Subgroups — #25101374

Order 11008.

1030 conjugacy classes of subgroups.

The subgroup lattice has 306 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	C₄×#13063284	C₂	1	1	yes	maximal, characteristic
3	#19135311	C₂	1	4	yes	maximal, complemented
4	#19134772	C₂	1	4	yes	maximal
5	C₄×Dic₄₃⋊C₈	C₂	1	4	yes	maximal, complemented
6	C₂×C₄₃×#431	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
8	C₂×#13063284	C₄	1	1	yes	characteristic
9	C₂×#13063284	C₄	1	1	yes	characteristic
10	C₂×Dic₄₃⋊C₈	C₂²	1	8	yes	—
11	C₂×C₄×C₄₃⋊C₈	—	2	4	no	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
14	C₄×C₄⋊Dic₄₃	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
16	C₄×C₂²⋊Dic₄₃	C₄	1	2	yes	—
17	C₄×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
18	#14582745	C₄	1	2	yes	—
19	C₄²×Dic₄₃	C₂²	1	2	yes	—
20	#14582653	C₂²	1	8	yes	—
21	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#431	—	2	4	no	—
23	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
26	C₂²×C₄₃⋊C₈	—	2	4	no	—
27	C₂²×C₄₃⋊C₈	—	2	8	no	—
28	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
29	#13063284	—	2	4	no	—
30	#13063284	C₈	1	8	yes	complemented
31	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
33	C₂×C₂²⋊Dic₄₃	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊Dic₄₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₄₃	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₄₃	C₂³	1	4	yes	—
39	Dic₄₃⋊C₈	—	2	16	no	—
40	C₄×C₄₃⋊C₈	—	2	4	no	—
41	C₄×C₄₃⋊C₈	—	2	4	no	—
42	C₄₃⋊#95	C₂×C₄	1	4	yes	—
43	C₄₃⋊#95	C₂×C₄	1	4	yes	—
44	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
48	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	D₄	1	4	yes	—
51	C₂×C₄²×C₄₃	Q₈	1	4	yes	—
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₃	—	2	4	no	—
55	C₂²⋊C₈×C₄₃	—	2	4	no	—
56	C₄×C₈×C₄₃	—	2	4	no	—
57	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	—	2	4	no	—
59	C₂²×C₄×C₄₃	—	2	4	no	—
60	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₄₃	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₄₃	M₁₆	1	4	yes	—
64	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
73	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₄₃	—	2	4	no	—
75	C₂×C₈×C₄₃	—	2	8	no	—
76	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
77	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
78	C₄²×C₄₃	C₂×D₄	1	2	yes	—
79	C₄²×C₄₃	C₂×D₄	1	2	yes	—
80	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
81	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
82	C₂²⋊Dic₄₃	—	2	4	no	—
83	C₂²⋊Dic₄₃	C₂×C₈	1	8	yes	—
84	C₄⋊Dic₄₃	—	2	4	no	—
85	C₄⋊Dic₄₃	C₂×C₈	1	8	yes	complemented
86	C₄×Dic₄₃	—	2	4	no	—
87	C₄×Dic₄₃	—	2	8	no	—
88	C₄×Dic₄₃	—	2	8	no	—
89	C₄×Dic₄₃	C₂×C₈	1	8	yes	complemented
90	C₂×C₄₃⋊C₈	—	2	16	no	—
91	C₂×C₄₃⋊C₈	—	2	4	no	—
92	C₂×C₄₃⋊C₈	—	2	4	no	—
93	C₂×C₄₃⋊C₈	#41	1	8	yes	—
94	C₂³×C₄₃	—	2	2	no	—
95	C₂³×C₄₃	—	2	2	no	—
96	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₄₃	C₂²×C₈	1	2	yes	—
98	C₂³×C₄₃	C₂×M₁₆	1	2	yes	—
99	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	2	4	no	—
103	C₂×C₄×C₄₃	—	2	4	no	—
