Subgroups — #25101280

Order 11008.

814 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	C₄²×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	#19135530	C₂	1	2	yes	maximal
3	#19135311	C₂	1	2	yes	maximal
4	#19135150	C₂	1	4	yes	maximal
5	C₄×Dic₄₃⋊C₈	C₂	1	2	yes	maximal
6	#19134340	C₂	1	2	yes	maximal
7	C₄²×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₈₆	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₈₆	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₃⋊C₈	C₂²	1	4	yes	—
13	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×Dic₄₃⋊C₄	C₄	1	2	yes	—
18	C₄×Dic₄₃⋊C₄	C₄	1	2	yes	—
19	C₄×Dic₄₃⋊C₄	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₄₃	C₄	1	1	yes	characteristic
21	C₄×C₄⋊Dic₄₃	C₄	1	1	yes	characteristic
22	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
23	C₄²×Dic₄₃	C₂²	1	2	yes	—
24	#14582653	C₂²	1	4	yes	—
25	#14582653	C₂²	1	4	yes	—
26	#14582653	C₂²	1	4	yes	—
27	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
29	C₄³×C₄₃	C₂²	1	1	yes	characteristic
30	C₄₃×#336	C₂²	1	2	yes	—
31	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
33	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
34	C₄×Dic₈₆	—	2	4	no	—
35	C₄×Dic₈₆	—	2	4	no	—
36	C₄×Dic₈₆	C₈	1	8	yes	complemented
37	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
40	C₂×Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
45	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
48	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
49	Dic₄₃⋊C₈	—	2	8	no	—
50	C₄×C₄₃⋊C₈	—	2	4	no	—
51	C₄×C₄₃⋊C₈	—	2	2	no	—
52	C₄×C₄₃⋊C₈	—	2	4	no	—
53	C₄×C₄₃⋊C₈	Q₈	1	4	yes	complemented
54	C₄₃⋊#95	C₂×C₄	1	2	yes	—
55	C₄₃⋊#95	C₂×C₄	1	2	yes	—
56	C₄₃⋊#95	C₂×C₄	1	2	yes	—
57	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
58	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₄₃	—	2	4	no	—
66	C₄×C₈×C₄₃	—	2	4	no	—
67	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₃	C₂²×C₄	1	2	yes	—
75	C₂²×Dic₄₃	C₂²×C₄	1	2	yes	—
76	C₂²×Dic₄₃	C₂²×C₄	1	2	yes	—
77	C₂²×Dic₄₃	C₂²×C₄	1	2	yes	—
78	C₂×Dic₈₆	—	2	2	no	—
79	C₂×Dic₈₆	—	2	2	no	—
80	C₂×Dic₈₆	C₂×C₈	1	4	yes	—
81	C₂×C₈×C₄₃	—	2	4	no	—
82	C₂×C₈×C₄₃	—	2	4	no	—
83	C₂×C₈×C₄₃	—	2	4	no	—
84	C₄²×C₄₃	—	2	2	no	—
85	C₄²×C₄₃	—	2	2	no	—
86	C₄²×C₄₃	—	2	2	no	—
87	C₄²×C₄₃	C₂×C₈	1	4	yes	—
88	C₄²×C₄₃	M₁₆	1	4	yes	—
89	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
90	C₄²×C₄₃	#41	1	4	yes	—
91	C₄²×C₄₃	#41	1	2	yes	—
92	C₄⋊Dic₄₃	—	2	2	no	—
93	C₄⋊Dic₄₃	—	2	2	no	—
94	C₄⋊Dic₄₃	C₂×C₈	1	4	yes	—
95	Dic₄₃⋊C₄	—	2	4	no	—
96	Dic₄₃⋊C₄	—	2	4	no	—
97	Dic₄₃⋊C₄	C₂×C₈	1	8	yes	—
98	C₄×Dic₄₃	—	2	4	no	—
99	C₄×Dic₄₃	—	2	4	no	—
100	C₄×Dic₄₃	—	2	4	no	—
101	C₄×Dic₄₃	—	2	4	no	—
102	C₄×Dic₄₃	—	2	4	no	—
103	C₄×Dic₄₃	C₂×C₈	1	8	yes	—
104	C₂×C₄₃⋊C₈	—	2	2	no	—
105	C₂×C₄₃⋊C₈	—	2	2	no	—
106	C₂×C₄₃⋊C₈	—	2	2	no	—
107	C₂×C₄₃⋊C₈	—	2	2	no	—
108	C₂×C₄₃⋊C₈	C₂×Q₈	1	4	yes	—
109	C₂×C₄₃⋊C₈	#41	1	4	yes	—
110	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	#117	1	2	yes	—
118	C₂×C₄×C₄₃	#117	1	2	yes	—
119	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₄₃	#131	1	2	yes	—
128	C₂×C₄×C₄₃	#131	1	2	yes	—
129	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
131	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
