Finite Group Databasebeta

← #34057050

Subgroups — #34057050

Order 19072.

388 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 125 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#23750335C211yesmaximal, characteristic, complemented
2#23750280C211yesmaximal, characteristic
3#23750023C211yesmaximal, characteristic, complemented
4#23749954C212yesmaximal, complemented
5#23749942C212yesmaximal, complemented
6#23749915C211yesmaximal, characteristic, complemented
7#23749910C211yesmaximal, characteristic, complemented
8#23749851C212yesmaximal, complemented
9C8×D596C211yesmaximal, characteristic, complemented
10#106×D149C212yesmaximal, complemented
11C4×SD16×C149C211yesmaximal, characteristic, complemented
12C2×#14558736—21no—
13C2×C8×D149C2212yes—
14#18011911C2211yescharacteristic
15#18011911C2211yescharacteristic
16#18011903—21no—
17#18011893—22no—
18#18011890—21no—
19#18011889—22no—
20#18011887C2212yes—
21#18011882C2212yes—
22C4×D596—21no—
23C4×D596C2211yescharacteristic
24C4⋊C4×D149C2212yes—
25C4⋊C4×D149C2212yes—
26#18011843C2212yes—
27#18011843C2212yes—
28#18011842C2212yes—
29#18011842C2212yes—
30C4⋊#11924746C2211yescharacteristic
31#18011835C2212yes—
32#18011832C2212yes—
33#18011823C2212yes—
34C2×SD16×C149C2211yescharacteristic
35C4×Q8×C149C2211yescharacteristic
36C4×D4×C149C2211yescharacteristic
37#106×C149C2211yescharacteristic
38#103×C149C2211yescharacteristic
39#102×C149C2211yescharacteristic
40C4×C8×C149C2211yescharacteristic
41#14558736—44no—
42C2×Dic298—21no—
43C2×Dic298—21no—
44C2×D596D411yescharacteristic
45C2×C22⋊D149—22no—
46C2×C4×D149—22no—
47C2×C4×D149—22no—
48C2×C4×D149C2312yes—
49C2×#11924746C2312yes—
50C8×D149—24no—
51C4⋊Dic149—21no—
52C4⋊Dic149—21no—
53C4⋊Dic149D411yescharacteristic
54C4⋊Dic149C2312yes—
55C4⋊Dic149C2312yes—
56#14558715—22no—
57#14558715—22no—
58#14558715—22no—
59C22⋊Dic149—22no—
60#14558713—22no—
61#14558713D412yes—
62C2×Q8×C149C2311yescharacteristic
63C2×D4×C149C2311yescharacteristic
64C22×C4×C149—21no—
65SD16×C149—22no—
66C2×C8×C149C2311yescharacteristic
67C2×C8×C149C2311yescharacteristic
68C4⋊C4×C149—21no—
69C4⋊C4×C149C2311yescharacteristic
70C4⋊C4×C149C2311yescharacteristic
71C22⋊C4×C149—21no—
72C42×C149—21no—
73C42×C149C2311yescharacteristic
74C22×D149C2×D412yes—
75Dic298—42no—
76Dic298—81no—
77D596SD1614yescomplemented
78C22⋊D149—44no—
79C2×Dic149—22no—
80C2×Dic149—22no—
81C2×Dic149C2×D412yes—
82C4×D149—22no—
83C4×D149—22no—
84C4×D149—44no—
85C4×D149—82no—
86#11924746—22no—
87C23×C149—21no—
88Q8×C149—21no—
89Q8×C149—21no—
90D4×C149—21no—
91D4×C149—21no—
92C2×C4×C149—21no—
93C2×C4×C149—21no—
94C2×C4×C149—21no—
95C2×C4×C149—42no—
96C2×C4×C149C2×D411yescharacteristic
97C2×C4×C149C2×D411yescharacteristic
98C2×C4×C149C2411yescharacteristic
99C8×C149—21no—
100C8×C149#4112yescomplemented
101C22×C149—41no—
102C22×C149—41no—
103C22×C149—41no—
104C22×C149C22×D411yescharacteristic
105C2×D149—22no—
106C2×D149—22no—
107C2×D149C2×SD1614yes—
108C4×C149—21no—
109C4×C149—41no—
110C4×C149—41no—
111C4×C149—41no—
112C4×C149C2×SD1612yes—
113C4×C149C2×#4111yescharacteristic
114C4×C1492−1+411yescharacteristic
115Dic149—22no—
116Dic149—42no—
117Dic149—42no—
118C2×C149—81no—
119C2×C149#54711yescharacteristic
120C2×C149C22×SD1611yescharacteristic
121C2×C149#57711yescharacteristic
122D149—24no—
