Finite Group Databasebeta

← #34057831

Subgroups — #34057831

Order 19072.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#23750427C211yesmaximal, characteristic, complemented
2#23750426C211yesmaximal, characteristic, complemented
3#23750063C211yesmaximal, characteristic, complemented
4#23750062C212yesmaximal, complemented
5#23750059C212yesmaximal, complemented
6#23750056C212yesmaximal, complemented
7#23750055C211yesmaximal, characteristic, complemented
8C2×#18011852C211yesmaximal, characteristic, complemented
9D16×D149C212yesmaximal, complemented
10#23750043C211yesmaximal, characteristic, complemented
11#509×C149C211yesmaximal, characteristic, complemented
12#18011957—21no—
13#18011956—21no—
14#18011938—22no—
15#18011937—21no—
16#18011936—21no—
17#18011935—22no—
18#18011934C2212yescomplemented
19#18011933C2211yescharacteristic, complemented
20#18011930C2212yescomplemented
21C2×D1192C2211yescharacteristic, complemented
22D8×D149C2212yescomplemented
23D8×D149C2212yescomplemented
24#18011917C2211yescharacteristic
25#18011857C2211yescharacteristic, complemented
26#18011856C2212yescomplemented
27#18011855C2211yescharacteristic, complemented
28#18011854C2211yescharacteristic, complemented
29D2384C2212yescomplemented
30#18011852C2211yescharacteristic, complemented
31#18011852C2212yescomplemented
32#18011852C2212yescomplemented
33#18011850C2212yescomplemented
34C2×#14558711C2211yescharacteristic, complemented
35C16×D149C2212yescomplemented
36#135×C149C2211yescharacteristic, complemented
37C2×D8×C149C2211yescharacteristic, complemented
38SD32×C149C2212yescomplemented
39D16×C149C2212yescomplemented
40M32×C149C2211yescharacteristic, complemented
41#14558737—22no—
42#14558736—21no—
43#14558736—41no—
44#14558736D411yescharacteristic
45#14558735—22no—
46#14558735—41no—
47C2×D596—21no—
48C2×D596—41no—
49C2×C22⋊D149—42no—
50D4×D149—22no—
51D4×D149—22no—
52D4×D149—41no—
53D4×D149—42no—
54Dic596C2311yescharacteristic
55#14558725—21no—
56#14558724—22no—
57#14558723—22no—
58D1192—21no—
59D1192C2311yescharacteristic
60D1192C2312yes—
61#14558721—22no—
62#14558721—22no—
63#14558720D411yescharacteristic
64#14558719D412yes—
65C8×D149C2312yes—
66#14558711C2311yescharacteristic
67#14558711C2311yescharacteristic
68#41×C149—21no—
69C2×D4×C149—21no—
70Q16×C149C2311yescharacteristic
71SD16×C149—21no—
72D8×C149—21no—
73D8×C149C2311yescharacteristic
74D8×C149C2312yes—
75C2×C8×C149C2311yescharacteristic
76C16×C149C2312yes—
77C22×D149—41no—
78C22×D149—42no—
79C22×D149—42no—
80Dic298—21no—
81Dic298D811yescharacteristic
82D596—21no—
83D596—22no—
84D596—42no—
85D596—41no—
86D596D811yescharacteristic
87C22⋊D149—41no—
88C22⋊D149—42no—
89C22⋊D149—42no—
90C22⋊D149—42no—
91C22⋊D149D812yes—
92C2×Dic149—42no—
93C4×D149—41no—
94C4×D149—42no—
95C4×D149C2×D412yes—
96#11924746C2×D412yes—
97C23×C149—41no—
98Q8×C149—21no—
99D4×C149—21no—
100D4×C149—22no—
101D4×C149—41no—
102D4×C149—41no—
103C2×C4×C149—41no—
104C2×C4×C149C2×D411yescharacteristic
105C8×C149C2×D411yescharacteristic
106C8×C149C2411yescharacteristic
107C22×C149—41no—
108C22×C149—42no—
109C22×C149—161no—
110C22×C149C2×D811yescharacteristic
111C2×D149—41no—
112C2×D149—42no—
113C2×D149—162no—
114C2×D149—162no—
115C2×D149—161no—
116C2×D149C2×D812yes—
117C4×C149—41no—
118C4×C149C2×D811yescharacteristic
119C4×C149C22×D411yescharacteristic
120Dic149—41no—
121Dic149C2×D812yes—
122C2×C149—21no—
123C2×C149—81no—
124C2×C149—82no—
125C2×C149C22×D811yescharacteristic
126D149—22no—
127D149—81no—
128D149—82no—
129C149#341511yescharacteristic, complemented, Sylow
130#3415—1491nomaximal, Sylow
131#576—1491no—
132#576—1491no—
133#509—1492no—
134#509—1492no—
135#509—1491no—
