Subgroups — C₂×#19143662 #25961369

Order 11008.

2274 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143662	C₂	1	16	yes	maximal, direct product factor
2	C₂×#14582977	C₂	1	4	yes	maximal, complemented
3	C₂×D₄×C₄₃⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₂×#14582952	C₂	1	2	yes	maximal, complemented
5	C₂×#14582866	C₂	1	4	yes	maximal, complemented
6	C₂×C₄₃×#532	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10621877	C₂²	1	4	yes	—
8	C₂³×C₄₃⋊C₈	C₂²	1	4	yes	—
9	#14582977	C₂²	1	32	yes	complemented
10	D₄×C₄₃⋊C₈	C₂²	1	32	yes	complemented
11	#14582952	C₂²	1	16	yes	complemented
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
14	#14582866	C₂²	1	32	yes	complemented
15	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
16	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	8	yes	—
17	C₂×C₄×C₄₃⋊C₈	C₂²	1	4	yes	—
18	C₂²×#41×C₄₃	C₂²	1	2	yes	—
19	C₄₃×#532	C₄	1	8	yes	—
20	C₄₃×#532	C₂²	1	8	yes	—
21	C₂×C₄⋊Q₈×C₄₃	C₄	1	1	yes	characteristic
22	C₂×#127×C₄₃	C₄	1	1	yes	characteristic
23	C₂×#124×C₄₃	C₄	1	2	yes	—
24	C₂×#121×C₄₃	C₄	1	4	yes	—
25	C₂×C₄×D₄×C₄₃	C₂²	1	4	yes	—
26	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#10621877	—	4	8	no	—
28	C₂×#10621877	C₂³	1	16	yes	—
29	C₂²×C₄₃⋊C₈	—	4	8	no	—
30	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
31	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
32	C₂²×C₄₃⋊C₈	C₂³	1	16	yes	—
33	C₄⋊(C₄₃⋊C₈)	C₂³	1	8	yes	—
34	C₄⋊(C₄₃⋊C₈)	C₂³	1	8	yes	—
35	C₂²⋊(C₄₃⋊C₈)	C₂³	1	16	yes	—
36	C₂²⋊(C₄₃⋊C₈)	C₂³	1	32	yes	—
37	C₄×C₄₃⋊C₈	C₂³	1	16	yes	—
38	C₂×#41×C₄₃	—	4	4	no	—
39	C₂×#41×C₄₃	—	4	4	no	—
40	C₂×#41×C₄₃	C₂×C₄	1	8	yes	—
41	C₂×#41×C₄₃	C₂³	1	8	yes	—
42	C₂²×Q₈×C₄₃	C₂×C₄	1	2	yes	—
43	C₂²×D₄×C₄₃	C₂×C₄	1	4	yes	—
44	C₂²×D₄×C₄₃	C₂×C₄	1	4	yes	—
45	C₂²×D₄×C₄₃	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
47	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
49	C₄⋊Q₈×C₄₃	C₂×C₄	1	8	yes	—
50	#127×C₄₃	C₂×C₄	1	8	yes	—
51	#124×C₄₃	C₂×C₄	1	16	yes	—
52	#121×C₄₃	C₂×C₄	1	32	yes	—
53	C₄×D₄×C₄₃	C₂×C₄	1	16	yes	—
54	C₄×D₄×C₄₃	C₂³	1	16	yes	—
55	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
56	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	8	yes	—
57	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
60	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
61	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
62	C₂⁴×C₄₃	C₂²×C₄	1	4	yes	—
63	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
64	#41×C₄₃	—	4	16	no	—
65	#41×C₄₃	—	4	8	no	—
66	#41×C₄₃	—	4	8	no	—
67	C₂×Q₈×C₄₃	—	4	4	no	—
68	C₂×Q₈×C₄₃	C₂²×C₄	1	8	yes	—
69	C₂×D₄×C₄₃	—	2	16	no	—
70	C₂×D₄×C₄₃	—	4	8	no	—
71	C₂×D₄×C₄₃	—	4	4	no	—
72	C₂×D₄×C₄₃	C₂²×C₄	1	16	yes	—
73	C₂×D₄×C₄₃	C₂²×C₄	1	16	yes	—
