Finite Group Databasebeta

← #38770055

Subgroups — #38770055

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×#20263935C211yesmaximal, characteristic
2C2×#20263880C211yesmaximal, characteristic, complemented
3C2×#20263878C211yesmaximal, characteristic, complemented
4C2×(C2×D199)⋊C8C211yesmaximal, characteristic
5#27163875C211yesmaximal, characteristic
6#27163780C211yesmaximal, characteristic, complemented
7C2×#107×C199C211yesmaximal, characteristic, complemented
8C22×Dic398C2211yescharacteristic
9C22×C4×D199C2211yescharacteristic
10C22×C199⋊C8C2211yescharacteristic
11#20263935—22no—
12#20263935—22no—
13C2×C4⋊Dic199—21no—
14C2×C4⋊Dic199C2211yescharacteristic
15C2×(C2×D199)⋊C4—21no—
16C2×(C2×D199)⋊C4—21no—
17#20263880—22no—
18#20263878—22no—
19(C2×D199)⋊C8—22no—
20C199⋊#95—21no—
21C22×C8×C199C2211yescharacteristic
22C2×C4⋊C4×C199C2211yescharacteristic
23C2×C4⋊C4×C199C2211yescharacteristic
24#107×C199—22no—
25C23×D199D411yescharacteristic
26C2×Dic398—21no—
27C2×Dic398—21no—
28C2×Dic398—21no—
29C2×Dic398—21no—
30C2×Dic398—21no—
31C2×Dic398D412yes—
32C22×Dic199—21no—
33C22×Dic199—21no—
34C22×Dic199D411yescharacteristic
35C2×C4×D199—21no—
36C2×C4×D199—21no—
37C2×C4×D199—21no—
38C2×C4×D199—21no—
39C2×C4×D199—21no—
40C2×C4×D199—21no—
41C2×C199⋊C8—21no—
42C2×C199⋊C8—21no—
43C2×C199⋊C8—21no—
44C4⋊Dic199—24no—
45C4⋊Dic199—21no—
46C4⋊Dic199—21no—
47(C2×D199)⋊C4—24no—
48(C2×D199)⋊C4—42no—
49C22×C4×C199—21no—
50C22×C4×C199—21no—
51C22×C4×C199C2311yescharacteristic
52C2×C8×C199—21no—
53C2×C8×C199—21no—
54C2×C8×C199D412yes—
55C4⋊C4×C199—21no—
56C4⋊C4×C199—21no—
57C4⋊C4×C199—21no—
58C4⋊C4×C199D412yes—
59C22×D199—21no—
60C22×D199—21no—
61C22×D199—21no—
62C22×D199—21no—
63C22×D199—21no—
64C22×D199—21no—
65C22×D199Q1612yes—
66Dic398—22no—
67Dic398—22no—
68Dic398—41no—
69Dic398—41no—
70C2×Dic199—22no—
71C2×Dic199—21no—
72C2×Dic199—21no—
73C2×Dic199—21no—
74C2×Dic199—41no—
75C2×Dic199—41no—
76C2×Dic199—41no—
77C2×Dic199—41no—
78C2×Dic199—41no—
79C4×D199—41no—
80C4×D199—41no—
81C4×D199—41no—
82C4×D199—41no—
83C199⋊C8—41no—
84C23×C199C2×D411yescharacteristic
85C2×C4×C199—22no—
86C2×C4×C199—41no—
87C2×C4×C199—41no—
88C2×C4×C199—41no—
89C2×C4×C199—41no—
90C2×C4×C199—41no—
91C2×C4×C199C2×D411yescharacteristic
92C2×C4×C199C2×D411yescharacteristic
93C2×C4×C199C2×D411yescharacteristic
94C2×C4×C199#4111yescharacteristic
95C2×C4×C199#4111yescharacteristic
96C2×C4×C199#4111yescharacteristic
97C8×C199—22no—
98C22×C199#12011yescharacteristic
99C22×C199#12111yescharacteristic
100C22×C199#12311yescharacteristic
101C22×C199C2×Q1611yescharacteristic
102C22×C199#13511yescharacteristic
103C22×C199AGL(1,ℤ/8)11yescharacteristic
104C22×C199#13711yescharacteristic
105C2×D199—22no—
106C2×D199—22no—
107C2×D199—22no—
108C2×D199—41no—
109C2×D199—41no—
110C2×D199—41no—
111C2×D199—41no—
112C4×C199—42no—
113C4×C199—81no—
114C4×C199#12111yescharacteristic
115C4×C199#12111yescharacteristic
116C4×C199#12311yescharacteristic
117C4×C199#12411yescharacteristic
118Dic199—41no—
119Dic199—42no—
120Dic199—81no—
121C2×C199#39411yescharacteristic
122C2×C199#44911yescharacteristic
123C2×C199#45111yescharacteristic
124C2×C199#46711yescharacteristic
125C2×C199#46811yescharacteristic
