Subgroups — C₂×#19143705 #25961643

Order 11008.

2434 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×C₄×#10621902	C₂	1	1	yes	maximal, characteristic
2	#19143705	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×D₁₇₂	C₂	1	3	yes	maximal, complemented
5	C₂×#14582964	C₂	1	3	yes	maximal, complemented
6	C₂×#14582950	C₂	1	3	yes	maximal, complemented
7	C₂×C₄⋊C₈×D₄₃	C₂	1	3	yes	maximal, complemented
8	C₂×C₈×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10621902	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₄₃	C₂²	1	6	yes	—
11	C₄×#10621902	C₄	1	8	yes	—
12	C₄×#10621902	C₂²	1	8	yes	—
13	C₂×Q₈×Dic₄₃	C₄	1	1	yes	characteristic
14	C₂×C₄×D₁₇₂	C₂²	1	3	yes	—
15	C₂×#13063287	C₄	1	3	yes	—
16	C₂×#13063283	C₄	1	3	yes	—
17	C₂×C₄²×D₄₃	C₂²	1	3	yes	—
18	Q₈×C₄₃⋊C₈	C₂²	1	8	yes	complemented
19	C₈×D₁₇₂	C₂²	1	24	yes	complemented
20	#14582964	C₂²	1	24	yes	complemented
21	#14582950	C₂²	1	24	yes	complemented
22	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	3	yes	—
23	C₂×Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×Dic₄₃⋊C₈	C₂²	1	3	yes	—
25	C₄⋊C₈×D₄₃	C₂²	1	24	yes	complemented
26	C₂×(C₂×D₄₃)⋊C₈	C₂²	1	3	yes	—
27	C₂×(C₂×D₄₃)⋊C₈	C₂²	1	3	yes	—
28	C₂×C₄×C₄₃⋊C₈	C₂²	1	3	yes	—
29	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₈×Q₈×C₄₃	C₂²	1	8	yes	complemented
31	C₂×C₄⋊C₈×C₄₃	C₂²	1	3	yes	—
32	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	—
33	C₂×#10621902	C₈	1	16	yes	complemented
34	C₂×#10621902	C₂×C₄	1	8	yes	—
35	C₂²×D₁₇₂	C₂×C₄	1	3	yes	—
36	C₂²×C₄×D₄₃	C₂×C₄	1	3	yes	—
37	C₂²×C₄×D₄₃	C₂×C₄	1	3	yes	—
38	C₂²×C₄×D₄₃	C₂³	1	3	yes	—
39	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₄₃⋊C₈	C₂³	1	3	yes	—
41	C₂×C₈×D₄₃	—	4	12	no	—
42	C₂×C₈×D₄₃	C₂³	1	24	yes	—
43	Q₈×Dic₄₃	C₂×C₄	1	8	yes	—
44	C₄×D₁₇₂	C₂×C₄	1	12	yes	—
45	C₄×D₁₇₂	C₂³	1	12	yes	—
46	#13063287	C₂×C₄	1	24	yes	—
47	#13063283	C₂×C₄	1	24	yes	—
48	C₂×C₄⋊Dic₄₃	C₂×C₄	1	3	yes	—
49	C₂×(C₂×D₄₃)⋊C₄	C₂×C₄	1	6	yes	—
50	C₂×C₄×Dic₄₃	C₂×C₄	1	3	yes	—
51	C₂×C₄×Dic₄₃	C₂×C₄	1	3	yes	—
52	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
53	C₄²×D₄₃	C₂×C₄	1	12	yes	—
54	C₄²×D₄₃	C₂³	1	12	yes	—
55	C₄⋊(C₄₃⋊C₈)	C₂³	1	12	yes	—
56	Dic₄₃⋊C₈	C₂³	1	12	yes	—
57	Dic₄₃⋊C₈	C₂³	1	4	yes	—
58	(C₂×D₄₃)⋊C₈	C₂³	1	12	yes	—
59	(C₂×D₄₃)⋊C₈	C₂³	1	12	yes	—