123C149#333511yescharacteristic, complemented, Sylow
124#3335—1491nomaximal, Sylow
125#554—1491no—
126#547—1491no—
127#498—1491no—
128#483—1492no—
129#481—1492no—
130#476—1491no—
131#475—1491no—
132#465—1492no—
133C4×SD16D14911yescharacteristic, complemented
134C8×D4—1491no—
135C2×#106—1492no—
136C2×#41—2981no—
137C2×SD16C2×D14911yescharacteristic
138C22×C8—1492no—
139C4⋊Q8—1491no—
140C4⋊Q8—1491no—
141#125—2981no—
142#123—2982no—
143C22⋊Q8—2981no—
144C22⋊Q8—2982no—
145C22⋊Q8—1492no—
146#121—1492no—
147C4×Q8C2×D14911yescharacteristic
148C4×D4—2981no—
149C4×D4—1491no—
150C4×D4C2×D14911yescharacteristic
151C2×C4⋊C4—1492no—
152C2×C4⋊C4—1492no—
153#106—1492no—
154#106—1492no—
155#106—1492no—
156#106—1492no—
157#106C2×D14911yescharacteristic
158C4⋊C8—1491no—
159#103—1492no—
160#103C2×D14911yescharacteristic
161#102—1492no—
162#102C2×D14911yescharacteristic
163C22⋊C8—1492no—
164C4×C8C2×D14911yescharacteristic
165#41—5964no—
166C2×Q8—2981no—
167C2×Q8—2981no—
168C2×Q8C22×D14911yescharacteristic
169C2×D4—2982no—
170C2×D4—1491no—
171C2×D4C22×D14911yescharacteristic
172C22×C4—21no—
173C22×C4—2982no—
174C22×C4—2982no—
175C22×C4—1492no—
176SD16—22no—
177C2×C8—2984no—
178C2×C8—1492no—
179C2×C8C22×D14911yescharacteristic
180C2×C8C22×D14911yescharacteristic
181C4⋊C4—21no—
182C4⋊C4—2982no—
183C4⋊C4—2982no—
184C4⋊C4—2982no—
185C4⋊C4—2981no—
186C4⋊C4—2981no—
187C4⋊C4—1492no—
188C4⋊C4—1492no—
189C4⋊C4—1491no—
190C4⋊C4C22×D14911yescharacteristic
191C4⋊C4C22×D14911yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—2982no—
194C22⋊C4—2982no—
195C22⋊C4—1492no—
196C42—21no—
197C42C22×D14911yescharacteristic
198C23—21no—
199C23—1492no—
200Q8—21no—
201Q8—21no—
202Q8—5962no—
203Q8—11921no—
204D4—21no—
205D4—21no—
206D4—5964no—
207D4—1494no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—42no—
212C2×C4—2982no—
213C2×C4—2982no—
214C2×C4—2982no—
215C2×C4—2982no—
216C2×C4—5964no—
217C2×C4—1492no—
218C2×C4—11922no—
219C2×C4D4×D14911yescharacteristic
220C2×C4D4×D14911yescharacteristic
221C2×C4C23×D14911yescharacteristic
222C8—21no—
223C8—2982no—
224C8#1455873612yescomplemented
225C22—41no—
226C22—41no—
227C22—41no—
228C22—2982no—
229C22—2982no—
230C22—1494no—
231C22C2×D4×D14911yescharacteristic
232C4—21no—
233C4—41no—
234C4—41no—
235C4—41no—
236C4—2982no—
237C4—5962no—
238C4—5962no—
239C4SD16×D14912yes—
240C4C2×#1455873611yescharacteristic
241C4#1801195811yescharacteristic
242C2—81no—
243C2—2984no—
244C2#2375028011yescharacteristic
245C2C2×SD16×D14911yescharacteristic
246C2#2375043211yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC4×SD16×C149C2
socleC22×C149C22×D4
prime radical#23750280C2
hypercenterC4×SD16D149
hypocenterC149#3335
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #23750335C2semidirect
3. #23750023C2semidirect
4. #23749954C2semidirect
5. #23749942C2semidirect
6. #23749915C2semidirect
7. #23749910C2semidirect
8. #23749851C2semidirect
9. C8×D596C2semidirect
10. #106×D149C2semidirect
11. C4×SD16×C149C2semidirect
77. D596SD16semidirect
100. C8×C149#41semidirect
123. C149#3335semidirect
133. C4×SD16D149semidirect
224. C8#14558736semidirect

Maximal subgroups

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

subgroupindex
#237503352
#237502802
#237500232
#237499542
#237499422
#237499152
#237499102
#237498512
C8×D5962
#106×D1492
C4×SD16×C1492
#3335149