Finite Group Databasebeta

← #34055764

Subgroups — #34055764

Order 19072.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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#23750238C211yesmaximal, characteristic, complemented
2#23750188C211yesmaximal, characteristic
3#23750019C211yesmaximal, characteristic, complemented
4#23749997C211yesmaximal, characteristic, complemented
5#23749982C211yesmaximal, characteristic, complemented
6#23749846C212yesmaximal, complemented
7#23749824C212yesmaximal, complemented
8#23749788C212yesmaximal, complemented
9#23749777C212yesmaximal, complemented
10#23749687C211yesmaximal, characteristic, complemented
11#490×C149C211yesmaximal, characteristic, complemented
12C2×#14558735C2211yescharacteristic
13C2×D4×D149C2211yescharacteristic
14C2×Dic596C2212yescomplemented
15C2×#14558725C2212yes—
16C2×#14558724C2212yes—
17C2×#14558724C2212yescomplemented
18#18011911C2211yescharacteristic, complemented
19#18011907C2211yescharacteristic
20#18011906—22no—
21#18011902—22no—
22#18011899C2211yescharacteristic
23#18011898C2211yescharacteristic
24#18011891C2212yes—
25#18011883—22no—
26#18011879—22no—
27#18011876C2212yescomplemented
28#18011868C2211yescharacteristic
29#18011840C2212yescomplemented
30#18011834C2212yescomplemented
31#18011831C2212yes—
32#18011823C2212yes—
33C4×#11924746C2211yescharacteristic, complemented
34#127×C149C2211yescharacteristic
35#125×C149C2211yescharacteristic, complemented
36#102×C149C2212yes—
37#102×C149C2212yescomplemented
38#97×C149C2211yescharacteristic, complemented
39C23×D149—21no—
40#14558735—42no—
41C2×Dic298C2312yes—
42C2×Dic298C2311yescharacteristic
43C2×D596C2311yescharacteristic
44C2×C22⋊D149—21no—
45C2×C22⋊D149—21no—
46C2×C22⋊D149—22no—
47D4×D149—42no—
48C22×Dic149—21no—
49C2×C4×D149C2311yescharacteristic
50Dic596—42no—
51#14558725—42no—
52#14558724—42no—
53#14558724—42no—
54C2×#11924746C2312yes—
55C4⋊Dic149—22no—
56C4⋊Dic149—22no—
57#14558715—22no—
58#14558715D412yes—
59C22⋊Dic149—22no—
60#14558713—22no—
61#14558713—22no—
62#14558713D412yes—
63C4×Dic149—22no—
64C4×Dic149C2311yescharacteristic
65C2×D4×C149—22no—
66C2×D4×C149C2311yescharacteristic
67C2×D4×C149C2311yescharacteristic
68C2×C8×C149C2312yes—
69C4⋊C4×C149—22no—
70C4⋊C4×C149C2312yes—
71C42×C149C2311yescharacteristic
72C22×D149—21no—
73C22×D149—42no—
74C22×D149—82no—
75C22×D149C2×D411yescharacteristic
76Dic298—22no—
77Dic298—22no—
78Dic298—22no—
79D596—22no—
80C22⋊D149—42no—
81C22⋊D149—42no—
82C22⋊D149—82no—
83C2×Dic149—21no—
84C2×Dic149—22no—
85C2×Dic149—42no—
86C2×Dic149C2×D411yescharacteristic
87C2×Dic149C2×D412yes—
88C4×D149—22no—
89#11924746—22no—
90C23×C149—21no—
91C23×C149—21no—
92D4×C149—22no—
93D4×C149—22no—
94D4×C149—82no—
95C2×C4×C149—22no—
96C2×C4×C149C2×D412yes—
97C2×C4×C149C2411yescharacteristic
98C8×C149—22no—
99C22×C149—41no—
100C22×C149—41no—
101C22×C149—42no—
102C22×C149—42no—
103C22×C149C22×D411yescharacteristic
104C2×D149—21no—
105C2×D149—22no—
106C2×D149—41no—
107C2×D149—42no—
108C2×D149—84no—
109C4×C149—22no—
110C4×C149—42no—
111C4×C149H(5,𝔽2)12yes—
112Dic149—21no—
113Dic149—22no—
114Dic149—41no—
115Dic149—42no—
116C2×C149—81no—
117C2×C149—81no—
118C2×C149#53511yescharacteristic
119C2×C149#57712yes—
120D149—41no—
121D149—81no—
122C149#322211yescharacteristic, complemented, Sylow
