Finite Group Databasebeta

← #34055156

Subgroups — #34055156

Order 19072.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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×#18011940C212yesmaximal, complemented
2C2×#18011935C212yesmaximal, complemented
3#23750173C211yesmaximal, characteristic, complemented
4#124×D149C211yesmaximal, characteristic, complemented
5#23750020C211yesmaximal, characteristic, complemented
6#23750019C211yesmaximal, characteristic, complemented
7#23750012C211yesmaximal, characteristic, complemented
8#23750008C212yesmaximal, complemented
9#23750006C212yesmaximal, complemented
10#23749577C211yesmaximal, characteristic, complemented
11#496×C149C211yesmaximal, characteristic, complemented
12C2×#14558735C2212yes—
13C2×Q8×D149C2211yescharacteristic
14C2×D4×D149C2211yescharacteristic
15#18011940—28no—
16#18011935—28no—
17C2×Dic596C2212yescomplemented
18C2×#14558726C2212yescomplemented
19C2×#14558725C2212yes—
20C2×#14558724C2212yes—
21C2×#14558724C2212yescomplemented
22C2×#14558721C2212yes—
23C2×#14558719C2212yes—
24#18011911C2211yescharacteristic, complemented
25#18011900C2211yescharacteristic
26#18011899C2211yescharacteristic
27#18011898C2211yescharacteristic
28#18011898C2212yescomplemented
29#18011897—22no—
30#18011882—22no—
31D4×Dic149—22no—
32C22⋊C4×D149—22no—
33C42×D149C2211yescharacteristic
34#18011822C2211yescharacteristic, complemented
35C8×Dic149C2212yescomplemented
36C2×SD16×C149C2212yescomplemented
37C2×D8×C149C2212yes—
38#127×C149C2211yescharacteristic, complemented
39#124×C149C2211yescharacteristic
40#97×C149C2211yescharacteristic, complemented
41C23×D149—21no—
42#14558735—24no—
43#14558735—24no—
44C2×Dic298—21no—
45C2×Dic298C2312yes—
46C2×Dic298C2311yescharacteristic
47Q8×D149—24no—
48C2×D596C2311yescharacteristic
49C2×C22⋊D149—22no—
50C2×C22⋊D149—21no—
51D4×D149—24no—
52C22×Dic149—22no—
53C2×C4×D149—21no—
54C2×C4×D149D412yes—
55C2×C4×D149C2311yescharacteristic
56Dic596—24no—
57#14558726—24no—
58#14558725—24no—
59#14558724—24no—
60#14558724—24no—
61#14558721—24no—
62#14558719D418yescomplemented
63C2×#11924746C2312yes—
64C4⋊Dic149—22no—
65C22⋊Dic149—22no—
66C22⋊Dic149—24no—
67#14558713—22no—
68#14558713—22no—
69C4×Dic149D412yes—
70C4×Dic149C2311yescharacteristic
71C2×Q8×C149C2311yescharacteristic
72C2×D4×C149—22no—
73C2×D4×C149C2312yes—
74C2×D4×C149C2311yescharacteristic
75SD16×C149—24no—
76D8×C149—24no—
77C2×C8×C149C2312yes—
78C22⋊C4×C149—22no—
79C42×C149C2311yescharacteristic
80C22×D149—21no—
81C22×D149—44no—
82C22×D149—42no—
83C22×D149C2×D411yescharacteristic
84Dic298—22no—
85Dic298—22no—
86Dic298—22no—
87Dic298—42no—
88D596—22no—
89C22⋊D149—44no—
90C22⋊D149—42no—
91C2×Dic149—21no—
92C2×Dic149—22no—
93C2×Dic149—44no—
94C2×Dic149C2×D412yes—
95C2×Dic149C2×D411yescharacteristic
96C4×D149—42no—
97C4×D149—42no—
98C4×D149C2×D414yes—
99#11924746C2×D414yes—
100C23×C149—22no—
101C23×C149—21no—
102Q8×C149—22no—
103D4×C149—22no—
104D4×C149—22no—
105D4×C149—22no—
106D4×C149—44no—
107C2×C4×C149—21no—
108C2×C4×C149C2×D412yes—
109C2×C4×C149C2411yescharacteristic
110C8×C149C2×D414yes—
111C22×C149—42no—
112C22×C149—41no—
113C22×C149—44no—
114C22×C149—42no—
115C22×C149C22×D411yescharacteristic
116C2×D149—22no—
117C2×D149—41no—
118C2×D149—42no—
119C2×D149—84no—
120C2×D149#12712yes—
121C4×C149—22no—
122C4×C149—41no—
123C4×C149C22×D412yes—
124Dic149—22no—
125Dic149—41no—
126Dic149—42no—
127Dic149#12712yes—
128C2×C149—81no—
129C2×C149—82no—
130C2×C149C2×#12711yescharacteristic
131C2×C149#57512yes—
132D149—22no—
133D149—81no—
