Subgroups — C₂×#19147478 #25977045

Order 11008.

1830 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583629	C₂	1	2	yes	maximal
2	#19147478	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14583303	C₂	1	2	yes	maximal
4	C₂×#14583193	C₂	1	4	yes	maximal
5	C₂×#14583191	C₂	1	2	yes	maximal
6	C₂×C₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
8	C₂×#14582993	C₂	1	2	yes	maximal
9	C₂×C₄₃×#500	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
11	#14583629	—	2	8	no	—
12	#14583629	C₂²	1	16	yes	—
13	C₂×#13063324	—	2	2	no	—
14	C₂×#13063324	C₂²	1	4	yes	—
15	C₂×#13063323	—	2	4	no	—
16	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
17	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
18	C₂×#13063290	—	2	2	no	—
19	C₂×#13063290	C₂²	1	4	yes	—
20	#14583303	C₂²	1	16	yes	—
21	#14583193	C₂²	1	32	yes	—
22	#14583191	C₂²	1	16	yes	—
23	C₈×Dic₈₆	C₂²	1	8	yes	—
24	C₄×Dic₁₇₂	C₂²	1	8	yes	—
25	#14582993	C₂²	1	16	yes	—
26	C₂×#13063259	C₂²	1	1	yes	characteristic
27	C₂×#13063258	C₂²	1	4	yes	—
28	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
29	C₂×Dic₄₃⋊C₈	C₂²	1	2	yes	—
30	C₂×#13063250	C₂²	1	2	yes	—
31	C₂×#13063248	C₂²	1	4	yes	—
32	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
33	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
34	C₄₃×#500	C₂²	1	8	yes	—
35	C₂×#107×C₄₃	C₂²	1	2	yes	—
36	C₂×#107×C₄₃	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
38	C₂²×Dic₈₆	—	2	4	no	—
39	C₂²×Dic₈₆	D₄	1	1	yes	characteristic
40	C₂²×Dic₈₆	C₂³	1	2	yes	—
41	C₂×Dic₁₇₂	Q₈	1	8	yes	complemented
42	C₂×Dic₁₇₂	C₂³	1	4	yes	—
43	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
44	#13063324	—	2	16	no	—
45	#13063324	—	2	8	no	—
46	#13063324	C₂³	1	16	yes	—
47	#13063323	—	2	32	no	—
48	C₄×Dic₈₆	—	2	4	no	—
49	C₄×Dic₈₆	D₄	1	4	yes	—
50	C₄×Dic₈₆	C₂³	1	8	yes	—
51	C₄×Dic₈₆	C₂³	1	4	yes	—
52	#13063290	—	2	16	no	—
53	#13063290	—	2	8	no	—
54	#13063290	C₂³	1	16	yes	—
55	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
56	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
57	C₂×Dic₄₃⋊C₄	—	2	2	no	—
58	C₂×Dic₄₃⋊C₄	—	2	4	no	—
59	C₂×Dic₄₃⋊C₄	—	2	4	no	—
60	C₂×Dic₄₃⋊C₄	D₄	1	2	yes	—
61	C₂×C₄×Dic₄₃	—	2	2	no	—
62	C₂×C₄×Dic₄₃	—	2	4	no	—
63	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
64	#13063259	C₂³	1	4	yes	—
65	#13063258	C₂³	1	16	yes	—
66	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
67	Dic₄₃⋊C₈	C₂³	1	8	yes	—
68	#13063250	C₂³	1	8	yes	—
69	#13063248	C₂³	1	16	yes	—
70	C₈×Dic₄₃	C₂³	1	8	yes	—
71	C₂²×Q₈×C₄₃	—	2	2	no	—
72	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈×C₄₃	—	2	4	no	—
75	C₄⋊Q₈×C₄₃	C₂³	1	8	yes	—
76	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
77	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
