C4—3981no—
178C4⋊C4—1991no—
179C4⋊C4C4×D19911yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—3984no—
182C22⋊C4—3981no—
183C22⋊C4—3981no—
184C22⋊C4—3981no—
185C22⋊C4—3981no—
186C42—7962no—
187C42—3981no—
188C23—21no—
189C23—7961no—
190C23—3982no—
191C23—3981no—
192C23—3981no—
193C23—1991no—
194C23—1991no—
195C23D4×D19911yescharacteristic
196Q8—7961no—
197D4—41no—
198D4—41no—
199D4—41no—
200D4—7962no—
201D4—7962no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—7961no—
208C2×C4—7961no—
209C2×C4—7961no—
210C2×C4—7961no—
211C2×C4—7962no—
212C2×C4—3982no—
213C2×C4—3982no—
214C2×C4—3982no—
215C2×C4—3981no—
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—41no—
227C22—41no—
228C22—41no—
229C22—7962no—
230C22—7962no—
231C22—3981no—
232C22—3981no—
233C22—3981no—
234C22—3981no—
235C22—3981no—
236C22—1992no—
237C22C22⋊C4×D19911yescharacteristic
238C22#2026397411yescharacteristic
239C22#2026397611yescharacteristic
240C4—21no—
241C4—21no—
242C4—42no—
243C4—41no—
244C4—7962no—
245C4—7961no—
246C4—7961no—
247C4—3981no—
248C4—1992no—
249C2—21no—
250C2—21no—
251C2—81no—
252C2—7961no—
253C2—3982no—
254C2#99×D19911yescharacteristic
255C2#2716401111yescharacteristic
256C2#104×D19911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D199
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#327×C199C2
socleC22×C199C2×C22⋊C4
prime radical#27164528C2
cosocleC22×C4×C199C23
hypercenter#327D199
hypocenterC199#1501
LCSG ▷ C2×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. C22⋊C8×D199C2semidirect
3. #27163865C2semidirect
4. #27163756C2semidirect
5. #27163754C2semidirect
6. #27163752C2semidirect
7. #327×C199C2semidirect
8. C2×#15691885C4semidirect
12. #20263935C4semidirect
14. #20263926C4semidirect
18. #20263913C4semidirect
128. C199#1501semidirect
136. #327D199semidirect

Maximal subgroups

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

subgroupindex
#271645282
C22⋊C8×D1992
#271638652
#271637562
#271637542
#271637522
#327×C1992
#1501199