Subgroups — #25100129

Order 11008.

866 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582947	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13063260	C₂	1	1	yes	maximal, characteristic
3	#19143159	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582926	C₂	1	1	yes	maximal, characteristic
5	#19135370	C₂	1	8	yes	maximal, complemented
6	#19134321	C₂	1	1	yes	maximal, characteristic
7	C₂×#14582663	C₂	1	1	yes	maximal, characteristic
8	C₄₃×#2873	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10621877	C₂²	1	1	yes	characteristic
10	C₂²×#10621877	C₂²	1	1	yes	characteristic
11	C₂²×#10621877	C₂²	1	1	yes	characteristic
12	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	#14582947	—	2	4	no	—
14	C₂×#13063260	—	2	4	no	—
15	C₂×#13063260	C₂²	1	4	yes	—
16	C₂×#13063260	C₂²	1	4	yes	—
17	#14582926	—	2	2	no	—
18	#14582926	—	2	2	no	—
19	#14582926	C₄	1	4	yes	—
20	#14582926	C₄	1	4	yes	—
21	#14582923	C₂²	1	4	yes	—
22	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
23	#14582663	C₂²	1	4	yes	—
24	#14582663	C₂²	1	4	yes	—
25	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
26	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#405	—	2	2	no	—
29	C₄₃×#405	—	2	2	no	—
30	C₄×M₁₆×C₄₃	C₂²	1	4	yes	—
31	C₂×#10621877	—	2	2	no	—
32	C₂×#10621877	—	2	2	no	—
33	C₂×#10621877	—	2	1	no	—
34	C₂×#10621877	—	2	1	no	—
35	C₂×#10621877	—	2	2	no	—
36	C₂×#10621877	—	2	2	no	—
37	C₂×#10621877	—	2	2	no	—
38	C₂×#10621877	—	2	2	no	—
39	C₂×#10621877	C₂×C₄	1	2	yes	—
40	C₂×#10621877	C₂×C₄	1	2	yes	—
41	C₂×#10621877	C₂³	1	2	yes	—
42	C₂×#10621877	C₂³	1	2	yes	—
43	C₂²×C₄₃⋊C₈	—	2	2	no	—
44	C₂²×C₄₃⋊C₈	—	2	2	no	—
45	C₂²×C₄₃⋊C₈	—	2	1	no	—
46	C₂²×C₄₃⋊C₈	—	2	1	no	—
47	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
48	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
49	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
50	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
51	#13063260	—	2	8	no	—
52	#13063260	—	2	8	no	—
53	C₄⋊(C₄₃⋊C₈)	—	2	8	no	—
54	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	16	yes	—
55	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₄₃	—	2	1	no	—
57	C₂×M₁₆×C₄₃	—	2	1	no	—
58	C₂×M₁₆×C₄₃	—	2	2	no	—
59	C₂×M₁₆×C₄₃	—	2	2	no	—
60	C₂×M₁₆×C₄₃	D₄	1	4	yes	—
61	C₂×M₁₆×C₄₃	Q₈	1	4	yes	—
62	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
63	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
64	C₂²×C₈×C₄₃	—	2	1	no	—
65	C₂²×C₈×C₄₃	—	2	1	no	—
66	#117×C₄₃	C₂×C₄	1	2	yes	—
67	#117×C₄₃	C₂×C₄	1	2	yes	—
68	#117×C₄₃	D₄	1	2	yes	—
69	#117×C₄₃	D₄	1	2	yes	—
70	C₂×C₄⋊C₄×C₄₃	Q₈	1	2	yes	—
71	C₂×C₂²⋊C₄×C₄₃	Q₈	1	2	yes	—
72	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
73	#97×C₄₃	—	2	4	no	—
74	C₄×C₈×C₄₃	—	2	4	no	—
75	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
83	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	#41	1	2	yes	—
