Subgroups — C₂×#24909217 #35557381

Order 21248.

798 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14581731	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₈₃	C₂	1	1	yes	maximal, characteristic
3	#24909217	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₈₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₈₃⋊C₁₆	C₂	1	2	yes	maximal, 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	C₂×#14581731	—	2	8	no	—
11	C₂×#14581731	C₂²	1	8	yes	—
12	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
13	C₂×C₄×C₈₃⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₈₃	C₄	1	4	yes	—
17	C₄×C₈×D₈₃	C₂²	1	4	yes	—
18	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	4	yes	complemented
19	Dic₈₃⋊C₁₆	C₂²	1	8	yes	complemented
20	C₂²×C₁₆×C₈₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆×C₈₃	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₈₃	C₈	1	1	yes	characteristic
24	C₂²×C₄×D₈₃	C₈	1	1	yes	characteristic
25	C₂²×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₈₃	—	2	4	no	—
29	C₂×C₈×D₈₃	C₂×C₄	1	4	yes	—
30	C₂×C₈×D₈₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₈₃	D₄	1	4	yes	—
32	C₂×C₈×D₈₃	Q₈	1	4	yes	—
33	C₂×C₈×D₈₃	C₂³	1	2	yes	—
34	C₂×C₄×Dic₈₃	C₈	1	1	yes	characteristic
35	C₂×C₄×Dic₈₃	C₈	1	1	yes	characteristic
36	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×D₈₃	C₈	1	4	yes	—
38	C₄²×D₈₃	C₂×C₄	1	4	yes	—
39	#14581731	—	2	16	no	—
40	C₂×C₈₃⋊C₁₆	—	2	4	no	—
41	C₂×C₈₃⋊C₁₆	C₂³	1	4	yes	—
42	C₄×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
43	C₈×Dic₈₃	C₂×C₄	1	2	yes	—
44	C₈×Dic₈₃	C₂×C₄	1	4	yes	—
45	C₈×Dic₈₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₈₃	—	2	4	no	—
50	C₂×C₁₆×C₈₃	C₂³	1	4	yes	—
51	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₈₃	C₂³	1	2	yes	—
53	C₂³×D₈₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
57	C₂×C₄×D₈₃	—	2	4	no	—
58	C₂×C₄×D₈₃	—	2	4	no	—
59	C₂×C₄×D₈₃	C₄⋊C₄	1	4	yes	—
60	C₂×C₄×D₈₃	C₂×C₈	1	4	yes	—
61	C₂×C₄×D₈₃	C₂×C₈	1	4	yes	—
62	C₂×C₄×D₈₃	C₂×C₈	1	2	yes	—
63	C₂×C₄×D₈₃	M₁₆	1	4	yes	—
64	C₂×C₄×D₈₃	C₂²×C₄	1	2	yes	—
65	C₂×C₈₃⋊C₈	—	2	2	no	—
66	C₂×C₈₃⋊C₈	C₄⋊C₄	1	4	yes	—
67	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
68	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
69	C₈×D₈₃	—	2	8	no	—
70	C₈×D₈₃	C₄⋊C₄	1	8	yes	—
71	C₈×D₈₃	C₂×D₄	1	4	yes	—
72	C₈×D₈₃	C₂×Q₈	1	4	yes	—
73	C₄×Dic₈₃	C₂×C₈	1	2	yes	—
74	C₄×Dic₈₃	C₂×C₈	1	4	yes	—
75	C₄×Dic₈₃	C₂×C₈	1	4	yes	—
76	C₄×Dic₈₃	C₂²×C₄	1	2	yes	—
77	C₈₃⋊C₁₆	—	2	4	no	—
78	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₈₃	—	2	2	no	—
82	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
87	C₄²×C₈₃	C₂×C₈	1	2	yes	—
88	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
89	C₁₆×C₈₃	—	2	4	no	—
90	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
91	C₂²×D₈₃	—	2	2	no	—
92	C₂²×D₈₃	—	2	2	no	—
93	C₂²×D₈₃	C₄⋊C₈	1	4	yes	—
94	C₂²×D₈₃	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₈₃	—	2	2	no	—
96	C₂×C₄×C₈₃	M₃₂	1	4	yes	complemented
97	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
102	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×Dic₈₃	—	2	2	no	—
104	C₂×Dic₈₃	C₄⋊C₈	1	4	yes	—
