Subgroups — #25001745

Order 11008.

1222 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	C₂²×#13063235	C₂	1	2	yes	maximal
2	C₂×C₄²×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582654	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582653	C₂	1	2	yes	maximal
5	C₂×C₄₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₄₃	C₄	1	4	yes	—
8	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
10	C₂×#13063235	—	2	4	no	—
11	C₂×#13063235	—	2	4	no	—
12	C₂×#13063235	C₄	1	8	yes	complemented
13	C₄²×Dic₄₃	—	2	4	no	—
14	#14582654	C₄	1	4	yes	complemented
15	#14582653	C₄	1	8	yes	complemented
16	C₂³×C₈×C₄₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#336	C₄	1	4	yes	complemented
19	C₂³×Dic₄₃	C₂×C₄	1	2	yes	—
20	C₂³×Dic₄₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄₃⋊C₈	—	2	2	no	—
22	C₂²×C₄₃⋊C₈	—	2	2	no	—
23	C₂²×C₄₃⋊C₈	—	2	2	no	—
24	C₂²×C₄₃⋊C₈	—	2	2	no	—
25	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
28	C₂×C₄×Dic₄₃	—	2	8	no	—
29	C₂×C₄×Dic₄₃	—	2	2	no	—
30	C₂×C₄×Dic₄₃	—	2	1	no	—
31	C₂×C₄×Dic₄₃	—	2	1	no	—
32	C₂×C₄×Dic₄₃	—	2	8	no	—
33	C₂×C₄×Dic₄₃	—	2	8	no	—
34	C₂×C₄×Dic₄₃	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₄₃	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
37	C₂×C₄×Dic₄₃	D₄	1	4	yes	—
38	C₂×C₄×Dic₄₃	Q₈	1	2	yes	—
39	#13063235	—	2	16	no	—
40	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₄₃	—	2	2	no	—
43	C₂²×C₈×C₄₃	—	2	2	no	—
44	C₂²×C₈×C₄₃	—	2	2	no	—
45	C₂²×C₈×C₄₃	—	2	2	no	—
46	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₃	—	2	2	no	—
50	C₂×C₄²×C₄₃	—	2	2	no	—
51	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
53	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	—	2	2	no	—
55	C₂²×C₄×C₄₃	—	2	2	no	—
56	C₂²×C₄×C₄₃	—	2	4	no	—
57	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
58	C₂²×C₄×C₄₃	M₁₆	1	8	yes	—
59	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	#41	1	2	yes	—
67	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
69	C₂²×Dic₄₃	—	2	2	no	—
70	C₂²×Dic₄₃	—	2	2	no	—
71	C₂²×Dic₄₃	—	2	4	no	—
72	C₂²×Dic₄₃	—	2	4	no	—
73	C₂²×Dic₄₃	—	2	4	no	—
74	C₂²×Dic₄₃	C₄²	1	4	yes	—
75	C₂²×Dic₄₃	C₄²	1	4	yes	—
76	C₂²×Dic₄₃	C₂²⋊C₄	1	4	yes	—
77	C₂²×Dic₄₃	C₄⋊C₄	1	4	yes	—
78	C₂²×Dic₄₃	M₁₆	1	8	yes	—
79	C₂×C₈×C₄₃	—	2	4	no	—
80	C₂×C₈×C₄₃	—	2	4	no	—
81	C₂×C₈×C₄₃	—	2	4	no	—
82	C₂×C₈×C₄₃	—	2	4	no	—
83	C₂×C₈×C₄₃	—	2	4	no	—
84	C₂×C₈×C₄₃	—	2	4	no	—
85	C₂×C₈×C₄₃	C₄²	1	8	yes	complemented
86	C₄²×C₄₃	—	2	4	no	—
87	C₄²×C₄₃	—	2	4	no	—
88	C₄×Dic₄₃	—	2	32	no	—
89	C₄×Dic₄₃	—	2	4	no	—
90	C₄×Dic₄₃	—	2	4	no	—
91	C₄×Dic₄₃	C₂²⋊C₄	1	8	yes	—
92	C₄×Dic₄₃	C₄⋊C₄	1	8	yes	—
93	C₂×C₄₃⋊C₈	—	2	4	no	—
