Subgroups — C₂×#19145135 #25966231

Order 11008.

1854 conjugacy classes of subgroups.

The subgroup lattice has 352 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	#19145135	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14583306	C₂	1	4	yes	maximal, complemented
4	C₂×#14583303	C₂	1	2	yes	maximal, complemented
5	C₂×#14583300	C₂	1	2	yes	maximal, complemented
6	C₂×#14582942	C₂	1	2	yes	maximal
7	C₂²×#13063259	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	#14583306	C₂²	1	32	yes	complemented
21	#14583303	C₂²	1	16	yes	complemented
22	#14583300	C₂²	1	16	yes	complemented
23	#14582942	C₂²	1	16	yes	—
24	C₂×#13063259	C₂²	1	2	yes	—
25	C₂×#13063259	C₂²	1	2	yes	—
26	C₂×#13063259	C₂²	1	8	yes	—
27	C₂×#13063258	C₂²	1	4	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	1	yes	characteristic
36	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
37	C₂×#106×C₄₃	C₂²	1	2	yes	—
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	1	yes	characteristic
56	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
57	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
58	C₂×C₄⋊Dic₄₃	C₂³	1	8	yes	—
59	C₂×Dic₄₃⋊C₄	—	2	4	no	—
60	C₂×Dic₄₃⋊C₄	—	2	4	no	—
61	C₂×C₂²⋊Dic₄₃	D₄	1	2	yes	—
62	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
63	#13063259	C₂³	1	8	yes	—
64	#13063259	C₂³	1	8	yes	—
65	#13063258	C₂³	1	16	yes	—
66	#13063256	C₂³	1	16	yes	—
67	#13063235	C₂³	1	8	yes	—
68	C₈×Dic₄₃	C₂³	1	8	yes	—
69	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₄₃	Q₈	1	8	yes	complemented
71	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
74	#117×C₄₃	—	2	4	no	—
75	#117×C₄₃	C₂³	1	8	yes	—
76	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
78	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
79	C₂×C₄²×C₄₃	—	2	2	no	—
80	#107×C₄₃	C₂³	1	4	yes	—
81	#107×C₄₃	C₂³	1	4	yes	—
82	#106×C₄₃	C₂³	1	8	yes	—
83	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	—	2	2	no	—
85	C₂²×C₄×C₄₃	—	2	2	no	—
86	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
89	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
90	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
91	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
92	C₂²×Dic₄₃	—	2	4	no	—
93	C₂²×Dic₄₃	—	2	8	no	—
94	C₂²×Dic₄₃	—	8	4	no	—
95	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
96	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
97	C₂×Dic₈₆	—	2	16	no	—
98	C₂×Dic₈₆	—	8	8	no	—
99	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
100	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
101	C₂×C₈×C₄₃	C₂×Q₈	1	16	yes	complemented
102	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
103	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
104	C₄⋊C₄×C₄₃	—	2	2	no	—
105	C₄⋊C₄×C₄₃	—	2	2	no	—
