Subgroups — C₂×#539×C₈₃ #24905136

Order 10624.

1144 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#539×C₈₃	C₂	1	16	yes	maximal, direct product factor
2	C₂×#126×C₈₃	C₂	1	2	yes	maximal, complemented
3	C₂×#125×C₈₃	C₂	1	2	yes	maximal, complemented
4	C₂×#123×C₈₃	C₂	1	2	yes	maximal, complemented
5	C₂×#123×C₈₃	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊Q₈×C₈₃	C₂	1	2	yes	maximal, complemented
8	C₂×#121×C₈₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×D₄×C₈₃	C₂	1	2	yes	maximal, complemented
10	C₂²×C₄⋊C₄×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₈₃	C₂²	1	2	yes	—
13	C₂²×D₄×C₈₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₈₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₈₃	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₈₃	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₈₃	C₂²	1	2	yes	—
18	#126×C₈₃	C₂²	1	16	yes	complemented
19	#125×C₈₃	C₂²	1	16	yes	complemented
20	#123×C₈₃	C₂²	1	16	yes	complemented
21	#123×C₈₃	C₂²	1	16	yes	complemented
22	C₂²⋊Q₈×C₈₃	C₂²	1	8	yes	complemented
23	C₂²⋊Q₈×C₈₃	C₂²	1	16	yes	complemented
24	#121×C₈₃	C₂²	1	8	yes	—
25	C₄×D₄×C₈₃	C₂²	1	16	yes	complemented
26	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
29	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
32	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
34	C₂×C₄⋊C₄×C₈₃	C₂²	1	8	yes	complemented
35	C₂×C₂²⋊C₄×C₈₃	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₄×C₈₃	C₂²	1	2	yes	—
37	C₂×C₂²⋊C₄×C₈₃	C₂²	1	2	yes	—
38	C₂×C₂²⋊C₄×C₈₃	C₂²	1	2	yes	—
39	C₂×C₂²⋊C₄×C₈₃	C₂²	1	2	yes	—
40	C₂×C₄²×C₈₃	C₂²	1	2	yes	—
41	C₂⁴×C₈₃	C₂³	1	2	yes	—
42	C₂⁴×C₈₃	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₈₃	—	4	2	no	—
44	C₂×Q₈×C₈₃	C₂³	1	4	yes	—
45	C₂×D₄×C₈₃	—	2	8	no	—
46	C₂×D₄×C₈₃	—	4	2	no	—
47	C₂×D₄×C₈₃	C₂³	1	8	yes	—
48	C₂×D₄×C₈₃	C₂³	1	4	yes	—
49	C₂²×C₄×C₈₃	—	2	4	no	—
50	C₂²×C₄×C₈₃	—	4	2	no	—
51	C₂²×C₄×C₈₃	—	4	4	no	—
52	C₂²×C₄×C₈₃	—	4	2	no	—
53	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
54	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
62	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
63	C₂²×C₄×C₈₃	C₂³	1	8	yes	—
64	C₄⋊C₄×C₈₃	C₂³	1	8	yes	—
65	C₄⋊C₄×C₈₃	C₂³	1	8	yes	—
66	C₄⋊C₄×C₈₃	C₂³	1	8	yes	—
67	C₄⋊C₄×C₈₃	C₂³	1	4	yes	—
68	C₄⋊C₄×C₈₃	C₂³	1	8	yes	—
69	C₄⋊C₄×C₈₃	C₂³	1	8	yes	—
70	C₄⋊C₄×C₈₃	C₂³	1	4	yes	—
71	C₄⋊C₄×C₈₃	C₂³	1	8	yes	—
72	C₂²⋊C₄×C₈₃	C₂³	1	8	yes	—
73	C₂²⋊C₄×C₈₃	C₂³	1	8	yes	—