126C2×C199#48411yescharacteristic
127C2×C199#48511yescharacteristic
128D199—42no—
129C199#203611yescharacteristic, complemented, Sylow
130#2036—1991nomaximal, Sylow
131C2×C22⋊Q8—1991no—
132C2×#107D19911yescharacteristic, complemented
133C2×#103—1991no—
134C2×#103—1991no—
135C2×C22⋊C8—1991no—
136#380—1991no—
137#340—1991no—
138C22×Q8—1991no—
139C23×C4—1991no—
140C22×C8—1991no—
141C22×C8C2×D19911yescharacteristic
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—3981no—
149C2×C22⋊C4—3981no—
150#107—22no—
151#103—3982no—
152#103—3982no—
153C22⋊C8—3982no—
154#95—3981no—
155C24—1991no—
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—3981no—
165C22×C4—3981no—
166C22×C4—3981no—
167C22×C4—3981no—
168C22×C4—3981no—
169C22×C4—3981no—
170C22×C4—3981no—
171C22×C4—3981no—
172C22×C4—1991no—
173C22×C4C22×D19911yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—3981no—
177C2×C8—3981no—
178C2×C8—3981no—
179C2×C8D79612yes—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—3984no—
184C4⋊C4—3981no—
185C4⋊C4—3981no—
186C4⋊C4C22⋊D19912yes—
187C22⋊C4—7962no—
188C22⋊C4—3984no—
189C23—3981no—
190C23—3981no—
191C23—3981no—
192C23—3981no—
193C23—3981no—
194C23—3981no—
195C23—1992no—
196C23D4×D19911yescharacteristic
197Q8—7961no—
198Q8—7961no—
199Q8—3982no—
200Q8—3982no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—7961no—
208C2×C4—7961no—
209C2×C4—7961no—
210C2×C4—7961no—
211C2×C4—7961no—
212C2×C4—7961no—
213C2×C4—7961no—
214C2×C4—7961no—
215C2×C4—7961no—
216C2×C4—3981no—
217C2×C4—3982no—
218C2×C4—3981no—
219C2×C4—3981no—
220C2×C4D4×D19911yescharacteristic
221C2×C4C2×C22⋊D19911yescharacteristic
222C2×C4C2×D79611yescharacteristic
223C2×C4#1569188611yescharacteristic
224C2×C4#1569188711yescharacteristic
225C2×C4#1569188711yescharacteristic
226C8—22no—
227C8—7961no—
228C22—7961no—
229C22—7961no—
230C22—7961no—
231C22—7961no—
232C22—3982no—
233C22—3982no—
234C22—3982no—
235C22#2026392311yescharacteristic
236C22#2026393011yescharacteristic
237C22#2026394011yescharacteristic
238C22Q16×D19911yescharacteristic
239C22#2026397411yescharacteristic
240C22#2026397911yescharacteristic
241C22#2026398711yescharacteristic
242C4—42no—
243C4—81no—
244C4—7961no—
245C4—7962no—
246C4—15921no—
247C4#2026392911yescharacteristic
248C4#2026393011yescharacteristic
249C4#2026394011yescharacteristic
250C4#2026394411yescharacteristic
251C2—7962no—
252C2#2716393511yescharacteristic
253C2#2716416011yescharacteristic
254C2#2716417211yescharacteristic
255C2#2716426011yescharacteristic
256C2#2716426511yescharacteristic
257C2#2716435511yescharacteristic
258C2#2716436111yescharacteristic
subgroupG/H
centerC23D4×D199
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
FittingC2×#107×C199C2
socleC23×C199C2×D4
prime radicalC23×D199D4
hypercenterC2×#107D199
hypocenterC199#2036
LCSG ▷ C22×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×#20263880C2semidirect
3. C2×#20263878C2semidirect
6. #27163780C2semidirect
7. C2×#107×C199C2semidirect
129. C199#2036semidirect
132. C2×#107D199semidirect

Maximal subgroups

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

subgroupindex
C2×#202639352
C2×#202638802
C2×#202638782
C2×(C2×D199)⋊C82
#271638752
#271637802
C2×#107×C1992
#2036199