133	C₂×Dic₄₃	—	2	2	no	—
134	C₂×Dic₄₃	—	2	2	no	—
135	C₂×Dic₄₃	—	2	2	no	—
136	C₂×Dic₄₃	—	2	2	no	—
137	C₂×Dic₄₃	—	2	2	no	—
138	C₂×Dic₄₃	—	2	2	no	—
139	C₂×Dic₄₃	—	2	2	no	—
140	C₂×Dic₄₃	—	2	2	no	—
141	C₂×Dic₄₃	—	2	2	no	—
142	C₂×Dic₄₃	—	2	2	no	—
143	C₂×Dic₄₃	C₂²×C₈	1	4	yes	—
144	C₂×Dic₄₃	C₂²×C₈	1	4	yes	—
145	Dic₈₆	—	2	8	no	—
146	C₈×C₄₃	—	4	4	no	—
147	C₄₃⋊C₈	—	2	4	no	—
148	C₄₃⋊C₈	—	2	4	no	—
149	#21339	—	43	1	no	maximal, Sylow
150	C₂²×C₄₃	C₂×#117	1	1	yes	characteristic
151	C₂²×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₂²×C₄₃	C₄×#41	1	1	yes	characteristic
153	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
154	C₂²×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
155	C₂²×C₄₃	C₂×#131	1	1	yes	characteristic
156	C₂²×C₄₃	C₂×#131	1	1	yes	characteristic
157	C₄×C₄₃	—	2	2	no	—
158	C₄×C₄₃	—	2	2	no	—
159	C₄×C₄₃	—	2	2	no	—
160	C₄×C₄₃	#431	1	4	yes	—
161	C₄×C₄₃	C₈×Q₈	1	4	yes	—
162	C₄×C₄₃	#446	1	4	yes	—
163	C₄×C₄₃	#529	1	2	yes	—
164	C₄×C₄₃	#533	1	2	yes	—
165	Dic₄₃	—	2	4	no	—
166	Dic₄₃	—	2	4	no	—
167	Dic₄₃	—	4	2	no	—
168	Dic₄₃	—	4	2	no	—
169	C₄²×Q₈	—	43	1	no	—
170	#1984	—	43	2	no	—
171	#1939	—	43	2	no	—
172	#1916	—	43	4	no	—
173	C₄×C₄⋊C₈	—	43	2	no	—
174	C₄×C₄⋊C₈	D₄₃	1	1	yes	characteristic
175	#1744	—	43	2	no	—
176	C₄²×C₈	—	43	1	no	—
177	C₂×C₄₃	#2295	1	1	yes	characteristic
178	C₂×C₄₃	C₂×#431	1	1	yes	characteristic
179	C₂×C₄₃	#2922	1	1	yes	characteristic
180	C₂×C₄₃	C₂×C₈×Q₈	1	1	yes	characteristic
181	C₂×C₄₃	C₂×#446	1	1	yes	characteristic
182	C₂×C₄₃	C₈×#41	1	1	yes	characteristic
183	C₂×C₄₃	#2965	1	1	yes	characteristic
184	C₂×C₄×Q₈	—	43	1	no	—
185	C₂×C₄×Q₈	—	43	1	no	—
186	C₂×C₄×Q₈	—	43	1	no	—
187	C₂×C₄⋊C₈	—	43	4	no	—
188	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
189	C₂×C₄×C₈	—	43	1	no	—
190	C₂×C₄×C₈	—	43	2	no	—
191	C₂×C₄×C₈	—	43	2	no	—
192	C₂×C₄×C₈	—	43	1	no	—
193	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
194	C₄×C₄⋊C₄	—	43	2	no	—
195	C₄×C₄⋊C₄	—	43	1	no	—
196	C₄×C₄⋊C₄	—	43	1	no	—
197	C₄×C₄⋊C₄	—	43	1	no	—
198	C₄×C₄⋊C₄	—	43	2	no	—
199	C₄×C₄⋊C₄	—	43	2	no	—
200	C₄³	—	43	2	no	—
201	C₄³	C₂×D₄₃	1	1	yes	characteristic
202	#336	—	43	4	no	—
203	#336	—	43	4	no	—
204	#336	—	43	4	no	—
205	#336	C₂×D₄₃	1	2	yes	—
206	C₄₃	#21339	1	1	yes	characteristic, complemented, Sylow
207	C₂²×Q₈	—	43	1	no	—
208	C₂²×C₈	—	43	2	no	—
209	C₂²×C₈	—	43	2	no	—
210	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
211	C₄×Q₈	—	43	8	no	—
212	C₄×Q₈	—	86	4	no	—
213	C₄×Q₈	—	86	4	no	—
214	C₂×C₄⋊C₄	—	43	2	no	—
215	C₂×C₄⋊C₄	—	43	1	no	—
216	C₂×C₄⋊C₄	—	43	2	no	—
217	C₂×C₄⋊C₄	—	43	2	no	—
218	C₂×C₄⋊C₄	—	43	1	no	—
219	C₂×C₄⋊C₄	—	43	1	no	—
220	C₂×C₄²	—	43	2	no	—
221	C₂×C₄²	—	43	2	no	—
222	C₂×C₄²	—	43	2	no	—
223	C₂×C₄²	—	43	2	no	—
224	C₂×C₄²	—	43	2	no	—
225	C₂×C₄²	—	43	2	no	—
226	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
227	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
228	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
229	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
233	C₄⋊C₈	—	2	4	no	—
234	C₄⋊C₈	—	86	8	no	—
235	C₄×C₈	—	2	4	no	—
236	C₄×C₈	—	43	4	no	—
237	C₄×C₈	—	86	4	no	—
238	C₄×C₈	—	86	4	no	—
239	C₄×C₈	—	86	2	no	—
240	#95	—	43	2	no	—
241	#95	—	43	2	no	—