134C149#315311yescharacteristic, complemented, Sylow
135#3153—1491nomaximal, Sylow
136C2×#137—1492no—
137C2×AGL(1,ℤ/8)—1492no—
138#532—1491no—
139C2×#124—1491no—
140#497—1491no—
141#497—1491no—
142#496—1491no—
143#496D14911yescharacteristic, complemented
144#495—1492no—
145#495—1492no—
146C4×M16—1491no—
147C2×#41—1492no—
148C22×Q8—1491no—
149C22×D4—1491no—
150#137—2988no—
151AGL(1,ℤ/8)—2988no—
152C2×Q16—1492no—
153C2×Q16—1492no—
154C2×SD16—1492no—
155C2×SD16—1492no—
156C2×SD16—1492no—
157C2×SD16C2×D14912yescomplemented
158C2×D8—1492no—
159C2×D8C2×D14912yes—
160C2×M16—1492no—
161C4⋊Q8—1491no—
162#127C2×D14911yescharacteristic, complemented
163#124—2982no—
164#124—1491no—
165#124—1491no—
166#124—1492no—
167#124—1491no—
168#124C2×D14911yescharacteristic
169#121—2982no—
170C4×D4—2982no—
171C2×C22⋊C4—2982no—
172C2×C42—1491no—
173#97—1491no—
174#97C2×D14911yescharacteristic, complemented
175C4×C8—1492no—
176C24—2981no—
177#41—2984no—
178#41—2984no—
179C2×Q8—2981no—
180C2×Q8—2984no—
181C2×Q8—1491no—
182C2×Q8—1492no—
183C2×Q8C22×D14911yescharacteristic
184C2×D4—22no—
185C2×D4—2982no—
186C2×D4—2984no—
187C2×D4—2981no—
188C2×D4—1491no—
189C2×D4C22×D14911yescharacteristic
190C2×D4C22×D14912yes—
191C22×C4—2981no—
192C22×C4—2982no—
193C22×C4—1492no—
194C22×C4—1491no—
195Q16—2984no—
196Q16—2984no—
197SD16—24no—
198SD16—2984no—
199SD16—2984no—
200SD16—2984no—
201D8—24no—
202D8—2984no—
203M16—1498no—
204C2×C8—1492no—
205C2×C8C22×D14912yes—
206C4⋊C4—2982no—
207C22⋊C4—22no—
208C22⋊C4—2982no—
209C22⋊C4—2982no—
210C22⋊C4—2984no—
211C22⋊C4—2982no—
212C42—1492no—
213C42—1491no—
214C42C22×D14911yescharacteristic
215C23—21no—
216C23—22no—
217C23—2981no—
218C23—5964no—
219C23—5962no—
220C23—1491no—
221Q8—22no—
222Q8—2982no—
223Q8—2982no—
224Q8—2982no—
225Q8—5962no—
226D4—22no—
227D4—22no—
228D4—22no—
229D4—44no—
230D4—2982no—
231D4—5964no—
232D4—5962no—
233C2×C4—21no—
234C2×C4—2981no—
235C2×C4—2982no—
236C2×C4—5962no—
237C2×C4—5962no—
238C2×C4—5964no—
239C2×C4—1492no—
240C2×C4—1491no—
241C2×C4—1494no—
242C2×C4D4×D14912yes—
243C2×C4C23×D14911yescharacteristic
244C8—1494no—
245C8D4×D14914yes—
246C22—42no—
247C22—41no—
248C22—44no—
249C22—42no—
250C22—2982no—
251C22—5961no—
252C22—5962no—
253C22—1492no—
254C22—11924no—
255C22C2×D4×D14911yescharacteristic
256C4—22no—
257C4—41no—
258C4—2982no—
259C4—5962no—
260C4—5961no—
261C4—1492no—
262C4C2×D4×D14912yes—
263C2—81no—
264C2—82no—
265C2—2982no—
266C2—11921no—
267C2#127×D14911yescharacteristic
268C2#2375042112yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#496×C149C2
socleC22×C149C22×D4
hypercenter#496D149
hypocenterC149#3153
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#18011940C2semidirect
2. C2×#18011935C2semidirect
3. #23750173C2semidirect
4. #124×D149C2semidirect
5. #23750020C2semidirect
6. #23750019C2semidirect
7. #23750012C2semidirect
8. #23750008C2semidirect
9. #23750006C2semidirect
10. #23749577C2semidirect
11. #496×C149C2semidirect
17. C2×Dic596C22semidirect
18. C2×#14558726C22semidirect
21. C2×#14558724C22semidirect
24. #18011911C22semidirect
28. #18011898C22semidirect
34. #18011822C22semidirect
35. C8×Dic149C22semidirect
36. C2×SD16×C149C22semidirect
38. #127×C149C22semidirect
40. #97×C149C22semidirect
62. #14558719D4semidirect
134. C149#3153semidirect
143. #496D149semidirect
157. C2×SD16C2×D149semidirect
162. #127C2×D149semidirect
174. #97C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119402
C2×#180119352
#237501732
#124×D1492
#237500202
#237500192
#237500122
#237500082
#237500062
#237495772
#496×C1492
#3153149