Subgroups — C₂×#19134450 #25094481

Order 11008.

1422 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13063235	C₂	1	2	yes	maximal
3	C₂×C₄²×Dic₄₃	C₂	1	1	yes	maximal, characteristic
4	#19134450	C₂	1	8	yes	maximal, direct product factor
5	C₂×#14582653	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₃	C₄	1	2	yes	—
9	C₂²×C₄×Dic₄₃	C₄	1	2	yes	—
10	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₄₃⋊C₈)	C₄	1	8	yes	complemented
13	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
14	C₂×#13063235	—	2	8	no	—
15	C₂×#13063235	C₂²	1	8	yes	—
16	C₄²×Dic₄₃	C₄	1	4	yes	—
17	C₄²×Dic₄₃	C₂²	1	4	yes	—
18	#14582653	C₂²	1	8	yes	—
19	C₂³×C₈×C₄₃	C₂²	1	2	yes	—
20	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₄₃	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₄₃	C₂²	1	4	yes	—
23	C₂³×Dic₄₃	C₂×C₄	1	2	yes	—
24	C₂³×Dic₄₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄₃⋊C₈	—	2	4	no	—
26	C₂²×C₄₃⋊C₈	—	2	4	no	—
27	C₂²×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
28	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
29	C₂×C₄×Dic₄₃	—	2	8	no	—
30	C₂×C₄×Dic₄₃	—	2	2	no	—
31	C₂×C₄×Dic₄₃	—	2	2	no	—
32	C₂×C₄×Dic₄₃	—	2	8	no	—
33	C₂×C₄×Dic₄₃	C₂×C₄	1	8	yes	—
34	C₂×C₄×Dic₄₃	C₂×C₄	1	8	yes	—
35	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₄₃	D₄	1	4	yes	—
38	C₂×C₄×Dic₄₃	Q₈	1	4	yes	—
39	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
40	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
41	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	16	yes	complemented
42	#13063235	—	2	16	no	—
43	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₄₃	—	2	4	no	—
47	C₂²×C₈×C₄₃	—	2	4	no	—
48	C₂²×C₈×C₄₃	C₂×C₄	1	8	yes	—
49	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
50	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
54	C₄⋊C₈×C₄₃	C₂×C₄	1	16	yes	complemented
55	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	—	2	4	no	—
57	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
58	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
59	C₂²×C₄×C₄₃	M₁₆	1	8	yes	—
60	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₃	#41	1	2	yes	—
68	C₂²×C₄×C₄₃	#41	1	2	yes	—
69	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₃	—	2	4	no	—
71	C₂²×Dic₄₃	—	2	4	no	—
72	C₂²×Dic₄₃	—	2	4	no	—
73	C₂²×Dic₄₃	—	2	4	no	—
74	C₂²×Dic₄₃	C₄⋊C₄	1	8	yes	—
75	C₂²×Dic₄₃	M₁₆	1	8	yes	—
76	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
77	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₄₃	—	2	16	no	—
79	C₂×C₈×C₄₃	—	2	4	no	—
80	C₂×C₈×C₄₃	—	2	4	no	—
81	C₂×C₈×C₄₃	C₂²×C₄	1	8	yes	—
82	C₄²×C₄₃	C₄²	1	8	yes	—
83	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
84	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
85	C₄×Dic₄₃	—	2	4	no	—
86	C₄×Dic₄₃	—	2	16	no	—
87	C₄×Dic₄₃	—	2	2	no	—
88	C₄×Dic₄₃	—	2	2	no	—
89	C₄×Dic₄₃	—	2	16	no	—
90	C₄×Dic₄₃	C₄⋊C₄	1	8	yes	—
91	C₄×Dic₄₃	C₂×D₄	1	4	yes	—
92	C₄×Dic₄₃	C₂×Q₈	1	4	yes	—
93	C₂×C₄₃⋊C₈	—	2	16	no	—
94	C₂×C₄₃⋊C₈	—	2	4	no	—
95	C₂×C₄₃⋊C₈	—	2	4	no	—
96	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
97	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
98	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₄₃	#117	1	2	yes	—
100	C₂³×C₄₃	C₂×M₁₆	1	4	yes	—
101	C₂³×C₄₃	C₂×M₁₆	1	4	yes	—
102	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₃	—	2	4	no	—
104	C₂×C₄×C₄₃	—	2	8	no	—
105	C₂×C₄×C₄₃	#97	1	16	yes	complemented
106	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
107	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
109	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₃	#117	1	2	yes	—
111	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
