Finite Group Databasebeta

← #38769206

Subgroups — #38769206

Order 25472.

374 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#15691875C211yesmaximal, characteristic
2C2×#20263920C211yesmaximal, characteristic, complemented
3C2×#20263878C211yesmaximal, characteristic, complemented
4C2×#20263875C211yesmaximal, characteristic
5#27163919C211yesmaximal, characteristic
6#27163772C211yesmaximal, characteristic, complemented
7#340×C199C211yesmaximal, characteristic, complemented
8C22×Dic398C2211yescharacteristic
9C22×D796C2211yescharacteristic
10C2×#15691875—22no—
11C2×#15691875—22no—
12C2×#15691875C414yescomplemented
13C22×C199⋊C8C2211yescharacteristic
14#20263920—22no—
15#20263920—22no—
16C2×Dic199⋊C4—21no—
17C2×(C2×D199)⋊C4—21no—
18C2×(C2×D199)⋊C4—21no—
19C2×C4×Dic199—21no—
20C2×C4×Dic199C2211yescharacteristic
21#20263878—22no—
22#20263875—22no—
23C22×C8×C199C2211yescharacteristic
24C2×C4⋊C4×C199C2211yescharacteristic
25C2×C4⋊C4×C199C2211yescharacteristic
26C23×D199—21no—
27C2×Dic398—21no—
28C2×Dic398—21no—
29C2×Dic398—21no—
30C2×Dic398C2×C412yes—
31C2×Dic398D412yes—
32C2×Dic398D412yes—
33C2×D796—21no—
34C2×D796—21no—
35C2×D796—21no—
36C2×D796—21no—
37C2×D796—21no—
38C2×D796C2×C412yes—
39C22×Dic199—21no—
40C22×Dic199D411yescharacteristic
41C22×Dic199D411yescharacteristic
42#15691875—28no—
43C2×C199⋊C8—21no—
44C2×C199⋊C8—21no—
45C2×C199⋊C8—21no—
46Dic199⋊C4—24no—
47(C2×D199)⋊C4—42no—
48(C2×D199)⋊C4—42no—
49C4×Dic199—24no—
50C4×Dic199—21no—
51C4×Dic199—21no—
52C22×C4×C199—21no—
53C22×C4×C199—21no—
54C22×C4×C199C2311yescharacteristic
55C2×C8×C199—21no—
56C2×C8×C199—21no—
57C2×C8×C199C2×C412yes—
58C4⋊C4×C199—21no—
59C4⋊C4×C199—21no—
60C4⋊C4×C199—21no—
61C4⋊C4×C199D412yes—
62C22×D199—22no—
63C22×D199—41no—
64C22×D199—41no—
65C22×D199—41no—
66C22×D199—41no—
67C22×D199—41no—
68C22×D199—41no—
69Dic398—22no—
70Dic398—22no—
71Dic398—22no—
72Dic398C22⋊C414yescomplemented
73D796—22no—
74D796—22no—
75D796—41no—
76D796—41no—
77C2×Dic199—22no—
78C2×Dic199—21no—
79C2×Dic199—22no—
80C2×Dic199—22no—
81C2×Dic199—21no—
82C2×Dic199—21no—
83C2×Dic199—21no—
84C2×Dic199—21no—
85C2×Dic199SD1612yescomplemented
86C199⋊C8—41no—
87C23×C199C2×D411yescharacteristic
88C2×C4×C199—22no—
89C2×C4×C199—41no—
90C2×C4×C199—41no—
91C2×C4×C199—41no—
92C2×C4×C199—41no—
93C2×C4×C199—41no—
94C2×C4×C199C22×C411yescharacteristic
95C2×C4×C199C2×D411yescharacteristic
96C2×C4×C199C2×D411yescharacteristic
97C2×C4×C199C2×D411yescharacteristic
98C2×C4×C199#4111yescharacteristic
99C2×C4×C199#4111yescharacteristic
100C8×C199—22no—
101C22×C199C4×D411yescharacteristic
102C22×C199#12011yescharacteristic
103C22×C199#12111yescharacteristic
104C22×C199C2×SD1611yescharacteristic
105C22×C199#13511yescharacteristic
106C22×C199AGL(1,ℤ/8)11yescharacteristic
107C22×C199#13711yescharacteristic
108C2×D199—42no—
109C2×D199—42no—
110C2×D199—42no—
111C2×D199—81no—
112C2×D199—81no—
113C2×D199—81no—
114C2×D199—81no—
115C4×C199—42no—
116C4×C199—81no—
117C4×C199C2×C22⋊C411yescharacteristic
118C4×C199C4×D411yescharacteristic
119C4×C199#12111yescharacteristic
120C4×C199#12311yescharacteristic
121Dic199—24no—
122Dic199—22no—
123Dic199—41no—
124C2×C199#38611yescharacteristic
125C2×C199C4×SD1611yescharacteristic
126C2×C199#44011yescharacteristic
127C2×C199#44811yescharacteristic
128C2×C199#44911yescharacteristic
129C2×C199#46011yescharacteristic
130C2×C199#46411yescharacteristic
131D199—82no—
132C199#188011yescharacteristic, complemented, Sylow