85	C₂²×C₄×C₄₃	#41	1	2	yes	—
86	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
87	M₁₆×C₄₃	—	2	4	no	—
88	M₁₆×C₄₃	—	2	4	no	—
89	M₁₆×C₄₃	—	2	2	no	—
90	M₁₆×C₄₃	—	2	2	no	—
91	M₁₆×C₄₃	C₂×D₄	1	4	yes	—
92	M₁₆×C₄₃	C₂×Q₈	1	4	yes	—
93	C₂×C₈×C₄₃	—	2	4	no	—
94	C₂×C₈×C₄₃	—	2	4	no	—
95	C₂×C₈×C₄₃	—	2	2	no	—
96	C₂×C₈×C₄₃	—	2	2	no	—
97	C₄⋊C₄×C₄₃	C₄⋊C₄	1	4	yes	—
98	C₄⋊C₄×C₄₃	C₄⋊C₄	1	4	yes	—
99	C₂²⋊C₄×C₄₃	C₄⋊C₄	1	4	yes	—
100	C₂²⋊C₄×C₄₃	C₄⋊C₄	1	4	yes	—
101	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
102	C₄²×C₄₃	C₂×D₄	1	4	yes	—
103	#10621877	—	2	4	no	—
104	#10621877	—	2	4	no	—
105	#10621877	—	2	2	no	—
106	#10621877	—	2	2	no	—
107	#10621877	—	2	2	no	—
108	#10621877	—	2	2	no	—
109	#10621877	—	4	2	no	—
110	#10621877	—	4	2	no	—
111	C₂×C₄₃⋊C₈	—	2	4	no	—
112	C₂×C₄₃⋊C₈	—	2	4	no	—
113	C₂×C₄₃⋊C₈	—	2	2	no	—
114	C₂×C₄₃⋊C₈	—	2	2	no	—
115	C₂×C₄₃⋊C₈	—	2	2	no	—
116	C₂×C₄₃⋊C₈	—	2	2	no	—
117	C₂×C₄₃⋊C₈	—	2	2	no	—
118	C₂×C₄₃⋊C₈	—	2	2	no	—
119	C₂×C₄₃⋊C₈	—	4	2	no	—
120	C₂×C₄₃⋊C₈	—	4	2	no	—
121	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
122	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
123	C₂³×C₄₃	—	4	2	no	—
124	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂³×C₄₃	#125	1	2	yes	—
127	C₂³×C₄₃	C₄⋊Q₈	1	2	yes	—
128	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₄₃	—	4	4	no	—
130	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	8	yes	—
131	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
136	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
137	C₂×C₄×C₄₃	#121	1	2	yes	—
138	C₂×C₄×C₄₃	#121	1	2	yes	—
139	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
140	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
141	C₂×C₄×C₄₃	#125	1	2	yes	—
142	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
143	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
149	C₈×C₄₃	—	2	4	no	—
150	C₈×C₄₃	—	2	4	no	—
151	C₄₃⋊C₈	—	2	4	no	—
152	C₄₃⋊C₈	—	2	4	no	—
153	C₄₃⋊C₈	—	4	2	no	—
154	C₄₃⋊C₈	—	4	2	no	—
155	#21178	—	43	1	no	maximal, Sylow
156	C₂²×C₄₃	—	2	2	no	—
157	C₂²×C₄₃	—	2	2	no	—
158	C₂²×C₄₃	—	2	2	no	—
159	C₂²×C₄₃	—	4	2	no	—
160	C₂²×C₄₃	—	4	2	no	—
161	C₂²×C₄₃	#389	1	2	yes	—
162	C₂²×C₄₃	#389	1	2	yes	—
163	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
164	C₂²×C₄₃	C₂×#125	1	1	yes	characteristic
165	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
166	C₄×C₄₃	—	2	8	no	—
167	C₄×C₄₃	#389	1	2	yes	—
168	C₄×C₄₃	#389	1	2	yes	—
169	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
171	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
172	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
173	C₂×#430	—	43	1	no	—
174	C₂²×#108	—	43	1	no	—
175	#2905	—	43	1	no	—
176	C₂×#424	—	43	1	no	—
177	#2873	D₄₃	1	1	yes	characteristic
178	#1950	—	43	8	no	—
179	#1738	—	43	1	no	—
180	C₂×#341	—	43	1	no	—