105	C₂×Dic₈₃	M₃₂	1	4	yes	complemented
106	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
107	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
108	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
109	C₄×D₈₃	—	2	8	no	—
110	C₄×D₈₃	—	2	8	no	—
111	C₄×D₈₃	C₄⋊C₈	1	8	yes	—
112	C₄×D₈₃	C₂×C₄⋊C₄	1	4	yes	—
113	C₄×D₈₃	C₂×M₁₆	1	4	yes	—
114	C₈×C₈₃	—	2	2	no	—
115	C₈×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₈×C₈₃	C₂²×D₄	1	1	yes	characteristic
117	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
118	C₈₃⋊C₈	—	2	2	no	—
119	C₈₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
120	C₂²×C₈₃	C₂×C₄⋊C₈	1	2	yes	—
121	C₂²×C₈₃	C₂×M₃₂	1	2	yes	—
122	C₂²×C₈₃	C₂×M₃₂	1	2	yes	—
123	C₂²×C₈₃	C₂³×C₈	1	1	yes	characteristic
124	C₂×D₈₃	—	2	2	no	—
125	C₂×D₈₃	—	2	2	no	—
126	C₂×D₈₃	—	2	8	no	—
127	C₂×D₈₃	C₂×C₄⋊C₈	1	4	yes	—
128	C₄×C₈₃	—	2	2	no	—
129	C₄×C₈₃	C₂×C₄⋊C₈	1	2	yes	—
130	C₄×C₈₃	C₂×M₃₂	1	4	yes	complemented
131	C₄×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
133	Dic₈₃	—	2	2	no	—
134	Dic₈₃	C₂×C₄⋊C₈	1	4	yes	—
135	Dic₈₃	C₂×M₃₂	1	4	yes	complemented
136	C₂×#2150	—	83	1	no	maximal, Sylow
137	C₂×C₈₃	#2150	1	4	yes	complemented
138	C₂×C₈₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
139	C₂×C₈₃	C₂²×M₃₂	1	1	yes	characteristic
140	C₂×C₈₃	C₂²×M₃₂	1	1	yes	characteristic
141	D₈₃	—	2	4	no	—
142	C₂²×M₃₂	—	83	2	no	—
143	C₂²×C₄×C₈	—	83	1	no	—
144	#2150	—	83	8	no	—
145	C₂×C₄⋊C₁₆	—	83	2	no	—
146	C₂×C₄⋊C₁₆	—	83	1	no	—
147	C₂×C₄⋊C₁₆	D₈₃	1	1	yes	characteristic, complemented
148	C₈₃	C₂×#2150	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₈	—	83	1	no	—
150	C₂³×C₈	—	83	1	no	—
151	C₂²×C₄²	—	83	1	no	—
152	C₂×M₃₂	—	83	8	no	—
153	C₂×M₃₂	—	166	8	no	—
154	C₂²×C₁₆	—	83	2	no	—
155	C₂²×C₁₆	C₂×D₈₃	1	2	yes	—
156	C₂×C₄×C₈	—	83	1	no	—
157	C₂×C₄×C₈	—	83	1	no	—
158	C₂×C₄×C₈	—	83	1	no	—
159	C₂×C₄×C₈	—	83	4	no	—
160	C₂×C₄×C₈	—	83	4	no	—
161	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
162	C₄⋊C₁₆	—	83	8	no	—
163	C₄⋊C₁₆	—	83	4	no	—
164	C₄⋊C₁₆	C₂×D₈₃	1	4	yes	complemented
165	C₂³×C₄	—	83	1	no	—
166	C₂³×C₄	—	83	1	no	—
167	C₂³×C₄	—	83	1	no	—
168	C₂²×C₈	—	83	1	no	—
169	C₂²×C₈	—	83	2	no	—
170	C₂²×C₈	—	83	4	no	—
171	C₂²×C₈	—	83	4	no	—
172	C₂²×C₈	—	83	2	no	—
173	C₂²×C₈	—	83	1	no	—
174	C₂²×C₈	—	83	4	no	—
175	C₂²×C₈	—	166	4	no	—
176	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
178	C₂×C₄²	—	83	1	no	—
179	C₂×C₄²	—	83	1	no	—
180	C₂×C₄²	—	83	1	no	—
181	C₂×C₄²	—	83	4	no	—
182	C₂×C₄²	—	83	4	no	—
183	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
184	M₃₂	—	166	16	no	—
185	C₂×C₁₆	—	2	4	no	—
186	C₂×C₁₆	—	83	4	no	—
187	C₂×C₁₆	—	166	4	no	—
188	C₂×C₁₆	C₂²×D₈₃	1	4	yes	—
189	C₄×C₈	—	83	4	no	—
190	C₄×C₈	—	83	2	no	—
191	C₄×C₈	—	83	2	no	—
192	C₄×C₈	—	83	4	no	—
193	C₄×C₈	C₄×D₈₃	1	2	yes	—
194	C₄×C₈	C₂²×D₈₃	1	2	yes	—
195	C₂⁴	—	83	1	no	—
196	C₂²×C₄	—	83	1	no	—
197	C₂²×C₄	—	83	1	no	—
198	C₂²×C₄	—	83	2	no	—
199	C₂²×C₄	—	83	4	no	—
200	C₂²×C₄	—	83	2	no	—
201	C₂²×C₄	—	83	4	no	—
202	C₂²×C₄	—	83	1	no	—
203	C₂²×C₄	—	83	4	no	—
204	C₂²×C₄	—	83	4	no	—
205	C₂²×C₄	—	166	4	no	—
206	C₂²×C₄	—	166	4	no	—
207	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
