Subgroups — C₂×#19133465 #25066143

Order 11008.

674 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583026	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583023	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
4	#19133465	C₂	1	8	yes	maximal, direct product factor
5	C₂×#14582629	C₂	1	1	yes	maximal, characteristic
6	C₂×#14582629	C₂	1	1	yes	maximal, characteristic
7	C₂×#14582621	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₃×#334	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₂	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
12	#14583026	—	2	2	no	—
13	#14583026	C₂²	1	4	yes	—
14	#14583023	—	2	2	no	—
15	#14583023	C₂²	1	4	yes	—
16	C₄×Dic₁₇₂	C₄	1	4	yes	—
17	C₄×Dic₁₇₂	C₂²	1	4	yes	—
18	C₂×#13063259	C₄	1	1	yes	characteristic
19	C₂×Dic₄₃⋊C₈	—	2	1	no	—
20	C₂×Dic₄₃⋊C₈	—	2	1	no	—
21	C₂×#13063250	C₄	1	1	yes	characteristic
22	C₂×#13063250	C₄	1	1	yes	characteristic
23	C₂×#13063235	—	2	1	no	—
24	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₄₃	—	2	1	no	—
27	#14582629	C₂²	1	4	yes	—
28	#14582629	C₂²	1	4	yes	—
29	#14582621	C₂²	1	4	yes	—
30	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₄₃×#334	C₂²	1	4	yes	—
34	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
36	C₂×Dic₁₇₂	—	2	4	no	—
37	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
38	C₂²×C₄₃⋊C₈	—	2	1	no	—
39	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
40	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
41	C₄×Dic₈₆	—	2	1	no	—
42	C₄×Dic₈₆	—	2	1	no	—
43	C₄×Dic₈₆	—	2	1	no	—
44	C₄×Dic₈₆	—	2	1	no	—
45	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
46	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
47	C₄×Dic₈₆	C₂³	1	2	yes	—
48	C₄×Dic₈₆	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂×Dic₄₃⋊C₄	—	2	1	no	—
52	C₂×Dic₄₃⋊C₄	—	2	1	no	—
53	C₂×C₄×Dic₄₃	—	2	1	no	—
54	C₂×C₄×Dic₄₃	—	2	1	no	—
55	#13063259	C₂×C₄	1	4	yes	—
56	Dic₄₃⋊C₈	—	2	4	no	—
57	Dic₄₃⋊C₈	—	2	4	no	—
58	#13063250	C₂×C₄	1	4	yes	—
59	#13063250	C₂×C₄	1	4	yes	—
60	#13063235	—	2	4	no	—
61	C₄×C₄₃⋊C₈	—	2	1	no	—
62	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
63	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
64	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
65	C₈×Dic₄₃	—	2	4	no	—
66	C₂²×C₈×C₄₃	—	2	1	no	—
67	C₂²×C₈×C₄₃	—	2	1	no	—
68	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₄₃	—	2	1	no	—
72	C₄⋊C₈×C₄₃	—	2	1	no	—
73	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
74	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
75	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
76	C₄×C₈×C₄₃	C₂³	1	2	yes	—
77	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
80	C₂²×Dic₄₃	—	2	1	no	—
81	C₂²×Dic₄₃	—	2	1	no	—
82	C₂²×Dic₄₃	—	2	1	no	—