181	C₂×C₄₃	—	2	1	no	—
182	C₂×C₄₃	—	2	1	no	—
183	C₂×C₄₃	—	2	2	no	—
184	C₂×C₄₃	—	2	2	no	—
185	C₂×C₄₃	C₂×#389	1	1	yes	characteristic
186	C₂×C₄₃	#2916	1	2	yes	—
187	C₂²×M₁₆	—	43	1	no	—
188	C₂²×M₁₆	—	43	1	no	—
189	C₂²×M₁₆	—	43	1	no	—
190	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
191	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
192	C₂³×C₈	—	43	1	no	—
193	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
194	#430	—	86	4	no	—
195	C₂×#108	—	43	4	no	—
196	C₂×#108	—	43	4	no	—
197	C₂×#108	—	86	4	no	—
198	#424	—	43	4	no	—
199	#424	—	43	4	no	—
200	#424	—	86	2	no	—
201	#424	—	86	2	no	—
202	#423	—	43	4	no	—
203	C₂×C₄⋊C₈	—	43	4	no	—
204	#405	—	2	2	no	—
205	#405	—	2	2	no	—
206	C₄×M₁₆	C₂×D₄₃	1	4	yes	—
207	#341	—	43	4	no	—
208	#341	—	43	4	no	—
209	C₄₃	#21178	1	1	yes	characteristic, complemented, Sylow
210	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
211	C₂×M₁₆	—	2	2	no	—
212	C₂×M₁₆	—	2	2	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	43	2	no	—
216	C₂×M₁₆	—	43	2	no	—
217	C₂×M₁₆	—	43	2	no	—
218	C₂×M₁₆	—	43	2	no	—
219	C₂×M₁₆	—	86	2	no	—
220	C₂×M₁₆	—	86	2	no	—
221	C₂×M₁₆	—	86	1	no	—
222	C₂×M₁₆	—	86	1	no	—
223	C₂×M₁₆	—	86	2	no	—
224	C₂×M₁₆	—	86	2	no	—
225	C₂×M₁₆	—	86	2	no	—
226	C₂×M₁₆	—	86	2	no	—
227	C₂×M₁₆	Dic₈₆	1	4	yes	—
228	C₂×M₁₆	D₁₇₂	1	4	yes	—
229	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
230	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
231	C₂²×C₈	—	2	1	no	—
232	C₂²×C₈	—	2	1	no	—
233	C₂²×C₈	—	43	2	no	—
234	C₂²×C₈	—	43	2	no	—
235	C₂²×C₈	—	43	2	no	—
236	C₂²×C₈	—	43	2	no	—
237	C₂²×C₈	—	86	2	no	—
238	C₂²×C₈	—	86	2	no	—
239	C₂²×C₈	—	86	1	no	—
240	C₂²×C₈	—	86	1	no	—
241	#117	C₄×D₄₃	1	2	yes	—
242	#117	C₄×D₄₃	1	2	yes	—
243	#117	C₂²⋊D₄₃	1	2	yes	—
244	#117	C₂²⋊D₄₃	1	2	yes	—
245	C₂×C₄⋊C₄	Dic₈₆	1	2	yes	—
246	C₂×C₂²⋊C₄	Dic₈₆	1	2	yes	—
247	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
248	#108	—	86	8	no	—
249	#108	—	86	8	no	—
250	C₄⋊C₈	—	43	16	no	—
251	C₄⋊C₈	—	86	8	no	—
252	#97	—	2	4	no	—
253	C₄×C₈	—	2	4	no	—
254	C₂⁴	C₂×Dic₈₆	1	1	yes	characteristic
255	C₂²×C₄	C₂×Dic₈₆	1	4	yes	—
256	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
257	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
258	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
261	C₂²×C₄	#10621898	1	2	yes	—
262	C₂²×C₄	#10621898	1	2	yes	—
263	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
265	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
266	M₁₆	—	2	4	no	—
267	M₁₆	—	2	4	no	—
268	M₁₆	—	2	2	no	—
269	M₁₆	—	2	2	no	—
270	M₁₆	—	86	4	no	—
271	M₁₆	—	86	4	no	—
272	M₁₆	—	86	2	no	—
273	M₁₆	—	86	2	no	—
274	M₁₆	—	86	2	no	—
275	M₁₆	—	86	2	no	—
276	M₁₆	—	172	2	no	—
277	M₁₆	—	172	2	no	—
278	M₁₆	C₂×Dic₈₆	1	4	yes	—
279	M₁₆	C₂×D₁₇₂	1	4	yes	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	43	4	no	—
285	C₂×C₈	—	43	4	no	—
