Finite Group Databasebeta

← #34047208

Subgroups — #34047208

Order 19072.

688 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011894C211yesmaximal, characteristic
2C2×#18011871C211yesmaximal, characteristic
3C22×C4×Dic149C211yesmaximal, characteristic, complemented
4#23749514C212yesmaximal, complemented
5#23749507C212yesmaximal, complemented
6#23749502C211yesmaximal, characteristic
7#23749497C211yesmaximal, characteristic, complemented
8#23749476C212yesmaximal, complemented
9C22⋊C4×Dic149C211yesmaximal, characteristic, complemented
10#23749472C212yesmaximal, complemented
11#381×C149C211yesmaximal, characteristic, complemented
12C22×C22⋊D149C2211yescharacteristic
13C23×Dic149C2211yescharacteristic
14C23×Dic149C2211yescharacteristic
15C22×C4×D149C2211yescharacteristic
16#18011894C418yescomplemented
17#18011871—24no—
18C2×#14558715C2212yes—
19C2×#14558715C2211yescharacteristic
20C2×#14558715C2212yes—
21C2×C22⋊Dic149C2211yescharacteristic
22C2×#14558713C2212yes—
23C2×#14558713C2211yescharacteristic
24C2×#14558713C2212yes—
25C2×C4×Dic149—24no—
26C2×C4×Dic149C2212yes—
27C2×C4×Dic149C2212yes—
28C2×C4×Dic149C2211yescharacteristic
29C2×C4×Dic149C2212yes—
30#18011818C2212yes—
31#18011818C2212yes—
32#18011818C2212yes—
33C23×C4×C149C2211yescharacteristic
34C2×C22⋊C4×C149C2211yescharacteristic
35C2×C22⋊C4×C149C2211yescharacteristic
36#95×C149C2212yes—
37#95×C149C2212yes—
38C23×D149C2311yescharacteristic
39C2×C22⋊D149—22no—
40C2×C22⋊D149—22no—
41C2×C22⋊D149C2×C414yes—
42C22×Dic149—24no—
43C22×Dic149—24no—
44C22×Dic149—22no—
45C22×Dic149D414yes—
46C22×Dic149C2311yescharacteristic
47C22×Dic149C2312yes—
48C22×Dic149C2311yescharacteristic
49C22×Dic149C2312yes—
50C22×Dic149C2311yescharacteristic
51C2×C4×D149—22no—
52C2×C4×D149—24no—
53#14558715—22no—
54#14558715—24no—
55#14558715C2×C418yescomplemented
56C22⋊Dic149C2×C414yescomplemented
57#14558713—22no—
58#14558713—24no—
59#14558713C2×C418yescomplemented
60C4×Dic149—24no—
61C4×Dic149—24no—
62C4×Dic149—22no—
63C4×Dic149—24no—
64C24×C149C2311yescharacteristic
65C22×C4×C149—22no—
66C22×C4×C149—22no—
67C22×C4×C149C2×C414yescomplemented
68C22×C4×C149C2311yescharacteristic
69C22×C4×C149C2312yes—
70C22×C4×C149C2311yescharacteristic
71C22×C4×C149C2312yes—
72C22⋊C4×C149—22no—
73C22⋊C4×C149—22no—
74C22×D149—21no—
75C22×D149—22no—
76C22×D149—21no—
77C22×D149—22no—
78C22×D149C22×C412yes—
79C22⋊D149—28no—
80C2×Dic149—28no—
81C2×Dic149—22no—
82C2×Dic149—21no—
83C2×Dic149—22no—
84C2×Dic149—22no—
85C2×Dic149—21no—
86C2×Dic149—22no—
87C2×Dic149—22no—
88C2×Dic149—28no—
89C2×Dic149C22×C412yes—
90C2×Dic149C22×C414yes—
91C2×Dic149C2×D412yes—
92C2×Dic149C2×D412yes—
93C2×Dic149#4114yes—
94C2×Dic149#4114yes—
95C4×D149—44no—
96C23×C149—22no—
97C23×C149—22no—
98C23×C149C22×C412yes—
99C23×C149C2×D412yes—
100C23×C149#4112yes—
101C23×C149C2411yescharacteristic
102C2×C4×C149—28no—
103C2×C4×C149—22no—
104C2×C4×C149—22no—
105C2×C4×C149—21no—
106C2×C4×C149—21no—
107C2×C4×C149—22no—
108C2×C4×C149—22no—
109C2×C4×C149—22no—
110C2×C4×C149—22no—
111C2×C4×C149—22no—
112C2×C4×C149C22×C414yes—
113C22×C149—24no—
114C22×C149—22no—
115C22×C149—22no—
116C22×C149—24no—
117C22×C149C4×D414yes—
118C22×C149C23×C411yescharacteristic
119C22×C149C22×D411yescharacteristic
120C22×C149C2×#4112yes—
121C22×C149C2×#4112yes—
122C22×C149C2×#4111yescharacteristic
123C2×D149—22no—
124C2×D149—24no—
125C2×D149—41no—
126C2×D149—41no—
127C2×D149—42no—
128C4×C149—24no—
129C4×C149—41no—
130C4×C149—41no—
131C4×C149—42no—
132Dic149—22no—
133Dic149—24no—
134Dic149—22no—
135Dic149—24no—
136Dic149#12314yescomplemented
137C2×C149—24no—
138C2×C149C2×C4×D411yescharacteristic
139C2×C149C4×#4112yes—
140C2×C149C2×#12311yescharacteristic
141C2×C149#52511yescharacteristic