133#1880—1991nomaximal, Sylow
134C22×SD16—1991no—
135C2×C4×Q8—1991no—
136C2×#103—1991no—
137C2×#102—1991no—
138#390—1991no—
139#340D19911yescharacteristic, complemented
140#336—1991no—
141C22×Q8—1991no—
142C22×D4—1991no—
143C2×SD16—3982no—
144C2×SD16—3982no—
145C2×SD16—1994no—
146C22×C8—1991no—
147C22×C8C2×D19911yescharacteristic
148C4×Q8—3982no—
149C4×Q8—3982no—
150C2×C4⋊C4—3981no—
151C2×C4⋊C4C2×D19911yescharacteristic
152C2×C4⋊C4C2×D19911yescharacteristic
153C2×C22⋊C4—3981no—
154C2×C22⋊C4—3981no—
155C2×C42—3981no—
156C2×C42—1991no—
157#103—3982no—
158#102—3982no—
159C24—3981no—
160C2×Q8—3981no—
161C2×Q8—3981no—
162C2×Q8—3981no—
163C2×Q8—1992no—
164C2×Q8—1992no—
165C2×Q8—1992no—
166C2×D4—3981no—
167C2×D4—3981no—
168C2×D4—3981no—
169C2×D4—3981no—
170C2×D4—3981no—
171C2×D4—1992no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—3981no—
175C22×C4—1991no—
176C22×C4—1991no—
177C22×C4C22×D19911yescharacteristic
178SD16—3988no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—3981no—
182C2×C8—3981no—
183C2×C8—3981no—
184C2×C8C4×D19912yes—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—3984no—
189C4⋊C4D79612yes—
190C22⋊C4—7962no—
191C22⋊C4—7962no—
192C42—3984no—
193C42—3981no—
194C42—3981no—
195C23—7961no—
196C23—7961no—
197C23—7961no—
198C23—7961no—
199C23—7961no—
200C23—7961no—
201C23—3982no—
202C23D4×D19911yescharacteristic
203Q8—3982no—
204Q8—3982no—
205Q8—3982no—
206Q8—1994no—
207D4—7961no—
208D4—7961no—
209D4—3982no—
210D4—3982no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—3981no—
218C2×C4—3981no—
219C2×C4—3981no—
220C2×C4—3981no—
221C2×C4—3982no—
222C2×C4—3981no—
223C2×C4—3982no—
224C2×C4—3982no—
225C2×C4—1992no—
226C2×C4C2×C4×D19911yescharacteristic
227C2×C4D4×D19911yescharacteristic
228C2×C4D4×D19911yescharacteristic
229C2×C4C2×D79611yescharacteristic
230C2×C4#1569188611yescharacteristic
231C2×C4#1569188711yescharacteristic
232C8—22no—
233C8—7961no—
234C22—7962no—
235C22—7962no—
236C22—7962no—
237C22—15921no—
238C22—15921no—
239C22—15921no—
240C22—15921no—
241C22#2026391711yescharacteristic
242C22#2026392511yescharacteristic
243C22#2026392811yescharacteristic
244C22SD16×D19911yescharacteristic
245C22#2026397611yescharacteristic
246C22#2026397911yescharacteristic
247C22#2026398711yescharacteristic
248C4—42no—
249C4—81no—
250C4—7961no—
251C4—3984no—
252C4—3982no—
253C4C22⋊C4×D19911yescharacteristic
254C4C4×D79611yescharacteristic
255C4#2026393011yescharacteristic
256C4#2026394011yescharacteristic
257C2—15922no—
258C2#2716389611yescharacteristic
259C2#2716410511yescharacteristic
260C2#2716411711yescharacteristic
261C2#2716415811yescharacteristic
262C2#2716416411yescharacteristic
263C2#2716421711yescharacteristic
264C2#2716424311yescharacteristic
subgroupG/H
centerC23D4×D199
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#340×C199C2
socleC23×C199C2×D4
prime radicalC22×D796C22
cosocleC22×C4×C199C23
hypercenter#340D199
hypocenterC199#1880
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#20263920C2semidirect
3. C2×#20263878C2semidirect
6. #27163772C2semidirect
7. #340×C199C2semidirect
12. C2×#15691875C4semidirect
72. Dic398C22⋊C4semidirect
85. C2×Dic199SD16semidirect
132. C199#1880semidirect
139. #340D199semidirect

Maximal subgroups

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

subgroupindex
C22×#156918752
C2×#202639202
C2×#202638782
C2×#202638752
#271639192
#271637722
#340×C1992
#1880199