112	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
113	C₂×C₄×C₄₃	C₄×Q₈	1	4	yes	—
114	C₂×C₄×C₄₃	C₄×Q₈	1	4	yes	—
115	C₂×C₄×C₄₃	C₂×M₁₆	1	8	yes	—
116	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
122	C₂×Dic₄₃	—	2	4	no	—
123	C₂×Dic₄₃	—	2	4	no	—
124	C₂×Dic₄₃	—	2	4	no	—
125	C₂×Dic₄₃	—	2	4	no	—
126	C₂×Dic₄₃	—	2	16	no	—
127	C₂×Dic₄₃	—	2	16	no	—
128	C₂×Dic₄₃	C₂×C₄⋊C₄	1	8	yes	—
129	C₂×Dic₄₃	C₂×M₁₆	1	8	yes	—
130	C₈×C₄₃	—	2	8	no	—
131	C₄₃⋊C₈	—	2	8	no	—
132	C₂×#1767	—	43	1	no	maximal, Sylow
133	C₂²×C₄₃	C₄×C₄⋊C₄	1	4	yes	—
134	C₂²×C₄₃	C₂×#97	1	4	yes	—
135	C₂²×C₄₃	C₄×M₁₆	1	4	yes	—
136	C₂²×C₄₃	#423	1	8	yes	—
137	C₂²×C₄₃	#432	1	8	yes	—
138	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
139	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₄₃	C₂×#117	1	1	yes	characteristic
141	C₂²×C₄₃	C₂²×M₁₆	1	2	yes	—
142	C₂²×C₄₃	C₂²×M₁₆	1	2	yes	—
143	C₄×C₄₃	—	2	4	no	—
144	C₄×C₄₃	C₄×C₄⋊C₄	1	4	yes	—
145	C₄×C₄₃	C₂×#97	1	8	yes	complemented
146	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
150	Dic₄₃	—	2	8	no	—
151	Dic₄₃	—	2	8	no	—
152	C₂²×C₄⋊C₈	—	43	1	no	—
153	C₂²×C₄⋊C₈	D₄₃	1	1	yes	characteristic
154	C₂²×#97	—	43	2	no	—
155	C₂×C₄³	—	43	1	no	—
156	#1767	—	43	8	no	—
157	C₂×#336	—	43	2	no	—
158	C₂×C₄₃	#1767	1	8	yes	complemented
159	C₂×C₄₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄₃	C₂²×#97	1	1	yes	characteristic
161	C₂×C₄₃	C₂×C₄×M₁₆	1	1	yes	characteristic
162	C₂×C₄₃	C₂×#423	1	2	yes	—
163	C₂×C₄₃	C₂×#432	1	2	yes	—
164	C₂³×C₈	—	43	2	no	—
165	C₂³×C₈	C₂×D₄₃	1	2	yes	—
166	C₂²×C₄²	—	43	1	no	—
167	C₂²×C₄²	—	43	1	no	—
168	C₂²×C₄²	—	43	2	no	—
169	C₂²×C₄²	—	43	2	no	—
170	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
171	C₂×C₄⋊C₈	—	43	4	no	—
172	C₂×C₄⋊C₈	—	43	8	no	—
173	C₂×C₄⋊C₈	Dic₄₃	1	8	yes	complemented
174	C₂×C₄⋊C₈	C₂×D₄₃	1	4	yes	—
175	C₂×#97	—	43	8	no	—
176	C₂×#97	—	86	8	no	—
177	C₄³	—	43	4	no	—
178	C₄³	—	43	4	no	—
179	#336	—	43	8	no	—
180	C₄₃	C₂×#1767	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₄	—	43	2	no	—
182	C₂³×C₄	—	43	2	no	—
183	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
184	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
185	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	—	2	4	no	—
188	C₂²×C₈	—	43	8	no	—
189	C₂²×C₈	—	43	4	no	—
190	C₂²×C₈	—	86	4	no	—
191	C₂²×C₈	—	86	4	no	—
192	C₂²×C₈	C₂×Dic₄₃	1	8	yes	—
193	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
194	C₂×C₄²	—	43	2	no	—
195	C₂×C₄²	—	43	4	no	—
196	C₂×C₄²	—	43	4	no	—
197	C₂×C₄²	—	43	2	no	—
198	C₂×C₄²	—	43	8	no	—
199	C₂×C₄²	—	43	2	no	—
200	C₂×C₄²	—	43	8	no	—
201	C₂×C₄²	—	43	2	no	—
202	C₂×C₄²	—	86	8	no	—
203	C₂×C₄²	—	86	8	no	—
204	C₂×C₄²	—	86	2	no	—
205	C₂×C₄²	—	86	2	no	—
206	C₂×C₄²	C₄×D₄₃	1	2	yes	—
207	C₂×C₄²	C₄×D₄₃	1	4	yes	—
208	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
209	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
210	C₄⋊C₈	—	43	16	no	—
211	C₄⋊C₈	C₂×Dic₄₃	1	16	yes	complemented
212	#97	—	86	16	no	—
213	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	43	8	no	—
216	C₂²×C₄	—	43	8	no	—
217	C₂²×C₄	—	43	4	no	—
218	C₂²×C₄	—	43	4	no	—
219	C₂²×C₄	—	86	4	no	—
220	C₂²×C₄	—	86	4	no	—
221	C₂²×C₄	—	86	4	no	—
222	C₂²×C₄	—	86	4	no	—
223	C₂²×C₄	#10621872	1	8	yes	—
224	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
225	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
226	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
227	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
229	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
230	C₂²×C₄	D₄×D₄₃	1	2	yes	—
231	C₂²×C₄	#10621900	1	2	yes	—
232	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
