Finite Group Databasebeta

← #34057021

Subgroups — #34057021

Order 19072.

430 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#23750295C211yesmaximal, characteristic, complemented
2#23750280C211yesmaximal, characteristic, complemented
3#23749971C211yesmaximal, characteristic, complemented
4#23749942C212yesmaximal, complemented
5#23749911C211yesmaximal, characteristic, complemented
6#23749875C212yesmaximal, complemented
7#23749767C212yesmaximal, complemented
8C4×#14558724C211yesmaximal, characteristic, complemented
9#23749702C211yesmaximal, characteristic, complemented
10#23749623C212yesmaximal, complemented
11#459×C149C211yesmaximal, characteristic, complemented
12C2×#14558736—21no—
13C2×#14558735C2212yes—
14C2×#14558724C2211yescharacteristic
15C2×#14558719C2212yes—
16#18011911C2211yescharacteristic
17#18011907—21no—
18#18011893—22no—
19#18011890—21no—
20#18011883—22no—
21#18011882C2212yes—
22#18011882C2212yes—
23C4×Dic298C2211yescharacteristic
24C4×D596C2211yescharacteristic
25D4×Dic149—21no—
26C4⋊C4×D149C2212yes—
27#18011843C2212yes—
28#18011842C2211yescharacteristic
29#18011836C2212yes—
30#18011834C2211yescharacteristic
31#18011832C2212yes—
32#18011832C2211yescharacteristic
33#18011832C2212yes—
34#18011823C2212yes—
35C4×#11924746C2211yescharacteristic
36C2×D8×C149C2212yes—
37#127×C149C2211yescharacteristic
38C4×D4×C149C2211yescharacteristic
39C4⋊C8×C149C2211yescharacteristic
40#102×C149C2212yes—
41#14558736—44no—
42#14558735—24no—
43#14558735—42no—
44C2×Dic298—21no—
45C2×Dic298C2311yescharacteristic
46C2×D596D411yescharacteristic
47C2×C22⋊D149—22no—
48C2×C22⋊D149—22no—
49C2×C22⋊D149—22no—
50C22×Dic149—22no—
51C2×C4×D149—22no—
52C2×C4×D149C2312yes—
53#14558724—22no—
54#14558719—42no—
55C2×#11924746C2311yescharacteristic
56C2×#11924746C2311yescharacteristic
57C4⋊Dic149—21no—
58C4⋊Dic149D411yescharacteristic
59C4⋊Dic149C2312yes—
60C4⋊Dic149C2311yescharacteristic
61#14558715—22no—
62#14558715—21no—
63C22⋊Dic149—22no—
64C22⋊Dic149—22no—
65#14558713D412yes—
66C4×Dic149—21no—
67C2×D4×C149—21no—
68C2×D4×C149—21no—
69C2×D4×C149C2311yescharacteristic
70C2×D4×C149C2312yes—
71C22×C4×C149—21no—
72D8×C149—42no—
73C2×C8×C149C2312yes—
74C4⋊C4×C149C2311yescharacteristic
75C22⋊C4×C149—21no—
76C42×C149C2311yescharacteristic
77C22×D149C2×D412yes—
78Dic298—21no—
79Dic298—42no—
80Dic298#4112yescomplemented
81D596—22no—
82C22⋊D149—44no—
83C22⋊D149—44no—
84C22⋊D149—82no—
85C2×Dic149—21no—
86C2×Dic149—22no—
87C2×Dic149—21no—
88C2×Dic149—44no—
89C2×Dic149C2×D412yes—
90C4×D149—22no—
91C4×D149—22no—
92C4×D149—44no—
93#11924746—21no—
94#11924746—21no—
95C23×C149—22no—
96C23×C149—21no—
97D4×C149—21no—
98D4×C149—22no—
99D4×C149—22no—
100D4×C149—21no—
101D4×C149—42no—
102D4×C149—81no—
103C2×C4×C149—21no—
104C2×C4×C149—42no—
105C2×C4×C149C2×D411yescharacteristic
106C2×C4×C149C2×D411yescharacteristic
107C2×C4×C149C2411yescharacteristic
108C8×C149—22no—
109C22×C149—41no—
110C22×C149—42no—
111C22×C149—41no—
112C22×C149—42no—
113C22×C149—42no—
114C22×C149—41no—
115C22×C149C22×D411yescharacteristic
116C2×D149—22no—
117C2×D149—22no—
118C2×D149—22no—
119C4×C149—21no—
120C4×C149—41no—
121C4×C149AGL(1,ℤ/8)12yes—
122C4×C149C2×#4111yescharacteristic
123C4×C149H(5,𝔽2)11yescharacteristic
124Dic149—22no—
125Dic149—22no—
126Dic149—42no—
127Dic149—41no—
128C2×C149—81no—
129C2×C149—82no—
130C2×C149#54611yescharacteristic