74	C₂²⋊C₄×C₈₃	C₂³	1	8	yes	—
75	C₂²⋊C₄×C₈₃	C₂³	1	8	yes	—
76	C₂²⋊C₄×C₈₃	C₂³	1	8	yes	—
77	C₄²×C₈₃	C₂³	1	8	yes	—
78	C₂³×C₈₃	—	2	2	no	—
79	C₂³×C₈₃	—	2	2	no	—
80	C₂³×C₈₃	—	2	4	no	—
81	C₂³×C₈₃	—	2	4	no	—
82	C₂³×C₈₃	—	2	4	no	—
83	C₂³×C₈₃	#41	1	4	yes	—
84	C₂³×C₈₃	C₂⁴	1	4	yes	—
85	C₂³×C₈₃	C₂⁴	1	2	yes	—
86	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
87	Q₈×C₈₃	—	4	4	no	—
88	D₄×C₈₃	—	2	16	no	—
89	D₄×C₈₃	—	4	4	no	—
90	C₂×C₄×C₈₃	—	2	8	no	—
91	C₂×C₄×C₈₃	—	2	4	no	—
92	C₂×C₄×C₈₃	—	2	4	no	—
93	C₂×C₄×C₈₃	—	2	4	no	—
94	C₂×C₄×C₈₃	—	2	2	no	—
95	C₂×C₄×C₈₃	—	2	4	no	—
96	C₂×C₄×C₈₃	—	2	2	no	—
97	C₂×C₄×C₈₃	—	2	4	no	—
98	C₂×C₄×C₈₃	—	4	4	no	—
99	C₂×C₄×C₈₃	—	4	8	no	—
100	C₂×C₄×C₈₃	—	4	4	no	—
101	C₂×C₄×C₈₃	C₂⁴	1	4	yes	—
102	C₂×C₄×C₈₃	C₂⁴	1	4	yes	—
103	C₂×C₄×C₈₃	C₂⁴	1	4	yes	—
104	C₂×C₄×C₈₃	C₂⁴	1	4	yes	—
105	C₂×C₄×C₈₃	C₂⁴	1	4	yes	—
106	C₂×C₄×C₈₃	C₂⁴	1	2	yes	—
107	C₂×C₄×C₈₃	C₂⁴	1	2	yes	—
108	C₂²×C₈₃	—	2	8	no	—
109	C₂²×C₈₃	—	2	2	no	—
110	C₂²×C₈₃	—	2	2	no	—
111	C₂²×C₈₃	—	2	4	no	—
112	C₂²×C₈₃	—	2	4	no	—
113	C₂²×C₈₃	—	2	4	no	—
114	C₂²×C₈₃	—	4	8	no	—
115	C₂²×C₈₃	C₂×#41	1	4	yes	—
116	C₂²×C₈₃	C₂×#41	1	2	yes	—
117	C₂²×C₈₃	H(5,𝔽₂)	1	2	yes	—
118	C₂²×C₈₃	2₋¹⁺⁴	1	2	yes	—
119	C₂²×C₈₃	C₂⁵	1	1	yes	characteristic
120	C₄×C₈₃	—	2	4	no	—
121	C₄×C₈₃	—	2	4	no	—
122	C₄×C₈₃	—	2	4	no	—
123	C₄×C₈₃	—	2	2	no	—
124	C₄×C₈₃	—	2	4	no	—
125	C₄×C₈₃	—	2	2	no	—
126	C₄×C₈₃	—	2	4	no	—
127	C₂×C₈₃	—	2	4	no	—
128	C₂×C₈₃	—	4	4	no	—
129	C₂×C₈₃	#539	1	4	yes	direct product factor
130	C₂×C₈₃	C₂²×#41	1	1	yes	characteristic
131	C₂×C₈₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₈₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×#539	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₈₃	C₂×#539	1	1	yes	characteristic, direct product factor, Sylow
135	#539	C₂×C₈₃	1	16	yes	direct product factor
136	C₂×#126	C₂×C₈₃	1	2	yes	complemented
137	C₂×#125	C₂×C₈₃	1	2	yes	complemented
138	C₂×#123	C₂×C₈₃	1	2	yes	complemented
139	C₂×#123	C₂×C₈₃	1	2	yes	complemented
140	C₂×C₂²⋊Q₈	C₂×C₈₃	1	1	yes	characteristic, complemented
141	C₂×C₂²⋊Q₈	C₂×C₈₃	1	2	yes	complemented
142	C₂×#121	C₂×C₈₃	1	1	yes	characteristic
143	C₂×C₄×D₄	C₂×C₈₃	1	2	yes	complemented
144	C₂²×C₄⋊C₄	C₂×C₈₃	1	1	yes	characteristic, complemented
145	C₂²×Q₈	C₂²×C₈₃	1	1	yes	characteristic
146	C₂²×D₄	C₂²×C₈₃	1	2	yes	—
147	C₂²×D₄	C₂²×C₈₃	1	1	yes	characteristic
148	C₂³×C₄	C₂²×C₈₃	1	1	yes	characteristic
149	C₂³×C₄	C₂²×C₈₃	1	1	yes	characteristic