74	C₂×D₄×C₄₃	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₄₃	—	2	8	no	—
76	C₂²×C₄×C₄₃	—	2	2	no	—
77	C₂²×C₄×C₄₃	—	2	2	no	—
78	C₂²×C₄×C₄₃	—	2	8	no	—
79	C₂²×C₄×C₄₃	—	2	8	no	—
80	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
84	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
85	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
86	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
88	C₂²×C₄×C₄₃	C₂⁴	1	8	yes	—
89	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₄₃	C₂²×C₄	1	16	yes	—
91	C₂²⋊C₄×C₄₃	C₂²×C₄	1	32	yes	—
92	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
93	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
94	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
95	C₄²×C₄₃	C₂⁴	1	4	yes	—
96	#10621877	—	4	16	no	—
97	C₂×C₄₃⋊C₈	—	2	8	no	—
98	C₂×C₄₃⋊C₈	—	2	8	no	—
99	C₂×C₄₃⋊C₈	—	4	16	no	—
100	C₂×C₄₃⋊C₈	C₂⁴	1	8	yes	—
101	C₂×C₄₃⋊C₈	C₂⁴	1	8	yes	—
102	C₂³×C₄₃	—	2	2	no	—
103	C₂³×C₄₃	—	2	8	no	—
104	C₂³×C₄₃	—	2	4	no	—
105	C₂³×C₄₃	—	2	16	no	—
106	C₂³×C₄₃	C₂³×C₄	1	8	yes	—
107	C₂³×C₄₃	C₂³×C₄	1	2	yes	—
108	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
109	Q₈×C₄₃	—	4	8	no	—
110	D₄×C₄₃	—	2	32	no	—
111	D₄×C₄₃	—	4	8	no	—
112	D₄×C₄₃	—	4	16	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	8	no	—
115	C₂×C₄×C₄₃	—	2	4	no	—
116	C₂×C₄×C₄₃	—	2	16	no	—
117	C₂×C₄×C₄₃	—	2	4	no	—
118	C₂×C₄×C₄₃	—	2	16	no	—
119	C₂×C₄×C₄₃	—	2	8	no	—
120	C₂×C₄×C₄₃	—	2	8	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	#131	1	8	yes	—
124	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₄₃	C₂³×C₄	1	4	yes	—
126	C₂×C₄×C₄₃	C₂³×C₄	1	8	yes	—
127	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂³×C₄	1	4	yes	—
129	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
130	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
131	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
132	C₄₃⋊C₈	—	2	8	no	—
133	C₄₃⋊C₈	—	2	8	no	—
134	C₂×#2979	—	43	1	no	maximal, Sylow
135	C₂²×C₄₃	—	2	8	no	—
136	C₂²×C₄₃	—	2	2	no	—
137	C₂²×C₄₃	—	2	4	no	—
138	C₂²×C₄₃	—	2	16	no	—
139	C₂²×C₄₃	—	4	4	no	—
140	C₂²×C₄₃	—	4	16	no	—
141	C₂²×C₄₃	#518	1	2	yes	—
142	C₂²×C₄₃	C₂×#131	1	4	yes	—
143	C₂²×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
144	C₄×C₄₃	—	2	2	no	—
145	C₄×C₄₃	—	2	8	no	—
146	C₄×C₄₃	—	2	4	no	—
147	C₄×C₄₃	#518	1	2	yes	—
148	C₄×C₄₃	C₂×#131	1	8	yes	—
149	C₄×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
150	C₄×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×#532	D₄₃	1	1	yes	characteristic
152	#2979	—	43	16	no	—
153	C₂×#436	—	43	4	no	—
154	C₂×C₈×D₄	—	43	4	no	—
155	C₂×#431	—	43	2	no	—
156	C₂×#408	—	43	4	no	—
157	C₂×C₄₃	—	4	8	no	—
158	C₂×C₄₃	—	4	2	no	—
159	C₂×C₄₃	#2979	1	4	yes	complemented
160	C₂×C₄₃	C₂×#518	1	1	yes	characteristic