60	C₄×C₄₃⋊C₈	C₂³	1	12	yes	—
61	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
64	C₄×Q₈×C₄₃	C₂×C₄	1	4	yes	—
65	C₄×Q₈×C₄₃	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	3	yes	—
67	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
68	C₄⋊C₈×C₄₃	C₂³	1	12	yes	—
69	C₄×C₈×C₄₃	C₂³	1	12	yes	—
70	C₂³×D₄₃	C₂²×C₄	1	3	yes	—
71	C₂×Q₈×C₄₃	C₂×C₈	1	8	yes	complemented
72	C₂×Q₈×C₄₃	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
74	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×Dic₄₃	C₂²×C₄	1	3	yes	—
78	C₂²×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
79	#10621902	C₂×C₈	1	64	yes	complemented
80	C₂×D₁₇₂	C₂×C₈	1	24	yes	complemented
81	C₂×D₁₇₂	C₂²×C₄	1	12	yes	—
82	C₂×C₄×D₄₃	—	2	6	no	—
83	C₂×C₄×D₄₃	—	2	6	no	—
84	C₂×C₄×D₄₃	—	4	6	no	—
85	C₂×C₄×D₄₃	C₂×C₈	1	24	yes	complemented
86	C₂×C₄×D₄₃	C₂²×C₄	1	6	yes	—
87	C₂×C₄×D₄₃	C₂²×C₄	1	12	yes	—
88	C₂×C₄×D₄₃	C₂²×C₄	1	12	yes	—
89	C₂×C₄×D₄₃	C₂⁴	1	6	yes	—
90	C₂×C₈×C₄₃	—	2	6	no	—
91	C₂×C₈×C₄₃	—	2	2	no	—
92	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
93	C₂×C₈×C₄₃	C₂⁴	1	6	yes	—
94	C₄⋊C₄×C₄₃	C₂²×C₄	1	12	yes	—
95	C₄²×C₄₃	C₂²×C₄	1	6	yes	—
96	C₄²×C₄₃	C₂⁴	1	6	yes	—
97	(C₂×D₄₃)⋊C₄	C₂²×C₄	1	24	yes	—
98	C₄⋊Dic₄₃	C₂²×C₄	1	12	yes	—
99	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
100	C₄×Dic₄₃	C₂²×C₄	1	12	yes	—
101	C₄×Dic₄₃	C₂²×C₄	1	12	yes	—
102	C₄×Dic₄₃	C₂⁴	1	2	yes	—
103	C₂×C₄₃⋊C₈	—	2	6	no	—
104	C₂×C₄₃⋊C₈	—	2	2	no	—
105	C₂×C₄₃⋊C₈	C₂⁴	1	2	yes	—
106	C₂×C₄₃⋊C₈	C₂⁴	1	6	yes	—
107	C₈×D₄₃	—	4	24	no	—
108	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
109	C₂²×D₄₃	—	2	6	no	—
110	C₂²×D₄₃	—	2	6	no	—
111	C₂²×D₄₃	C₂²×C₈	1	12	yes	—
112	C₂²×D₄₃	C₂³×C₄	1	6	yes	—
113	Q₈×C₄₃	C₂²×C₈	1	16	yes	complemented
114	C₂×C₄×C₄₃	—	2	6	no	—
115	C₂×C₄×C₄₃	C₂²×C₈	1	12	yes	—
116	C₂×C₄×C₄₃	C₂³×C₄	1	6	yes	—
117	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂³×C₄	1	6	yes	—
119	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
120	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
121	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
122	C₂×Dic₄₃	—	2	6	no	—
123	C₂×Dic₄₃	—	2	2	no	—
124	C₂×Dic₄₃	C₂²×C₈	1	4	yes	—
125	C₂×Dic₄₃	C₂³×C₄	1	2	yes	—
126	C₂×Dic₄₃	C₂³×C₄	1	6	yes	—
127	C₂×Dic₄₃	C₂³×C₄	1	2	yes	—
128	D₁₇₂	C₂²×C₈	1	48	yes	complemented
129	C₄×D₄₃	—	2	12	no	—