142C2×C149#52912yes—
143D149—42no—
144C149C4×#12311yescharacteristic, complemented, Sylow
145C4×#123—1491nomaximal, Sylow
146C2×#123—1491no—
147C2×C4×D4—1491no—
148C22×C42—1491no—
149#388—1492no—
150#387—1492no—
151#386—1491no—
152#385—1491no—
153#381D14911yescharacteristic, complemented
154C4×C4⋊C4—1492no—
155C4×C22⋊C4—1491no—
156C4×C22⋊C4—1492no—
157C22×D4—1491no—
158C23×C4—1491no—
159C23×C4—1491no—
160C23×C4—1491no—
161C23×C4C2×D14911yescharacteristic
162#123—1498no—
163C4×D4—2984no—
164C2×C4⋊C4—1491no—
165C2×C4⋊C4—1492no—
166C2×C4⋊C4—1492no—
167C2×C22⋊C4—1491no—
168C2×C22⋊C4—1492no—
169C2×C22⋊C4—1491no—
170C2×C22⋊C4—1492no—
171C2×C22⋊C4C2×D14911yescharacteristic
172C2×C22⋊C4C2×D14911yescharacteristic
173C2×C42—2984no—
174C2×C42—1491no—
175C2×C42—1492no—
176C2×C42—1492no—
177C2×C42—1492no—
178#95—1492no—
179#95—1492no—
180#95—1492no—
181#95C2×D14912yes—
182#95C2×D14912yes—
183C24—1491no—
184C24C22×D14911yescharacteristic
185C2×D4—2982no—
186C2×D4—2982no—
187C2×D4—1494no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—2982no—
191C22×C4—2984no—
192C22×C4—2982no—
193C22×C4—2984no—
194C22×C4—2984no—
195C22×C4—1494no—
196C22×C4—1492no—
197C22×C4—1491no—
198C22×C4—1492no—
199C22×C4—1491no—
200C22×C4—1491no—
201C22×C4C4×D14914yescomplemented
202C22×C4C22×D14912yes—
203C22×C4C22×D14911yescharacteristic
204C22×C4C22×D14911yescharacteristic
205C22×C4C22×D14912yes—
206C4⋊C4—2982no—
207C4⋊C4—2984no—
208C4⋊C4—1498no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—2982no—
212C22⋊C4—2984no—
213C22⋊C4—1494no—
214C22⋊C4—1498no—
215C42—2984no—
216C42—2984no—
217C42—2982no—
218C42—2984no—
219C23—22no—
220C23—22no—
221C23—2981no—
222C23—2982no—
223C23—2982no—
224C23—2981no—
225C23—1492no—
226C23C2×C4×D14912yes—
227C23D4×D14912yes—
228C23#1455873512yes—
229C23C23×D14911yescharacteristic
230D4—2988no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—28no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—2988no—
242C2×C4—2982no—
243C2×C4—2981no—
244C2×C4—2982no—
245C2×C4—2982no—
246C2×C4—2982no—
247C2×C4—2982no—
248C2×C4—2981no—
249C2×C4—2988no—
250C2×C4—5964no—
251C2×C4—1494no—
252C2×C4—1492no—
253C2×C4—1492no—
254C2×C4—1494no—
255C2×C4—1492no—
256C2×C4—1494no—
257C2×C4C2×C4×D14914yes—
258C22—24no—
259C22—24no—
260C22—22no—
261C22—22no—
262C22—2982no—
263C22—2984no—
264C22—5962no—
265C22—5961no—
266C22—5961no—
267C22#1801187114yes—
268C22C22×C4×D14911yescharacteristic
269C22C2×D4×D14911yescharacteristic
270C22#41×D14912yes—
271C22#41×D14912yes—
272C22C2×#1455873511yescharacteristic
273C4—24no—
274C4—42no—
275C4—41no—
276C4—41no—
277C4—2982no—
278C4—2984no—
279C4—2982no—
280C4—2984no—
281C4—1494no—
282C2—24no—
283C2—5962no—
284C2C2×#1801187111yescharacteristic
285C2#2375009312yes—
286C2#123×D14911yescharacteristic
287C2#2375013311yescharacteristic
288C2#2375015212yes—
subgroupG/H
center, FrattiniC23C23×D149
commutatorC22×C149C23×C4
Fitting#381×C149C2
socle, cosocleC23×C149C24
prime radicalC22×C22⋊D149C22
hypercenter#381D149
hypocenterC149C4×#123
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. C22×C4×Dic149C2semidirect
4. #23749514C2semidirect
5. #23749507C2semidirect
7. #23749497C2semidirect
8. #23749476C2semidirect
9. C22⋊C4×Dic149C2semidirect
10. #23749472C2semidirect
11. #381×C149C2semidirect
16. #18011894C4semidirect
55. #14558715C2×C4semidirect
56. C22⋊Dic149C2×C4semidirect
59. #14558713C2×C4semidirect
67. C22×C4×C149C2×C4semidirect
136. Dic149#123semidirect
144. C149C4×#123semidirect
153. #381D149semidirect
201. C22×C4C4×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180118942
C2×#180118712
C22×C4×Dic1492
#237495142
#237495072
#237495022
#237494972
#237494762
C22⋊C4×Dic1492
#237494722
#381×C1492
C4×#123149