Finite Group Databasebeta

← #34053969

Subgroups — #34053969

Order 19072.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#18011956C211yesmaximal, characteristic
2C2×#18011944C211yesmaximal, characteristic, complemented
3C22×#14558724C211yesmaximal, characteristic, complemented
4#23750201C211yesmaximal, characteristic, complemented
5#23749788C214yesmaximal, complemented
6#23749777C214yesmaximal, complemented
7#23749619C211yesmaximal, characteristic, complemented
8#23749593C211yesmaximal, characteristic, complemented
9#418×C149C211yesmaximal, characteristic, complemented
10#18011956—24no—
11#18011956—24no—
12C2×#14558736C2211yescharacteristic
13C2×#14558736C2211yescharacteristic
14C2×#14558735—21no—
15C2×#14558735C2212yes—
16C22×Dic298C2211yescharacteristic, complemented
17C22×C22⋊D149—22no—
18C2×D4×D149—21no—
19C2×D4×D149C2212yes—
20#18011944—24no—
21C2×Dic596C2212yescomplemented
22C2×#14558725C2212yescomplemented
23C2×#14558724—24no—
24C2×#14558724C2214yescomplemented
25C22×#11924746C2211yescharacteristic, complemented
26#18011911—21no—
27#18011899—21no—
28#18011895—22no—
29#18011891C2214yescomplemented
30#18011834C2212yescomplemented
31#18011831C2212yescomplemented
32#18011823C2214yescomplemented
33C22×D4×C149C2211yescharacteristic
34C2×M16×C149C2211yescharacteristic, complemented
35#117×C149C2211yescharacteristic, complemented
36#102×C149C2214yescomplemented
37C23×D149—22no—
38#14558736—22no—
39#14558736—22no—
40#14558736—22no—
41#14558736D414yes—
42#14558735—28no—
43#14558735—24no—
44#14558735—24no—
45C2×Dic298—21no—
46C2×Dic298—22no—
47C2×Dic298—22no—
48C2×Dic298D411yescharacteristic
49C2×Dic298C2312yes—
50C2×Dic298C2311yescharacteristic
51C2×D596D411yescharacteristic
52C2×D596C2311yescharacteristic
53C2×C22⋊D149—216no—
54C2×C22⋊D149—22no—
55C2×C22⋊D149—22no—
56C2×C22⋊D149—21no—
57C2×C22⋊D149D412yes—
58D4×D149—28no—
59D4×D149—24no—
60D4×D149—24no—
61C22×Dic149—22no—
62C22×Dic149—21no—
63C2×C4×D149—21no—
64C2×C4×D149C2312yes—
65Dic596—24no—
66#14558725—24no—
67#14558724—28no—
68C2×#11924746—22no—
69C2×#11924746C2312yes—
70C4⋊Dic149—24no—
71#14558713—24no—
72C24×C149—21no—
73C2×D4×C149—21no—
74C2×D4×C149—22no—
75C2×D4×C149D414yes—
76C2×D4×C149C2312yes—
77C22×C4×C149C2311yescharacteristic
78M16×C149—22no—
79C2×C8×C149C2312yes—
80C4⋊C4×C149C2312yes—
81C22⋊C4×C149—21no—
82C42×C149—21no—
83C22×D149—21no—
84C22×D149—28no—
85C22×D149—84no—
86C22×D149C2×D412yes—
87Dic298—22no—
88Dic298—21no—
89Dic298—22no—
90Dic298—22no—
91Dic298—42no—
92Dic298C2×D412yes—
93D596—22no—
94D596—21no—
95D596C2×D412yes—
96C22⋊D149—28no—
97C22⋊D149—28no—
98C22⋊D149—24no—
99C22⋊D149—24no—
100C2×Dic149—28no—
101C2×Dic149—21no—
102C2×Dic149—22no—
103C2×Dic149—42no—
104C2×Dic149C2×D412yes—
105C4×D149—24no—
106C4×D149—22no—
107C4×D149—22no—
108#11924746—22no—
109C23×C149—22no—
110C23×C149—24no—
111C23×C149—82no—
112C23×C149C2×D411yescharacteristic
113D4×C149—24no—
114D4×C149—22no—
115D4×C149C2×D414yes—
116C2×C4×C149—22no—
117C2×C4×C149C2×D411yescharacteristic
118C2×C4×C149C2×D412yes—
119C2×C4×C149C2×D412yes—
120C2×C4×C149C2411yescharacteristic
121C8×C149—22no—
122C22×C149—22no—
123C22×C149—24no—
124C22×C149—44no—
125C22×C149—84no—
126C22×C149#12012yes—
127C22×C149C22×D411yescharacteristic
128C2×D149—22no—
129C2×D149—22no—
130C2×D149—24no—
131C2×D149—42no—
132C2×D149—88no—
133C4×C149—42no—
134C4×C149#12012yes—
135C4×C149C22×D411yescharacteristic
136C4×C149C22×D411yescharacteristic
137Dic149—22no—
138Dic149—22no—
139Dic149—42no—
140C2×C149—22no—
141C2×C149—44no—
142C2×C149C2×#12011yescharacteristic
143C2×C149#57712yes—
144D149—42no—
145D149—42no—
146C149#300911yescharacteristic, complemented, Sylow
147#3009—1491nomaximal, Sylow
