Subgroups — #405×D₈₃ #24915873

Order 10624.

592 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₈₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×D₈₃	C₂	1	1	yes	maximal, characteristic
3	#117×D₈₃	C₂	1	1	yes	maximal, characteristic
4	#19127659	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127658	C₂	1	2	yes	maximal, complemented
6	#19127657	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127656	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127655	C₂	1	2	yes	maximal, complemented
9	#97×D₈₃	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₈₃	C₂	1	2	yes	maximal, complemented
11	#405×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₈₃	C₂²	1	1	yes	characteristic
13	C₂×#13050574	C₂²	1	1	yes	characteristic
14	C₂×#13050573	C₂²	1	2	yes	—
15	M₁₆×D₈₃	C₄	1	8	yes	complemented
16	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₈₃	C₄	1	8	yes	complemented
18	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₃	C₂²	1	2	yes	—
20	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
23	#14581751	C₂²	1	1	yes	characteristic
24	#14581750	C₄	1	2	yes	—
25	#14581749	C₂²	1	2	yes	—
26	#14581748	C₄	1	2	yes	—
27	C₄⋊C₄×D₈₃	C₄	1	2	yes	—
28	C₂²⋊C₄×D₈₃	C₄	1	2	yes	—
29	C₄²×D₈₃	C₂²	1	2	yes	—
30	#14581703	C₂²	1	2	yes	complemented
31	#14581702	C₂²	1	2	yes	complemented
32	#14581702	C₂²	1	2	yes	complemented
33	C₄×C₈₃⋊C₈	C₂²	1	2	yes	complemented
34	C₈×Dic₈₃	C₂²	1	2	yes	complemented
35	C₈×Dic₈₃	C₂²	1	2	yes	complemented
36	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
38	#117×C₈₃	C₂²	1	1	yes	characteristic
39	#97×C₈₃	C₂²	1	2	yes	complemented
40	C₄×C₈×C₈₃	C₂²	1	2	yes	complemented
41	C₂³×D₈₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₈₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₈₃	C₂×C₄	1	4	yes	—
44	C₂×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₈₃	C₂×C₄	1	4	yes	—
47	C₂×C₄×D₈₃	C₂×C₄	1	4	yes	—
48	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
50	#13050574	C₂×C₄	1	4	yes	complemented
51	#13050573	C₂×C₄	1	8	yes	complemented
52	C₂×C₈₃⋊C₈	C₂×C₄	1	4	yes	complemented
53	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
54	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
55	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
56	C₈×D₈₃	C₂×C₄	1	4	yes	—
57	C₈×D₈₃	C₂×C₄	1	4	yes	—
58	C₈×D₈₃	C₂×C₄	1	8	yes	—
59	C₈×D₈₃	C₂×C₄	1	4	yes	complemented
60	C₈×D₈₃	C₂×C₄	1	4	yes	complemented
61	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
62	Dic₈₃⋊C₄	C₂×C₄	1	4	yes	—
63	C₂²⋊Dic₈₃	C₂×C₄	1	2	yes	—
64	(C₂×D₈₃)⋊C₄	C₂×C₄	1	4	yes	—
65	C₄×Dic₈₃	C₂×C₄	1	4	yes	—
66	C₄×Dic₈₃	C₂³	1	2	yes	—
67	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
68	M₁₆×C₈₃	C₂×C₄	1	4	yes	complemented
69	C₂×C₈×C₈₃	C₂×C₄	1	4	yes	complemented
70	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₈₃	C₂³	1	2	yes	—
72	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
74	C₂²⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
75	C₄²×C₈₃	C₂³	1	2	yes	—
76	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₈₃	—	2	4	no	—
78	C₂²×D₈₃	C₄²	1	4	yes	—
