Finite Group Databasebeta

← #34054051

Subgroups — #34054051

Order 19072.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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, complemented
2#23750425C212yesmaximal, complemented
3#23750192C211yesmaximal, characteristic, complemented
4#23749805C214yesmaximal, complemented
5#23749795C214yesmaximal, complemented
6#23749641C212yesmaximal, complemented
7C2×#100×C149C211yesmaximal, characteristic, complemented
8#18011956—22no—
9#18011956—24no—
10#18011956C2214yes—
11C2×#14558736C2212yes—
12C2×#14558736C2211yescharacteristic, complemented
13C2×#14558735—22no—
14C22×C22⋊D149—22no—
15C2×D4×D149—22no—
16#18011944—22no—
17#18011944C2214yescomplemented
18#18011939—22no—
19#18011939C2214yescomplemented
20#18011933—24no—
21#18011907C2212yescomplemented
22#18011898C2212yescomplemented
23#18011882—22no—
24#18011880—22no—
25#18011870C2211yescharacteristic, complemented
26#18011837C2218yescomplemented
27C22×D4×C149C2211yescharacteristic
28C2×M16×C149C2212yescomplemented
29#100×C149C2214yescomplemented
30C23×D149—22no—
31#14558736—22no—
32#14558736—22no—
33#14558736—22no—
34#14558736—22no—
35#14558736D414yescomplemented
36#14558736C2314yes—
37#14558735—28no—
38#14558735—28no—
39C2×Dic298D412yescomplemented
40C2×Dic298C2311yescharacteristic, complemented
41C2×D596D412yescomplemented
42C2×D596C2311yescharacteristic, complemented
43C2×C22⋊D149—24no—
44C2×C22⋊D149—28no—
45C2×C22⋊D149—22no—
46C2×C22⋊D149—28no—
47C2×C22⋊D149—21no—
48C2×C22⋊D149—24no—
49D4×D149—28no—
50D4×D149—28no—
51C22×Dic149—22no—
52C2×C4×D149—22no—
53C2×C4×D149—21no—
54Dic596—24no—
55#14558725—24no—
56#14558725—24no—
57D1192—24no—
58C4⋊Dic149—21no—
59C22⋊Dic149—21no—
60C22⋊Dic149—24no—
61C4×Dic149C2312yescomplemented
62C24×C149D412yes—
63C2×D4×C149—22no—
64C2×D4×C149—22no—
65C2×D4×C149D412yes—
66C2×D4×C149C2312yes—
67C22×C4×C149C2311yescharacteristic
68M16×C149—22no—
69M16×C149C2314yescomplemented
70C2×C8×C149—22no—
71C22×D149—22no—
72C22×D149—28no—
73C22×D149—21no—
74C22×D149—84no—
75Dic298—22no—
76Dic298—22no—
77Dic298C2×D414yes—
78D596—22no—
79D596—22no—
80D596C2×D414yes—
81C22⋊D149—216no—
82C22⋊D149—24no—
83C22⋊D149—24no—
84C22⋊D149—41no—
85C22⋊D149—41no—
86C22⋊D149—84no—
87C2×Dic149—28no—
88C2×Dic149—22no—
89C2×Dic149—21no—
90C2×Dic149—22no—
91C4×D149—24no—
92C4×D149—24no—
93C4×D149—42no—
94C23×C149—22no—
95C23×C149—22no—
96C23×C149—82no—
97C23×C149C2×D414yes—
98C23×C149C2×D411yescharacteristic
99D4×C149—24no—
100D4×C149—24no—
101C2×C4×C149C2×D412yes—
102C2×C4×C149C2×D412yes—
103C2×C4×C149C2×D411yescharacteristic
104C2×C4×C149C2411yescharacteristic
105C8×C149—24no—
106C22×C149—24no—
107C22×C149—41no—
108C22×C149—44no—
109C22×C149—44no—
110C22×C149—82no—
111C22×C149#12011yescharacteristic
112C22×C149#12011yescharacteristic
113C22×C149C22×D411yescharacteristic
114C2×D149—24no—
115C2×D149—44no—
116C2×D149—42no—
117C2×D149—41no—
118C2×D149—88no—
119C4×C149#12012yes—
120C4×C149C22×D412yes—
121Dic149—24no—
122Dic149—42no—
123Dic149—41no—
124C2×C149—21no—
125C2×C149—21no—
126C2×C149—21no—
127C2×C149—44no—
128C2×C149C2×#12011yescharacteristic
129D149—44no—
130D149—81no—
131C149#301911yescharacteristic, complemented, Sylow
132#3019—1491nomaximal, Sylow
133C2×H(5,𝔽2)—1491no—
134#575—1492no—
135#535—1491no—
136#455—1494no—
137#453—1494no—
138#421—1492no—
