Finite Group Databasebeta

← #34047360

Subgroups — #34047360

Order 19072.

632 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×C4×C22⋊D149C211yesmaximal, characteristic
3#23749520C212yesmaximal, complemented
4#23749515C212yesmaximal, complemented
5#23749506C212yesmaximal, complemented
6#23749504C211yesmaximal, characteristic
7#23749498C211yesmaximal, characteristic, complemented
8#23749496C212yesmaximal, complemented
9#23749488C211yesmaximal, characteristic, complemented
10C4×C22⋊Dic149C211yesmaximal, characteristic, complemented
11C22×C4⋊C4×C149C211yesmaximal, characteristic, complemented
12C22×C22⋊D149C2211yescharacteristic
13C22×C4×D149C2211yescharacteristic
14#18011894C418yescomplemented
15C4×C22⋊D149—24no—
16C2×#14558715C2211yescharacteristic
17C2×#14558715C2212yes—
18C2×#14558715C2212yes—
19C2×C22⋊Dic149C2211yescharacteristic
20C2×C22⋊Dic149C2211yescharacteristic
21C2×C22⋊Dic149C2211yescharacteristic
22C2×#14558713C2211yescharacteristic
23C2×#14558713C2212yes—
24C2×#14558713C2212yes—
25C2×C4×Dic149C2211yescharacteristic
26C2×C4×Dic149C2212yes—
27#18011818C2212yes—
28#18011818C2212yes—
29#18011818C2212yes—
30#18011818C2212yes—
31#18011818C2212yes—
32C23×C4×C149C2211yescharacteristic
33C23×C4×C149C2211yescharacteristic
34C23×C4×C149C2211yescharacteristic
35C2×C4⋊C4×C149—24no—
36C2×C4⋊C4×C149C2212yes—
37C2×C4⋊C4×C149C2212yes—
38C23×D149C2311yescharacteristic
39C2×C22⋊D149—22no—
40C2×C22⋊D149—22no—
41C2×C22⋊D149C2×C414yes—
42C22×Dic149C2311yescharacteristic
43C22×Dic149C2311yescharacteristic
44C22×Dic149C2312yes—
45C22×Dic149C2311yescharacteristic
46C22×Dic149C2312yes—
47C2×C4×D149—22no—
48C2×C4×D149—42no—
49#14558715—22no—
50#14558715—42no—
51#14558715C2×C418yescomplemented
52C22⋊Dic149—22no—
53C22⋊Dic149—22no—
54C22⋊Dic149C2×C414yescomplemented
55#14558713—22no—
56#14558713—42no—
57#14558713C2×C418yescomplemented
58C4×Dic149—22no—
59C4×Dic149—42no—
60C24×C149C2311yescharacteristic
61C22×C4×C149—22no—
62C22×C4×C149—22no—
63C22×C4×C149—22no—
64C22×C4×C149—42no—
65C22×C4×C149—42no—
66C22×C4×C149C2×C414yescomplemented
67C22×C4×C149D414yes—
68C22×C4×C149C2312yes—
69C22×C4×C149C2312yes—
70C22×C4×C149C2311yescharacteristic
71C22×C4×C149C2311yescharacteristic
72C4⋊C4×C149—24no—
73C4⋊C4×C149—24no—
74C22×D149—21no—
75C22×D149—21no—
76C22×D149—22no—
77C22×D149—22no—
78C22×D149C22×C412yes—
79C22⋊D149—28no—
80C2×Dic149—21no—
81C2×Dic149—21no—
82C2×Dic149—22no—
83C2×Dic149—22no—
84C2×Dic149—21no—
85C2×Dic149—21no—
86C2×Dic149—22no—
87C2×Dic149—22no—
88C2×Dic149—22no—
89C2×Dic149—22no—
90C2×Dic149—22no—
91C2×Dic149C22×C412yes—
92C2×Dic149C22×C414yes—
93C4×D149—44no—
94C23×C149—22no—
95C23×C149—22no—
96C23×C149C22×C412yes—
97C23×C149C2×D412yes—
98C23×C149#4112yes—
99C23×C149C2411yescharacteristic
100C2×C4×C149—28no—
101C2×C4×C149—22no—
102C2×C4×C149—22no—
103C2×C4×C149—22no—
104C2×C4×C149—22no—
105C2×C4×C149—22no—
106C2×C4×C149—22no—
107C2×C4×C149—22no—
108C2×C4×C149—44no—
109C2×C4×C149—44no—
110C2×C4×C149C22×C414yes—
111C2×C4×C149C2×D412yes—
112C2×C4×C149C2×D412yes—
113C22×C149—24no—
114C22×C149—22no—
115C22×C149—22no—
116C22×C149—24no—
117C22×C149C4×D414yes—
118C22×C149C23×C411yescharacteristic
119C22×C149C22×D411yescharacteristic
120C22×C149C2×#4111yescharacteristic
121C22×C149H(5,𝔽2)12yes—
122C22×C1492−1+412yes—
123C2×D149—22no—
124C2×D149—24no—
125C2×D149—41no—
126C2×D149—41no—
127C2×D149—42no—
128C4×C149—24no—
129C4×C149—22no—
130C4×C149—22no—
131C4×C149—42no—
132Dic149—22no—
133Dic149—24no—
134Dic149—41no—
135Dic149—41no—
136Dic149—42no—
137C2×C149—24no—
138C2×C149C2×C4×D411yescharacteristic
139C2×C149#52012yes—
140C2×C149#53411yescharacteristic
141C2×C149#53611yescharacteristic