161	C₂×C₄₃	C₂²×#131	1	2	yes	—
162	C₂²×#41	C₂×D₄₃	1	2	yes	—
163	C₂²×M₁₆	—	43	4	no	—
164	C₂³×C₈	—	43	4	no	—
165	#532	Dic₄₃	1	8	yes	—
166	#532	C₂×D₄₃	1	8	yes	—
167	C₂×C₄⋊Q₈	Dic₄₃	1	1	yes	characteristic
168	C₂×#127	Dic₄₃	1	1	yes	characteristic
169	C₂×#124	Dic₄₃	1	2	yes	—
170	C₂×#121	Dic₄₃	1	4	yes	—
171	C₂×C₄×D₄	C₂×D₄₃	1	4	yes	—
172	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
173	#436	—	43	32	no	—
174	C₈×D₄	—	43	32	no	—
175	#431	—	43	16	no	—
176	C₂×C₄⋊C₈	—	43	2	no	—
177	C₂×C₄⋊C₈	—	43	2	no	—
178	#408	—	43	32	no	—
179	C₂×C₂²⋊C₈	—	43	4	no	—
180	C₂×C₂²⋊C₈	—	43	8	no	—
181	C₂×C₄×C₈	—	43	4	no	—
182	C₄₃	C₂×#2979	1	1	yes	characteristic, complemented, Sylow
183	C₂×#41	—	4	4	no	—
184	C₂×#41	—	4	4	no	—
185	C₂×#41	C₂×Dic₄₃	1	8	yes	—
186	C₂×#41	C₂²×D₄₃	1	8	yes	—
187	C₂²×Q₈	C₂×Dic₄₃	1	2	yes	—
188	C₂²×D₄	C₂×Dic₄₃	1	4	yes	—
189	C₂²×D₄	C₂×Dic₄₃	1	4	yes	—
190	C₂²×D₄	C₂×Dic₄₃	1	2	yes	—
191	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
192	C₂³×C₄	C₂²×D₄₃	1	4	yes	—
193	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
194	C₂×M₁₆	—	43	16	no	—
195	C₂×M₁₆	—	172	8	no	—
196	C₂²×C₈	—	43	4	no	—
197	C₂²×C₈	—	43	4	no	—
198	C₂²×C₈	—	43	16	no	—
199	C₂²×C₈	—	172	8	no	—
200	C₄⋊Q₈	C₂×Dic₄₃	1	8	yes	—
201	#127	C₂×Dic₄₃	1	8	yes	—
202	#124	C₂×Dic₄₃	1	16	yes	—
203	#121	C₂×Dic₄₃	1	32	yes	—
204	C₄×D₄	C₂×Dic₄₃	1	16	yes	—
205	C₄×D₄	C₂²×D₄₃	1	16	yes	—
206	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
207	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	8	yes	—
208	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
209	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
211	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
212	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
213	C₄⋊C₈	—	43	8	no	—
214	C₄⋊C₈	—	43	8	no	—
215	C₂²⋊C₈	—	43	16	no	—
216	C₂²⋊C₈	—	43	32	no	—
217	C₄×C₈	—	43	16	no	—
218	C₂⁴	C₂²×Dic₄₃	1	4	yes	—
219	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
220	#41	—	4	8	no	—
221	#41	—	4	8	no	—
222	#41	—	4	16	no	—
223	C₂×Q₈	—	4	4	no	—
224	C₂×Q₈	C₂²×Dic₄₃	1	8	yes	—
225	C₂×D₄	—	2	16	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	—	4	8	no	—
228	C₂×D₄	C₂²×Dic₄₃	1	16	yes	—
229	C₂×D₄	C₂²×Dic₄₃	1	16	yes	—
230	C₂×D₄	C₂²×Dic₄₃	1	8	yes	—
231	C₂²×C₄	—	2	8	no	—
232	C₂²×C₄	—	2	8	no	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
237	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
238	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
240	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
241	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
242	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
243	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
244	C₂²×C₄	C₂³×D₄₃	1	8	yes	—