106	C₄⋊C₄×C₄₃	C₂⁴	1	4	yes	—
107	C₄⋊C₄×C₄₃	C₂⁴	1	4	yes	—
108	C₂²⋊C₄×C₄₃	—	2	8	no	—
109	C₄²×C₄₃	—	2	8	no	—
110	C₂²⋊Dic₄₃	C₂×D₄	1	8	yes	—
111	C₄⋊Dic₄₃	—	2	4	no	—
112	C₄⋊Dic₄₃	—	2	8	no	—
113	C₄⋊Dic₄₃	—	2	16	no	—
114	C₄⋊Dic₄₃	C₂×D₄	1	4	yes	—
115	C₄⋊Dic₄₃	C₂×D₄	1	4	yes	—
116	C₄⋊Dic₄₃	C₂⁴	1	8	yes	—
117	Dic₄₃⋊C₄	—	2	16	no	—
118	Dic₄₃⋊C₄	—	2	16	no	—
119	C₄×Dic₄₃	C₂×D₄	1	4	yes	—
120	C₄×Dic₄₃	C₂⁴	1	4	yes	—
121	#10621877	C₂×Q₈	1	16	yes	complemented
122	C₂×C₄₃⋊C₈	C₂×Q₈	1	8	yes	—
123	C₂×C₄₃⋊C₈	C₂⁴	1	4	yes	—
124	C₂³×C₄₃	—	2	2	no	—
125	C₂³×C₄₃	—	2	2	no	—
126	C₂³×C₄₃	C₄⋊Q₈	1	4	yes	—
127	C₂³×C₄₃	C₂²×D₄	1	2	yes	—
128	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	—	2	4	no	—
130	C₂×C₄×C₄₃	—	2	4	no	—
131	C₂×C₄×C₄₃	—	4	4	no	—
132	C₂×C₄×C₄₃	—	4	4	no	—
133	C₂×C₄×C₄₃	C₄⋊Q₈	1	8	yes	—
134	C₂×C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
135	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
137	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
138	C₂×C₄×C₄₃	C₂²×Q₈	1	4	yes	—
139	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
140	C₂×C₄×C₄₃	C₂²×Q₈	1	4	yes	—
141	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
142	C₂×Dic₄₃	—	2	4	no	—
143	C₂×Dic₄₃	—	2	4	no	—
144	C₂×Dic₄₃	—	2	8	no	—
145	C₂×Dic₄₃	—	4	8	no	—
146	C₂×Dic₄₃	—	8	8	no	—
147	C₂×Dic₄₃	C₂²×D₄	1	4	yes	—
148	C₂×Dic₄₃	C₂²×D₄	1	4	yes	—
149	Dic₈₆	—	8	16	no	—
150	C₈×C₄₃	C₂²×Q₈	1	4	yes	—
151	C₈×C₄₃	C₂²×Q₈	1	4	yes	—
152	C₄₃⋊C₈	C₂²×Q₈	1	8	yes	—
153	C₂×#3164	—	43	1	no	maximal, Sylow
154	C₂²×C₄₃	—	2	8	no	—
155	C₂²×C₄₃	—	2	2	no	—
156	C₂²×C₄₃	—	2	2	no	—
157	C₂²×C₄₃	C₂×C₄⋊Q₈	1	4	yes	—
158	C₂²×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
159	C₂²×C₄₃	#576	1	2	yes	—
160	C₂²×C₄₃	#578	1	2	yes	—
161	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
162	C₄×C₄₃	—	4	4	no	—
163	C₄×C₄₃	—	4	4	no	—
164	C₄×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
165	C₄×C₄₃	C₂×C₄⋊Q₈	1	4	yes	—
166	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
167	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
168	Dic₄₃	—	2	4	no	—
169	Dic₄₃	—	2	4	no	—
170	Dic₄₃	—	4	8	no	—
171	C₂×#544	—	43	2	no	—
172	#3164	—	43	16	no	—
173	C₂×#501	—	43	4	no	—
174	C₂×#500	—	43	2	no	—
175	C₂×#499	—	43	2	no	—
176	C₂×#428	—	43	2	no	—
177	C₂×#427	D₄₃	1	1	yes	characteristic
178	C₂²×#107	—	43	1	no	—
179	C₂×#405	—	43	1	no	—
180	C₂×C₄₃	—	2	4	no	—
181	C₂×C₄₃	#3164	1	4	yes	complemented
182	C₂×C₄₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
183	C₂×C₄₃	C₂×#576	1	1	yes	characteristic
184	C₂×C₄₃	C₂×#578	1	1	yes	characteristic
185	C₂²×M₁₆	—	43	1	no	—
186	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
187	#544	—	43	16	no	—
188	#544	—	86	8	no	—
189	C₂×C₄⋊Q₈	—	43	4	no	—
190	C₂×C₄⋊Q₈	—	86	2	no	—
191	C₂×#125	—	43	4	no	—
