Subgroups — C₂×#24909260 #35557467

Order 21248.

726 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₈₃	C₂	1	1	yes	maximal, characteristic
2	#24909260	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₈₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×(C₂×D₈₃)⋊C₁₆	C₂	1	2	yes	maximal
5	C₂×C₄×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₈₃	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₈₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₈₃	C₄	1	1	yes	characteristic
10	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
11	C₂×C₄×C₈₃⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×D₈₃	C₄	1	4	yes	—
15	C₄×C₈×D₈₃	C₂²	1	4	yes	—
16	Dic₈₃⋊C₁₆	C₂²	1	8	yes	complemented
17	(C₂×D₈₃)⋊C₁₆	C₂²	1	8	yes	—
18	C₄×C₈₃⋊C₁₆	C₂²	1	4	yes	complemented
19	C₂²×C₁₆×C₈₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₄×C₁₆×C₈₃	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₈₃	C₈	1	1	yes	characteristic
23	C₂²×C₄×D₈₃	C₈	1	1	yes	characteristic
24	C₂²×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₈₃	—	2	4	no	—
28	C₂×C₈×D₈₃	—	2	4	no	—
29	C₂×C₈×D₈₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₈₃	C₂×C₄	1	4	yes	—
31	C₂×C₈×D₈₃	C₂³	1	2	yes	—
32	C₂×C₄×Dic₈₃	C₈	1	1	yes	characteristic
33	C₂×C₄×Dic₈₃	C₈	1	1	yes	characteristic
34	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₈₃	C₈	1	4	yes	—
36	C₄²×D₈₃	C₂×C₄	1	4	yes	—
37	C₂×C₈₃⋊C₁₆	—	2	4	no	—
38	C₂×C₈₃⋊C₁₆	C₂³	1	4	yes	—
39	C₄×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
40	C₈×Dic₈₃	C₂×C₄	1	2	yes	—
41	C₈×Dic₈₃	C₂×C₄	1	4	yes	—
42	C₈×Dic₈₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₆×C₈₃	—	2	4	no	—
47	C₂×C₁₆×C₈₃	C₂³	1	4	yes	—
48	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₈₃	C₂³	1	2	yes	—
50	C₂³×D₈₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
54	C₂×C₄×D₈₃	—	2	4	no	—
55	C₂×C₄×D₈₃	—	2	2	no	—
56	C₂×C₄×D₈₃	—	2	2	no	—
57	C₂×C₄×D₈₃	C₁₆	1	8	yes	complemented
58	C₂×C₄×D₈₃	C₂×C₈	1	4	yes	—
59	C₂×C₄×D₈₃	C₂×C₈	1	2	yes	—
60	C₂×C₄×D₈₃	C₂×C₈	1	4	yes	—
61	C₂×C₄×D₈₃	C₂²×C₄	1	2	yes	—
62	C₂×C₈₃⋊C₈	—	2	2	no	—
63	C₂×C₈₃⋊C₈	—	2	2	no	—
64	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
65	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
66	C₈×D₈₃	—	2	4	no	—
67	C₈×D₈₃	—	2	8	no	—
68	C₈×D₈₃	—	2	4	no	—
69	C₄×Dic₈₃	C₂×C₈	1	2	yes	—
70	C₄×Dic₈₃	C₂×C₈	1	4	yes	—
71	C₄×Dic₈₃	C₂×C₈	1	4	yes	—
72	C₄×Dic₈₃	C₂²×C₄	1	2	yes	—
73	C₈₃⋊C₁₆	—	2	4	no	—
74	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₈₃	M₁₆	1	4	yes	—
78	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₈₃	#41	1	4	yes	—
81	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
82	C₄²×C₈₃	C₂×C₈	1	2	yes	—
83	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
84	C₁₆×C₈₃	—	2	4	no	—
85	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
86	C₂²×D₈₃	—	2	2	no	—
87	C₂²×D₈₃	—	2	2	no	—
88	C₂²×D₈₃	C₂×C₁₆	1	4	yes	—
89	C₂²×D₈₃	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₈₃	C₂×C₁₆	1	4	yes	—
91	C₂×C₄×C₈₃	M₃₂	1	4	yes	—
92	C₂×C₄×C₈₃	#117	1	2	yes	—
