Subgroups — C₂×#24905763 #35370028

Order 21248.

802 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127754	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127754	C₂	1	1	yes	maximal, characteristic
3	#24905763	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19127423	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19127423	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19127422	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#431×C₈₃	C₂	1	1	yes	maximal, characteristic
9	#19127754	C₄	1	4	yes	—
10	#19127754	C₄	1	4	yes	—
11	#19127754	C₂²	1	4	yes	—
12	#19127754	C₂²	1	4	yes	—
13	C₂×C₄⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
14	C₂×C₄⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
15	C₂×C₂²⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
16	C₂×C₂²⋊(C₈₃⋊C₈)	C₄	1	2	yes	—
17	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
21	#19127423	C₂²	1	4	yes	complemented
22	#19127423	C₂²	1	4	yes	complemented
23	#19127422	C₂²	1	4	yes	complemented
24	C₈×C₈₃⋊C₈	C₂²	1	4	yes	complemented
25	C₂²×C₄²×C₈₃	C₂²	1	1	yes	characteristic
26	#431×C₈₃	C₄	1	4	yes	—
27	#431×C₈₃	C₂²	1	4	yes	—
28	C₂×C₄⋊C₈×C₈₃	C₄	1	2	yes	—
29	C₂×C₂²⋊C₈×C₈₃	C₄	1	2	yes	—
30	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
32	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	8	yes	—
37	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	8	yes	—
38	C₂²⋊(C₈₃⋊C₈)	C₂×C₄	1	8	yes	—
39	C₂²⋊(C₈₃⋊C₈)	C₂×C₄	1	8	yes	—
40	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
41	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
42	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
44	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
45	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
46	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
47	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
48	C₂³×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₈₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₈₃	C₂×C₄	1	8	yes	—
62	C₂²⋊C₈×C₈₃	C₂×C₄	1	8	yes	—
63	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₈₃	C₂³	1	2	yes	—
66	C₄×C₈×C₈₃	C₂³	1	2	yes	—
67	C₂×C₈₃⋊C₈	—	2	4	no	—
68	C₂×C₈₃⋊C₈	—	2	4	no	—
69	C₂×C₈₃⋊C₈	—	2	4	no	—
70	C₂×C₈₃⋊C₈	—	2	4	no	—
71	C₂×C₈₃⋊C₈	C₂²×C₄	1	4	yes	—
72	C₂×C₈₃⋊C₈	C₂²×C₄	1	4	yes	—
73	C₂×C₈₃⋊C₈	C₂²×C₄	1	4	yes	—
74	C₂×C₈₃⋊C₈	C₂²×C₄	1	4	yes	—
75	C₂⁴×C₈₃	C₄²	1	1	yes	characteristic
76	C₂²×C₄×C₈₃	—	2	4	no	—
77	C₂²×C₄×C₈₃	—	2	4	no	—
78	C₂²×C₄×C₈₃	—	2	4	no	—
79	C₂²×C₄×C₈₃	C₄²	1	1	yes	characteristic
80	C₂²×C₄×C₈₃	C₄²	1	2	yes	—
81	C₂²×C₄×C₈₃	C₄²	1	1	yes	characteristic
82	C₂²×C₄×C₈₃	C₄²	1	1	yes	characteristic
83	C₂²×C₄×C₈₃	C₄²	1	2	yes	—
84	C₂²×C₄×C₈₃	C₄²	1	2	yes	—
85	C₂²×C₄×C₈₃	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₈₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₃	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₃	—	2	4	no	—
92	C₂×C₈×C₈₃	—	2	4	no	—
93	C₂×C₈×C₈₃	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₈₃	C₂²×C₄	1	4	yes	—
95	C₄²×C₈₃	C₄²	1	2	yes	—
96	C₄²×C₈₃	C₄²	1	2	yes	—
97	C₄²×C₈₃	C₄²	1	2	yes	—
98	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
101	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
105	C₂³×C₈₃	—	2	2	no	—
106	C₂³×C₈₃	—	2	2	no	—
107	C₂³×C₈₃	—	2	2	no	—
108	C₂³×C₈₃	C₂×C₄²	1	2	yes	—