142C2×C149#53912yes—
143D149—42no—
144C149#232211yescharacteristic, complemented, Sylow
145#2322—1491nomaximal, Sylow
146C2×#123—1491no—
147C2×C4×D4—1491no—
148C22×C4⋊C4D14911yescharacteristic, complemented
149#389—1492no—
150#388—1492no—
151#387—1492no—
152#386—1491no—
153#385—1491no—
154#385—1492no—
155#381—1491no—
156C4×C22⋊C4—1491no—
157C22×D4—1491no—
158C23×C4—1491no—
159C23×C4C2×D14911yescharacteristic
160C23×C4C2×D14911yescharacteristic
161C23×C4C2×D14911yescharacteristic
162#123—1498no—
163C4×D4—2984no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—1491no—
166C2×C4⋊C4—1492no—
167C2×C4⋊C4—1492no—
168C2×C4⋊C4C2×D14912yes—
169C2×C4⋊C4C2×D14912yes—
170C2×C22⋊C4—1491no—
171C2×C22⋊C4—1491no—
172C2×C22⋊C4—1491no—
173C2×C22⋊C4—1492no—
174C2×C22⋊C4—1491no—
175C2×C22⋊C4—1492no—
176C2×C42—1491no—
177C2×C42—1492no—
178#95—1492no—
179#95—1492no—
180#95—1492no—
181#95—1492no—
182#95—1492no—
183C24—1491no—
184C24C22×D14911yescharacteristic
185C2×D4—2982no—
186C2×D4—2982no—
187C2×D4—1494no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—2982no—
194C22×C4—5962no—
195C22×C4—1491no—
196C22×C4—1492no—
197C22×C4—1491no—
198C22×C4—1492no—
199C22×C4—1491no—
200C22×C4C4×D14914yescomplemented
201C22×C4C22⋊D14914yes—
202C22×C4C22×D14912yes—
203C22×C4C22×D14912yes—
204C22×C4C22×D14911yescharacteristic
205C22×C4C22×D14911yescharacteristic
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—2982no—
209C4⋊C4—5962no—
210C4⋊C4—1498no—
211C22⋊C4—2982no—
212C22⋊C4—2982no—
213C22⋊C4—2982no—
214C22⋊C4—5962no—
215C22⋊C4—1494no—
216C22⋊C4—1498no—
217C42—2982no—
218C42—5962no—
219C23—22no—
220C23—22no—
221C23—2981no—
222C23—2981no—
223C23—2982no—
224C23—2982no—
225C23—1492no—
226C23C2×C4×D14912yes—
227C23C2×C22⋊D14912yes—
228C23#1455873612yes—
229C23C23×D14911yescharacteristic
230D4—2988no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—28no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—2981no—
242C2×C4—2981no—
243C2×C4—2982no—
244C2×C4—2982no—
245C2×C4—2981no—
246C2×C4—2981no—
247C2×C4—2982no—
248C2×C4—2982no—
249C2×C4—2982no—
250C2×C4—2982no—
251C2×C4—2982no—
252C2×C4—5964no—
253C2×C4—1494no—
254C2×C4—1492no—
255C2×C4C2×C4×D14914yes—
256C2×C4C2×C22⋊D14912yes—
257C2×C4C2×C22⋊D14912yes—
258C22—22no—
259C22—22no—
260C22—24no—
261C22—24no—
262C22—2982no—
263C22—2984no—
264C22—5961no—
265C22—5961no—
266C22—5962no—
267C22C4×C22⋊D14914yes—
268C22C22×C4×D14911yescharacteristic
269C22C22×C22⋊D14911yescharacteristic
270C22C2×#1455873611yescharacteristic
271C22#1801195612yes—
272C22#1801195912yes—
273C4—22no—
274C4—22no—
275C4—24no—
276C4—42no—
277C4—2982no—
278C4—2984no—
279C4—5961no—
280C4—5961no—
281C4—5962no—
282C2—24no—
283C2—5962no—
284C2C2×C4×C22⋊D14911yescharacteristic
285C2#2375010612yes—
286C2#2375018411yescharacteristic
287C2#2375019911yescharacteristic
288C2#2375022212yes—
subgroupG/H
center, FrattiniC23C23×D149
commutatorC22×C149C23×C4
FittingC22×C4⋊C4×C149C2
socle, cosocleC23×C149C24
prime radicalC22×C22⋊D149C22
hypercenterC22×C4⋊C4D149
hypocenterC149#2322
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #23749520C2semidirect
4. #23749515C2semidirect
5. #23749506C2semidirect
7. #23749498C2semidirect
8. #23749496C2semidirect
9. #23749488C2semidirect
10. C4×C22⋊Dic149C2semidirect
11. C22×C4⋊C4×C149C2semidirect
14. #18011894C4semidirect
51. #14558715C2×C4semidirect
54. C22⋊Dic149C2×C4semidirect
57. #14558713C2×C4semidirect
66. C22×C4×C149C2×C4semidirect
144. C149#2322semidirect
148. C22×C4⋊C4D149semidirect
200. 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×C4×C22⋊D1492
#237495202
#237495152
#237495062
#237495042
#237494982
#237494962
#237494882
C4×C22⋊Dic1492
C22×C4⋊C4×C1492
#2322149