130	C₄×D₄₃	—	2	6	no	—
131	C₄×D₄₃	—	2	6	no	—
132	C₄×D₄₃	—	4	12	no	—
133	C₄×D₄₃	C₂²×C₈	1	48	yes	complemented
134	C₈×C₄₃	—	2	6	no	—
135	C₈×C₄₃	—	2	2	no	—
136	C₄₃⋊C₈	—	2	6	no	—
137	C₄₃⋊C₈	—	2	2	no	—
138	C₂×#2988	—	43	1	no	maximal, Sylow
139	C₂²×C₄₃	#520	1	2	yes	—
140	C₂²×C₄₃	C₂³×C₈	1	2	yes	—
141	C₂²×C₄₃	#568	1	2	yes	—
142	C₂²×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
143	C₂×D₄₃	—	2	6	no	—
144	C₂×D₄₃	—	2	24	no	—
145	C₂×D₄₃	—	2	6	no	—
146	C₂×D₄₃	C₂³×C₈	1	12	yes	—
147	C₄×C₄₃	—	2	6	no	—
148	C₄×C₄₃	#520	1	2	yes	—
149	C₄×C₄₃	C₂³×C₈	1	12	yes	—
150	C₄×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
151	C₄×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
152	Dic₄₃	—	2	6	no	—
153	Dic₄₃	—	2	2	no	—
154	Dic₄₃	C₂³×C₈	1	4	yes	—
155	C₂×C₄×#41	—	43	1	no	—
156	#2988	—	43	16	no	—
157	C₂×C₈×Q₈	—	43	1	no	—
158	C₂×C₈×Q₈	D₄₃	1	1	yes	characteristic, complemented
159	C₂×C₈×D₄	—	43	3	no	—
160	C₂×C₈×D₄	—	43	3	no	—
161	C₂×#431	—	43	3	no	—
162	C₂²×C₄⋊C₈	—	43	3	no	—
163	C₂×C₄₃	#2988	1	4	yes	complemented
164	C₂×C₄₃	C₂×#520	1	1	yes	characteristic
165	C₂×C₄₃	C₂⁴×C₈	1	1	yes	characteristic
166	C₂×C₄₃	C₂×#568	1	1	yes	characteristic
167	D₄₃	—	2	12	no	—
168	C₂²×#41	—	43	1	no	—
169	C₂³×C₈	—	43	6	no	—
170	C₄×#41	—	43	8	no	—
171	C₄×#41	—	43	8	no	—
172	C₂×C₄×Q₈	—	43	1	no	—
173	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
174	C₂×C₄×D₄	—	43	3	no	—
175	C₂×C₄×D₄	—	43	3	no	—
176	C₂×#117	—	43	3	no	—
177	C₂²×C₄²	—	43	3	no	—
178	C₈×Q₈	—	43	8	no	—
179	C₈×Q₈	C₂×D₄₃	1	8	yes	complemented
180	C₈×D₄	—	43	24	no	—
181	C₈×D₄	—	43	24	no	—
182	#431	—	43	24	no	—
183	C₂×C₄⋊C₈	—	43	3	no	—
184	C₂×C₄⋊C₈	—	43	1	no	—
185	C₂×C₄⋊C₈	—	43	3	no	—
186	C₂×C₄⋊C₈	—	43	24	no	—
187	C₂×C₄⋊C₈	C₂×D₄₃	1	3	yes	—
188	C₂×C₂²⋊C₈	—	43	3	no	—
189	C₂×C₂²⋊C₈	—	43	3	no	—
190	C₂×C₄×C₈	—	43	3	no	—
191	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	—
192	C₄₃	C₂×#2988	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	—	43	16	no	—
194	C₂×#41	—	43	8	no	—
195	C₂²×Q₈	C₄×D₄₃	1	1	yes	characteristic
196	C₂²×D₄	—	43	3	no	—
197	C₂³×C₄	—	43	3	no	—
198	C₂³×C₄	—	43	3	no	—
199	C₂³×C₄	—	43	3	no	—
200	C₂²×C₈	—	43	3	no	—
201	C₂²×C₈	—	43	1	no	—
202	C₂²×C₈	—	43	24	no	—
203	C₂²×C₈	—	172	12	no	—
204	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