78	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
79	#107×C₄₃	C₂³	1	8	yes	—
80	#107×C₄₃	C₂³	1	8	yes	—
81	C₄×C₈×C₄₃	C₂³	1	4	yes	—
82	C₂×Q₈×C₄₃	—	2	8	no	—
83	C₂×Q₈×C₄₃	—	8	4	no	—
84	C₂²×C₄×C₄₃	—	2	4	no	—
85	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₂²×Dic₄₃	—	2	2	no	—
89	C₂²×Dic₄₃	—	2	2	no	—
90	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
91	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
92	C₂×Dic₈₆	—	2	2	no	—
93	C₂×Dic₈₆	—	2	4	no	—
94	C₂×Dic₈₆	—	2	16	no	—
95	C₂×Dic₈₆	—	8	8	no	—
96	C₂×Dic₈₆	Q₁₆	1	8	yes	complemented
97	C₂×Dic₈₆	C₂×D₄	1	4	yes	—
98	C₂×Dic₈₆	C₂×Q₈	1	8	yes	—
99	C₂×Dic₈₆	C₂⁴	1	4	yes	—
100	C₂×C₈×C₄₃	—	2	2	no	—
101	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
102	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
103	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
104	C₄⋊C₄×C₄₃	—	2	8	no	—
105	C₄⋊C₄×C₄₃	—	2	4	no	—
106	C₄⋊C₄×C₄₃	C₂⁴	1	8	yes	—
107	C₄²×C₄₃	C₂×D₄	1	2	yes	—
108	C₄²×C₄₃	C₂⁴	1	2	yes	—
109	C₄⋊Dic₄₃	—	2	2	no	—
110	C₄⋊Dic₄₃	C₂×D₄	1	4	yes	—
111	C₄⋊Dic₄₃	C₂⁴	1	4	yes	—
112	Dic₄₃⋊C₄	—	2	8	no	—
113	Dic₄₃⋊C₄	—	2	16	no	—
114	Dic₄₃⋊C₄	—	2	16	no	—
115	Dic₄₃⋊C₄	C₂×D₄	1	8	yes	—
116	C₄×Dic₄₃	—	2	8	no	—
117	C₄×Dic₄₃	—	2	16	no	—
118	C₄×Dic₄₃	C₂×D₄	1	4	yes	—
119	C₄×Dic₄₃	C₂⁴	1	4	yes	—
120	C₂×C₄₃⋊C₈	—	2	4	no	—
121	C₂×C₄₃⋊C₈	C₂⁴	1	4	yes	—
122	Dic₁₇₂	C₂×Q₈	1	16	yes	complemented
123	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
124	Q₈×C₄₃	—	8	8	no	—
125	C₂×C₄×C₄₃	—	2	2	no	—
126	C₂×C₄×C₄₃	—	2	8	no	—
127	C₂×C₄×C₄₃	—	4	8	no	—
128	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
129	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
133	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
134	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
135	C₂×Dic₄₃	—	2	8	no	—
136	C₂×Dic₄₃	—	2	4	no	—
137	C₂×Dic₄₃	—	2	4	no	—
138	C₂×Dic₄₃	—	2	4	no	—
139	C₂×Dic₄₃	—	4	4	no	—
140	C₂×Dic₄₃	C₂×Q₁₆	1	8	yes	—
141	C₂×Dic₄₃	C₂²×D₄	1	4	yes	—
142	C₂×Dic₄₃	C₂²×D₄	1	4	yes	—
143	Dic₈₆	—	2	8	no	—
144	Dic₈₆	—	2	4	no	—
145	Dic₈₆	—	8	16	no	—
146	Dic₈₆	C₂×Q₁₆	1	16	yes	complemented
147	Dic₈₆	C₂²×Q₈	1	8	yes	—
148	C₈×C₄₃	—	2	2	no	—
149	C₈×C₄₃	C₂²×Q₈	1	4	yes	—
150	C₄₃⋊C₈	—	2	4	no	—
151	C₂×Q₈×Q₁₆	—	43	1	no	maximal, Sylow
152	C₂²×C₄₃	D₄×Q₈	1	2	yes	—
153	C₂²×C₄₃	C₂²×Q₁₆	1	2	yes	—
154	C₂²×C₄₃	#576	1	2	yes	—
155	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
156	C₄×C₄₃	—	2	2	no	—
157	C₄×C₄₃	—	4	8	no	—
158	C₄×C₄₃	D₄×Q₈	1	2	yes	—
159	C₄×C₄₃	C₂²×Q₁₆	1	4	yes	—
160	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
161	C₄×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
162	Dic₄₃	—	2	8	no	—
163	Dic₄₃	—	2	4	no	—
164	Dic₄₃	—	4	4	no	—
165	Dic₄₃	C₂²×Q₁₆	1	8	yes	—
166	C₂×Q₈²	—	43	2	no	—
167	Q₈×Q₁₆	—	43	16	no	—
168	C₂×#500	—	43	2	no	—