79	C₂²×D₈₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×D₈₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₈₃	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
86	C₂×Dic₈₃	C₄²	1	4	yes	—
87	C₂×Dic₈₃	C₂²×C₄	1	4	yes	—
88	C₂×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
90	C₄×D₈₃	—	2	8	no	—
91	C₄×D₈₃	C₄²	1	4	yes	—
92	C₄×D₈₃	C₄²	1	4	yes	—
93	C₄×D₈₃	C₂²×C₄	1	2	yes	—
94	C₄×D₈₃	C₂²×C₄	1	2	yes	—
95	C₄×D₈₃	C₂²×C₄	1	2	yes	—
96	C₄×D₈₃	C₂²×C₄	1	2	yes	—
97	C₈×C₈₃	C₂²×C₄	1	2	yes	—
98	C₈×C₈₃	C₂²×C₄	1	2	yes	—
99	C₈×C₈₃	C₂²×C₄	1	4	yes	—
100	C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
101	C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
102	C₈₃⋊C₈	C₂²×C₄	1	4	yes	—
103	C₂²×C₈₃	—	2	2	no	—
104	C₂²×C₈₃	C₂×C₄²	1	2	yes	—
105	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×D₈₃	—	2	2	no	—
107	C₂×D₈₃	—	2	8	no	—
108	C₂×D₈₃	C₂×C₄²	1	2	yes	—
109	C₂×D₈₃	C₂×C₄²	1	2	yes	—
110	C₂×D₈₃	#131	1	4	yes	—
111	C₄×C₈₃	—	2	4	no	—
112	C₄×C₈₃	C₂×C₄²	1	2	yes	—
113	C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
114	C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
115	Dic₈₃	—	2	4	no	—
116	Dic₈₃	C₂×C₄²	1	2	yes	—
117	Dic₈₃	C₂×C₄²	1	2	yes	—
118	C₂×C₈₃	—	2	2	no	—
119	C₂×C₈₃	C₂²×C₄²	1	1	yes	characteristic
120	C₂×C₈₃	C₂×#131	1	2	yes	—
121	D₈₃	—	2	2	no	—
122	D₈₃	#405	1	4	yes	direct product factor
123	C₂×#405	—	83	1	no	maximal, Sylow
124	C₈₃	C₂×#405	1	1	yes	characteristic, complemented, Sylow
125	C₂²×M₁₆	—	83	1	no	—
126	C₂³×C₈	—	83	1	no	—
127	C₂×#117	—	83	1	no	—
128	#405	—	83	1	no	—
129	#405	—	83	1	no	—
130	#405	—	83	1	no	—
131	#405	—	83	2	no	—
132	#405	—	83	2	no	—
133	#405	D₈₃	1	1	yes	characteristic, direct product factor
134	C₂×#97	—	83	2	no	—
135	C₂×C₄×C₈	—	83	2	no	—
136	C₂³×C₄	—	83	1	no	—
137	C₂×M₁₆	—	83	1	no	—
138	C₂×M₁₆	—	83	8	no	—
139	C₂×M₁₆	—	83	2	no	—
140	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
141	C₂²×C₈	—	83	1	no	—
142	C₂²×C₈	—	83	8	no	—
143	C₂²×C₈	—	83	1	no	—
144	C₂²×C₈	—	83	1	no	—
145	C₂²×C₈	—	83	2	no	—
146	C₂²×C₈	—	83	1	no	—
147	C₂²×C₈	—	83	1	no	—
148	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
149	#117	—	83	1	no	—
150	#117	—	83	2	no	—
151	#117	—	83	2	no	—
152	#117	—	83	2	no	—
153	#117	C₂×D₈₃	1	1	yes	characteristic
154	C₂×C₄⋊C₄	—	83	2	no	—
155	C₂×C₂²⋊C₄	—	83	2	no	—
156	C₂×C₄²	—	83	2	no	—
157	#97	—	83	2	no	—
158	#97	—	83	2	no	—
159	#97	—	83	2	no	—
160	#97	C₂×D₈₃	1	2	yes	complemented
161	C₄×C₈	—	83	2	no	—
162	C₄×C₈	—	83	2	no	—
163	C₄×C₈	—	83	2	no	—
164	C₄×C₈	C₂×D₈₃	1	2	yes	complemented
165	C₂⁴	—	83	1	no	—
166	C₂²×C₄	—	83	1	no	—
167	C₂²×C₄	—	83	1	no	—
168	C₂²×C₄	—	83	1	no	—
169	C₂²×C₄	—	83	4	no	—
170	C₂²×C₄	—	83	4	no	—
171	C₂²×C₄	—	83	1	no	—
172	C₂²×C₄	—	83	1	no	—
173	C₂²×C₄	—	83	4	no	—
174	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
175	M₁₆	—	83	4	no	—
176	M₁₆	—	83	8	no	—
177	M₁₆	C₄×D₈₃	1	4	yes	complemented
178	C₂×C₈	—	83	4	no	—
179	C₂×C₈	—	83	1	no	—
180	C₂×C₈	—	83	2	no	—
181	C₂×C₈	—	83	1	no	—
182	C₂×C₈	—	83	4	no	—
183	C₂×C₈	—	83	4	no	—
184	C₂×C₈	—	83	4	no	—
185	C₂×C₈	—	83	4	no	—
186	C₂×C₈	—	83	8	no	—
