Finite Group Databasebeta

← #508×D149 #34057805

Subgroups — #508×D149 #34057805

Order 19072.

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

normal subgroups only

This group has 368 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#135×D149C212yesmaximal, complemented
2#23750058C211yesmaximal, characteristic, complemented
3#23750057C211yesmaximal, characteristic, complemented
4#23750056C212yesmaximal, complemented
5#23750055C212yesmaximal, complemented
6#23750054C211yesmaximal, characteristic, complemented
7Q32×D149C211yesmaximal, characteristic, complemented
8SD32×D149C212yesmaximal, complemented
9D16×D149C211yesmaximal, characteristic, complemented
10C2×C16×D149C211yesmaximal, characteristic, complemented
11#508×C149C211yesmaximal, characteristic, complemented
12#41×D149—22no—
13#18011934C2212yescomplemented
14#18011933C2212yescomplemented
15#18011932—22no—
16#18011931—22no—
17#18011930C2212yescomplemented
18Q16×D149C2212yes—
19SD16×D149—22no—
20D8×D149C2212yes—
21C2×C8×D149C2211yescharacteristic
22Dic1192C2211yescharacteristic, complemented
23#18011857C2212yescomplemented
24#18011856C2212yescomplemented
25#18011855C2212yescomplemented
26#18011854C2212yescomplemented
27D2384C2211yescharacteristic, complemented
28#18011852C2212yescomplemented
29C2×#14558711C2211yescharacteristic, complemented
30C16×D149C2211yescharacteristic
31C16×D149C2211yescharacteristic
32C16×D149C2211yescharacteristic, complemented
33C16×D149C2211yescharacteristic, complemented
34#135×C149C2212yescomplemented
35Q32×C149C2211yescharacteristic, complemented
36SD32×C149C2212yescomplemented
37D16×C149C2211yescharacteristic, complemented
38C2×C16×C149C2211yescharacteristic, complemented
39#14558737—22no—
40#14558736—22no—
41#14558736—42no—
42#14558735—22no—
43#14558735—42no—
44Q8×D149—22no—
45D4×D149—22no—
46D4×D149—42no—
47C2×C4×D149—42no—
48C2×C4×D149D411yescharacteristic
49Dic596C2312yes—
50#14558726—22no—
51#14558725—22no—
52#14558724—22no—
53#14558723—22no—
54D1192C2312yes—
55#14558721—22no—
56C2×#11924746D411yescharacteristic
57C8×D149D411yescharacteristic
58C8×D149D411yescharacteristic
59C8×D149C2311yescharacteristic
60C8×D149C2311yescharacteristic
61#14558711C2311yescharacteristic
62#14558711C2311yescharacteristic
63#41×C149—22no—
64Q16×C149C2312yes—
65SD16×C149—22no—
66D8×C149C2312yes—
67C2×C8×C149C2311yescharacteristic
68C16×C149C2311yescharacteristic
69C16×C149C2311yescharacteristic
70C22×D149—42no—
71C22×D149D811yescharacteristic
72Dic298—22no—
73Dic298—42no—
74D596—22no—
75D596—42no—
76C22⋊D149—42no—
77C22⋊D149—42no—
78C22⋊D149—42no—
79C2×Dic149—42no—
80C2×Dic149D811yescharacteristic
81C4×D149—42no—
82C4×D149—42no—
83C4×D149—42no—
84C4×D149D811yescharacteristic
85C4×D149D811yescharacteristic
86C4×D149C2×D411yescharacteristic
87C4×D149C2×D411yescharacteristic
88#11924746C2×D411yescharacteristic
89#11924746C2×D411yescharacteristic
90Q8×C149—22no—
91D4×C149—22no—
92D4×C149—42no—
93C2×C4×C149—42no—
94C2×C4×C149C2×D411yescharacteristic
95C8×C149C2×D411yescharacteristic
96C8×C149C2411yescharacteristic
97C22×C149—42no—
98C22×C149C2×D811yescharacteristic
99C2×D149—22no—
100C2×D149—42no—
101C2×D149—84no—
102C2×D149C2×D811yescharacteristic
103C2×D149C2×D811yescharacteristic
104C4×C149—42no—
105C4×C149C2×D811yescharacteristic
106C4×C149C22×D411yescharacteristic
107Dic149—42no—
108Dic149C2×D811yescharacteristic
109Dic149C2×D811yescharacteristic
110C2×C149—21no—
111C2×C149—82no—
112C2×C149C22×D811yescharacteristic
113D149—21no—
114D149—82no—
115D149#50812yesdirect product factor
116C149C2×#50811yescharacteristic, complemented, Sylow