83	C₂²×Dic₄₃	—	2	1	no	—
84	C₂×Dic₈₆	—	2	1	no	—
85	C₂×Dic₈₆	—	2	2	no	—
86	C₂×Dic₈₆	—	2	2	no	—
87	C₂×Dic₈₆	—	2	2	no	—
88	C₂×Dic₈₆	—	2	1	no	—
89	C₂×Dic₈₆	M₁₆	1	4	yes	—
90	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
91	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₄₃	—	2	2	no	—
93	C₂×C₈×C₄₃	—	2	2	no	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	—	2	2	no	—
96	C₂×C₈×C₄₃	—	4	2	no	—
97	C₂×C₈×C₄₃	—	4	2	no	—
98	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₄₃	#41	1	1	yes	characteristic
103	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
104	C₄⋊Dic₄₃	—	2	1	no	—
105	C₄⋊Dic₄₃	—	2	1	no	—
106	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
107	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
108	Dic₄₃⋊C₄	—	2	2	no	—
109	Dic₄₃⋊C₄	—	2	2	no	—
110	Dic₄₃⋊C₄	—	2	2	no	—
111	Dic₄₃⋊C₄	—	2	2	no	—
112	C₄×Dic₄₃	—	2	2	no	—
113	C₄×Dic₄₃	—	2	2	no	—
114	C₄×Dic₄₃	—	2	2	no	—
115	C₄×Dic₄₃	—	2	2	no	—
116	C₂×C₄₃⋊C₈	—	2	4	no	—
117	C₂×C₄₃⋊C₈	—	2	2	no	—
118	C₂×C₄₃⋊C₈	—	2	2	no	—
119	C₂×C₄₃⋊C₈	—	2	2	no	—
120	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
121	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
122	Dic₁₇₂	—	2	8	no	—
123	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	—	2	2	no	—
125	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₄₃	#131	1	2	yes	—
130	C₂×C₄×C₄₃	#137	1	2	yes	—
131	C₂×C₄×C₄₃	#137	1	2	yes	—
132	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₄₃	—	2	2	no	—
136	C₂×Dic₄₃	—	2	4	no	—
137	C₂×Dic₄₃	—	2	2	no	—
138	C₂×Dic₄₃	—	2	2	no	—
139	C₂×Dic₄₃	—	2	2	no	—
140	C₂×Dic₄₃	—	4	2	no	—
141	C₂×Dic₄₃	—	4	2	no	—
142	C₂×Dic₄₃	—	4	2	no	—
143	Dic₈₆	—	2	2	no	—
144	Dic₈₆	—	2	4	no	—
145	Dic₈₆	—	2	2	no	—
146	Dic₈₆	—	2	2	no	—
147	Dic₈₆	—	4	2	no	—
148	Dic₈₆	C₂×M₁₆	1	4	yes	—
149	C₈×C₄₃	—	2	2	no	—
150	C₈×C₄₃	—	2	2	no	—
151	C₈×C₄₃	—	4	2	no	—
152	C₈×C₄₃	—	4	2	no	—
153	C₄₃⋊C₈	—	2	4	no	—
154	C₄₃⋊C₈	—	2	2	no	—
155	C₄₃⋊C₈	—	2	2	no	—
156	C₂×#1587	—	43	1	no	maximal, Sylow
157	C₂²×C₄₃	#435	1	2	yes	—
158	C₂²×C₄₃	#441	1	2	yes	—
159	C₂²×C₄₃	#443	1	2	yes	—
160	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₄₃	—	2	1	no	—
162	C₄×C₄₃	—	2	1	no	—
163	C₄×C₄₃	#435	1	2	yes	—
164	C₄×C₄₃	#441	1	2	yes	—
165	C₄×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
166	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
167	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
168	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
169	Dic₄₃	—	2	4	no	—
170	Dic₄₃	—	4	2	no	—
171	Dic₄₃	—	4	2	no	—
172	Dic₄₃	—	4	2	no	—
173	C₂×#446	—	43	1	no	—
174	C₂×C₈×Q₈	—	43	1	no	—
175	C₂×C₄×Q₁₆	—	43	1	no	—
176	#1587	—	43	8	no	—
177	C₂×#334	D₄₃	1	1	yes	characteristic
178	C₂×Q₈⋊C₈	—	43	1	no	—
179	C₂×Q₈⋊C₈	—	43	1	no	—