123#3222—1491nomaximal, Sylow
124#541—1491no—
125#535—1491no—
126#497—1491no—
127#492—1491no—
128#490D14911yescharacteristic, complemented
129#488—1491no—
130#464—1492no—
131#460—1492no—
132#452—1492no—
133#450—1492no—
134#433—1491no—
135C2×#41—1491no—
136C22×D4—1491no—
137C2×Q16—1492no—
138C2×SD16—1492no—
139C2×SD16—1492no—
140C2×SD16—1492no—
141C4⋊Q8—1491no—
142#127—1491no—
143#127C2×D14911yescharacteristic
144#126—2982no—
145#125—2982no—
146#125C2×D14911yescharacteristic, complemented
147#124—1491no—
148#124—1491no—
149C22⋊Q8—1492no—
150#121—2982no—
151#120—2982no—
152C4×Q8—1492no—
153#117—1491no—
154C4⋊C8—1492no—
155#103—1492no—
156#102—1492no—
157#102C2×D14912yes—
158#102C2×D14912yescomplemented
159C22⋊C8—1492no—
160#97C2×D14911yescharacteristic, complemented
161C4×C8—1491no—
162C24—2981no—
163#41—5962no—
164C2×Q8—1491no—
165C2×Q8—1492no—
166C2×D4—22no—
167C2×D4—2981no—
168C2×D4—2981no—
169C2×D4—2982no—
170C2×D4—5962no—
171C2×D4—1491no—
172C2×D4C22×D14911yescharacteristic
173C2×D4C22×D14911yescharacteristic
174C22×C4—2981no—
175C22×C4—1491no—
176Q16—5962no—
177SD16—5962no—
178SD16—5962no—
179SD16—5962no—
180C2×C8—1492no—
181C2×C8C22×D14912yes—
182C4⋊C4—22no—
183C4⋊C4—2982no—
184C4⋊C4—2982no—
185C4⋊C4—2982no—
186C4⋊C4—1492no—
187C4⋊C4C22×D14912yes—
188C22⋊C4—2982no—
189C22⋊C4—2982no—
190C22⋊C4—2982no—
191C22⋊C4—1492no—
192C42—2982no—
193C42—1491no—
194C42C22×D14911yescharacteristic
195C23—21no—
196C23—21no—
197C23—2981no—
198C23—5962no—
199C23—1491no—
200C23—11922no—
201Q8—2982no—
202Q8—2982no—
203Q8—2982no—
204D4—22no—
205D4—22no—
206D4—82no—
207D4—2982no—
208D4—5962no—
209D4—5962no—
210D4—11922no—
211C2×C4—22no—
212C2×C4—2982no—
213C2×C4—2982no—
214C2×C4—2981no—
215C2×C4—5962no—
216C2×C4—1492no—
217C2×C4—1491no—
218C2×C4D4×D14912yes—
219C2×C4C23×D14911yescharacteristic
220C8—22no—
221C8—2982no—
222C22—42no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22—2982no—
227C22—2981no—
228C22—5962no—
229C22—5961no—
230C22—11924no—
231C22C2×D4×D14911yescharacteristic
232C4—22no—
233C4—42no—
234C4—2982no—
235C4—2981no—
236C4—5961no—
237C4—5962no—
238C4#1801195712yes—
239C2—81no—
240C2—81no—
241C2—5961no—
242C2—11921no—
243C2#2375019311yescharacteristic
244C2#2375043512yes—
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#490×C149C2
socleC22×C149C22×D4
prime radical#23750188C2
hypercenter#490D149
hypocenterC149#3222
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #23750238C2semidirect
3. #23750019C2semidirect
4. #23749997C2semidirect
5. #23749982C2semidirect
6. #23749846C2semidirect
7. #23749824C2semidirect
8. #23749788C2semidirect
9. #23749777C2semidirect
10. #23749687C2semidirect
11. #490×C149C2semidirect
14. C2×Dic596C22semidirect
17. C2×#14558724C22semidirect
18. #18011911C22semidirect
27. #18011876C22semidirect
29. #18011840C22semidirect
30. #18011834C22semidirect
33. C4×#11924746C22semidirect
35. #125×C149C22semidirect
37. #102×C149C22semidirect
38. #97×C149C22semidirect
122. C149#3222semidirect
128. #490D149semidirect
146. #125C2×D149semidirect
158. #102C2×D149semidirect
160. #97C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237502382
#237501882
#237500192
#237499972
#237499822
#237498462
#237498242
#237497882
#237497772
#237496872
#490×C1492
#3222149