Subgroups — C₂×#19145127 #25966222

Order 11008.

1854 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583527	C₂	1	2	yes	maximal
2	#19145127	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14583307	C₂	1	4	yes	maximal, complemented
4	C₂×#14583299	C₂	1	2	yes	maximal, complemented
5	C₂×#14583296	C₂	1	2	yes	maximal, complemented
6	C₂×#14582942	C₂	1	2	yes	maximal
7	C₂²×#13063257	C₂	1	1	yes	maximal, characteristic
8	C₂×#14582854	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₃×#427	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10621877	C₂²	1	1	yes	characteristic
11	#14583527	—	2	8	no	—
12	#14583527	C₂²	1	16	yes	—
13	C₂×#13063325	—	2	2	no	—
14	C₂×#13063324	C₂²	1	4	yes	—
15	C₂×#13063317	—	2	2	no	—
16	C₂×#13063316	C₂²	1	4	yes	—
17	C₂×#13063302	—	2	4	no	—
18	C₂×#13063284	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
20	#14583307	C₂²	1	32	yes	complemented
21	#14583299	C₂²	1	16	yes	complemented
22	#14583296	C₂²	1	16	yes	complemented
23	#14582942	C₂²	1	16	yes	—
24	C₂×#13063258	C₂²	1	4	yes	—
25	C₂×#13063257	C₂²	1	2	yes	—
26	C₂×#13063257	C₂²	1	2	yes	—
27	C₂×#13063257	C₂²	1	8	yes	—
28	C₂×#13063256	C₂²	1	4	yes	—
29	#14582854	C₂²	1	8	yes	—
30	C₂×#13063235	C₂²	1	2	yes	—
31	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
32	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×#117×C₄₃	C₂²	1	2	yes	—
34	C₄₃×#427	C₂²	1	8	yes	—
35	C₂×#107×C₄₃	C₂²	1	2	yes	—
36	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
37	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
38	C₂²×Dic₈₆	—	2	4	no	—
39	C₂³×Dic₄₃	—	2	2	no	—
40	C₂×#10621877	Q₈	1	8	yes	complemented
41	C₂×#10621877	C₂³	1	4	yes	—
42	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
43	#13063325	—	2	16	no	—
44	#13063324	—	2	8	no	—
45	#13063324	C₂³	1	16	yes	—
46	#13063317	—	2	16	no	—
47	#13063316	—	2	8	no	—
48	#13063316	C₂³	1	16	yes	—
49	#13063302	—	2	32	no	—
50	#13063284	D₄	1	4	yes	—
51	#13063284	C₂³	1	4	yes	—
52	C₂×C₄⋊Dic₄₃	—	2	2	no	—
53	C₂×C₄⋊Dic₄₃	—	2	4	no	—
54	C₂×C₄⋊Dic₄₃	—	2	4	no	—
55	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
56	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
57	C₂×C₄⋊Dic₄₃	C₂³	1	8	yes	—
58	C₂×Dic₄₃⋊C₄	—	2	4	no	—
59	C₂×Dic₄₃⋊C₄	—	2	4	no	—
60	C₂×C₂²⋊Dic₄₃	D₄	1	2	yes	—
61	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
62	#13063258	C₂³	1	16	yes	—
63	#13063257	C₂³	1	8	yes	—
64	#13063257	C₂³	1	8	yes	—
65	#13063256	C₂³	1	16	yes	—
66	#13063235	C₂³	1	8	yes	—
67	C₈×Dic₄₃	C₂³	1	8	yes	—
68	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₄₃	Q₈	1	8	yes	complemented
70	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
73	#117×C₄₃	—	2	4	no	—
74	#117×C₄₃	C₂³	1	8	yes	—
75	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
77	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
78	C₂×C₄²×C₄₃	—	2	2	no	—
79	#107×C₄₃	C₂³	1	8	yes	—