94	C₂×C₄₃⋊C₈	—	2	4	no	—
95	C₂×C₄₃⋊C₈	—	2	4	no	—
96	C₂×C₄₃⋊C₈	—	2	4	no	—
97	C₂×C₄₃⋊C₈	—	2	4	no	—
98	C₂×C₄₃⋊C₈	—	2	4	no	—
99	C₂×C₄₃⋊C₈	C₄²	1	8	yes	complemented
100	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
101	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
102	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
103	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₄₃	#117	1	1	yes	characteristic
106	C₂³×C₄₃	#117	1	1	yes	characteristic
107	C₂³×C₄₃	C₂×M₁₆	1	4	yes	—
108	C₂³×C₄₃	C₂×M₁₆	1	4	yes	—
109	C₂×C₄×C₄₃	—	2	4	no	—
110	C₂×C₄×C₄₃	—	2	4	no	—
111	C₂×C₄×C₄₃	—	2	8	no	—
112	C₂×C₄×C₄₃	—	2	8	no	—
113	C₂×C₄×C₄₃	#97	1	8	yes	complemented
114	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₃	#117	1	2	yes	—
119	C₂×C₄×C₄₃	#117	1	2	yes	—
120	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
130	C₂×Dic₄₃	—	2	4	no	—
131	C₂×Dic₄₃	—	2	4	no	—
132	C₂×Dic₄₃	—	2	4	no	—
133	C₂×Dic₄₃	—	2	4	no	—
134	C₂×Dic₄₃	—	2	8	no	—
135	C₂×Dic₄₃	—	2	8	no	—
136	C₂×Dic₄₃	—	2	8	no	—
137	C₂×Dic₄₃	—	2	8	no	—
138	C₂×Dic₄₃	#95	1	8	yes	—
139	C₂×Dic₄₃	#97	1	8	yes	complemented
140	C₈×C₄₃	—	2	8	no	—
141	C₄₃⋊C₈	—	2	8	no	—
142	#10693	—	43	1	no	maximal, Sylow
143	C₂²×C₄₃	C₄³	1	1	yes	characteristic
144	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
145	C₂²×C₄₃	#376	1	1	yes	characteristic
146	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₄₃	C₂×#97	1	2	yes	—
151	C₂²×C₄₃	C₂×#97	1	2	yes	—
152	C₂²×C₄₃	C₄×M₁₆	1	2	yes	—
153	C₂²×C₄₃	C₄×M₁₆	1	2	yes	—
154	C₂²×C₄₃	C₄×M₁₆	1	2	yes	—
155	C₂²×C₄₃	C₄×M₁₆	1	2	yes	—
156	C₂²×C₄₃	#407	1	4	yes	—
157	C₂²×C₄₃	#423	1	4	yes	—
158	C₂²×C₄₃	#432	1	4	yes	—
159	C₂²×C₄₃	#432	1	4	yes	—
160	C₄×C₄₃	—	2	8	no	—
161	C₄×C₄₃	C₄×C₂²⋊C₄	1	2	yes	—
162	C₄×C₄₃	C₄×C₂²⋊C₄	1	2	yes	—
163	C₄×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
164	C₄×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
165	Dic₄₃	—	2	8	no	—
166	Dic₄₃	—	2	8	no	—
167	C₂²×#97	—	43	2	no	—
168	C₂×C₄³	—	43	1	no	—
169	C₂×#336	—	43	1	no	—
170	C₂×#336	—	43	2	no	—
171	C₂×#336	D₄₃	1	1	yes	characteristic
172	C₂×C₄₃	C₄×#95	1	1	yes	characteristic
173	C₂×C₄₃	C₄×#97	1	1	yes	characteristic
174	C₂×C₄₃	C₄×#97	1	1	yes	characteristic
175	C₂×C₄₃	#1728	1	2	yes	—
176	C₂×C₄₃	#1745	1	2	yes	—
177	C₂×C₄₃	#1749	1	2	yes	—
178	C₂×C₄₃	#1749	1	2	yes	—
179	C₂×C₄₃	#1767	1	2	yes	—
180	C₂×C₄₃	#1767	1	2	yes	—
181	C₂³×C₈	—	43	2	no	—
182	C₂³×C₈	C₂×D₄₃	1	2	yes	—
183	C₂²×C₄²	—	43	1	no	—
184	C₂²×C₄²	—	43	1	no	—
185	C₂²×C₄²	—	43	4	no	—
186	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
187	C₂×#97	—	43	8	no	—
188	C₂×#97	—	86	4	no	—
189	C₂×#97	—	86	4	no	—
190	C₄³	—	86	4	no	—
191	#336	—	43	8	no	—