169	C₂×#500	D₄₃	1	1	yes	characteristic
170	C₂×#477	—	43	4	no	—
171	C₂×#477	—	43	2	no	—
172	C₂×C₈×Q₈	—	43	1	no	—
173	C₂×C₄×Q₁₆	—	43	2	no	—
174	C₂×C₄×Q₁₆	—	43	1	no	—
175	C₂×C₄₃	Q₈×Q₁₆	1	4	yes	complemented
176	C₂×C₄₃	C₂×D₄×Q₈	1	1	yes	characteristic
177	C₂×C₄₃	C₂³×Q₁₆	1	1	yes	characteristic
178	C₂×C₄₃	C₂×#576	1	1	yes	characteristic
179	C₂²×Q₁₆	—	43	1	no	—
180	Q₈²	—	43	16	no	—
181	Q₈²	—	86	8	no	—
182	C₂×C₄⋊Q₈	—	43	4	no	—
183	C₂×C₄⋊Q₈	—	86	2	no	—
184	C₂×C₄⋊Q₈	—	86	4	no	—
185	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
186	C₂×C₄×Q₈	—	43	1	no	—
187	C₂×C₄×Q₈	—	43	4	no	—
188	C₂×C₄×Q₈	—	43	2	no	—
189	C₂×C₄×Q₈	—	86	2	no	—
190	#500	—	43	16	no	—
191	#500	C₂×D₄₃	1	8	yes	—
192	#477	—	43	16	no	—
193	#477	—	43	32	no	—
194	C₈×Q₈	—	43	8	no	—
195	C₄×Q₁₆	—	43	16	no	—
196	C₄×Q₁₆	—	43	8	no	—
197	C₂×#107	—	43	1	no	—
198	C₂×#107	—	43	4	no	—
199	C₂×#107	C₂×D₄₃	1	2	yes	—
200	C₂×#107	C₂×D₄₃	1	2	yes	—
201	C₂×C₄⋊C₈	—	43	2	no	—
202	C₂×C₄⋊C₈	—	43	1	no	—
203	C₂×#103	—	43	2	no	—
204	C₂×#103	—	43	4	no	—
205	C₂×C₄×C₈	—	43	2	no	—
206	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
207	C₄₃	C₂×Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
208	C₂²×Q₈	—	2	2	no	—
209	C₂²×Q₈	—	43	1	no	—
210	C₂²×Q₈	—	43	2	no	—
211	C₂²×Q₈	—	86	4	no	—
212	C₂×Q₁₆	—	43	8	no	—
213	C₂×Q₁₆	—	43	4	no	—
214	C₂²×C₈	—	43	2	no	—
215	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	4	no	—
218	C₄⋊Q₈	—	43	16	no	—
219	C₄⋊Q₈	—	86	8	no	—
220	C₄⋊Q₈	—	86	16	no	—
221	C₄⋊Q₈	—	86	32	no	—
222	C₄⋊Q₈	C₂²×D₄₃	1	8	yes	—
223	C₄×Q₈	—	43	4	no	—
224	C₄×Q₈	—	43	4	no	—
225	C₄×Q₈	—	43	16	no	—
226	C₄×Q₈	—	43	8	no	—
227	C₄×Q₈	—	86	8	no	—
228	C₄×Q₈	—	86	16	no	—
229	C₄×Q₈	—	86	4	no	—
230	C₂×C₄⋊C₄	—	2	2	no	—
231	C₂×C₄⋊C₄	—	43	2	no	—
232	C₂×C₄⋊C₄	—	43	1	no	—
233	C₂×C₄⋊C₄	—	43	2	no	—
234	C₂×C₄⋊C₄	—	86	4	no	—
235	C₂×C₄⋊C₄	—	86	4	no	—
236	C₂×C₄⋊C₄	—	86	2	no	—
237	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
238	C₂×C₄²	—	43	2	no	—
239	C₂×C₄²	—	86	4	no	—
240	C₂×C₄²	—	86	2	no	—
241	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
242	#107	—	43	4	no	—
243	#107	—	43	16	no	—
244	#107	C₂²×D₄₃	1	8	yes	—
245	#107	C₂²×D₄₃	1	8	yes	—
246	C₄⋊C₈	—	43	8	no	—
247	C₄⋊C₈	—	43	4	no	—
248	#103	—	43	8	no	—
249	#103	—	43	16	no	—
250	C₄×C₈	—	43	8	no	—
251	C₄×C₈	C₂²×D₄₃	1	4	yes	—
252	C₂×Q₈	—	2	8	no	—
253	C₂×Q₈	—	8	4	no	—
254	C₂×Q₈	—	43	8	no	—
255	C₂×Q₈	—	43	4	no	—
256	C₂×Q₈	—	43	8	no	—
257	C₂×Q₈	—	43	4	no	—
258	C₂×Q₈	—	86	16	no	—
259	C₂×Q₈	—	86	2	no	—
260	C₂×Q₈	—	86	4	no	—
261	C₂×Q₈	—	344	8	no	—
262	C₂²×C₄	—	2	4	no	—
263	C₂²×C₄	—	43	2	no	—
264	C₂²×C₄	—	43	2	no	—
265	C₂²×C₄	—	86	2	no	—
266	C₂²×C₄	—	86	2	no	—
267	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
268	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