80	#106×C₄₃	C₂³	1	4	yes	—
81	#106×C₄₃	C₂³	1	4	yes	—
82	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	—	2	2	no	—
84	C₂²×C₄×C₄₃	—	2	2	no	—
85	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
88	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
89	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
90	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
91	C₂²×Dic₄₃	—	2	4	no	—
92	C₂²×Dic₄₃	—	2	8	no	—
93	C₂²×Dic₄₃	—	8	4	no	—
94	C₂²×Dic₄₃	C₂×D₄	1	4	yes	—
95	C₂×Dic₈₆	—	2	16	no	—
96	C₂×Dic₈₆	—	8	8	no	—
97	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
98	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
99	C₂×C₈×C₄₃	C₂×Q₈	1	16	yes	complemented
100	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
101	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
102	C₄⋊C₄×C₄₃	—	2	2	no	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	C₂⁴	1	4	yes	—
105	C₄⋊C₄×C₄₃	C₂⁴	1	4	yes	—
106	C₂²⋊C₄×C₄₃	—	2	8	no	—
107	C₄²×C₄₃	—	2	8	no	—
108	C₂²⋊Dic₄₃	C₂×D₄	1	8	yes	—
109	C₄⋊Dic₄₃	—	2	4	no	—
110	C₄⋊Dic₄₃	—	2	8	no	—
111	C₄⋊Dic₄₃	—	2	16	no	—
112	C₄⋊Dic₄₃	C₂×D₄	1	8	yes	—
113	C₄⋊Dic₄₃	C₂⁴	1	8	yes	—
114	Dic₄₃⋊C₄	—	2	16	no	—
115	Dic₄₃⋊C₄	—	2	16	no	—
116	C₄×Dic₄₃	C₂×D₄	1	4	yes	—
117	C₄×Dic₄₃	C₂⁴	1	4	yes	—
118	#10621877	C₂×Q₈	1	16	yes	complemented
119	C₂×C₄₃⋊C₈	C₂×Q₈	1	8	yes	—
120	C₂×C₄₃⋊C₈	C₂⁴	1	4	yes	—
121	C₂³×C₄₃	—	2	4	no	—
122	C₂³×C₄₃	C₄⋊Q₈	1	4	yes	—
123	C₂³×C₄₃	C₂²×D₄	1	2	yes	—
124	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	—	2	4	no	—
126	C₂×C₄×C₄₃	—	2	4	no	—
127	C₂×C₄×C₄₃	—	4	4	no	—
128	C₂×C₄×C₄₃	—	4	4	no	—
129	C₂×C₄×C₄₃	C₄⋊Q₈	1	8	yes	—
130	C₂×C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
131	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
133	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
134	C₂×C₄×C₄₃	C₂²×Q₈	1	4	yes	—
135	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
136	C₂×C₄×C₄₃	C₂²×Q₈	1	4	yes	—
137	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
138	C₂×Dic₄₃	—	2	8	no	—
139	C₂×Dic₄₃	—	2	8	no	—
140	C₂×Dic₄₃	—	4	8	no	—
141	C₂×Dic₄₃	—	8	8	no	—
142	C₂×Dic₄₃	C₂²×D₄	1	8	yes	—
143	Dic₈₆	—	8	16	no	—
144	C₈×C₄₃	C₂²×Q₈	1	4	yes	—
145	C₈×C₄₃	C₂²×Q₈	1	4	yes	—
146	C₄₃⋊C₈	C₂²×Q₈	1	8	yes	—
147	C₂×#3163	—	43	1	no	maximal, Sylow
148	C₂²×C₄₃	—	2	8	no	—
149	C₂²×C₄₃	—	2	4	no	—
150	C₂²×C₄₃	C₂×C₄⋊Q₈	1	4	yes	—
151	C₂²×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
152	C₂²×C₄₃	#577	1	4	yes	—
153	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
154	C₄×C₄₃	—	4	4	no	—
155	C₄×C₄₃	—	4	4	no	—
156	C₄×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
157	C₄×C₄₃	C₂×C₄⋊Q₈	1	4	yes	—
158	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
159	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
160	Dic₄₃	—	2	8	no	—
161	Dic₄₃	—	4	8	no	—
162	C₂×#544	—	43	2	no	—
163	#3163	—	43	16	no	—