136#509D14911yescharacteristic, complemented
137#508—1491no—
138#508—1492no—
139C2×D16—1491no—
140C2×D16—1492no—
141#504—1491no—
142H(5,𝔽2)—2981no—
143H(5,𝔽2)—2981no—
144AGL(1,ℤ/8)—2982no—
145AGL(1,ℤ/8)—2982no—
146AGL(1,ℤ/8)—2981no—
147AGL(1,ℤ/8)—2981no—
148#135—1492no—
149#135—1492no—
150#135—1491no—
151#135C2×D14911yescharacteristic, complemented
152C2×D8—1492no—
153C2×D8—1492no—
154C2×D8—1491no—
155C2×D8C2×D14911yescharacteristic, complemented
156#131—1491no—
157Q32—1491no—
158SD32—1491no—
159SD32—1491no—
160SD32—1492no—
161SD32C2×D14912yescomplemented
162D16—1492no—
163D16—1491no—
164D16—1492no—
165D16—1492no—
166D16C2×D14912yescomplemented
167M32—1492no—
168M32C2×D14911yescharacteristic, complemented
169C2×C16—1491no—
170C2×C16—1492no—
171#41—21no—
172#41—2982no—
173#41—2982no—
174#41—2981no—
175#41—5961no—
176#41—5961no—
177#41—1491no—
178C2×D4—21no—
179C2×D4—2982no—
180C2×D4—2982no—
181C2×D4—2981no—
182C2×D4—5961no—
183C2×D4—5962no—
184C2×D4—5961no—
185C2×D4—5962no—
186Q16—1491no—
187Q16C22×D14911yescharacteristic
188SD16—21no—
189SD16—2982no—
190SD16—2982no—
191SD16—2981no—
192D8—21no—
193D8—2982no—
194D8—2982no—
195D8—2981no—
196D8—1491no—
197D8—1492no—
198D8C22×D14912yes—
199D8C22×D14911yescharacteristic
200M16—1491no—
201M16—1492no—
202C2×C8—1492no—
203C2×C8C22×D14911yescharacteristic
204C16—1491no—
205C16—1491no—
206C16C22×D14912yes—
207C23—41no—
208C23—5962no—
209C23—5962no—
210C23—5961no—
211Q8—21no—
212Q8—2981no—
213Q8—1491no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—2981no—
219D4—2982no—
220D4—5962no—
221D4—5962no—
222D4—5961no—
223D4—5962no—
224D4—5962no—
225D4—5961no—
226D4—1492no—
227D4—1491no—
228C2×C4—41no—
229C2×C4—5962no—
230C2×C4—5962no—
231C2×C4—5961no—
232C2×C4—1492no—
233C2×C4D4×D14911yescharacteristic
234C8—1492no—
235C8D4×D14911yescharacteristic
236C8C23×D14911yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—23841no—
241C22—23842no—
242C22—23842no—
243C22—5962no—
244C22—5961no—
245C22—1492no—
246C22D8×D14911yescharacteristic
247C4—41no—
248C4—5961no—
249C4—1492no—
250C4D8×D14911yescharacteristic
251C4C2×D4×D14911yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—2982no—
256C2—11922no—
257C2—11921no—
258C2C2×D8×D14911yescharacteristic
subgroupG/H
centerC2C2×D8×D149
commutator, cosocleC8×C149C24
FrattiniC8C23×D149
Fitting#509×C149C2
socleC2×C149C22×D8
hypercenter#509D149
hypocenterC149#3415
LCSG ▷ C8×C149 ▷ C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #23750427C2semidirect
2. #23750426C2semidirect
3. #23750063C2semidirect
4. #23750062C2semidirect
5. #23750059C2semidirect
6. #23750056C2semidirect
7. #23750055C2semidirect
8. C2×#18011852C2semidirect
9. D16×D149C2semidirect
10. #23750043C2semidirect
11. #509×C149C2semidirect
18. #18011934C22semidirect
19. #18011933C22semidirect
20. #18011930C22semidirect
21. C2×D1192C22semidirect
22. D8×D149C22semidirect
23. D8×D149C22semidirect
25. #18011857C22semidirect
26. #18011856C22semidirect
27. #18011855C22semidirect
28. #18011854C22semidirect
29. D2384C22semidirect
30. #18011852C22semidirect
31. #18011852C22semidirect
32. #18011852C22semidirect
33. #18011850C22semidirect
34. C2×#14558711C22semidirect
35. C16×D149C22semidirect
36. #135×C149C22semidirect
37. C2×D8×C149C22semidirect
38. SD32×C149C22semidirect
39. D16×C149C22semidirect
40. M32×C149C22semidirect
129. C149#3415semidirect
136. #509D149semidirect
151. #135C2×D149semidirect
155. C2×D8C2×D149semidirect
161. SD32C2×D149semidirect
166. D16C2×D149semidirect
168. M32C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237504272
#237504262
#237500632
#237500622
#237500592
#237500562
#237500552
C2×#180118522
D16×D1492
#237500432
#509×C1492
#3415149