148C2×H(5,𝔽2)—1491no—
149C2×#137—1491no—
150C22×SD16—1491no—
151#536—1491no—
152#452—1494no—
153#450—1494no—
154#418D14911yescharacteristic, complemented
155#416—1491no—
156#408—1491no—
157H(5,𝔽2)—2984no—
158H(5,𝔽2)—2984no—
159C2×#41—2981no—
160C2×#41—1492no—
161C2×#41—1491no—
162C2×#41—1491no—
163C22×Q8—1491no—
164C22×D4—2981no—
165C22×D4—2982no—
166C22×D4—1492no—
167C22×D4C2×D14911yescharacteristic
168#137—2984no—
169C2×Q16—1492no—
170C2×SD16—2984no—
171C2×SD16—1494no—
172C2×SD16—1492no—
173C2×M16C2×D14911yescharacteristic, complemented
174C22×C8—1491no—
175C4⋊Q8—2981no—
176#124—2981no—
177#123—2982no—
178C22⋊Q8—1494no—
179#117C2×D14911yescharacteristic, complemented
180#103—1492no—
181#102—1492no—
182#102C2×D14914yescomplemented
183C22⋊C8—1494no—
184C24—21no—
185C24—2982no—
186#41—2988no—
187#41—2984no—
188#41—2984no—
189#41—2982no—
190#41—2982no—
191#41—2982no—
192#41—1494no—
193C2×Q8—2981no—
194C2×Q8—2982no—
195C2×Q8—2982no—
196C2×Q8—1492no—
197C2×Q8—1491no—
198C2×Q8—1491no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—2988no—
202C2×D4—2984no—
203C2×D4—29816no—
204C2×D4—2982no—
205C2×D4—2981no—
206C2×D4—2984no—
207C2×D4—2982no—
208C2×D4—1492no—
209C2×D4—1491no—
210C2×D4—1491no—
211C2×D4D59614yes—
212C2×D4C22×D14912yes—
213C22×C4—2982no—
214C22×C4—2981no—
215C22×C4—2981no—
216C22×C4—1492no—
217C22×C4C22×D14911yescharacteristic
218Q16—2984no—
219SD16—2984no—
220SD16—2988no—
221M16—22no—
222C2×C8—2982no—
223C2×C8—1492no—
224C2×C8C22×D14912yes—
225C4⋊C4—2984no—
226C4⋊C4C22×D14912yes—
227C22⋊C4—21no—
228C22⋊C4—2984no—
229C42—21no—
230C23—22no—
231C23—24no—
232C23—82no—
233C23—2988no—
234C23—2981no—
235C23—1492no—
236C23—11924no—
237C23D4×D14911yescharacteristic
238Q8—2981no—
239Q8—2982no—
240Q8—2982no—
241Q8—2982no—
242Q8—5962no—
243Q8—1492no—
244D4—22no—
245D4—24no—
246D4—2988no—
247D4—2988no—
248D4—2984no—
249D4—2984no—
250D4—2981no—
251D4—2982no—
252D4—1492no—
253D4C2×D59614yes—
254C2×C4—22no—
255C2×C4—2988no—
256C2×C4—2981no—
257C2×C4—2982no—
258C2×C4—2982no—
259C2×C4—2984no—
260C2×C4—2982no—
261C2×C4—5962no—
262C2×C4—1492no—
263C2×C4D4×D14912yes—
264C2×C4D4×D14911yescharacteristic
265C2×C4C2×D59612yes—
266C2×C4C23×D14911yescharacteristic
267C8—22no—
268C8—2982no—
269C22—22no—
270C22—24no—
271C22—44no—
272C22—84no—
273C22—2984no—
274C22—2982no—
275C22—2982no—
276C22—5962no—
277C22—11928no—
278C22#1801188112yes—
279C22C2×D4×D14911yescharacteristic
280C4—42no—
281C4—2982no—
282C4—2982no—
283C4—5962no—
284C4#1801188112yes—
285C4C2×D4×D14911yescharacteristic
286C4C22×D59611yescharacteristic
287C2—22no—
288C2—44no—
289C2—5962no—
290C2—5962no—
291C2C2×#1801188111yescharacteristic
292C2#2375043512yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#418×C149C2
socleC22×C149C22×D4
prime radicalC2×#18011956C2
hypercenter#418D149
hypocenterC149#3009
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C2×#18011944C2semidirect
3. C22×#14558724C2semidirect
4. #23750201C2semidirect
5. #23749788C2semidirect
6. #23749777C2semidirect
7. #23749619C2semidirect
8. #23749593C2semidirect
9. #418×C149C2semidirect
16. C22×Dic298C22semidirect
21. C2×Dic596C22semidirect
22. C2×#14558725C22semidirect
24. C2×#14558724C22semidirect
25. C22×#11924746C22semidirect
29. #18011891C22semidirect
30. #18011834C22semidirect
31. #18011831C22semidirect
32. #18011823C22semidirect
34. C2×M16×C149C22semidirect
35. #117×C149C22semidirect
36. #102×C149C22semidirect
146. C149#3009semidirect
154. #418D149semidirect
173. C2×M16C2×D149semidirect
179. #117C2×D149semidirect
182. #102C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119562
C2×#180119442
C22×#145587242
#237502012
#237497882
#237497772
#237496192
#237495932
#418×C1492
#3009149