104	C₂×C₄×C₄₃	—	2	8	no	—
105	C₂×C₄×C₄₃	C₄⋊C₈	1	16	yes	complemented
106	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₄₃	#131	1	4	yes	—
116	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
119	C₂×Dic₄₃	—	2	4	no	—
120	C₂×Dic₄₃	—	2	4	no	—
121	C₂×Dic₄₃	—	2	4	no	—
122	C₂×Dic₄₃	—	2	4	no	—
123	C₂×Dic₄₃	—	2	4	no	—
124	C₂×Dic₄₃	C₂²×C₈	1	8	yes	—
125	C₈×C₄₃	—	4	4	no	—
126	C₄₃⋊C₈	—	2	8	no	—
127	#21349	—	43	1	no	maximal, Sylow
128	C₂²×C₄₃	—	2	2	no	—
129	C₂²×C₄₃	—	2	4	no	—
130	C₂²×C₄₃	—	2	4	no	—
131	C₂²×C₄₃	—	2	2	no	—
132	C₂²×C₄₃	C₂×C₄⋊C₈	1	4	yes	—
133	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₃	C₄×#41	1	2	yes	—
135	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
136	C₂²×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
137	C₂²×C₄₃	C₂×#131	1	2	yes	—
138	C₄×C₄₃	—	2	2	no	—
139	C₄×C₄₃	—	2	2	no	—
140	C₄×C₄₃	C₂×C₄⋊C₈	1	4	yes	—
141	C₄×C₄₃	C₂×C₄⋊C₈	1	4	yes	—
142	C₄×C₄₃	#424	1	4	yes	—
143	C₄×C₄₃	#525	1	2	yes	—
144	C₄×C₄₃	#533	1	2	yes	—
145	Dic₄₃	—	2	8	no	—
146	Dic₄₃	—	4	4	no	—
147	C₂×#431	D₄₃	1	1	yes	characteristic
148	C₂²×C₄×C₈	—	43	1	no	—
149	C₄×#117	—	43	1	no	—
150	#1939	—	43	4	no	—
151	#1850	—	43	4	no	—
152	C₄×C₄⋊C₈	—	43	4	no	—
153	C₂×C₄₃	—	2	4	no	—
154	C₂×C₄₃	#2296	1	1	yes	characteristic
155	C₂×C₄₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
156	C₂×C₄₃	C₂×#424	1	1	yes	characteristic
157	C₂×C₄₃	C₈×#41	1	2	yes	—
158	C₂×C₄₃	#2965	1	2	yes	—
159	C₂³×C₈	—	43	2	no	—
160	C₂×#117	—	43	1	no	—
161	C₂×#117	—	43	1	no	—
162	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
163	#431	—	2	4	no	—
164	C₂×C₄⋊C₈	—	43	8	no	—
165	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
166	C₂×C₂²⋊C₈	C₂×D₄₃	1	2	yes	—
167	C₂×C₄×C₈	—	43	2	no	—
168	C₂×C₄×C₈	—	43	2	no	—
169	C₂×C₄×C₈	—	86	4	no	—
170	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
171	C₄×C₄⋊C₄	—	43	2	no	—
172	C₄×C₄⋊C₄	—	43	2	no	—
173	C₄×C₂²⋊C₄	—	43	2	no	—
174	C₄×C₂²⋊C₄	—	43	2	no	—
175	#376	—	43	2	no	—
176	C₄³	—	43	2	no	—
177	#336	—	43	8	no	—
178	C₄₃	#21349	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
180	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
181	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
182	C₂²×C₈	—	43	4	no	—
183	C₂²×C₈	—	86	8	no	—
184	C₂²×C₈	—	86	4	no	—
185	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
186	#117	—	43	8	no	—
187	#117	—	86	4	no	—
188	C₂×C₄⋊C₄	—	43	2	no	—
189	C₂×C₄⋊C₄	—	43	2	no	—
190	C₂×C₂²⋊C₄	—	43	2	no	—
191	C₂×C₂²⋊C₄	—	43	2	no	—
192	C₂×C₄²	—	43	2	no	—
193	C₂×C₄²	—	43	2	no	—
194	C₂×C₄²	—	43	4	no	—
195	C₂×C₄²	—	43	4	no	—
196	C₂×C₄²	Dic₈₆	1	4	yes	—
197	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
198	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₄₃	1	4	yes	—
200	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
202	C₄⋊C₈	—	2	4	no	—
203	C₄⋊C₈	—	86	16	no	—
204	C₂²⋊C₈	—	2	4	no	—
205	C₄×C₈	—	2	4	no	—
206	C₄×C₈	—	86	4	no	—
207	C₄×C₈	—	86	4	no	—
208	#95	—	43	4	no	—
209	#95	—	43	4	no	—
210	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	43	4	no	—
214	C₂²×C₄	—	43	4	no	—