109	C₂³×C₈₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₈₃	—	2	2	no	—
111	C₂×C₄×C₈₃	—	2	2	no	—
112	C₂×C₄×C₈₃	—	2	2	no	—
113	C₂×C₄×C₈₃	—	2	4	no	—
114	C₂×C₄×C₈₃	—	2	4	no	—
115	C₂×C₄×C₈₃	—	2	4	no	—
116	C₂×C₄×C₈₃	—	2	4	no	—
117	C₂×C₄×C₈₃	—	2	4	no	—
118	C₂×C₄×C₈₃	—	2	4	no	—
119	C₂×C₄×C₈₃	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₈₃	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₈₃	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₈₃	#131	1	4	yes	—
126	C₂×C₄×C₈₃	#131	1	4	yes	—
127	C₂×C₄×C₈₃	#131	1	4	yes	—
128	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
131	C₈×C₈₃	—	2	4	no	—
132	C₈×C₈₃	—	2	4	no	—
133	C₈₃⋊C₈	—	2	4	no	—
134	C₈₃⋊C₈	—	2	4	no	—
135	C₈₃⋊C₈	—	2	4	no	—
136	C₈₃⋊C₈	—	2	4	no	—
137	C₂²×C₈₃	—	2	2	no	—
138	C₂²×C₈₃	—	2	2	no	—
139	C₂²×C₈₃	—	2	2	no	—
140	C₂²×C₈₃	—	4	4	no	—
141	C₂²×C₈₃	#405	1	2	yes	—
142	C₂²×C₈₃	#405	1	2	yes	—
143	C₂²×C₈₃	#405	1	2	yes	—
144	C₂²×C₈₃	C₂²×C₄²	1	1	yes	characteristic
145	C₄×C₈₃	—	2	2	no	—
146	C₄×C₈₃	—	2	2	no	—
147	C₄×C₈₃	—	2	2	no	—
148	C₄×C₈₃	#405	1	2	yes	—
149	C₄×C₈₃	#405	1	2	yes	—
150	C₄×C₈₃	#405	1	2	yes	—
151	C₄×C₈₃	C₂×#131	1	2	yes	—
152	C₄×C₈₃	C₂×#131	1	2	yes	—
153	C₄×C₈₃	C₂×#131	1	2	yes	—
154	C₂×#1454	—	83	1	no	maximal, Sylow
155	C₂×C₈₃	—	4	2	no	—
156	C₂×C₈₃	#1454	1	4	yes	complemented
157	C₂×C₈₃	C₂×#405	1	1	yes	characteristic
158	C₂×C₈₃	C₂×#405	1	1	yes	characteristic
159	C₂×C₈₃	C₂×#405	1	1	yes	characteristic
160	C₂×#431	—	83	1	no	—
161	C₂×#431	—	83	1	no	—
162	C₂×#431	D₈₃	1	1	yes	characteristic
163	#1454	—	83	8	no	—
164	C₂×#322	—	83	1	no	—
165	C₂×#322	—	83	1	no	—
166	C₂×#322	—	83	1	no	—
167	C₂×C₈²	—	83	1	no	—
168	C₈₃	C₂×#1454	1	1	yes	characteristic, complemented, Sylow
169	C₂²×C₄²	C₂×D₈₃	1	1	yes	characteristic
170	#431	—	83	4	no	—
171	#431	—	83	4	no	—
172	#431	—	83	4	no	—
173	#431	—	83	4	no	—
174	#431	Dic₈₃	1	4	yes	—
175	#431	C₂×D₈₃	1	4	yes	—
176	C₂×C₄⋊C₈	—	83	2	no	—
177	C₂×C₄⋊C₈	—	83	2	no	—
178	C₂×C₄⋊C₈	Dic₈₃	1	2	yes	—
179	C₂×C₂²⋊C₈	—	83	2	no	—
180	C₂×C₂²⋊C₈	—	83	2	no	—
181	C₂×C₂²⋊C₈	Dic₈₃	1	2	yes	—
182	C₂×C₄×C₈	—	83	1	no	—
183	C₂×C₄×C₈	—	83	1	no	—
184	C₂×C₄×C₈	—	83	1	no	—
185	C₂×C₄×C₈	—	83	1	no	—
186	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
187	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
188	#322	—	83	4	no	—
189	#322	—	83	4	no	—
190	#322	—	83	4	no	—
191	C₈²	—	83	4	no	—
192	C₂³×C₄	C₄×D₈₃	1	1	yes	characteristic
193	C₂³×C₄	C₄×D₈₃	1	1	yes	characteristic
194	C₂³×C₄	C₂×Dic₈₃	1	1	yes	characteristic
195	C₂²×C₈	—	83	2	no	—
196	C₂²×C₈	—	83	2	no	—
197	C₂²×C₈	—	83	2	no	—
198	C₂²×C₈	—	83	2	no	—
199	C₂²×C₈	C₂×Dic₈₃	1	2	yes	—
200	C₂²×C₈	C₂×Dic₈₃	1	2	yes	—
201	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
202	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
203	C₂×C₄²	C₄×D₈₃	1	2	yes	—
204	C₂×C₄²	C₄×D₈₃	1	2	yes	—
205	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
206	C₂×C₄²	C₂×Dic₈₃	1	2	yes	—
207	C₂×C₄²	C₂²×D₈₃	1	2	yes	—
208	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
209	C₄⋊C₈	—	83	8	no	—
210	C₄⋊C₈	—	83	8	no	—
211	C₄⋊C₈	C₂×Dic₈₃	1	8	yes	—
212	C₂²⋊C₈	—	83	8	no	—
213	C₂²⋊C₈	—	83	8	no	—
214	C₂²⋊C₈	C₂×Dic₈₃	1	8	yes	—
215	C₄×C₈	—	83	2	no	—
216	C₄×C₈	—	83	2	no	—
217	C₄×C₈	—	83	2	no	—
218	C₄×C₈	—	83	2	no	—
219	C₄×C₈	—	83	2	no	—
220	C₄×C₈	—	83	2	no	—
221	C₄×C₈	—	83	2	no	—
222	C₄×C₈	—	83	2	no	—
223	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