164	C₂×#501	—	43	4	no	—
165	C₂×#499	—	43	2	no	—
166	C₂×#498	—	43	2	no	—
167	C₂×#428	—	43	2	no	—
168	C₂×#427	D₄₃	1	1	yes	characteristic
169	C₂²×#106	—	43	1	no	—
170	C₂×#405	—	43	1	no	—
171	C₂×C₄₃	—	2	4	no	—
172	C₂×C₄₃	#3163	1	4	yes	complemented
173	C₂×C₄₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
174	C₂×C₄₃	C₂×#577	1	2	yes	—
175	C₂²×M₁₆	—	43	1	no	—
176	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
177	#544	—	43	16	no	—
178	#544	—	86	8	no	—
179	C₂×C₄⋊Q₈	—	43	4	no	—
180	C₂×C₄⋊Q₈	—	86	2	no	—
181	C₂×#125	—	43	4	no	—
182	C₂×#125	—	86	2	no	—
183	C₂×C₂²⋊Q₈	—	86	4	no	—
184	C₂×#117	—	43	1	no	—
185	C₂×#117	C₂×D₄₃	1	2	yes	—
186	C₂²×C₄⋊C₄	—	43	2	no	—
187	#501	—	43	32	no	—
188	#499	—	43	16	no	—
189	#498	—	43	16	no	—
190	#428	—	43	16	no	—
191	#427	C₂×D₄₃	1	8	yes	—
192	C₂×#107	—	43	4	no	—
193	C₂×#107	C₂×D₄₃	1	2	yes	—
194	C₂×#106	—	43	2	no	—
195	C₂×#106	—	43	2	no	—
196	C₂×#106	—	43	8	no	—
197	C₂×#106	—	43	4	no	—
198	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
199	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
200	#405	—	43	8	no	—
201	C₂×#97	—	43	2	no	—
202	C₂×C₄×C₈	—	43	2	no	—
203	C₄₃	C₂×#3163	1	1	yes	characteristic, complemented, Sylow
204	C₂²×Q₈	—	86	4	no	—
205	C₂³×C₄	—	86	2	no	—
206	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
207	C₂×M₁₆	—	43	8	no	—
208	C₂×M₁₆	—	43	4	no	—
209	C₂²×C₈	—	43	2	no	—
210	C₂²×C₈	Dic₈₆	1	8	yes	complemented
211	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
214	C₄⋊Q₈	—	43	16	no	—
215	C₄⋊Q₈	—	86	16	no	—
216	C₄⋊Q₈	—	86	8	no	—
217	#125	—	43	16	no	—
218	#125	—	86	16	no	—
219	#125	—	86	8	no	—
220	C₂²⋊Q₈	—	86	32	no	—
221	#117	—	2	4	no	—
222	#117	—	43	4	no	—
223	#117	—	43	4	no	—
224	#117	C₂²×D₄₃	1	8	yes	—
225	C₂×C₄⋊C₄	—	43	2	no	—
226	C₂×C₄⋊C₄	—	43	4	no	—
227	C₂×C₄⋊C₄	—	43	8	no	—
228	C₂×C₄⋊C₄	—	86	4	no	—
229	C₂×C₄⋊C₄	—	86	4	no	—
230	C₂×C₄⋊C₄	—	86	4	no	—
231	C₂×C₄⋊C₄	—	86	2	no	—
232	C₂×C₄⋊C₄	—	86	4	no	—
233	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
234	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₂²⋊C₄	—	43	2	no	—
237	C₂×C₄²	—	2	2	no	—
238	C₂×C₄²	—	43	2	no	—
239	#107	—	43	16	no	—
240	#107	C₂²×D₄₃	1	8	yes	—
241	#106	—	43	8	no	—
242	#106	—	43	8	no	—
243	#106	—	43	16	no	—
244	#106	C₂²×D₄₃	1	4	yes	—
245	#106	C₂²×D₄₃	1	4	yes	—
246	#97	—	43	8	no	—
247	C₄×C₈	—	43	8	no	—
248	C₂⁴	D₄×D₄₃	1	1	yes	characteristic
249	C₂×Q₈	—	86	16	no	—
250	C₂×Q₈	—	344	8	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	43	4	no	—
254	C₂²×C₄	—	86	8	no	—
255	C₂²×C₄	—	86	4	no	—
256	C₂²×C₄	—	344	4	no	—
257	C₂²×C₄	C₂×Dic₈₆	1	4	yes	—
258	C₂²×C₄	D₄×D₄₃	1	2	yes	—
259	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
260	C₂²×C₄	Q₈×D₄₃	1	4	yes	—
261	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
262	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
263	M₁₆	—	43	16	no	—