117C2×#508—1491nomaximal, Sylow
118C2×#135—1492no—
119#508—1492no—
120#508—1491no—
121#508—1492no—
122#508—1491no—
123#508—1491no—
124#508D14911yescharacteristic, direct product factor
125C2×Q32—1491no—
126C2×SD32—1492no—
127C2×D16—1491no—
128C22×C16—1491no—
129C2×#41—2982no—
130#135—2982no—
131#135—2982no—
132#135—1492no—
133#135—1492no—
134#135—1492no—
135#135C2×D14912yescomplemented
136C2×Q16—1492no—
137C2×SD16—2982no—
138C2×D8—1492no—
139C22×C8—1491no—
140Q32—1492no—
141Q32—1491no—
142Q32C2×D14911yescharacteristic, complemented
143SD32—1492no—
144SD32—1492no—
145SD32—1492no—
146SD32C2×D14912yescomplemented
147D16—1492no—
148D16—1491no—
149D16C2×D14911yescharacteristic, complemented
150C2×C16—1491no—
151C2×C16—1491no—
152C2×C16—1491no—
153C2×C16—1491no—
154C2×C16—1491no—
155C2×C16C2×D14911yescharacteristic, complemented
156#41—22no—
157#41—2982no—
158#41—2982no—
159#41—2982no—
160#41—5962no—
161#41—5962no—
162C2×Q8—2982no—
163C2×D4—2982no—
164C2×D4—5962no—
165C22×C4—5962no—
166C22×C4—1491no—
167Q16—2982no—
168Q16—1492no—
169Q16C22×D14912yes—
170SD16—22no—
171SD16—2982no—
172SD16—2982no—
173SD16—2982no—
174D8—2982no—
175D8—1492no—
176D8C22×D14912yes—
177C2×C8—1491no—
178C2×C8—1491no—
179C2×C8—1491no—
180C2×C8—1491no—
181C2×C8—1491no—
182C2×C8C22×D14911yescharacteristic
183C16—1491no—
184C16—1491no—
185C16C22×D14911yescharacteristic
186C16C22×D14911yescharacteristic
187C23—5962no—
188C23—1491no—
189Q8—22no—
190Q8—2982no—
191Q8—5962no—
192D4—22no—
193D4—42no—
194D4—2982no—
195D4—5962no—
196D4—5962no—
197D4—5962no—
198D4—5962no—
199C2×C4—42no—
200C2×C4—5962no—
201C2×C4—5962no—
202C2×C4—5962no—
203C2×C4—5962no—
204C2×C4—1491no—
205C2×C4—1491no—
206C2×C4—1491no—
207C2×C4—1491no—
208C2×C4—1491no—
209C2×C4D4×D14911yescharacteristic
210C8—1491no—
211C8—1491no—
212C8D4×D14911yescharacteristic
213C8C23×D14911yescharacteristic
214C22—42no—
215C22—2982no—
216C22—5962no—
217C22—1491no—
218C22—1491no—
219C22—11924no—
220C22D8×D14911yescharacteristic
221C4—42no—
222C4—5962no—
223C4—1491no—
224C4—1491no—
225C4D8×D14911yescharacteristic
226C4C2×D4×D14911yescharacteristic
227C2—21no—
228C2—82no—
229C2—2981no—
230C2—1492no—
231C2—11922no—
232C2C2×D8×D14911yescharacteristic
subgroupG/H
centerC4D8×D149
commutator, cosocleC8×C149C24
FrattiniC8C23×D149
Fitting#508×C149C2
socleC2×C149C22×D8
hypercenter#508D149
hypocenterC149C2×#508
LCSG ▷ C8×C149 ▷ C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D149C2semidirect
2. #23750058C2semidirect
3. #23750057C2semidirect
4. #23750056C2semidirect
5. #23750055C2semidirect
6. #23750054C2semidirect
7. Q32×D149C2semidirect
8. SD32×D149C2semidirect
9. D16×D149C2semidirect
10. C2×C16×D149C2semidirect
11. #508×C149C2semidirect
13. #18011934C22semidirect
14. #18011933C22semidirect
17. #18011930C22semidirect
22. Dic1192C22semidirect
23. #18011857C22semidirect
24. #18011856C22semidirect
25. #18011855C22semidirect
26. #18011854C22semidirect
27. D2384C22semidirect
28. #18011852C22semidirect
29. C2×#14558711C22semidirect
32. C16×D149C22semidirect
33. C16×D149C22semidirect
34. #135×C149C22semidirect
35. Q32×C149C22semidirect
36. SD32×C149C22semidirect
37. D16×C149C22semidirect
38. C2×C16×C149C22semidirect
115. D149#508direct (N × H)
116. C149C2×#508semidirect
124. #508D149direct (N × H)
135. #135C2×D149semidirect
142. Q32C2×D149semidirect
146. SD32C2×D149semidirect
149. D16C2×D149semidirect
155. C2×C16C2×D149semidirect

Maximal subgroups

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

subgroupindex
#135×D1492
#237500582
#237500572
#237500562
#237500552
#237500542
Q32×D1492
SD32×D1492
D16×D1492
C2×C16×D1492
#508×C1492
C2×#508149