187	C₂×C₈	C₄×D₈₃	1	4	yes	complemented
188	C₂×C₈	C₂²×D₈₃	1	2	yes	—
189	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
190	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
191	C₄⋊C₄	—	83	4	no	—
192	C₄⋊C₄	—	83	2	no	—
193	C₄⋊C₄	C₄×D₈₃	1	2	yes	—
194	C₂²⋊C₄	—	83	2	no	—
195	C₂²⋊C₄	—	83	4	no	—
196	C₂²⋊C₄	C₄×D₈₃	1	2	yes	—
197	C₄²	—	83	4	no	—
198	C₄²	—	83	2	no	—
199	C₄²	C₂²×D₈₃	1	2	yes	—
200	C₂³	—	83	1	no	—
201	C₂³	—	83	1	no	—
202	C₂³	—	83	4	no	—
203	C₂³	—	166	4	no	—
204	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
205	C₂×C₄	—	83	1	no	—
206	C₂×C₄	—	83	1	no	—
207	C₂×C₄	—	83	4	no	—
208	C₂×C₄	—	83	2	no	—
209	C₂×C₄	—	83	2	no	—
210	C₂×C₄	—	83	2	no	—
211	C₂×C₄	—	83	2	no	—
212	C₂×C₄	—	83	4	no	—
213	C₂×C₄	—	83	4	no	—
214	C₂×C₄	—	83	4	no	—
215	C₂×C₄	—	166	8	no	—
216	C₂×C₄	C₂×C₄×D₈₃	1	4	yes	—
217	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
221	C₈	—	83	2	no	—
222	C₈	—	83	2	no	—
223	C₈	—	83	4	no	—
224	C₈	C₂×C₄×D₈₃	1	2	yes	—
225	C₈	C₂×C₄×D₈₃	1	2	yes	—
226	C₈	C₂×C₄×D₈₃	1	4	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	83	2	no	—
229	C₂²	—	83	4	no	—
230	C₂²	—	83	2	no	—
231	C₂²	—	166	8	no	—
232	C₂²	—	166	2	no	—
233	C₂²	C₄²×D₈₃	1	2	yes	—
234	C₂²	C₂²×C₄×D₈₃	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	83	2	no	—
237	C₄	—	83	2	no	—
238	C₄	—	166	4	no	—
239	C₄	C₄²×D₈₃	1	2	yes	—
240	C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
241	C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	83	4	no	—
244	C₂	—	166	2	no	—
245	C₂	C₂×C₄²×D₈₃	1	1	yes	characteristic
246	C₂	#131×D₈₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₈₃
commutator	C₂×C₈₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#405×C₈₃	C₂
socle	C₂²×C₈₃	C₂³×C₄
prime radical	C₂³×D₈₃	C₂×C₄
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	#405	D₈₃
hypocenter	C₈₃	C₂×#405
LCS	G ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
4. #19127659	C₂	semidirect
5. #19127658	C₂	semidirect
6. #19127657	C₂	semidirect
7. #19127656	C₂	semidirect
8. #19127655	C₂	semidirect
9. #97×D₈₃	C₂	semidirect
10. C₄×C₈×D₈₃	C₂	semidirect
11. #405×C₈₃	C₂	semidirect
15. M₁₆×D₈₃	C₄	semidirect
17. C₂×C₈×D₈₃	C₄	semidirect
30. #14581703	C₂²	semidirect
31. #14581702	C₂²	semidirect
32. #14581702	C₂²	semidirect
33. C₄×C₈₃⋊C₈	C₂²	semidirect
34. C₈×Dic₈₃	C₂²	semidirect
35. C₈×Dic₈₃	C₂²	semidirect
39. #97×C₈₃	C₂²	semidirect
40. C₄×C₈×C₈₃	C₂²	semidirect
50. #13050574	C₂×C₄	semidirect
51. #13050573	C₂×C₄	semidirect
52. C₂×C₈₃⋊C₈	C₂×C₄	semidirect
59. C₈×D₈₃	C₂×C₄	semidirect
60. C₈×D₈₃	C₂×C₄	semidirect
68. M₁₆×C₈₃	C₂×C₄	semidirect
69. C₂×C₈×C₈₃	C₂×C₄	semidirect
122. D₈₃	#405	direct (N × H)
124. C₈₃	C₂×#405	semidirect
133. #405	D₈₃	direct (N × H)
160. #97	C₂×D₈₃	semidirect
164. C₄×C₈	C₂×D₈₃	semidirect
177. M₁₆	C₄×D₈₃	semidirect
187. C₂×C₈	C₄×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₈₃	2
C₂²×C₈×D₈₃	2
#117×D₈₃	2
#19127659	2
#19127658	2
#19127657	2
#19127656	2
#19127655	2
#97×D₈₃	2
C₄×C₈×D₈₃	2
#405×C₈₃	2
C₂×#405	83

Subgroups — #405×D83 #24915873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #405×D₈₃ #24915873