264	C₂×C₈	—	43	8	no	—
265	C₂×C₈	—	43	4	no	—
266	C₂×C₈	C₂×Dic₈₆	1	4	yes	—
267	C₂×C₈	C₂×Dic₈₆	1	4	yes	—
268	C₂×C₈	C₂×Dic₈₆	1	16	yes	complemented
269	C₂×C₈	C₂³×D₄₃	1	2	yes	—
270	C₂×C₈	C₂³×D₄₃	1	2	yes	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	43	8	no	—
274	C₄⋊C₄	—	43	8	no	—
275	C₄⋊C₄	—	86	16	no	—
276	C₄⋊C₄	—	86	16	no	—
277	C₄⋊C₄	—	86	16	no	—
278	C₄⋊C₄	—	86	8	no	—
279	C₄⋊C₄	—	86	4	no	—
280	C₄⋊C₄	C₂³×D₄₃	1	4	yes	—
281	C₄⋊C₄	C₂³×D₄₃	1	4	yes	—
282	C₂²⋊C₄	—	2	8	no	—
283	C₂²⋊C₄	—	43	8	no	—
284	C₄²	—	2	8	no	—
285	C₄²	—	43	4	no	—
286	C₄²	—	43	4	no	—
287	C₂³	—	2	4	no	—
288	C₂³	#13063324	1	4	yes	—
289	C₂³	C₂×D₄×D₄₃	1	2	yes	—
290	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
291	Q₈	—	344	16	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	43	8	no	—
297	C₂×C₄	—	86	8	no	—
298	C₂×C₄	—	86	8	no	—
299	C₂×C₄	—	344	8	no	—
300	C₂×C₄	—	172	8	no	—
301	C₂×C₄	#13063324	1	8	yes	—
302	C₂×C₄	#13063324	1	4	yes	—
303	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
304	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
305	C₂×C₄	C₂×Q₈×D₄₃	1	2	yes	—
306	C₂×C₄	C₂×Q₈×D₄₃	1	4	yes	—
307	C₂×C₄	C₂²×Dic₈₆	1	2	yes	—
308	C₂×C₄	C₂²×Dic₈₆	1	4	yes	—
309	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
310	C₈	—	43	8	no	—
311	C₈	C₂²×Dic₈₆	1	4	yes	—
312	C₈	C₂²×Dic₈₆	1	4	yes	—
313	C₂²	—	2	4	no	—
314	C₂²	—	2	8	no	—
315	C₂²	C₂×#13063324	1	4	yes	—
316	C₂²	C₂×#13063324	1	2	yes	—
317	C₂²	#14583706	1	4	yes	—
318	C₂²	C₂²×D₄×D₄₃	1	1	yes	characteristic
319	C₄	—	4	4	no	—
320	C₄	—	4	4	no	—
321	C₄	—	86	8	no	—
322	C₄	—	172	8	no	—
323	C₄	C₂×#13063324	1	4	yes	—
324	C₄	C₂×#13063324	1	2	yes	—
325	C₄	C₂²×Q₈×D₄₃	1	1	yes	characteristic
326	C₄	C₂³×Dic₈₆	1	1	yes	characteristic
327	C₂	—	2	4	no	—
328	C₂	#19145127	1	4	yes	direct product factor
329	C₂	C₂²×#13063324	1	1	yes	characteristic
330	C₂	C₂×#14583706	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂×C₄₃×#427	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
hypercenter	C₂×#427	D₄₃
hypocenter	C₄₃	C₂×#3163
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. #19145127	C₂	direct (N × H)
3. C₂×#14583307	C₂	semidirect
4. C₂×#14583299	C₂	semidirect
5. C₂×#14583296	C₂	semidirect
20. #14583307	C₂²	semidirect
21. #14583299	C₂²	semidirect
22. #14583296	C₂²	semidirect
40. C₂×#10621877	Q₈	semidirect
69. C₂²×C₈×C₄₃	Q₈	semidirect
99. C₂×C₈×C₄₃	C₂×Q₈	semidirect
118. #10621877	C₂×Q₈	semidirect
172. C₂×C₄₃	#3163	semidirect
203. C₄₃	C₂×#3163	semidirect
210. C₂²×C₈	Dic₈₆	semidirect
268. C₂×C₈	C₂×Dic₈₆	semidirect
328. C₂	#19145127	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×#14583527	2
#19145127	2
C₂×#14583307	2
C₂×#14583299	2
C₂×#14583296	2
C₂×#14582942	2
C₂²×#13063257	2
C₂×#14582854	2
C₂×C₄₃×#427	2
C₂×#3163	43

Subgroups — C2×#19145127 #25966222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19145127 #25966222