208	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	83	2	no	—
212	C₂×C₈	—	83	4	no	—
213	C₂×C₈	—	83	4	no	—
214	C₂×C₈	—	83	4	no	—
215	C₂×C₈	—	83	8	no	—
216	C₂×C₈	—	83	2	no	—
217	C₂×C₈	—	166	8	no	—
218	C₂×C₈	—	166	2	no	—
219	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
220	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
221	C₂×C₈	D₄×D₈₃	1	2	yes	—
222	C₂×C₈	Q₈×D₈₃	1	2	yes	—
223	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
224	C₄²	—	83	4	no	—
225	C₄²	—	83	4	no	—
226	C₄²	—	83	2	no	—
227	C₄²	—	83	2	no	—
228	C₄²	C₈×D₈₃	1	2	yes	—
229	C₄²	C₂×C₄×D₈₃	1	2	yes	—
230	C₁₆	—	2	4	no	—
231	C₁₆	—	166	4	no	—
232	C₂³	—	83	2	no	—
233	C₂³	—	83	4	no	—
234	C₂³	—	166	2	no	—
235	C₂³	—	166	2	no	—
236	C₂³	C₂×C₈×D₈₃	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	83	4	no	—
239	C₂×C₄	—	83	2	no	—
240	C₂×C₄	—	83	2	no	—
241	C₂×C₄	—	83	4	no	—
242	C₂×C₄	—	83	4	no	—
243	C₂×C₄	—	83	4	no	—
244	C₂×C₄	—	83	8	no	—
245	C₂×C₄	—	83	2	no	—
246	C₂×C₄	—	166	8	no	—
247	C₂×C₄	—	166	2	no	—
248	C₂×C₄	—	166	8	no	—
249	C₂×C₄	#14581731	1	4	yes	complemented
250	C₂×C₄	C₄⋊C₄×D₈₃	1	2	yes	—
251	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
252	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
253	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
254	C₂×C₄	M₁₆×D₈₃	1	2	yes	—
255	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	83	4	no	—
258	C₈	—	166	2	no	—
259	C₈	C₄⋊C₄×D₈₃	1	2	yes	—
260	C₈	C₂×D₄×D₈₃	1	1	yes	characteristic
261	C₈	C₂×Q₈×D₈₃	1	1	yes	characteristic
262	C₂²	—	83	4	no	—
263	C₂²	—	166	8	no	—
264	C₂²	—	166	2	no	—
265	C₂²	—	166	2	no	—
266	C₂²	C₄⋊C₈×D₈₃	1	2	yes	—
267	C₂²	M₃₂×D₈₃	1	2	yes	—
268	C₂²	C₂×#14581731	1	2	yes	—
269	C₂²	C₂²×C₈×D₈₃	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	83	4	no	—
272	C₄	—	83	4	no	—
273	C₄	—	166	2	no	—
274	C₄	C₄⋊C₈×D₈₃	1	2	yes	—
275	C₄	C₂×#14581731	1	4	yes	complemented
276	C₄	C₂×C₄⋊C₄×D₈₃	1	1	yes	characteristic
277	C₄	C₂×M₁₆×D₈₃	1	1	yes	characteristic
278	C₂	—	166	4	no	—
279	C₂	#24909217	1	4	yes	direct product factor
280	C₂	C₂×C₄⋊C₈×D₈₃	1	1	yes	characteristic
281	C₂	C₂×M₃₂×D₈₃	1	1	yes	characteristic
282	C₂	C₂²×#14581731	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂³×C₈
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×#2150
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #24909217	C₂	direct (N × H)
4. C₂×C₄⋊(C₈₃⋊C₁₆)	C₂	semidirect
5. C₂×Dic₈₃⋊C₁₆	C₂	semidirect
6. C₂×C₄⋊C₁₆×C₈₃	C₂	semidirect
18. C₄⋊(C₈₃⋊C₁₆)	C₂²	semidirect
19. Dic₈₃⋊C₁₆	C₂²	semidirect
22. C₄⋊C₁₆×C₈₃	C₂²	semidirect
96. C₂×C₄×C₈₃	M₃₂	semidirect
105. C₂×Dic₈₃	M₃₂	semidirect
130. C₄×C₈₃	C₂×M₃₂	semidirect
135. Dic₈₃	C₂×M₃₂	semidirect
137. C₂×C₈₃	#2150	semidirect
147. C₂×C₄⋊C₁₆	D₈₃	semidirect
148. C₈₃	C₂×#2150	semidirect
164. C₄⋊C₁₆	C₂×D₈₃	semidirect
249. C₂×C₄	#14581731	semidirect
275. C₄	C₂×#14581731	semidirect
279. C₂	#24909217	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14581731	2
C₂×C₄×C₈×D₈₃	2
#24909217	2
C₂×C₄⋊(C₈₃⋊C₁₆)	2
C₂×Dic₈₃⋊C₁₆	2
C₂×C₄⋊C₁₆×C₈₃	2
C₂×#2150	83

Subgroups — C2×#24909217 #35557381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24909217 #35557381