150	C₂³×C₄	C₂²×C₈₃	1	1	yes	characteristic
151	C₂³×C₄	C₂²×C₈₃	1	2	yes	—
152	#126	C₂²×C₈₃	1	16	yes	complemented
153	#125	C₂²×C₈₃	1	16	yes	complemented
154	#123	C₂²×C₈₃	1	16	yes	complemented
155	#123	C₂²×C₈₃	1	16	yes	complemented
156	C₂²⋊Q₈	C₂²×C₈₃	1	8	yes	complemented
157	C₂²⋊Q₈	C₂²×C₈₃	1	16	yes	complemented
158	#121	C₂²×C₈₃	1	8	yes	—
159	C₄×D₄	C₂²×C₈₃	1	16	yes	complemented
160	C₂×C₄⋊C₄	C₂²×C₈₃	1	2	yes	—
161	C₂×C₄⋊C₄	C₂²×C₈₃	1	2	yes	—
162	C₂×C₄⋊C₄	C₂²×C₈₃	1	2	yes	—
163	C₂×C₄⋊C₄	C₂²×C₈₃	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂²×C₈₃	1	2	yes	—
165	C₂×C₄⋊C₄	C₂²×C₈₃	1	2	yes	—
166	C₂×C₄⋊C₄	C₂²×C₈₃	1	1	yes	characteristic
167	C₂×C₄⋊C₄	C₂²×C₈₃	1	2	yes	—
168	C₂×C₄⋊C₄	C₂²×C₈₃	1	8	yes	complemented
169	C₂×C₂²⋊C₄	C₂²×C₈₃	1	2	yes	—
170	C₂×C₂²⋊C₄	C₂²×C₈₃	1	2	yes	—
171	C₂×C₂²⋊C₄	C₂²×C₈₃	1	2	yes	—
172	C₂×C₂²⋊C₄	C₂²×C₈₃	1	2	yes	—
173	C₂×C₂²⋊C₄	C₂²×C₈₃	1	2	yes	—
174	C₂×C₄²	C₂²×C₈₃	1	2	yes	—
175	C₂⁴	C₂³×C₈₃	1	2	yes	—
176	C₂⁴	C₂³×C₈₃	1	1	yes	characteristic
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	C₂³×C₈₃	1	4	yes	—
179	C₂×D₄	—	2	8	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	C₂³×C₈₃	1	8	yes	—
182	C₂×D₄	C₂³×C₈₃	1	4	yes	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	4	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	C₂³×C₈₃	1	2	yes	—
188	C₂²×C₄	C₂³×C₈₃	1	2	yes	—
189	C₂²×C₄	C₂³×C₈₃	1	2	yes	—
190	C₂²×C₄	C₂³×C₈₃	1	2	yes	—
191	C₂²×C₄	C₂³×C₈₃	1	2	yes	—
192	C₂²×C₄	C₂³×C₈₃	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₈₃	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₈₃	1	4	yes	—
195	C₂²×C₄	C₂³×C₈₃	1	4	yes	—
196	C₂²×C₄	C₂³×C₈₃	1	4	yes	—
197	C₂²×C₄	C₂³×C₈₃	1	8	yes	—
198	C₄⋊C₄	C₂³×C₈₃	1	8	yes	—
199	C₄⋊C₄	C₂³×C₈₃	1	8	yes	—
200	C₄⋊C₄	C₂³×C₈₃	1	8	yes	—
201	C₄⋊C₄	C₂³×C₈₃	1	4	yes	—
202	C₄⋊C₄	C₂³×C₈₃	1	8	yes	—
203	C₄⋊C₄	C₂³×C₈₃	1	8	yes	—
204	C₄⋊C₄	C₂³×C₈₃	1	4	yes	—
205	C₄⋊C₄	C₂³×C₈₃	1	8	yes	—
206	C₂²⋊C₄	C₂³×C₈₃	1	8	yes	—
207	C₂²⋊C₄	C₂³×C₈₃	1	8	yes	—
208	C₂²⋊C₄	C₂³×C₈₃	1	8	yes	—
209	C₂²⋊C₄	C₂³×C₈₃	1	8	yes	—
210	C₂²⋊C₄	C₂³×C₈₃	1	8	yes	—
211	C₄²	C₂³×C₈₃	1	8	yes	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	4	no	—
215	C₂³	—	2	4	no	—
216	C₂³	—	2	4	no	—
217	C₂³	#41×C₈₃	1	4	yes	—
218	C₂³	C₂⁴×C₈₃	1	4	yes	—
219	C₂³	C₂⁴×C₈₃	1	2	yes	—
220	C₂³	C₂⁴×C₈₃	1	1	yes	characteristic
221	Q₈	—	4	4	no	—
222	D₄	—	2	16	no	—
223	D₄	—	4	4	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	8	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	C₂⁴×C₈₃	1	4	yes	—