224	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
225	C₄×C₈	C₂²×D₈₃	1	2	yes	—
226	C₄×C₈	C₂²×D₈₃	1	2	yes	—
227	C₂⁴	C₄×Dic₈₃	1	1	yes	characteristic
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	C₄×Dic₈₃	1	1	yes	characteristic
232	C₂²×C₄	C₄×Dic₈₃	1	2	yes	—
233	C₂²×C₄	C₄×Dic₈₃	1	1	yes	characteristic
234	C₂²×C₄	C₄×Dic₈₃	1	1	yes	characteristic
235	C₂²×C₄	C₄×Dic₈₃	1	2	yes	—
236	C₂²×C₄	C₄×Dic₈₃	1	2	yes	—
237	C₂²×C₄	C₂×C₄×D₈₃	1	2	yes	—
238	C₂²×C₄	C₂×C₄×D₈₃	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²×Dic₈₃	1	2	yes	—
242	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	83	4	no	—
246	C₂×C₈	—	83	4	no	—
247	C₂×C₈	—	83	4	no	—
248	C₂×C₈	—	83	4	no	—
249	C₂×C₈	—	166	4	no	—
250	C₂×C₈	—	166	4	no	—
251	C₂×C₈	—	166	4	no	—
252	C₂×C₈	—	166	4	no	—
253	C₂×C₈	C₂²×Dic₈₃	1	4	yes	—
254	C₂×C₈	C₂²×Dic₈₃	1	4	yes	—
255	C₄²	C₄×Dic₈₃	1	2	yes	—
256	C₄²	C₄×Dic₈₃	1	2	yes	—
257	C₄²	C₄×Dic₈₃	1	2	yes	—
258	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
260	C₄²	C₂×C₄×D₈₃	1	2	yes	—
261	C₄²	C₂×C₄×D₈₃	1	2	yes	—
262	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
263	C₄²	C₂²×Dic₈₃	1	2	yes	—
264	C₄²	C₂³×D₈₃	1	1	yes	characteristic
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	C₂×C₄×Dic₈₃	1	2	yes	—
269	C₂³	C₂×C₄×Dic₈₃	1	1	yes	characteristic
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	C₂×C₄×Dic₈₃	1	2	yes	—
280	C₂×C₄	C₂×C₄×Dic₈₃	1	2	yes	—
281	C₂×C₄	C₂×C₄×Dic₈₃	1	2	yes	—
282	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
283	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
285	C₂×C₄	#14581798	1	4	yes	—
286	C₂×C₄	#14581799	1	4	yes	—
287	C₂×C₄	#14581800	1	4	yes	—
288	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
290	C₂×C₄	C₂³×Dic₈₃	1	1	yes	characteristic
291	C₈	—	2	4	no	—
292	C₈	—	2	4	no	—
293	C₈	—	166	4	no	—
294	C₈	—	166	4	no	—
295	C₈	—	166	4	no	—
296	C₈	—	166	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	4	4	no	—
301	C₂²	#19127656	1	2	yes	—
302	C₂²	#19127657	1	2	yes	—
303	C₂²	#19127659	1	2	yes	—
304	C₂²	C₂²×C₄×Dic₈₃	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	#19127656	1	2	yes	—
309	C₄	#19127657	1	2	yes	—
310	C₄	#19127659	1	2	yes	—
311	C₄	C₂×#14581798	1	2	yes	—
312	C₄	C₂×#14581799	1	2	yes	—
313	C₄	C₂×#14581800	1	2	yes	—
314	C₂	—	4	2	no	—
315	C₂	#24905763	1	4	yes	direct product factor
316	C₂	C₂×#19127656	1	1	yes	characteristic
317	C₂	C₂×#19127657	1	1	yes	characteristic
318	C₂	C₂×#19127659	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₈₃
Fitting	C₂×#431×C₈₃	C₂
socle	C₂³×C₈₃	C₂×C₄²
prime radical	C₂⁴×C₈₃	C₄²
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	C₂×#431	D₈₃
hypocenter	C₈₃	C₂×#1454
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #24905763	C₂	direct (N × H)
4. C₂×#19127423	C₂	semidirect
5. C₂×#19127423	C₂	semidirect
6. C₂×#19127422	C₂	semidirect
7. C₂×C₈×C₈₃⋊C₈	C₂	semidirect
21. #19127423	C₂²	semidirect
22. #19127423	C₂²	semidirect
23. #19127422	C₂²	semidirect
24. C₈×C₈₃⋊C₈	C₂²	semidirect
156. C₂×C₈₃	#1454	semidirect
168. C₈₃	C₂×#1454	semidirect
315. C₂	#24905763	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×#19127754	2
#24905763	2
C₂×#19127423	2
C₂×#19127422	2
C₂×C₈×C₈₃⋊C₈	2
C₂×#431×C₈₃	2
C₂×#1454	83

Subgroups — C2×#24905763 #35370028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24905763 #35370028