205	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
206	C₄×Q₈	—	43	8	no	—
207	C₄×Q₈	C₄×D₄₃	1	4	yes	—
208	C₄×Q₈	C₂²×D₄₃	1	4	yes	—
209	C₄×D₄	—	43	24	no	—
210	C₄×D₄	—	43	12	no	—
211	C₄×D₄	—	43	12	no	—
212	#117	—	43	24	no	—
213	C₂×C₄⋊C₄	—	43	3	no	—
214	C₂×C₄⋊C₄	C₄×D₄₃	1	3	yes	—
215	C₂×C₂²⋊C₄	—	43	6	no	—
216	C₂×C₄²	—	43	3	no	—
217	C₂×C₄²	—	43	3	no	—
218	C₂×C₄²	—	43	1	no	—
219	C₂×C₄²	—	43	12	no	—
220	C₂×C₄²	—	43	12	no	—
221	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
222	C₄⋊C₈	—	43	12	no	—
223	C₄⋊C₈	—	43	4	no	—
224	C₄⋊C₈	—	43	12	no	—
225	C₄⋊C₈	C₂²×D₄₃	1	12	yes	—
226	C₂²⋊C₈	—	43	12	no	—
227	C₂²⋊C₈	—	43	12	no	—
228	C₄×C₈	—	43	12	no	—
229	C₄×C₈	C₂²×D₄₃	1	12	yes	—
230	C₂⁴	—	43	3	no	—
231	#41	—	43	64	no	—
232	C₂×Q₈	C₈×D₄₃	1	8	yes	complemented
233	C₂×Q₈	C₂×C₄×D₄₃	1	4	yes	—
234	C₂×D₄	—	43	24	no	—
235	C₂×D₄	—	43	12	no	—
236	C₂²×C₄	—	43	1	no	—
237	C₂²×C₄	—	43	24	no	—
238	C₂²×C₄	—	43	12	no	—
239	C₂²×C₄	—	43	1	no	—
240	C₂²×C₄	—	43	3	no	—
241	C₂²×C₄	—	43	6	no	—
242	C₂²×C₄	—	43	6	no	—
243	C₂²×C₄	—	43	12	no	—
244	C₂²×C₄	—	86	6	no	—
245	C₂²×C₄	—	86	6	no	—
246	C₂²×C₄	—	172	6	no	—
247	C₂²×C₄	C₂×C₄×D₄₃	1	3	yes	—
248	C₂²×C₄	C₂×C₄×D₄₃	1	3	yes	—
249	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
250	C₂×C₈	—	2	6	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	43	2	no	—
253	C₂×C₈	—	43	6	no	—
254	C₂×C₈	—	86	6	no	—
255	C₂×C₈	—	86	2	no	—
256	C₂×C₈	—	172	24	no	—
257	C₂×C₈	C₂³×D₄₃	1	2	yes	—
258	C₂×C₈	C₂³×D₄₃	1	6	yes	—
259	C₄⋊C₄	—	43	12	no	—
260	C₄⋊C₄	C₂×C₄×D₄₃	1	12	yes	—
261	C₂²⋊C₄	—	43	24	no	—
262	C₄²	—	43	2	no	—
263	C₄²	—	43	12	no	—
264	C₄²	—	43	12	no	—
265	C₄²	—	43	2	no	—
266	C₄²	C₂×C₄×D₄₃	1	6	yes	—
267	C₄²	C₂³×D₄₃	1	6	yes	—
268	C₂³	—	43	12	no	—
269	C₂³	—	43	6	no	—
270	C₂³	—	86	6	no	—
271	C₂³	—	86	6	no	—
272	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
273	Q₈	C₂×C₈×D₄₃	1	16	yes	complemented
274	D₄	—	43	48	no	—
275	C₂×C₄	—	2	6	no	—
276	C₂×C₄	—	43	4	no	—
277	C₂×C₄	—	43	2	no	—
278	C₂×C₄	—	43	48	no	—
279	C₂×C₄	—	43	2	no	—
280	C₂×C₄	—	43	6	no	—
281	C₂×C₄	—	86	6	no	—
282	C₂×C₄	—	86	2	no	—
283	C₂×C₄	—	86	12	no	—
284	C₂×C₄	—	86	6	no	—
285	C₂×C₄	—	86	6	no	—
286	C₂×C₄	—	172	12	no	—
287	C₂×C₄	C₂×C₈×D₄₃	1	12	yes	—
288	C₂×C₄	C₂²×C₄×D₄₃	1	6	yes	—
289	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₄×D₄₃	1	6	yes	—