233	C₂²×C₄	#10621902	1	2	yes	—
234	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
235	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
236	C₂×C₈	—	2	16	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	43	8	no	—
240	C₂×C₈	—	86	16	no	—
241	C₂×C₈	—	86	4	no	—
242	C₂×C₈	—	86	4	no	—
243	C₂×C₈	C₂²×Dic₄₃	1	8	yes	—
244	C₄²	—	43	8	no	—
245	C₄²	—	43	4	no	—
246	C₄²	—	43	4	no	—
247	C₄²	—	86	16	no	—
248	C₄²	—	86	4	no	—
249	C₄²	—	86	16	no	—
250	C₄²	—	86	2	no	—
251	C₄²	—	86	2	no	—
252	C₄²	C₄×Dic₄₃	1	8	yes	—
253	C₄²	C₂×C₄×D₄₃	1	4	yes	—
254	C₄²	C₂²×Dic₄₃	1	4	yes	—
255	C₂³	C₂×C₄×Dic₄₃	1	2	yes	—
256	C₂³	C₄⋊C₄×D₄₃	1	2	yes	—
257	C₂³	#13063283	1	2	yes	—
258	C₂³	M₁₆×D₄₃	1	4	yes	—
259	C₂³	C₂×#10621872	1	4	yes	—
260	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	43	8	no	—
264	C₂×C₄	—	43	8	no	—
265	C₂×C₄	—	86	4	no	—
266	C₂×C₄	—	86	4	no	—
267	C₂×C₄	—	86	4	no	—
268	C₂×C₄	—	86	4	no	—
269	C₂×C₄	—	86	16	no	—
270	C₂×C₄	—	86	16	no	—
271	C₂×C₄	#13063235	1	16	yes	complemented
272	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
273	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
274	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
275	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
276	C₂×C₄	#13063283	1	2	yes	—
277	C₂×C₄	#13063287	1	4	yes	—
278	C₂×C₄	D₄×Dic₄₃	1	4	yes	—
279	C₂×C₄	#13063290	1	4	yes	—
280	C₂×C₄	Q₈×Dic₄₃	1	4	yes	—
281	C₂×C₄	C₂×#10621872	1	8	yes	—
282	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
283	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
284	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
285	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
286	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
287	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
288	C₈	—	2	8	no	—
289	C₈	—	86	8	no	—
290	C₂²	C₄⋊C₄×Dic₄₃	1	4	yes	—
291	C₂²	C₂×#13063235	1	4	yes	—
292	C₂²	M₁₆×Dic₄₃	1	4	yes	—
293	C₂²	#14582921	1	8	yes	—
294	C₂²	#14582955	1	8	yes	—
295	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
296	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
297	C₂²	C₂×#13063283	1	1	yes	characteristic
298	C₂²	C₂×M₁₆×D₄₃	1	2	yes	—
299	C₂²	C₂²×#10621872	1	2	yes	—
300	C₄	—	2	4	no	—
301	C₄	—	86	8	no	—
302	C₄	—	86	8	no	—
303	C₄	C₄⋊C₄×Dic₄₃	1	4	yes	—
304	C₄	C₂×#13063235	1	8	yes	complemented
305	C₄	C₂×#13063287	1	1	yes	characteristic
306	C₄	C₂×D₄×Dic₄₃	1	1	yes	characteristic
307	C₄	C₂×#13063290	1	1	yes	characteristic
308	C₄	C₂×Q₈×Dic₄₃	1	1	yes	characteristic
309	C₂	#19134450	1	8	yes	direct product factor
310	C₂	C₂×C₄⋊C₄×Dic₄₃	1	1	yes	characteristic
311	C₂	C₂²×#13063235	1	1	yes	characteristic
312	C₂	C₂×M₁₆×Dic₄₃	1	1	yes	characteristic
313	C₂	C₂×#14582921	1	2	yes	—
314	C₂	C₂×#14582955	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂²×C₄⋊C₈×C₄₃	C₂
socle, prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₄₃
hypocenter	C₄₃	C₂×#1767
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #19134450	C₂	direct (N × H)
12. C₂×C₄⋊(C₄₃⋊C₈)	C₄	semidirect
21. C₂×C₄⋊C₈×C₄₃	C₄	semidirect
41. C₄⋊(C₄₃⋊C₈)	C₂×C₄	semidirect
54. C₄⋊C₈×C₄₃	C₂×C₄	semidirect
105. C₂×C₄×C₄₃	#97	semidirect
145. C₄×C₄₃	C₂×#97	semidirect
158. C₂×C₄₃	#1767	semidirect
173. C₂×C₄⋊C₈	Dic₄₃	semidirect
180. C₄₃	C₂×#1767	semidirect
211. C₄⋊C₈	C₂×Dic₄₃	semidirect
271. C₂×C₄	#13063235	semidirect
304. C₄	C₂×#13063235	semidirect
309. C₂	#19134450	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₄₃⋊C₈)	2
C₂²×#13063235	2
C₂×C₄²×Dic₄₃	2
#19134450	2
C₂×#14582653	2
C₂²×C₄⋊C₈×C₄₃	2
C₂×#1767	43

Subgroups — C2×#19134450 #25094481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19134450 #25094481