93	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×Dic₈₃	—	2	2	no	—
99	C₂×Dic₈₃	—	2	2	no	—
100	C₂×Dic₈₃	C₂×C₁₆	1	4	yes	—
101	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
102	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
103	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
104	C₄×D₈₃	—	2	2	no	—
105	C₄×D₈₃	—	2	2	no	—
106	C₄×D₈₃	—	2	4	no	—
107	C₄×D₈₃	—	2	8	no	—
108	C₄×D₈₃	C₂×C₁₆	1	16	yes	complemented
109	C₈×C₈₃	#117	1	2	yes	—
110	C₈×C₈₃	C₂×M₁₆	1	4	yes	—
111	C₈×C₈₃	C₂×#41	1	2	yes	—
112	C₈₃⋊C₈	—	2	2	no	—
113	C₈₃⋊C₈	—	2	2	no	—
114	C₂²×C₈₃	#431	1	2	yes	—
115	C₂²×C₈₃	C₂²×C₁₆	1	2	yes	—
116	C₂²×C₈₃	C₂×M₃₂	1	2	yes	—
117	C₂²×C₈₃	C₂³×C₈	1	1	yes	characteristic
118	C₂×D₈₃	—	2	2	no	—
119	C₂×D₈₃	—	2	2	no	—
120	C₂×D₈₃	—	2	8	no	—
121	C₂×D₈₃	C₂²×C₁₆	1	4	yes	—
122	C₄×C₈₃	#431	1	2	yes	—
123	C₄×C₈₃	C₂²×C₁₆	1	4	yes	—
124	C₄×C₈₃	C₂×M₃₂	1	4	yes	—
125	C₄×C₈₃	C₂×#117	1	1	yes	characteristic
126	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
127	Dic₈₃	—	2	2	no	—
128	Dic₈₃	—	2	2	no	—
129	Dic₈₃	C₂²×C₁₆	1	4	yes	—
130	C₂×#2162	—	83	1	no	maximal, Sylow
131	C₂×C₈₃	#2162	1	4	yes	complemented
132	C₂×C₈₃	C₂×#431	1	1	yes	characteristic
133	C₂×C₈₃	C₂³×C₁₆	1	1	yes	characteristic
134	C₂×C₈₃	C₂²×M₃₂	1	1	yes	characteristic
135	D₈₃	—	2	4	no	—
136	C₂²×C₄×C₈	—	83	1	no	—
137	#2162	—	83	8	no	—
138	C₂×C₄⋊C₁₆	—	83	2	no	—
139	C₂×C₂²⋊C₁₆	—	83	2	no	—
140	C₂×C₄×C₁₆	—	83	1	no	—
141	C₂×C₄×C₁₆	D₈₃	1	1	yes	characteristic, complemented
142	C₈₃	C₂×#2162	1	1	yes	characteristic, complemented, Sylow
143	C₂³×C₈	—	83	1	no	—
144	C₂³×C₈	—	83	1	no	—
145	C₂²×C₄²	—	83	1	no	—
146	C₂²×C₁₆	—	83	2	no	—
147	C₂²×C₁₆	C₂×D₈₃	1	2	yes	—
148	C₂×C₄×C₈	—	83	1	no	—
149	C₂×C₄×C₈	—	83	1	no	—
150	C₂×C₄×C₈	—	83	1	no	—
151	C₂×C₄×C₈	—	83	4	no	—
152	C₂×C₄×C₈	—	83	4	no	—
153	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
154	C₄⋊C₁₆	—	83	8	no	—
155	C₂²⋊C₁₆	—	83	8	no	—
156	C₄×C₁₆	—	83	4	no	—
157	C₄×C₁₆	C₂×D₈₃	1	4	yes	complemented
158	C₂³×C₄	—	83	1	no	—
159	C₂³×C₄	—	83	1	no	—
160	C₂³×C₄	—	83	1	no	—
161	C₂²×C₈	—	83	1	no	—
162	C₂²×C₈	—	83	1	no	—
163	C₂²×C₈	—	83	4	no	—
164	C₂²×C₈	—	83	2	no	—
165	C₂²×C₈	—	83	2	no	—
166	C₂²×C₈	—	166	4	no	—
167	C₂²×C₈	—	166	4	no	—
168	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
170	C₂×C₄²	—	83	1	no	—
171	C₂×C₄²	—	83	1	no	—
172	C₂×C₄²	—	83	1	no	—
173	C₂×C₄²	—	83	4	no	—
174	C₂×C₄²	—	83	4	no	—
175	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	—	83	4	no	—
178	C₂×C₁₆	—	166	4	no	—
179	C₂×C₁₆	C₂²×D₈₃	1	4	yes	—
180	C₄×C₈	—	83	4	no	—
181	C₄×C₈	—	83	2	no	—
182	C₄×C₈	—	83	2	no	—
183	C₄×C₈	—	83	4	no	—
184	C₄×C₈	C₄×D₈₃	1	2	yes	—
185	C₄×C₈	C₂²×D₈₃	1	2	yes	—
186	C₂⁴	—	83	1	no	—
187	C₂²×C₄	—	83	1	no	—
188	C₂²×C₄	—	83	4	no	—
189	C₂²×C₄	—	83	8	no	—
190	C₂²×C₄	—	83	1	no	—
191	C₂²×C₄	—	83	1	no	—
192	C₂²×C₄	—	83	4	no	—
193	C₂²×C₄	—	83	2	no	—
194	C₂²×C₄	—	83	2	no	—
195	C₂²×C₄	—	166	2	no	—
196	C₂²×C₄	—	166	4	no	—
197	C₂²×C₄	—	166	2	no	—
198	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
199	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
201	C₂×C₈	—	83	2	no	—
202	C₂×C₈	—	83	2	no	—