131C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C149#57711yescharacteristic
133D149—42no—
134C149#333211yescharacteristic, complemented, Sylow
135#3332—1491nomaximal, Sylow
136#550—1491no—
137#547—1491no—
138#486—1491no—
139#481—1492no—
140#475—1491no—
141#469—1492no—
142#459D14911yescharacteristic, complemented
143#449—1492no—
144C4×SD16—1491no—
145#436—1491no—
146#417—1492no—
147C2×#41—2981no—
148C2×#41—1492no—
149C2×SD16—1491no—
150C2×D8C2×D14912yes—
151C2×M16—1492no—
152C4⋊Q8—1491no—
153#127—2981no—
154#127C2×D14911yescharacteristic
155#123—2982no—
156C22⋊Q8—2981no—
157#121—2982no—
158#121—1492no—
159#121—1492no—
160C4×Q8—1491no—
161C4×D4—2981no—
162C4×D4—1491no—
163C4×D4C2×D14911yescharacteristic
164C2×C4⋊C4—1492no—
165#106—1492no—
166#106—1491no—
167C4⋊C8C2×D14911yescharacteristic
168#103—1492no—
169#103—1491no—
170#102—1492no—
171#102—1491no—
172#102—1492no—
173#102C2×D14912yes—
174C22⋊C8—1492no—
175C4×C8—1491no—
176#41—2984no—
177#41—5964no—
178#41—5962no—
179C2×Q8—2981no—
180C2×Q8—1491no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—2982no—
184C2×D4—2982no—
185C2×D4—2982no—
186C2×D4—1491no—
187C2×D4C22×D14912yes—
188C2×D4C22×D14911yescharacteristic
189C22×C4—21no—
190C22×C4—2982no—
191C22×C4—2982no—
192C22×C4—1492no—
193SD16—2982no—
194D8—42no—
195M16—5962no—
196C2×C8—1491no—
197C2×C8—1491no—
198C2×C8C22×D14912yes—
199C4⋊C4—2982no—
200C4⋊C4—2981no—
201C4⋊C4—2981no—
202C4⋊C4—1492no—
203C4⋊C4—1491no—
204C4⋊C4—1491no—
205C4⋊C4C22×D14911yescharacteristic
206C22⋊C4—21no—
207C22⋊C4—2982no—
208C22⋊C4—2982no—
209C22⋊C4—1492no—
210C42—2981no—
211C42C22×D14911yescharacteristic
212C23—21no—
213C23—22no—
214C23—1492no—
215Q8—2981no—
216Q8—5962no—
217Q8—1492no—
218D4—21no—
219D4—22no—
220D4—21no—
221D4—22no—
222D4—42no—
223D4—81no—
224D4—2982no—
225D4—5964no—
226D4—5964no—
227D4—11922no—
228C2×C4—21no—
229C2×C4—42no—
230C2×C4—2981no—
231C2×C4—2982no—
232C2×C4—2982no—
233C2×C4—2981no—
234C2×C4—2982no—
235C2×C4—5964no—
236C2×C4—5964no—
237C2×C4—1492no—
238C2×C4D4×D14911yescharacteristic
239C2×C4D4×D14911yescharacteristic
240C2×C4C23×D14911yescharacteristic
241C8—22no—
242C8—2981no—
243C8—2981no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—41no—
248C22—42no—
249C22—41no—
250C22—2982no—
251C22—2982no—
252C22—2982no—
253C22C2×D4×D14911yescharacteristic
254C4—21no—
255C4—41no—
256C4—2982no—
257C4—2982no—
258C4—5962no—
259C4—5961no—
260C4#1801193512yes—
261C4#41×D14911yescharacteristic
262C4#1801195611yescharacteristic
263C2—81no—
264C2—82no—
265C2—5962no—
266C2#2375027211yescharacteristic
267C2C2×#1801193511yescharacteristic
268C2#2375043511yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#459×C149C2
socleC22×C149C22×D4
hypercenter#459D149
hypocenterC149#3332
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #23750295C2semidirect
2. #23750280C2semidirect
3. #23749971C2semidirect
4. #23749942C2semidirect
5. #23749911C2semidirect
6. #23749875C2semidirect
7. #23749767C2semidirect
8. C4×#14558724C2semidirect
9. #23749702C2semidirect
10. #23749623C2semidirect
11. #459×C149C2semidirect
80. Dic298#41semidirect
134. C149#3332semidirect
142. #459D149semidirect

Maximal subgroups

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

subgroupindex
#237502952
#237502802
#237499712
#237499422
#237499112
#237498752
#237497672
C4×#145587242
#237497022
#237496232
#459×C1492
#3332149