192	C₂×#125	—	86	2	no	—
193	C₂×C₂²⋊Q₈	—	86	4	no	—
194	C₂×#117	—	43	1	no	—
195	C₂×#117	C₂×D₄₃	1	2	yes	—
196	C₂²×C₄⋊C₄	—	43	2	no	—
197	#501	—	43	32	no	—
198	#500	—	43	16	no	—
199	#499	—	43	16	no	—
200	#428	—	43	16	no	—
201	#427	C₂×D₄₃	1	8	yes	—
202	C₂×#107	—	43	2	no	—
203	C₂×#107	—	43	2	no	—
204	C₂×#107	—	43	8	no	—
205	C₂×#107	—	43	4	no	—
206	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
207	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
208	C₂×#106	—	43	4	no	—
209	C₂×#106	C₂×D₄₃	1	2	yes	—
210	#405	—	43	8	no	—
211	C₂×#97	—	43	2	no	—
212	C₂×C₄×C₈	—	43	2	no	—
213	C₄₃	C₂×#3164	1	1	yes	characteristic, complemented, Sylow
214	C₂²×Q₈	—	86	4	no	—
215	C₂³×C₄	—	86	2	no	—
216	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
217	C₂×M₁₆	—	43	8	no	—
218	C₂×M₁₆	—	43	4	no	—
219	C₂²×C₈	—	43	2	no	—
220	C₂²×C₈	Dic₈₆	1	8	yes	complemented
221	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
222	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
223	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
224	C₄⋊Q₈	—	43	16	no	—
225	C₄⋊Q₈	—	86	16	no	—
226	C₄⋊Q₈	—	86	8	no	—
227	#125	—	43	16	no	—
228	#125	—	86	16	no	—
229	#125	—	86	8	no	—
230	C₂²⋊Q₈	—	86	32	no	—
231	#117	—	2	4	no	—
232	#117	—	43	4	no	—
233	#117	—	43	4	no	—
234	#117	C₂²×D₄₃	1	8	yes	—
235	C₂×C₄⋊C₄	—	43	1	no	—
236	C₂×C₄⋊C₄	—	43	1	no	—
237	C₂×C₄⋊C₄	—	43	4	no	—
238	C₂×C₄⋊C₄	—	43	8	no	—
239	C₂×C₄⋊C₄	—	86	4	no	—
240	C₂×C₄⋊C₄	—	86	4	no	—
241	C₂×C₄⋊C₄	—	86	4	no	—
242	C₂×C₄⋊C₄	—	86	2	no	—
243	C₂×C₄⋊C₄	—	86	4	no	—
244	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
245	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
246	C₂×C₂²⋊C₄	—	2	2	no	—
247	C₂×C₂²⋊C₄	—	43	2	no	—
248	C₂×C₄²	—	2	2	no	—
249	C₂×C₄²	—	43	2	no	—
250	#107	—	43	8	no	—
251	#107	—	43	8	no	—
252	#107	—	43	16	no	—
253	#107	C₂²×D₄₃	1	4	yes	—
254	#107	C₂²×D₄₃	1	4	yes	—
255	#106	—	43	16	no	—
256	#106	C₂²×D₄₃	1	8	yes	—
257	#97	—	43	8	no	—
258	C₄×C₈	—	43	8	no	—
259	C₂⁴	D₄×D₄₃	1	1	yes	characteristic
260	C₂×Q₈	—	86	16	no	—
261	C₂×Q₈	—	344	8	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	43	2	no	—
265	C₂²×C₄	—	43	2	no	—
266	C₂²×C₄	—	86	8	no	—
267	C₂²×C₄	—	86	4	no	—
268	C₂²×C₄	—	344	4	no	—
269	C₂²×C₄	C₂×Dic₈₆	1	4	yes	—
270	C₂²×C₄	D₄×D₄₃	1	2	yes	—
271	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
272	C₂²×C₄	Q₈×D₄₃	1	4	yes	—
273	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
274	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
275	M₁₆	—	43	16	no	—
276	C₂×C₈	—	43	8	no	—
277	C₂×C₈	—	43	4	no	—
278	C₂×C₈	C₂×Dic₈₆	1	4	yes	—
279	C₂×C₈	C₂×Dic₈₆	1	4	yes	—
280	C₂×C₈	C₂×Dic₈₆	1	16	yes	complemented
281	C₂×C₈	C₂³×D₄₃	1	2	yes	—
282	C₂×C₈	C₂³×D₄₃	1	2	yes	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	43	4	no	—
286	C₄⋊C₄	—	43	4	no	—