215	C₂²×C₄	C₈×D₄₃	1	4	yes	—
216	C₂²×C₄	#10621872	1	4	yes	—
217	C₂²×C₄	Dic₄₃⋊C₄	1	4	yes	—
218	C₂²×C₄	Dic₄₃⋊C₄	1	4	yes	—
219	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
220	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
226	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	8	no	—
229	C₂×C₈	—	43	8	no	—
230	C₂×C₈	—	86	16	no	—
231	C₂×C₈	—	86	4	no	—
232	C₂×C₈	—	86	4	no	—
233	C₄⋊C₄	—	43	8	no	—
234	C₄⋊C₄	—	86	4	no	—
235	C₂²⋊C₄	—	43	8	no	—
236	C₂²⋊C₄	—	86	4	no	—
237	C₄²	—	43	8	no	—
238	C₄²	—	86	4	no	—
239	C₄²	—	86	8	no	—
240	C₄²	—	86	8	no	—
241	C₄²	Dic₄₃⋊C₄	1	4	yes	—
242	C₄²	Dic₄₃⋊C₄	1	4	yes	—
243	C₄²	C₂×Dic₈₆	1	2	yes	—
244	C₄²	C₂×Dic₈₆	1	2	yes	—
245	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
246	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	C₂×Dic₄₃⋊C₄	1	2	yes	—
250	C₂³	C₂×C₈×D₄₃	1	2	yes	—
251	C₂³	C₂×#10621872	1	2	yes	—
252	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	43	8	no	—
259	C₂×C₄	—	86	4	no	—
260	C₂×C₄	—	86	4	no	—
261	C₂×C₄	—	86	4	no	—
262	C₂×C₄	—	86	4	no	—
263	C₂×C₄	—	86	4	no	—
264	C₂×C₄	Dic₄₃⋊C₈	1	16	yes	complemented
265	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
266	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
267	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
268	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
269	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
270	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
271	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
272	C₂×C₄	C₂×#10621872	1	2	yes	—
273	C₂×C₄	C₂×#10621872	1	2	yes	—
274	C₂×C₄	#13063333	1	4	yes	—
275	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
277	C₂×C₄	#41×D₄₃	1	4	yes	—
278	C₈	—	4	4	no	—
279	C₈	—	86	8	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	2	no	—
284	C₂²	C₂×Dic₄₃⋊C₈	1	4	yes	—
285	C₂²	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
286	C₂²	#14583367	1	2	yes	—
287	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
288	C₂²	C₂²×#10621872	1	1	yes	characteristic
289	C₂²	C₂×#13063333	1	2	yes	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	86	8	no	—
293	C₄	—	172	4	no	—
294	C₄	C₂×Dic₄₃⋊C₈	1	4	yes	—
295	C₄	C₂×Dic₄₃⋊C₈	1	4	yes	—
296	C₄	#14582927	1	4	yes	—
297	C₄	#14583406	1	2	yes	—
298	C₄	#14583452	1	2	yes	—
299	C₂	—	2	4	no	—
300	C₂	#19137077	1	1	yes	characteristic
301	C₂	C₂²×Dic₄₃⋊C₈	1	1	yes	characteristic
302	C₂	C₂×#14582927	1	1	yes	characteristic
303	C₂	#19143551	1	2	yes	—
304	C₂	#19143556	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#431	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#431	D₄₃
hypocenter	C₄₃	#21349
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #19135311	C₂	semidirect
5. C₄×Dic₄₃⋊C₈	C₂	semidirect
30. #13063284	C₈	semidirect
85. C₄⋊Dic₄₃	C₂×C₈	semidirect
89. C₄×Dic₄₃	C₂×C₈	semidirect
105. C₂×C₄×C₄₃	C₄⋊C₈	semidirect
178. C₄₃	#21349	semidirect
264. C₂×C₄	Dic₄₃⋊C₈	semidirect

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
C₄×#13063284	2
#19135311	2
#19134772	2
C₄×Dic₄₃⋊C₈	2
C₂×C₄₃×#431	2
#21349	43