192	#336	—	43	4	no	—
193	#336	Dic₄₃	1	4	yes	complemented
194	C₄₃	#10693	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₄	—	43	2	no	—
196	C₂³×C₄	—	43	2	no	—
197	C₂³×C₄	C₄×D₄₃	1	2	yes	—
198	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	—	43	4	no	—
204	C₂²×C₈	—	43	4	no	—
205	C₂²×C₈	—	43	4	no	—
206	C₂²×C₈	—	86	2	no	—
207	C₂²×C₈	—	86	2	no	—
208	C₂²×C₈	—	86	2	no	—
209	C₂²×C₈	—	86	2	no	—
210	C₂²×C₈	C₄×D₄₃	1	4	yes	—
211	C₂²×C₈	C₄×D₄₃	1	4	yes	—
212	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	43	4	no	—
216	C₂×C₄²	—	43	4	no	—
217	C₂×C₄²	—	43	2	no	—
218	C₂×C₄²	—	43	2	no	—
219	C₂×C₄²	—	43	4	no	—
220	C₂×C₄²	—	86	8	no	—
221	C₂×C₄²	—	86	8	no	—
222	C₂×C₄²	—	86	8	no	—
223	C₂×C₄²	—	86	2	no	—
224	C₂×C₄²	—	86	1	no	—
225	C₂×C₄²	—	86	1	no	—
226	C₂×C₄²	C₄×D₄₃	1	2	yes	—
227	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
228	#97	—	86	16	no	—
229	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	43	4	no	—
234	C₂²×C₄	—	43	4	no	—
235	C₂²×C₄	—	43	4	no	—
236	C₂²×C₄	—	43	4	no	—
237	C₂²×C₄	—	43	8	no	—
238	C₂²×C₄	—	86	2	no	—
239	C₂²×C₄	—	86	2	no	—
240	C₂²×C₄	—	86	4	no	—
241	C₂²×C₄	—	86	4	no	—
242	C₂²×C₄	—	86	4	no	—
243	C₂²×C₄	#10621872	1	8	yes	—
244	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
245	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
247	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
248	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
249	C₂²×C₄	D₄×D₄₃	1	2	yes	—
250	C₂²×C₄	#10621900	1	1	yes	characteristic
251	C₂²×C₄	#10621900	1	2	yes	—
252	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
253	C₂²×C₄	#10621902	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	43	8	no	—
262	C₂×C₈	—	86	4	no	—
263	C₂×C₈	—	86	4	no	—
264	C₂×C₈	—	86	4	no	—
265	C₂×C₈	—	86	4	no	—
266	C₂×C₈	—	86	4	no	—
267	C₂×C₈	—	86	4	no	—
268	C₂×C₈	C₄×Dic₄₃	1	8	yes	complemented
269	C₄²	—	2	4	no	—
270	C₄²	—	2	4	no	—
271	C₄²	—	43	8	no	—
272	C₄²	—	43	8	no	—
273	C₄²	—	86	32	no	—
274	C₄²	—	86	4	no	—
275	C₄²	—	86	4	no	—
276	C₂³	C₄²×D₄₃	1	1	yes	characteristic
277	C₂³	C₄²×D₄₃	1	1	yes	characteristic
278	C₂³	C₂×C₄×Dic₄₃	1	1	yes	characteristic
279	C₂³	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
280	C₂³	C₄⋊C₄×D₄₃	1	1	yes	characteristic
281	C₂³	#13063281	1	1	yes	characteristic
282	C₂³	#13063283	1	1	yes	characteristic
283	C₂³	M₁₆×D₄₃	1	4	yes	—
284	C₂³	C₂×#10621872	1	4	yes	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	8	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	43	8	no	—
290	C₂×C₄	—	43	8	no	—
291	C₂×C₄	—	86	4	no	—
292	C₂×C₄	—	86	4	no	—
293	C₂×C₄	—	86	4	no	—
294	C₂×C₄	—	86	8	no	—
295	C₂×C₄	—	86	8	no	—
296	C₂×C₄	—	86	8	no	—