180	C₂×#322	—	43	1	no	—
181	C₂×C₄₃	#1587	1	4	yes	complemented
182	C₂×C₄₃	C₂×#435	1	1	yes	characteristic
183	C₂×C₄₃	C₂×#441	1	1	yes	characteristic
184	C₂×C₄₃	C₂×#443	1	1	yes	characteristic
185	C₂²×Q₁₆	—	43	1	no	—
186	C₂×C₄×Q₈	—	43	1	no	—
187	C₂×C₄×Q₈	—	43	1	no	—
188	#446	—	43	4	no	—
189	#446	—	86	2	no	—
190	C₈×Q₈	—	43	4	no	—
191	C₈×Q₈	—	86	2	no	—
192	C₄×Q₁₆	—	43	4	no	—
193	C₄×Q₁₆	—	43	4	no	—
194	C₂×#107	—	43	1	no	—
195	C₂×C₄⋊C₈	—	86	1	no	—
196	C₂×C₄⋊C₈	—	86	1	no	—
197	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
198	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
199	C₂×#103	—	43	1	no	—
200	C₂×#103	—	43	1	no	—
201	C₂×#97	—	86	1	no	—
202	C₂×C₄×C₈	—	43	1	no	—
203	C₂×C₄×C₈	—	43	1	no	—
204	C₂×C₄×C₈	—	86	1	no	—
205	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
206	#334	C₂×D₄₃	1	4	yes	—
207	Q₈⋊C₈	—	43	4	no	—
208	Q₈⋊C₈	—	43	4	no	—
209	#322	—	43	4	no	—
210	C₄₃	C₂×#1587	1	1	yes	characteristic, complemented, Sylow
211	C₂²×Q₈	—	43	1	no	—
212	C₂²×Q₈	—	43	1	no	—
213	C₂×Q₁₆	—	43	4	no	—
214	C₂×Q₁₆	—	86	4	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	43	1	no	—
218	C₂²×C₈	—	43	1	no	—
219	C₂²×C₈	—	86	1	no	—
220	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
221	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
222	C₄×Q₈	—	43	2	no	—
223	C₄×Q₈	—	43	2	no	—
224	C₄×Q₈	—	43	2	no	—
225	C₄×Q₈	—	43	2	no	—
226	C₄×Q₈	—	86	1	no	—
227	C₄×Q₈	—	86	1	no	—
228	C₄×Q₈	—	86	1	no	—
229	C₄×Q₈	—	86	1	no	—
230	C₂×C₄⋊C₄	—	43	1	no	—
231	C₂×C₄⋊C₄	—	43	1	no	—
232	C₂×C₄⋊C₄	—	86	1	no	—
233	C₂×C₄⋊C₄	—	86	1	no	—
234	C₂×C₄²	—	86	1	no	—
235	C₂×C₄²	—	86	1	no	—
236	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
237	#107	—	43	4	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	86	4	no	—
241	C₄⋊C₈	—	86	4	no	—
242	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
243	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
244	#103	—	43	4	no	—
245	#103	—	43	4	no	—
246	#97	—	86	4	no	—
247	C₄×C₈	—	43	2	no	—
248	C₄×C₈	—	43	2	no	—
249	C₄×C₈	—	43	2	no	—
250	C₄×C₈	—	86	4	no	—
251	C₄×C₈	—	86	1	no	—
252	C₄×C₈	C₄×D₄₃	1	2	yes	—
253	C₄×C₈	C₂²×D₄₃	1	2	yes	—
254	C₂×Q₈	—	43	4	no	—
255	C₂×Q₈	—	43	2	no	—
256	C₂×Q₈	—	43	2	no	—
257	C₂×Q₈	—	86	1	no	—
258	C₂×Q₈	—	86	1	no	—
259	C₂×Q₈	—	86	2	no	—
260	C₂×Q₈	—	86	2	no	—
261	C₂×Q₈	—	86	2	no	—
262	C₂²×C₄	—	86	1	no	—
263	C₂²×C₄	—	86	1	no	—
264	C₂²×C₄	—	86	1	no	—
265	C₂²×C₄	—	86	1	no	—
266	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
267	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
268	C₂²×C₄	#10621902	1	1	yes	characteristic
269	Q₁₆	—	86	8	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	—	4	2	no	—
276	C₂×C₈	—	43	2	no	—
277	C₂×C₈	—	43	2	no	—
278	C₂×C₈	—	86	4	no	—
279	C₂×C₈	—	86	2	no	—