291	C₂×C₄	#13063371	1	2	yes	—
292	C₂×C₄	#13063373	1	2	yes	—
293	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
294	C₈	—	2	6	no	—
295	C₈	—	2	2	no	—
296	C₈	—	86	6	no	—
297	C₈	—	86	2	no	—
298	C₂²	—	43	12	no	—
299	C₂²	—	86	6	no	—
300	C₂²	—	86	24	no	—
301	C₂²	—	86	6	no	—
302	C₂²	#14583381	1	2	yes	—
303	C₂²	C₂²×C₈×D₄₃	1	2	yes	—
304	C₂²	#14583664	1	2	yes	—
305	C₂²	C₂³×C₄×D₄₃	1	1	yes	characteristic
306	C₄	—	2	6	no	—
307	C₄	—	43	4	no	—
308	C₄	—	86	6	no	—
309	C₄	—	86	2	no	—
310	C₄	#14583381	1	2	yes	—
311	C₄	C₂²×C₈×D₄₃	1	12	yes	—
312	C₄	C₂×#13063371	1	1	yes	characteristic
313	C₄	C₂×#13063373	1	1	yes	characteristic
314	C₂	—	86	12	no	—
315	C₂	#19143705	1	4	yes	direct product factor
316	C₂	C₂×#14583381	1	1	yes	characteristic
317	C₂	C₂³×C₈×D₄₃	1	1	yes	characteristic
318	C₂	C₂×#14583664	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₃
commutator	C₂×C₄₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂×C₈×Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂²×#10621902	C₄
cosocle	C₂×C₄×C₄₃	C₂⁵
hypercenter	C₂×C₈×Q₈	D₄₃
hypocenter	C₄₃	C₂×#2988
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #19143705	C₂	direct (N × H)
3. C₂×Q₈×C₄₃⋊C₈	C₂	semidirect
4. C₂×C₈×D₁₇₂	C₂	semidirect
5. C₂×#14582964	C₂	semidirect
6. C₂×#14582950	C₂	semidirect
7. C₂×C₄⋊C₈×D₄₃	C₂	semidirect
8. C₂×C₈×Q₈×C₄₃	C₂	semidirect
18. Q₈×C₄₃⋊C₈	C₂²	semidirect
19. C₈×D₁₇₂	C₂²	semidirect
20. #14582964	C₂²	semidirect
21. #14582950	C₂²	semidirect
25. C₄⋊C₈×D₄₃	C₂²	semidirect
30. C₈×Q₈×C₄₃	C₂²	semidirect
33. C₂×#10621902	C₈	semidirect
71. C₂×Q₈×C₄₃	C₂×C₈	semidirect
79. #10621902	C₂×C₈	semidirect
80. C₂×D₁₇₂	C₂×C₈	semidirect
85. C₂×C₄×D₄₃	C₂×C₈	semidirect
113. Q₈×C₄₃	C₂²×C₈	semidirect
128. D₁₇₂	C₂²×C₈	semidirect
133. C₄×D₄₃	C₂²×C₈	semidirect
158. C₂×C₈×Q₈	D₄₃	semidirect
163. C₂×C₄₃	#2988	semidirect
179. C₈×Q₈	C₂×D₄₃	semidirect
192. C₄₃	C₂×#2988	semidirect
232. C₂×Q₈	C₈×D₄₃	semidirect
273. Q₈	C₂×C₈×D₄₃	semidirect
315. C₂	#19143705	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×C₄×#10621902	2
#19143705	2
C₂×Q₈×C₄₃⋊C₈	2
C₂×C₈×D₁₇₂	2
C₂×#14582964	2
C₂×#14582950	2
C₂×C₄⋊C₈×D₄₃	2
C₂×C₈×Q₈×C₄₃	2
C₂×#2988	43

Subgroups — C2×#19143705 #25961643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19143705 #25961643