139C2×#100D14911yescharacteristic, complemented
140H(5,𝔽2)—2984no—
141H(5,𝔽2)—2982no—
142H(5,𝔽2)—1494no—
143C2×#41—2982no—
144C2×#41—1491no—
145C2×#41—1492no—
146C22×D4—2982no—
147C22×D4—2982no—
148C22×D4C2×D14911yescharacteristic
149#137—2982no—
150#137—1494no—
151AGL(1,ℤ/8)—2982no—
152AGL(1,ℤ/8)—1494no—
153#135—2984no—
154C2×M16C2×D14912yescomplemented
155#127—1492no—
156#124—1492no—
157#121—2982no—
158#120—2982no—
159#117—1491no—
160#104—1498no—
161#100C2×D14914yescomplemented
162C24—2982no—
163C24D59612yes—
164#41—2988no—
165#41—2988no—
166#41—2982no—
167#41—2982no—
168#41—2982no—
169#41—2982no—
170#41—1494no—
171#41—1494no—
172C2×Q8—1492no—
173C2×Q8—1491no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—2988no—
177C2×D4—2988no—
178C2×D4—2984no—
179C2×D4—2988no—
180C2×D4—2982no—
181C2×D4—2988no—
182C2×D4—2981no—
183C2×D4—2984no—
184C2×D4—1492no—
185C2×D4—1491no—
186C2×D4D59612yes—
187C2×D4C22×D14912yes—
188C22×C4—2982no—
189C22×C4—2982no—
190C22×C4—2981no—
191C22×C4C22×D14911yescharacteristic
192Q16—2984no—
193SD16—2984no—
194SD16—2984no—
195D8—2984no—
196M16—22no—
197M16C22×D14914yescomplemented
198C2×C8—22no—
199C4⋊C4—2981no—
200C22⋊C4—2981no—
201C22⋊C4—2984no—
202C42—1492no—
203C23—22no—
204C23—22no—
205C23—82no—
206C23—2988no—
207C23—2982no—
208C23—2981no—
209C23—11924no—
210C23C2×D59614yes—
211C23C2×D59611yescharacteristic
212Q8—2982no—
213Q8—2982no—
214Q8—1494no—
215D4—24no—
216D4—24no—
217D4—29816no—
218D4—2984no—
219D4—2984no—
220D4—2982no—
221D4—2982no—
222D4—5961no—
223D4—5961no—
224D4—1494no—
225D4—11924no—
226C2×C4—2988no—
227C2×C4—2982no—
228C2×C4—2984no—
229C2×C4—2984no—
230C2×C4—2982no—
231C2×C4—2981no—
232C2×C4—5962no—
233C2×C4D4×D14912yes—
234C2×C4D4×D14912yes—
235C2×C4C2×D59611yescharacteristic
236C2×C4C23×D14911yescharacteristic
237C8—24no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—82no—
243C22—2984no—
244C22—5964no—
245C22—5962no—
246C22—5961no—
247C22—11928no—
248C22#1801188111yescharacteristic
249C22#1801188111yescharacteristic
250C22C22×D59611yescharacteristic
251C4—2984no—
252C4—5962no—
253C4—5961no—
254C4#1801188112yes—
255C4C2×D4×D14912yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—5964no—
261C2—11921no—
262C2C2×#1801188111yescharacteristic
subgroupG/H
centerC2C2×#18011881
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC2×#100×C149C2
socleC2×C149C2×#120
hypercenterC2×#100D149
hypocenterC149#3019
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#18011956C2semidirect
2. #23750425C2semidirect
3. #23750192C2semidirect
4. #23749805C2semidirect
5. #23749795C2semidirect
6. #23749641C2semidirect
7. C2×#100×C149C2semidirect
12. C2×#14558736C22semidirect
17. #18011944C22semidirect
19. #18011939C22semidirect
21. #18011907C22semidirect
22. #18011898C22semidirect
25. #18011870C22semidirect
26. #18011837C22semidirect
28. C2×M16×C149C22semidirect
29. #100×C149C22semidirect
35. #14558736D4semidirect
39. C2×Dic298D4semidirect
40. C2×Dic298C23semidirect
41. C2×D596D4semidirect
42. C2×D596C23semidirect
61. C4×Dic149C23semidirect
69. M16×C149C23semidirect
131. C149#3019semidirect
139. C2×#100D149semidirect
154. C2×M16C2×D149semidirect
161. #100C2×D149semidirect
197. M16C22×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119562
#237504252
#237501922
#237498052
#237497952
#237496412
C2×#100×C1492
#3019149