280	C₂×C₈	—	86	2	no	—
281	C₂×C₈	—	86	2	no	—
282	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
283	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
284	C₄⋊C₄	—	43	2	no	—
285	C₄⋊C₄	—	43	2	no	—
286	C₄⋊C₄	—	86	1	no	—
287	C₄⋊C₄	—	86	1	no	—
288	C₄⋊C₄	—	86	2	no	—
289	C₄⋊C₄	—	86	2	no	—
290	C₄⋊C₄	—	86	2	no	—
291	C₄⋊C₄	—	86	2	no	—
292	C₄²	—	86	2	no	—
293	C₄²	—	86	2	no	—
294	C₄²	—	86	2	no	—
295	C₄²	—	86	2	no	—
296	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
297	C₄²	D₄×D₄₃	1	1	yes	characteristic
298	C₄²	#10621902	1	1	yes	characteristic
299	C₄²	C₂³×D₄₃	1	1	yes	characteristic
300	C₂³	#13063287	1	1	yes	characteristic
301	Q₈	—	43	4	no	—
302	Q₈	—	86	2	no	—
303	Q₈	—	86	4	no	—
304	Q₈	—	86	2	no	—
305	Q₈	—	86	2	no	—
306	Q₈	—	172	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	86	2	no	—
309	C₂×C₄	—	86	4	no	—
310	C₂×C₄	—	86	2	no	—
311	C₂×C₄	—	86	2	no	—
312	C₂×C₄	—	86	2	no	—
313	C₂×C₄	—	172	2	no	—
314	C₂×C₄	—	172	2	no	—
315	C₂×C₄	—	172	2	no	—
316	C₂×C₄	#13063287	1	1	yes	characteristic
317	C₂×C₄	#13063287	1	1	yes	characteristic
318	C₂×C₄	#13063287	1	1	yes	characteristic
319	C₂×C₄	M₁₆×D₄₃	1	2	yes	—
320	C₂×C₄	#13063331	1	2	yes	—
321	C₂×C₄	#13063354	1	2	yes	—
322	C₂×C₄	#13063354	1	2	yes	—
323	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
324	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
325	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	—	4	2	no	—
329	C₈	—	4	2	no	—
330	C₈	—	86	4	no	—
331	C₈	—	86	2	no	—
332	C₈	—	86	2	no	—
333	C₂²	#14582969	1	2	yes	—
334	C₂²	#14583005	1	2	yes	—
335	C₂²	#14583015	1	2	yes	—
336	C₂²	C₂×#13063287	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	86	4	no	—
340	C₄	—	172	2	no	—
341	C₄	—	172	2	no	—
342	C₄	—	172	2	no	—
343	C₄	#14582969	1	2	yes	—
344	C₄	#14583005	1	2	yes	—
345	C₄	C₂×M₁₆×D₄₃	1	1	yes	characteristic
346	C₄	C₂×#13063331	1	1	yes	characteristic
347	C₄	C₂×#13063354	1	1	yes	characteristic
348	C₄	C₂×#13063354	1	1	yes	characteristic
349	C₂	#19133465	1	4	yes	direct product factor
350	C₂	C₂×#14582969	1	1	yes	characteristic
351	C₂	C₂×#14583005	1	1	yes	characteristic
352	C₂	C₂×#14583015	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#334	C₂
socle, prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	C₂×#334	D₄₃
hypocenter	C₄₃	C₂×#1587
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
4. #19133465	C₂	direct (N × H)
181. C₂×C₄₃	#1587	semidirect
210. C₄₃	C₂×#1587	semidirect
349. C₂	#19133465	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×#14583026	2
C₂×#14583023	2
C₂×C₄×Dic₁₇₂	2
#19133465	2
C₂×#14582629	2
C₂×#14582621	2
C₂×C₄₃×#334	2
C₂×#1587	43

Subgroups — C2×#19133465 #25066143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19133465 #25066143