203	C₂×C₈	—	166	2	no	—
204	C₂×C₈	—	166	2	no	—
205	C₂×C₈	—	166	8	no	—
206	C₂×C₈	—	166	4	no	—
207	C₂×C₈	—	166	4	no	—
208	C₂×C₈	#13050573	1	4	yes	—
209	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
210	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
211	C₂×C₈	#13050590	1	4	yes	—
212	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
213	C₄²	—	83	4	no	—
214	C₄²	—	83	2	no	—
215	C₄²	—	83	4	no	—
216	C₄²	—	83	2	no	—
217	C₄²	C₈×D₈₃	1	2	yes	—
218	C₄²	C₂×C₄×D₈₃	1	2	yes	—
219	C₁₆	—	2	4	no	—
220	C₁₆	—	166	4	no	—
221	C₂³	—	83	4	no	—
222	C₂³	—	83	2	no	—
223	C₂³	—	166	2	no	—
224	C₂³	—	166	2	no	—
225	C₂³	C₂×C₈×D₈₃	1	1	yes	characteristic
226	C₂×C₄	—	83	4	no	—
227	C₂×C₄	—	83	2	no	—
228	C₂×C₄	—	83	16	no	—
229	C₂×C₄	—	83	2	no	—
230	C₂×C₄	—	83	2	no	—
231	C₂×C₄	—	166	2	no	—
232	C₂×C₄	—	166	2	no	—
233	C₂×C₄	—	166	8	no	—
234	C₂×C₄	—	166	2	no	—
235	C₂×C₄	—	166	2	no	—
236	C₂×C₄	—	166	4	no	—
237	C₂×C₄	C₁₆×D₈₃	1	4	yes	—
238	C₂×C₄	#14581731	1	4	yes	—
239	C₂×C₄	#14581749	1	2	yes	—
240	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
241	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
242	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
243	C₂×C₄	C₂×#13050573	1	2	yes	—
244	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
245	C₈	—	166	2	no	—
246	C₈	—	166	2	no	—
247	C₈	#14581749	1	2	yes	—
248	C₈	C₂×#13050573	1	4	yes	—
249	C₈	C₂×#13050590	1	2	yes	—
250	C₂²	—	83	4	no	—
251	C₂²	—	166	2	no	—
252	C₂²	—	166	8	no	—
253	C₂²	—	166	2	no	—
254	C₂²	#19127752	1	2	yes	—
255	C₂²	C₂×C₁₆×D₈₃	1	2	yes	—
256	C₂²	C₂×#14581731	1	2	yes	—
257	C₂²	C₂²×C₈×D₈₃	1	1	yes	characteristic
258	C₄	—	83	4	no	—
259	C₄	—	166	2	no	—
260	C₄	—	166	2	no	—
261	C₄	#19127752	1	2	yes	—
262	C₄	C₂×C₁₆×D₈₃	1	4	yes	—
263	C₄	C₂×#14581731	1	4	yes	—
264	C₄	C₂×#14581749	1	1	yes	characteristic
265	C₄	C₂²×#13050573	1	1	yes	characteristic
266	C₂	—	166	4	no	—
267	C₂	#24909260	1	4	yes	direct product factor
268	C₂	C₂×#19127752	1	1	yes	characteristic
269	C₂	C₂²×C₁₆×D₈₃	1	1	yes	characteristic
270	C₂	C₂²×#14581731	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₈₃
commutator	C₂×C₈₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×C₄×C₁₆×C₈₃	C₂
socle	C₂³×C₈₃	C₂²×C₈
prime radical	C₂³×D₈₃	C₂×C₈
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₈₃
hypocenter	C₈₃	C₂×#2162
LCS	G ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #24909260	C₂	direct (N × H)
3. C₂×Dic₈₃⋊C₁₆	C₂	semidirect
5. C₂×C₄×C₈₃⋊C₁₆	C₂	semidirect
6. C₂×C₄×C₁₆×C₈₃	C₂	semidirect
16. Dic₈₃⋊C₁₆	C₂²	semidirect
18. C₄×C₈₃⋊C₁₆	C₂²	semidirect
21. C₄×C₁₆×C₈₃	C₂²	semidirect
57. C₂×C₄×D₈₃	C₁₆	semidirect
108. C₄×D₈₃	C₂×C₁₆	semidirect
131. C₂×C₈₃	#2162	semidirect
141. C₂×C₄×C₁₆	D₈₃	semidirect
142. C₈₃	C₂×#2162	semidirect
157. C₄×C₁₆	C₂×D₈₃	semidirect
267. C₂	#24909260	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₈₃	2
#24909260	2
C₂×Dic₈₃⋊C₁₆	2
C₂×(C₂×D₈₃)⋊C₁₆	2
C₂×C₄×C₈₃⋊C₁₆	2
C₂×C₄×C₁₆×C₈₃	2
C₂×#2162	83

Subgroups — C2×#24909260 #35557467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24909260 #35557467