Finite Group Databasebeta

← #38770053

Subgroups — #38770053

Order 25472.

316 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#20263934C211yesmaximal, characteristic
2C2×#20263888C211yesmaximal, characteristic, complemented
3C2×#20263878C211yesmaximal, characteristic, complemented
4C2×#20263878C211yesmaximal, characteristic, complemented
5C2×C22⋊(C199⋊C8)C211yesmaximal, characteristic
6#27163780C211yesmaximal, characteristic, complemented
7#380×C199C211yesmaximal, characteristic
8C22×Dic398C2211yescharacteristic
9C22×C199⋊C8C2211yescharacteristic
10C22×C199⋊C8C2211yescharacteristic
11#20263934—22no—
12#20263934—22no—
13C2×C4⋊Dic199C2211yescharacteristic
14C2×Dic199⋊C4—21no—
15C2×C22⋊Dic199—21no—
16#20263888—22no—
17#20263878—22no—
18#20263878—22no—
19C22⋊(C199⋊C8)—22no—
20C23×C4×C199C2211yescharacteristic
21C2×C4⋊C4×C199C2211yescharacteristic
22C2×C4⋊C4×C199C2211yescharacteristic
23C2×C22⋊C4×C199—21no—
24#95×C199—21no—
25C2×Dic398—21no—
26C2×Dic398—21no—
27C2×Dic398—21no—
28C2×Dic398—21no—
29C2×Dic398—21no—
30C2×Dic398D412yes—
31C22×Dic199—21no—
32C22×Dic199—21no—
33C2×C199⋊C8—21no—
34C2×C199⋊C8—21no—
35C2×C199⋊C8—21no—
36C2×C199⋊C8—21no—
37C2×C199⋊C8—21no—
38C2×C199⋊C8D412yes—
39C4⋊Dic199—21no—
40C4⋊Dic199D412yes—
41Dic199⋊C4—24no—
42C22⋊Dic199—24no—
43C24×C199D411yescharacteristic
44C22×C4×C199—21no—
45C22×C4×C199—21no—
46C22×C4×C199—21no—
47C22×C4×C199—21no—
48C22×C4×C199—21no—
49C22×C4×C199—21no—
50C22×C4×C199—21no—
51C22×C4×C199—21no—
52C22×C4×C199D411yescharacteristic
53C22×C4×C199C2311yescharacteristic
54C4⋊C4×C199—21no—
55C4⋊C4×C199—21no—
56C4⋊C4×C199—21no—
57C4⋊C4×C199—21no—
58C22⋊C4×C199—42no—
59Dic398—22no—
60Dic398—22no—
61Dic398—41no—
62Dic398—41no—
63C2×Dic199—22no—
64C2×Dic199—22no—
65C2×Dic199—41no—
66C2×Dic199—41no—
67C2×Dic199—41no—
68C2×Dic199—41no—
69C199⋊C8—22no—
70C199⋊C8—41no—
71C23×C199—21no—
72C23×C199—21no—
73C23×C199—21no—
74C23×C199—21no—
75C23×C199—21no—
76C23×C199—21no—
77C23×C199Q1612yes—
78C23×C199C2×D411yescharacteristic
79C2×C4×C199—21no—
80C2×C4×C199—21no—
81C2×C4×C199—21no—
82C2×C4×C199—41no—
83C2×C4×C199—41no—
84C2×C4×C199—41no—
85C2×C4×C199—41no—
86C2×C4×C199—41no—
87C2×C4×C199—41no—
88C2×C4×C199—41no—
89C2×C4×C199—41no—
90C2×C4×C199—41no—
91C2×C4×C199—41no—
92C2×C4×C199C2×D411yescharacteristic
93C2×C4×C199C2×D411yescharacteristic
94C2×C4×C199C2×D411yescharacteristic
95C2×C4×C199#4111yescharacteristic
96C2×C4×C199#4111yescharacteristic
97C2×C4×C199#4111yescharacteristic
98C22×C199—22no—
99C22×C199—22no—
100C22×C199—22no—
101C22×C199—41no—
102C22×C199—41no—
103C22×C199—41no—
104C22×C199—41no—
105C22×C199#12011yescharacteristic
106C22×C199#12111yescharacteristic
107C22×C199#12311yescharacteristic
108C22×C199C2×Q1611yescharacteristic
109C22×C199#13511yescharacteristic
110C22×C199AGL(1,ℤ/8)11yescharacteristic
111C22×C199#13711yescharacteristic
112C4×C199—41no—
113C4×C199—81no—
114C4×C199—81no—
115C4×C199#12111yescharacteristic
116C4×C199#12111yescharacteristic
117C4×C199#12311yescharacteristic
118C4×C199#12411yescharacteristic
119Dic199—42no—
120Dic199—42no—
121C2×C199—42no—
122C2×C199#39411yescharacteristic
123C2×C199#44911yescharacteristic