287	C₄⋊C₄	—	43	8	no	—
288	C₄⋊C₄	—	86	16	no	—
289	C₄⋊C₄	—	86	16	no	—
290	C₄⋊C₄	—	86	16	no	—
291	C₄⋊C₄	—	86	8	no	—
292	C₄⋊C₄	—	86	4	no	—
293	C₄⋊C₄	C₂³×D₄₃	1	4	yes	—
294	C₄⋊C₄	C₂³×D₄₃	1	4	yes	—
295	C₂²⋊C₄	—	2	8	no	—
296	C₂²⋊C₄	—	43	8	no	—
297	C₄²	—	2	8	no	—
298	C₄²	—	43	4	no	—
299	C₄²	—	43	4	no	—
300	C₂³	—	2	2	no	—
301	C₂³	—	2	2	no	—
302	C₂³	#13063324	1	4	yes	—
303	C₂³	C₂×D₄×D₄₃	1	2	yes	—
304	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
305	Q₈	—	344	16	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	43	4	no	—
311	C₂×C₄	—	43	4	no	—
312	C₂×C₄	—	86	4	no	—
313	C₂×C₄	—	86	4	no	—
314	C₂×C₄	—	86	8	no	—
315	C₂×C₄	—	344	8	no	—
316	C₂×C₄	—	172	8	no	—
317	C₂×C₄	#13063324	1	8	yes	—
318	C₂×C₄	#13063324	1	4	yes	—
319	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
320	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
321	C₂×C₄	C₂×Q₈×D₄₃	1	2	yes	—
322	C₂×C₄	C₂×Q₈×D₄₃	1	4	yes	—
323	C₂×C₄	C₂²×Dic₈₆	1	2	yes	—
324	C₂×C₄	C₂²×Dic₈₆	1	4	yes	—
325	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
326	C₈	—	43	8	no	—
327	C₈	C₂²×Dic₈₆	1	4	yes	—
328	C₈	C₂²×Dic₈₆	1	4	yes	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	8	no	—
332	C₂²	C₂×#13063324	1	4	yes	—
333	C₂²	C₂×#13063324	1	2	yes	—
334	C₂²	#14583701	1	2	yes	—
335	C₂²	#14583713	1	2	yes	—
336	C₂²	C₂²×D₄×D₄₃	1	1	yes	characteristic
337	C₄	—	4	4	no	—
338	C₄	—	4	4	no	—
339	C₄	—	86	4	no	—
340	C₄	—	86	4	no	—
341	C₄	—	172	8	no	—
342	C₄	C₂×#13063324	1	4	yes	—
343	C₄	C₂×#13063324	1	2	yes	—
344	C₄	C₂²×Q₈×D₄₃	1	1	yes	characteristic
345	C₄	C₂³×Dic₈₆	1	1	yes	characteristic
346	C₂	—	2	4	no	—
347	C₂	#19145135	1	4	yes	direct product factor
348	C₂	C₂²×#13063324	1	1	yes	characteristic
349	C₂	C₂×#14583701	1	1	yes	characteristic
350	C₂	C₂×#14583713	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×#427	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
hypercenter	C₂×#427	D₄₃
hypocenter	C₄₃	C₂×#3164
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. #19145135	C₂	direct (N × H)
3. C₂×#14583306	C₂	semidirect
4. C₂×#14583303	C₂	semidirect
5. C₂×#14583300	C₂	semidirect
20. #14583306	C₂²	semidirect
21. #14583303	C₂²	semidirect
22. #14583300	C₂²	semidirect
40. C₂×#10621877	Q₈	semidirect
70. C₂²×C₈×C₄₃	Q₈	semidirect
101. C₂×C₈×C₄₃	C₂×Q₈	semidirect
121. #10621877	C₂×Q₈	semidirect
181. C₂×C₄₃	#3164	semidirect
213. C₄₃	C₂×#3164	semidirect
220. C₂²×C₈	Dic₈₆	semidirect
280. C₂×C₈	C₂×Dic₈₆	semidirect
347. C₂	#19145135	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
#19145135	2
C₂×#14583306	2
C₂×#14583303	2
C₂×#14583300	2
C₂×#14582942	2
C₂²×#13063259	2
C₂×#14582854	2
C₂×C₄₃×#427	2
C₂×#3164	43

Subgroups — C2×#19145135 #25966231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19145135 #25966231