245	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
246	M₁₆	—	172	16	no	—
247	C₂×C₈	—	43	8	no	—
248	C₂×C₈	—	43	8	no	—
249	C₂×C₈	—	86	8	no	—
250	C₂×C₈	—	86	8	no	—
251	C₂×C₈	—	172	16	no	—
252	C₄⋊C₄	C₂²×Dic₄₃	1	16	yes	—
253	C₂²⋊C₄	C₂²×Dic₄₃	1	32	yes	—
254	C₄²	C₂²×Dic₄₃	1	4	yes	—
255	C₄²	C₂²×Dic₄₃	1	4	yes	—
256	C₄²	C₂²×Dic₄₃	1	4	yes	—
257	C₄²	C₂³×D₄₃	1	4	yes	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	8	no	—
260	C₂³	—	2	4	no	—
261	C₂³	—	2	16	no	—
262	C₂³	C₂³×Dic₄₃	1	8	yes	—
263	C₂³	C₂³×Dic₄₃	1	2	yes	—
264	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
265	Q₈	—	4	8	no	—
266	D₄	—	2	32	no	—
267	D₄	—	4	8	no	—
268	D₄	—	4	16	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	16	no	—
274	C₂×C₄	—	2	16	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	#13063332	1	8	yes	—
280	C₂×C₄	C₂³×Dic₄₃	1	2	yes	—
281	C₂×C₄	C₂³×Dic₄₃	1	4	yes	—
282	C₂×C₄	C₂³×Dic₄₃	1	8	yes	—
283	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
284	C₂×C₄	C₂³×Dic₄₃	1	4	yes	—
285	C₂×C₄	#13063370	1	2	yes	—
286	C₂×C₄	#13063370	1	2	yes	—
287	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
288	C₈	—	86	8	no	—
289	C₈	—	86	8	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	8	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	16	no	—
294	C₂²	—	4	4	no	—
295	C₂²	—	4	16	no	—
296	C₂²	#14583374	1	2	yes	—
297	C₂²	C₂×#13063332	1	4	yes	—
298	C₂²	C₂⁴×Dic₄₃	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	2	8	no	—
301	C₄	—	2	4	no	—
302	C₄	#14583374	1	2	yes	—
303	C₄	C₂×#13063332	1	8	yes	—
304	C₄	C₂×#13063370	1	1	yes	characteristic
305	C₄	C₂×#13063370	1	1	yes	characteristic
306	C₂	—	4	2	no	—
307	C₂	—	4	8	no	—
308	C₂	#19143662	1	4	yes	direct product factor
309	C₂	C₂×#14583374	1	1	yes	characteristic
310	C₂	C₂²×#13063332	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting, prime radical	C₂×C₄₃×#532	C₂
socle	C₂³×C₄₃	C₂³×C₄
cosocle	C₂×C₄×C₄₃	C₂⁵
hypercenter	C₂×#532	D₄₃
hypocenter	C₄₃	C₂×#2979
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#532

Complements

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

N	H ≅ G/N	type
1. #19143662	C₂	direct (N × H)
2. C₂×#14582977	C₂	semidirect
3. C₂×D₄×C₄₃⋊C₈	C₂	semidirect
4. C₂×#14582952	C₂	semidirect
5. C₂×#14582866	C₂	semidirect
9. #14582977	C₂²	semidirect
10. D₄×C₄₃⋊C₈	C₂²	semidirect
11. #14582952	C₂²	semidirect
14. #14582866	C₂²	semidirect
159. C₂×C₄₃	#2979	semidirect
182. C₄₃	C₂×#2979	semidirect
308. C₂	#19143662	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19143662	2
C₂×#14582977	2
C₂×D₄×C₄₃⋊C₈	2
C₂×#14582952	2
C₂×#14582866	2
C₂×C₄₃×#532	2
C₂×#2979	43

Subgroups — C2×#19143662 #25961369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19143662 #25961369