124C2×C199#45111yescharacteristic
125C2×C199#46711yescharacteristic
126C2×C199#46811yescharacteristic
127C2×C199#48411yescharacteristic
128C2×C199#48511yescharacteristic
129C199#203611yescharacteristic, complemented, Sylow
130#2036—1991nomaximal, Sylow
131C2×C22⋊Q8—1991no—
132C2×#107—1991no—
133C2×#103—1991no—
134C2×#103—1991no—
135C2×C22⋊C8—1991no—
136#380D19911yescharacteristic
137#340—1991no—
138C22×Q8—1991no—
139C23×C4C2×D19911yescharacteristic
140C22×C8—1991no—
141C22×C8—1991no—
142C22⋊Q8—3982no—
143C22⋊Q8—3982no—
144C2×C4⋊C4—3981no—
145C2×C4⋊C4—1991no—
146C2×C4⋊C4C2×D19911yescharacteristic
147C2×C4⋊C4C2×D19911yescharacteristic
148C2×C22⋊C4—21no—
149C2×C22⋊C4—3981no—
150#107—3982no—
151#103—3982no—
152#103—3982no—
153C22⋊C8—3982no—
154#95—21no—
155C24C22⋊D19911yescharacteristic
156C2×Q8—3981no—
157C2×Q8—3981no—
158C2×Q8—3981no—
159C2×Q8—3981no—
160C2×Q8—3981no—
161C2×Q8—1992no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—3981no—
171C22×C4—3981no—
172C22×C4C22⋊D19911yescharacteristic
173C22×C4C22×D19911yescharacteristic
174C2×C8—3981no—
175C2×C8—3981no—
176C2×C8—3981no—
177C2×C8—3981no—
178C2×C8—3981no—
179C2×C8—1992no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—3984no—
185C4⋊C4—3981no—
186C4⋊C4—1992no—
187C22⋊C4—42no—
188C22⋊C4—3984no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23#1569187612yes—
196C23C2×C22⋊D19911yescharacteristic
197Q8—7961no—
198Q8—7961no—
199Q8—3982no—
200Q8—3982no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—7961no—
215C2×C4—7961no—
216C2×C4—7961no—
217C2×C4—7961no—
218C2×C4—3982no—
219C2×C4—3982no—
220C2×C4D4×D19911yescharacteristic
221C2×C4D4×D19911yescharacteristic
222C2×C4D4×D19911yescharacteristic
223C2×C4#1569188511yescharacteristic
224C2×C4#1569188611yescharacteristic
225C2×C4#1569188611yescharacteristic
226C8—7961no—
227C8—3982no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22#2026392311yescharacteristic
236C22#2026392711yescharacteristic
237C22#2026393811yescharacteristic
238C22C2×#1569187611yescharacteristic
239C22#2026397511yescharacteristic
240C22#2026398111yescharacteristic
241C22#2026398511yescharacteristic
242C4—41no—
243C4—81no—
244C4—81no—
245C4—7962no—
246C4—7962no—
247C4#2026392811yescharacteristic
248C4#2026392811yescharacteristic
249C4#2026393711yescharacteristic
250C4#2026394111yescharacteristic
251C2—42no—
252C2#2716393311yescharacteristic
253C2#2716416111yescharacteristic
254C2#2716417311yescharacteristic
255C2#2716425711yescharacteristic
256C2#2716426211yescharacteristic
257C2#2716435311yescharacteristic
258C2#2716435911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D199
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#380×C199C2
socleC23×C199C2×D4
prime radicalC24×C199D4
hypercenter#380D199
hypocenterC199#2036
LCSG ▷ C22×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#20263888C2semidirect
3. C2×#20263878C2semidirect
4. C2×#20263878C2semidirect
6. #27163780C2semidirect
129. C199#2036semidirect

Maximal subgroups

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

subgroupindex
C2×#202639342
C2×#202638882
C2×#202638782
C2×C22⋊(C199⋊C8)2
#271637802
#380×C1992
#2036199