236	C₂×C₄	C₂⁴×C₈₃	1	4	yes	—
237	C₂×C₄	C₂⁴×C₈₃	1	4	yes	—
238	C₂×C₄	C₂⁴×C₈₃	1	4	yes	—
239	C₂×C₄	C₂⁴×C₈₃	1	4	yes	—
240	C₂×C₄	C₂⁴×C₈₃	1	2	yes	—
241	C₂×C₄	C₂⁴×C₈₃	1	2	yes	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	4	8	no	—
249	C₂²	C₂×#41×C₈₃	1	4	yes	—
250	C₂²	C₂×#41×C₈₃	1	2	yes	—
251	C₂²	H(5,𝔽₂)×C₈₃	1	2	yes	—
252	C₂²	2₋¹⁺⁴×C₈₃	1	2	yes	—
253	C₂²	C₂⁵×C₈₃	1	1	yes	characteristic
254	C₄	—	2	4	no	—
255	C₄	—	2	4	no	—
256	C₄	—	2	4	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	4	no	—
261	C₂	—	2	4	no	—
262	C₂	—	4	4	no	—
263	C₂	#539×C₈₃	1	4	yes	direct product factor
264	C₂	C₂²×#41×C₈₃	1	1	yes	characteristic
265	C₂	C₂×H(5,𝔽₂)×C₈₃	1	1	yes	characteristic
266	C₂	C₂×2₋¹⁺⁴×C₈₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₈₃
prime radical	C₂×#121×C₈₃	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #539×C₈₃	C₂	direct (N × H)
2. C₂×#126×C₈₃	C₂	semidirect
3. C₂×#125×C₈₃	C₂	semidirect
4. C₂×#123×C₈₃	C₂	semidirect
5. C₂×#123×C₈₃	C₂	semidirect
6. C₂×C₂²⋊Q₈×C₈₃	C₂	semidirect
7. C₂×C₂²⋊Q₈×C₈₃	C₂	semidirect
9. C₂×C₄×D₄×C₈₃	C₂	semidirect
10. C₂²×C₄⋊C₄×C₈₃	C₂	semidirect
18. #126×C₈₃	C₂²	semidirect
19. #125×C₈₃	C₂²	semidirect
20. #123×C₈₃	C₂²	semidirect
21. #123×C₈₃	C₂²	semidirect
22. C₂²⋊Q₈×C₈₃	C₂²	semidirect
23. C₂²⋊Q₈×C₈₃	C₂²	semidirect
25. C₄×D₄×C₈₃	C₂²	semidirect
34. C₂×C₄⋊C₄×C₈₃	C₂²	semidirect
129. C₂×C₈₃	#539	direct (N × H)
133. C₂×#539	C₈₃	direct (N × H)
134. C₈₃	C₂×#539	direct (N × H)
135. #539	C₂×C₈₃	direct (N × H)
136. C₂×#126	C₂×C₈₃	semidirect
137. C₂×#125	C₂×C₈₃	semidirect
138. C₂×#123	C₂×C₈₃	semidirect
139. C₂×#123	C₂×C₈₃	semidirect
140. C₂×C₂²⋊Q₈	C₂×C₈₃	semidirect
141. C₂×C₂²⋊Q₈	C₂×C₈₃	semidirect
143. C₂×C₄×D₄	C₂×C₈₃	semidirect
144. C₂²×C₄⋊C₄	C₂×C₈₃	semidirect
152. #126	C₂²×C₈₃	semidirect
153. #125	C₂²×C₈₃	semidirect
154. #123	C₂²×C₈₃	semidirect
155. #123	C₂²×C₈₃	semidirect
156. C₂²⋊Q₈	C₂²×C₈₃	semidirect
157. C₂²⋊Q₈	C₂²×C₈₃	semidirect
159. C₄×D₄	C₂²×C₈₃	semidirect
168. C₂×C₄⋊C₄	C₂²×C₈₃	semidirect
263. C₂	#539×C₈₃	direct (N × H)

Maximal subgroups

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

subgroup	index
#539×C₈₃	2
C₂×#126×C₈₃	2
C₂×#125×C₈₃	2
C₂×#123×C₈₃	2
C₂×C₂²⋊Q₈×C₈₃	2
C₂×#121×C₈₃	2
C₂×C₄×D₄×C₈₃	2
C₂²×C₄⋊C₄×C₈₃	2
C₂×#539	83

Subgroups — C2×#539×C83 #24905136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#539×C₈₃ #24905136