242	#95	—	43	2	no	—
243	C₂×Q₈	—	43	4	no	—
244	C₂×Q₈	—	86	2	no	—
245	C₂×Q₈	—	86	2	no	—
246	C₂²×C₄	—	43	2	no	—
247	C₂²×C₄	—	43	2	no	—
248	C₂²×C₄	—	43	2	no	—
249	C₂²×C₄	—	43	2	no	—
250	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	43	4	no	—
261	C₂×C₈	—	43	4	no	—
262	C₂×C₈	—	86	2	no	—
263	C₂×C₈	—	86	2	no	—
264	C₂×C₈	—	86	2	no	—
265	C₂×C₈	—	86	2	no	—
266	C₄⋊C₄	—	43	8	no	—
267	C₄⋊C₄	—	43	4	no	—
268	C₄⋊C₄	—	86	4	no	—
269	C₄⋊C₄	—	86	2	no	—
270	C₄⋊C₄	—	86	4	no	—
271	C₄⋊C₄	—	86	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	43	8	no	—
276	C₄²	—	86	4	no	—
277	C₄²	—	86	4	no	—
278	C₄²	—	86	4	no	—
279	C₄²	—	86	4	no	—
280	C₄²	—	86	4	no	—
281	C₄²	C₈×D₄₃	1	4	yes	—
282	C₄²	#10621872	1	4	yes	—
283	C₄²	#10621898	1	4	yes	—
284	C₄²	#10621900	1	2	yes	—
285	C₄²	Q₈×D₄₃	1	2	yes	—
286	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
287	Q₈	—	86	8	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	43	4	no	—
295	C₂×C₄	—	43	4	no	—
296	C₂×C₄	—	86	2	no	—
297	C₂×C₄	—	86	2	no	—
298	C₂×C₄	—	86	2	no	—
299	C₂×C₄	—	86	2	no	—
300	C₂×C₄	—	86	2	no	—
301	C₂×C₄	—	86	2	no	—
302	C₂×C₄	—	86	2	no	—
303	C₂×C₄	—	86	2	no	—
304	C₂×C₄	—	86	2	no	—
305	C₂×C₄	—	86	2	no	—
306	C₂×C₄	#13063282	1	2	yes	—
307	C₂×C₄	#13063282	1	2	yes	—
308	C₂×C₄	#13063290	1	2	yes	—
309	C₂×C₄	#13063290	1	2	yes	—
310	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
311	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
312	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
313	C₂×C₄	C₂×#10621872	1	2	yes	—
314	C₂×C₄	C₂×#10621872	1	2	yes	—
315	C₂×C₄	C₂×#10621872	1	2	yes	—
316	C₂×C₄	#13063333	1	2	yes	—
317	C₂×C₄	#13063333	1	2	yes	—
318	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
319	C₂×C₄	#41×D₄₃	1	2	yes	—
320	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
321	C₂×C₄	C₂×#10621898	1	2	yes	—
322	C₈	—	4	4	no	—
323	C₈	—	86	4	no	—
324	C₈	—	86	4	no	—
325	C₂²	C₂×#13063282	1	1	yes	characteristic
326	C₂²	C₂×#13063290	1	1	yes	characteristic
327	C₂²	#14583367	1	1	yes	characteristic
328	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
329	C₂²	C₂²×#10621872	1	1	yes	characteristic
330	C₂²	C₂×#13063333	1	1	yes	characteristic
331	C₂²	C₂×#13063333	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	86	4	no	—
336	C₄	—	86	4	no	—
337	C₄	—	172	2	no	—
338	C₄	—	172	2	no	—
339	C₄	#14582950	1	4	yes	—
340	C₄	#14583023	1	4	yes	—
341	C₄	#14583026	1	4	yes	—
342	C₄	#14583431	1	2	yes	—
343	C₄	#14583453	1	2	yes	—
344	C₂	#19137069	1	1	yes	characteristic
345	C₂	C₂×#14582950	1	1	yes	characteristic
346	C₂	#19143240	1	1	yes	characteristic
347	C₂	C₂×#14583023	1	1	yes	characteristic
348	C₂	C₂×#14583026	1	1	yes	characteristic
349	C₂	#19143551	1	1	yes	characteristic
350	C₂	#19143556	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×C₄⋊C₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂³×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#21339
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
36. C₄×Dic₈₆	C₈	semidirect
53. C₄×C₄₃⋊C₈	Q₈	semidirect
206. C₄₃	#21339	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Dic₈₆	2
#19135530	2
#19135311	2
#19135150	2
C₄×Dic₄₃⋊C₈	2
#19134340	2
C₄²×C₄₃⋊C₈	2
C₄×C₄⋊C₈×C₄₃	2
#21339	43