269	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
270	Q₁₆	—	43	16	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	43	4	no	—
273	C₂×C₈	—	86	4	no	—
274	C₂×C₈	Q₈×D₄₃	1	4	yes	—
275	C₂×C₈	C₂³×D₄₃	1	2	yes	—
276	C₂×C₈	C₂³×D₄₃	1	2	yes	—
277	C₄⋊C₄	—	2	8	no	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	—	43	8	no	—
280	C₄⋊C₄	—	43	4	no	—
281	C₄⋊C₄	—	43	4	no	—
282	C₄⋊C₄	—	86	16	no	—
283	C₄⋊C₄	—	86	16	no	—
284	C₄⋊C₄	—	86	2	no	—
285	C₄⋊C₄	—	86	8	no	—
286	C₄⋊C₄	C₂³×D₄₃	1	8	yes	—
287	C₄²	—	43	4	no	—
288	C₄²	—	43	4	no	—
289	C₄²	—	86	16	no	—
290	C₄²	—	86	8	no	—
291	C₄²	D₄×D₄₃	1	2	yes	—
292	C₄²	C₂³×D₄₃	1	2	yes	—
293	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
294	Q₈	—	8	8	no	—
295	Q₈	—	43	16	no	—
296	Q₈	—	43	8	no	—
297	Q₈	—	86	8	no	—
298	Q₈	—	86	4	no	—
299	Q₈	—	344	16	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	8	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	—	43	8	no	—
304	C₂×C₄	—	43	4	no	—
305	C₂×C₄	—	43	4	no	—
306	C₂×C₄	—	86	8	no	—
307	C₂×C₄	—	86	4	no	—
308	C₂×C₄	—	86	4	no	—
309	C₂×C₄	—	86	4	no	—
310	C₂×C₄	—	172	4	no	—
311	C₂×C₄	Q₁₆×D₄₃	1	4	yes	—
312	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
313	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
314	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
315	C₂×C₄	C₂×Q₈×D₄₃	1	2	yes	—
316	C₂×C₄	#13063373	1	2	yes	—
317	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	86	4	no	—
320	C₈	C₂×Q₈×D₄₃	1	4	yes	—
321	C₂²	#14583564	1	2	yes	—
322	C₂²	C₂×Q₁₆×D₄₃	1	2	yes	—
323	C₂²	#14583700	1	2	yes	—
324	C₂²	C₂²×D₄×D₄₃	1	1	yes	characteristic
325	C₄	—	2	2	no	—
326	C₄	—	4	8	no	—
327	C₄	—	43	8	no	—
328	C₄	—	86	4	no	—
329	C₄	—	86	8	no	—
330	C₄	—	172	4	no	—
331	C₄	#14583564	1	2	yes	—
332	C₄	C₂×Q₁₆×D₄₃	1	4	yes	—
333	C₄	C₂²×Q₈×D₄₃	1	1	yes	characteristic
334	C₄	C₂×#13063373	1	1	yes	characteristic
335	C₂	#19147478	1	4	yes	direct product factor
336	C₂	C₂×#14583564	1	1	yes	characteristic
337	C₂	C₂²×Q₁₆×D₄₃	1	1	yes	characteristic
338	C₂	C₂×#14583700	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂×C₄₃×#500	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	C₂×#500	D₄₃
hypocenter	C₄₃	C₂×Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #19147478	C₂	direct (N × H)
41. C₂×Dic₁₇₂	Q₈	semidirect
96. C₂×Dic₈₆	Q₁₆	semidirect
122. Dic₁₇₂	C₂×Q₈	semidirect
146. Dic₈₆	C₂×Q₁₆	semidirect
175. C₂×C₄₃	Q₈×Q₁₆	semidirect
207. C₄₃	C₂×Q₈×Q₁₆	semidirect
335. C₂	#19147478	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×#14583629	2
#19147478	2
C₂×#14583303	2
C₂×#14583193	2
C₂×#14583191	2
C₂×C₈×Dic₈₆	2
C₂×C₄×Dic₁₇₂	2
C₂×#14582993	2
C₂×C₄₃×#500	2
C₂×Q₈×Q₁₆	43

Subgroups — C2×#19147478 #25977045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19147478 #25977045