286	C₂×C₈	—	86	4	no	—
287	C₂×C₈	—	86	4	no	—
288	C₂×C₈	—	86	2	no	—
289	C₂×C₈	—	86	2	no	—
290	C₂×C₈	—	86	2	no	—
291	C₂×C₈	—	86	2	no	—
292	C₂×C₈	—	86	2	no	—
293	C₂×C₈	—	86	2	no	—
294	C₂×C₈	—	172	2	no	—
295	C₂×C₈	—	172	2	no	—
296	C₄⋊C₄	Dic₄₃⋊C₄	1	4	yes	—
297	C₄⋊C₄	Dic₄₃⋊C₄	1	4	yes	—
298	C₂²⋊C₄	Dic₄₃⋊C₄	1	4	yes	—
299	C₂²⋊C₄	Dic₄₃⋊C₄	1	4	yes	—
300	C₄²	C₂×C₄×D₄₃	1	4	yes	—
301	C₄²	C₂×C₂²⋊D₄₃	1	4	yes	—
302	C₂³	—	4	2	no	—
303	C₂³	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
304	C₂³	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
305	C₂³	#13063317	1	2	yes	—
306	C₂³	#13063325	1	2	yes	—
307	C₂³	C₂²×Dic₈₆	1	1	yes	characteristic
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	C₂×Dic₄₃⋊C₄	1	8	yes	—
310	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
311	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
312	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
313	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
314	C₂×C₄	C₄×Dic₈₆	1	2	yes	—
315	C₂×C₄	C₄×Dic₈₆	1	2	yes	—
316	C₂×C₄	#13063299	1	2	yes	—
317	C₂×C₄	#13063299	1	2	yes	—
318	C₂×C₄	#13063304	1	2	yes	—
319	C₂×C₄	#13063304	1	2	yes	—
320	C₂×C₄	#13063317	1	2	yes	—
321	C₂×C₄	#13063325	1	2	yes	—
322	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
324	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
325	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
326	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
327	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
328	C₈	—	2	4	no	—
329	C₈	—	2	4	no	—
330	C₈	—	86	4	no	—
331	C₈	—	86	4	no	—
332	C₈	—	172	2	no	—
333	C₈	—	172	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	#14582797	1	2	yes	—
340	C₂²	#14582797	1	2	yes	—
341	C₂²	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
342	C₂²	C₂×#13063317	1	1	yes	characteristic
343	C₂²	C₂×#13063325	1	1	yes	characteristic
344	C₄	—	2	8	no	—
345	C₄	#14582797	1	2	yes	—
346	C₄	#14582797	1	2	yes	—
347	C₄	C₂×C₄×D₁₇₂	1	1	yes	characteristic
348	C₄	C₂×C₄×Dic₈₆	1	1	yes	characteristic
349	C₄	C₂×#13063299	1	1	yes	characteristic
350	C₄	C₂×#13063304	1	1	yes	characteristic
351	C₂	—	2	2	no	—
352	C₂	—	2	2	no	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	C₂×#14582797	1	1	yes	characteristic
356	C₂	#19143204	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#2873	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₃	C₂×Q₈
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#2873	D₄₃
hypocenter	C₄₃	#21178
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ #2873

Complements

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

N	H ≅ G/N	type
5. #19135370	C₂	semidirect
209. C₄₃	#21178	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582947	2
C₂²×#13063260	2
#19143159	2
C₂×#14582926	2
#19135370	2
#19134321	2
C₂×#14582663	2
C₄₃×#2873	2
#21178	43