297	C₂×C₄	—	86	8	no	—
298	C₂×C₄	—	86	4	no	—
299	C₂×C₄	#13063235	1	8	yes	complemented
300	C₂×C₄	C₄²×D₄₃	1	2	yes	—
301	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
302	C₂×C₄	C₂²⋊C₄×D₄₃	1	2	yes	—
303	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
304	C₂×C₄	#13063281	1	2	yes	—
305	C₂×C₄	#13063283	1	2	yes	—
306	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
307	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
308	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
309	C₂×C₄	#13063287	1	1	yes	characteristic
310	C₂×C₄	#13063287	1	2	yes	—
311	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
312	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
313	C₂×C₄	#13063290	1	1	yes	characteristic
314	C₂×C₄	#13063290	1	2	yes	—
315	C₂×C₄	Q₈×Dic₄₃	1	1	yes	characteristic
316	C₈	—	2	8	no	—
317	C₈	—	86	8	no	—
318	C₂²	C₄²×Dic₄₃	1	1	yes	characteristic
319	C₂²	#95×D₄₃	1	1	yes	characteristic
320	C₂²	#14582744	1	1	yes	characteristic
321	C₂²	#14582746	1	1	yes	characteristic
322	C₂²	C₂²⋊C₄×Dic₄₃	1	1	yes	characteristic
323	C₂²	#14582750	1	1	yes	characteristic
324	C₂²	C₄⋊C₄×Dic₄₃	1	1	yes	characteristic
325	C₂²	#97×D₄₃	1	2	yes	—
326	C₂²	C₂×#13063235	1	2	yes	—
327	C₂²	C₄×#10621872	1	2	yes	—
328	C₂²	C₄×#10621872	1	2	yes	—
329	C₂²	M₁₆×Dic₄₃	1	2	yes	—
330	C₂²	#14582851	1	2	yes	—
331	C₂²	#14582861	1	4	yes	—
332	C₂²	#14582921	1	4	yes	—
333	C₂²	#14582953	1	4	yes	—
334	C₂²	#14582955	1	4	yes	—
335	C₄	—	2	8	no	—
336	C₄	—	86	8	no	—
337	C₄	—	86	8	no	—
338	C₄	#14582746	1	2	yes	—
339	C₄	C₂²⋊C₄×Dic₄₃	1	2	yes	—
340	C₄	#14582750	1	2	yes	—
341	C₄	C₄⋊C₄×Dic₄₃	1	2	yes	—
342	C₂	#95×Dic₄₃	1	1	yes	characteristic
343	C₂	#97×Dic₄₃	1	1	yes	characteristic
344	C₂	C₄×#13063235	1	1	yes	characteristic
345	C₂	#19134287	1	2	yes	—
346	C₂	#19134342	1	2	yes	—
347	C₂	#19134356	1	2	yes	—
348	C₂	#19134359	1	2	yes	—
349	C₂	#19134447	1	2	yes	—
350	C₂	#19134450	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₄₃
commutator	C₂²×C₄₃	C₄³
Fitting	C₂×C₄₃×#336	C₂
socle, prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×#336	D₄₃
hypocenter	C₄₃	#10693
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
12. C₂×#13063235	C₄	semidirect
14. #14582654	C₄	semidirect
15. #14582653	C₄	semidirect
18. C₄₃×#336	C₄	semidirect
85. C₂×C₈×C₄₃	C₄²	semidirect
99. C₂×C₄₃⋊C₈	C₄²	semidirect
113. C₂×C₄×C₄₃	#97	semidirect
139. C₂×Dic₄₃	#97	semidirect
193. #336	Dic₄₃	semidirect
194. C₄₃	#10693	semidirect
268. C₂×C₈	C₄×Dic₄₃	semidirect
299. C₂×C₄	#13063235	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063235	2
C₂×C₄²×Dic₄₃	2
C₂×#14582654	2
C₂×#14582653	2
C₂×C₄₃×#336	2
#10693	43